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Agile Mind (2016)

Agile Mind | HS Traditional | 2016 Edition

High School

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Report Published: October 18, 2017

Edition:

Algebra I: http://www.agilemind.com/programs/mathematics/algebra-i/

Geometry: http://www.agilemind.com/programs/mathematics/geometry/

Algebra II: http://www.agilemind.com/programs/mathematics/algebra-ii/

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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for focusing on the non-plus standards of the CCSSM and exhibiting coherence within and across courses that is consistent with a logical structure of mathematics. Overall, the instructional materials attend to the full intent of the non-plus standards and allow students to fully learn each non-plus standard, but they do not attend to the full intent of the modeling process when applied to the modeling standards. Although the materials regularly use age-appropriate contexts and apply key takeaways from Grades 6-8, they do not vary the types of numbers being used. The materials do not explicitly identify and build on knowledge from Grades 6-8 although they do foster coherence through meaningful connections in a single course and throughout the series. The instructional materials spend a majority of time on the widely applicable prerequisites from the CCSSM.

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The instructional materials reviewed for the Agile Mind Traditional series meet the expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. Although there are a few instances where all of the aspects of the standards are not addressed, most non-plus standards are addressed to the full intent of the mathematical content by the instructional materials.

The following are examples of standards that are fully addressed:

  • A-APR.3: Algebra I Topic 18 Roots, Factors, and Zeros connects x-intercepts to zeros to factors. In Algebra II Topic 5 Polynomial Functions there are two lessons, Long Term Behavior and Zeros and Higher Degree Polynomials, where students factor polynomials to find zeros and use zeros to construct polynomial functions. Also in Algebra II Topic 6 Polynomial equations - Theorems of Algebra, students use Theorems of Algebra (such as The Fundamental Theorem of Algebra and Remainder Theorem), and factorizations to find zeros in order to graph the polynomial function.
  • F-BF.4a: In Algebra II Topic 2 Understanding Inverse Relationships students find equations of inverses of linear, exponential, and quadratic functions and give restrictions where needed.
  • G-SRT.8: In Topic 14 Pythagorean Theorem and the Distance Formula and Topic 15 Right Triangle and Trig Relationships of Geometry, students use the Pythagorean Theorem as well as the trigonometric ratios to solve right triangles. In addition to the lesson demonstrations, student activity sheets, practice, and assessment items, both topics include MARS tasks which fully address the intent of this standard by providing students opportunities to solve right triangles using the trigonometric ratios and Pythagorean Theorem in applied problems.
  • S-ID.2: In Algebra I Topic 7 Descriptive Statistics students compare data sets using mean and median in the lesson Measures of Center. In the next lesson, Measures of Spread, students compare data sets using range and standard deviation.

The following standard is partially addressed:

  • G-CO.13: While no instruction was provided on G-CO.13, there is one instance where this standard is assessed, in Topic 15 of Geometry Constructed Response Assessment #1. Students inscribe an equilateral triangle in a circle, but students are not provided an opportunity to practice this concept in the lesson materials. Constructions can be found in Geometry Topic 11 Compass and Straightedge Constructions; Geometry Topic 19 Chords, Arcs, and Inscribed Angles; and Geometry Topic 20 Lines and Segments on Circles; however, students are not given an opportunity to construct a shape inside of a circle.
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The instructional materials reviewed for the Agile Mind Traditional series partially meet the expectations for attending to the full intent of the modeling process when applied to the modeling standards. Overall, most of the modeling standards are addressed with various aspects of the modeling process present in isolation or combination. However, opportunities for the full modeling process are absent.

The materials often allow students to incorporate their own solution method to find a predetermined quantity. Modeling opportunities in the materials are thus “closed” in the beginning and the end while “open” in the middle. In many instances, materials step students through the modeling process using a series of questions and/or prompts. In addition, students are rarely given the opportunity to question their reasoning and “cycle” through the modeling process by validating their conclusions and potentially making improvements to their model.

The following examples allow students to engage in only a part of the modeling process:

  • In Algebra I Topic 18 Student Activity Sheet 2 (A-SSE.3a), students are given a function rule of a real-world context and guided through a series of questions, mostly directed by the teacher with questions and/or Exploring “Solving by graphing.” The same is true on Student Activity Sheet 4, which follows along with Exploring “Roots, factors, and zeros.” Students are given a real-world context and taken through a series of questions, as posed on the Student Activity Sheet. Students use a variety of tools to find a solution to a quadratic equation. Students do not define their own variables or formulate the equation or function needed to work the problem. The materials provide students with a graph with predefined axes and scale as well as the function they are to graph.
  • In Algebra I Topic 14 (A-SSE.3c, F-BF.1a), there are two sections in the Exploring part of the Topic that starts out with the word “Modeling.” Students follow step-by-step directions on how to apply the modeling context (make a table, graph the data, answer questions, etc.). The Constructed Response assessment item for the topic gives students exact measurements when starting the problem, steps them through by telling them which tools to use, and does not have them justify their solution; therefore, students do not complete the modeling cycle. Student Activity Sheets 2-5 have application problems where students are asked to do things such as explain, describe, and discuss; students are also asked to check the validity of their answer. However, there isn’t an opportunity for students to complete the full modeling process in any one problem.
  • In Algebra I Topic 8 (A-CED.2) Student Activity Sheet 3 Question 26 is an example of an application problem where students choose their tools to use in order to solve the problem. However, there are exact values given to students leading to one correct answer. In addition, students are not required to provide any justification for or validate their solution. The same thing happens in Student Activity Sheet Question 31.
  • In Algebra 1 Topic 10 (A-CED.3) there is a MARS task which asks students to explain their work. However, all quantities are fixed, and students are not asked to check the validity of their solution or to adjust as necessary. There is a Constructed Response question on the Assessment that is an application where students are required to identify the variables, write a system, use a graph or table to solve, and then show how to check the answer. They are taken step-by-step through the process.
  • In Algebra II Topic 11 (F-IF.5) a Constructed Response Assessment question has students find the domain and range, in context, and relate it to the context of the situation. Students are also asked to justify their answer in another part of the problem; however, students do not develop the model.
  • In Geometry Topic 15 (G-SRT.8) Student Activity Sheet 3 Question 19 is also found in Exploring Right triangle and trig relationships. Students are presented with an open-ended question but given specific variables to use in order to solve it. Students are not given an opportunity to define the variables. In addition, students are not asked to validate or interpret their solutions. All application problems in this Topic are routine and require one or two steps. An example of this can be found in Student Activity Sheet 3 Question 21 where students are given the context and a labeled picture to find the solution to the problem which asks students to find the height of the cliff. Students are required to use trigonometric ratios to solve for the height given the angle of elevation and the horizontal distance from the cliff to the boat. A second example is found in Assessment for this Topic; Constructed Response 3 has students solve a few problems using the context of a lighthouse used to orient ships.
  • In Geometry Topic 14 (G-GPE.7) Question 12 on Student Activity Sheet 2 gives measurements and asks students if the door frame is rectangular. Students are directed to justify their response. They are not afforded the opportunity to complete the modeling process. Student Activity Sheet 3 requires students to support their response but requires that support be in a diagram.
  • In Algebra I Topic 7 (S-ID.1, S-ID.2, and S-ID.3) on Student Activity Sheet 2 Question 12 students are asked to predict, explain, and check their prediction by calculating the mean and median. Students are given the tools to use throughout the step-by-step questions. Question 13 and 14 are contextual problems where students create histograms. Students explain their reasoning if something was changed based on the histograms. Students use a context but are not able to formulate the variables. The data set to be used is given at the beginning, and students are given questions to guide them through. In Student Activity Sheet 4 Question 17 students create a survey for the class, conduct it, and do specific things to interpret the survey. However, students are not asked to verify their responses.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectations for allowing students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers (WAPs). (Those standards that were not fully attended to by the materials, as noted in indicator 1ai, are not mentioned here.)

In the Algebra I course, students spend most of their time working with WAPs from the Algebra, Functions, and Statistics and Probability categories. During the Geometry Course, students spend most of their time working with WAPs from the Geometry category. The Algebra II course focuses on the WAPs in the Functions, Algebra, and Geometry categories. Within the Algebra I and Algebra II courses, students also spend time on the Number and Quantity WAPs.

Examples of students engaging with the WAPs include:

  • Algebra I: In Topic 13 Law of Exponents students are provided with multiple opportunities to explore and interpret laws of exponents using scenarios such as fuel consumption and distance from the sun to the Milky Way Galactic Center (N-RN.1,2). Topic 13 covers the general rules for exponents as well as scientific notation. Topics 16 and 18, Operations on Polynomials and Solving Quadratic Equations, provide several opportunities to explore the structure of an expression to identify ways to rewrite it and to factor a quadratic expression to reveal the zeros of the function it defines. The topics provide practical illustrations using blueprints from a construction site to illustrate finding sums and differences of two polynomials and a water balloon launch to illustrate solving quadratic equations (A-SSE.2,3a).
  • Geometry: Topics 9, 10, 12, and 13 address similarity and congruence as referred to in G-SRT.5. Topics 9 and 10 focus on congruence, and Topics 12 and 13 focus on similarity.
  • Geometry: In Topics 4, 5, and 6 students prove theorems about lines and angles (G-CO.9). Proofs begin in Topic 4 on Student Activity Sheet 2 with algebraic proofs. The topic then progresses in Student Activity Sheet 3 as materials provide multiple proofs for students to “fill in the blank” for the missing part. In Student Activity Sheet 4 there are multiple cases where students are expected to complete the majority of an entire proof. In Topic 5 proofs continue in Student Activity Sheet 1 as well as indirect proofs in Student Activity Sheet 4.
  • Algebra II: In Topic 1 Student Activity Sheet 3 students work with geometric series in word problems and are asked to write a function rule that models the given situation (A-SSE.4). Throughout Student Activity Sheet 3 students are exposed to finite, geometric series by using the general formula and finding sums.
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The instructional materials reviewed for the Agile Mind Traditional series, when used as designed, meet the expectation for allowing students to fully learn each non-plus standard. Overall, there are multiple opportunities for students to fully learn the non-plus standards by engaging with all aspects of the standards and not distracting students with prerequisite or additional topics. Examples of the standards where students have multiple opportunities to fully learn the standard include, but are not limited to:

  • A-SSE.1a: Algebra I Topic 2 Exploring Tiling Square Pools offers students the opportunity to interpret parts of expressions as they examine different representations for determining the number of tiles needed in a pattern to create a border around a pool. Constructed Response 1c of Topic 2 offers students the opportunity to interpret parts of an expression as they create a symbolic representation of the relationship between the length of the side of a square flower bed and the perimeter of the flower bed. Throughout the remainder of Algebra I and into Algebra II, there are multiple opportunities for students to interpret parts of an expression, and some of those opportunities are:
    • Algebra I Topic 6 More Practice Problem 23 has students interpret parts of an equation in order to determine which conclusion can be made based on the equation and its accompanying graph.
    • Algebra I Topic 14 includes multiple opportunities for students to interpret parts of expressions, equations, and functions in different contexts that represent exponential growth and decay.
    • Algebra II Topic 1 Exploring Arithmetic Sequences and Series has opportunities for students to interpret parts of expressions that represent the same arithmetic sequence, and Algebra II Topic 1 Exploring Geometric Sequences and Series has students interpret parts of expressions while comparing different expressions that represent the same geometric sequence.
    • Algebra II Topic 13 Guided Practice Problems 11 and 12 have students interpret parts of a general exponential equation in order to determine how to substitute numerical values into the equation, and More Practice Problems 3 and 6 has students selecting which exponential equation models a situation which means the students interpret parts of the exponential expression to choose the correct equation.
    • Algebra II Topic 21 Automatically Scored Problem 10 has students interpret parts of a trigonometric expression in order to choose which trigonometric equation best represents a given situation.
  • A-APR.6: In Algebra 1 Topic 16 students are introduced to dividing polynomials, and the problems include dividing by monomials with remainders. In Algebra II Topic 6 Exploring Theorems of Algebra students divide polynomials by linear binomials as they engage with The Remainder Theorem, and in Algebra II Topic 9 Exploring Rational Expressions, students use polynomials to build rational expressions by dividing polynomials using factoring techniques with no remainders. In the remainder of Algebra II Topic 9, students further develop their skills in rewriting simple rational expressions as they use long division with expressions that involve remainders in order to analyze the graphs of rational functions that correspond to the rational expressions.
  • G-GPE.5: In Algebra 1 Topic 5 Student Activity Sheet 3 Problems 18 and 19 students informally use the slope criteria for parallel and perpendicular lines to solve geometric problems by writing equations of lines that are parallel and perpendicular to given lines, and they do the same thing in More Practice Problem 16 of the same topic. In Geometry Topic 6 Exploring Lines and Algebra, students formally derive the slope criteria for parallel and perpendicular lines. Student Activity Sheet 3 of the same topic, along with More Practice Problems 17 and 19 and Automatically Scored Problems 12 and 13, gives students opportunities to write the equations of lines parallel and perpendicular to given lines. Geometry Topic 8 Constructed Response Problems 2 and 3 have students use the slope criteria for parallel and perpendicular lines to solve geometric problems by having the students find the centroid, orthocenter, and Euler line for triangles with given coordinates.

There are non-plus standards where the materials provide students an opportunity to fully learn the standard, and the materials could solidify the students’ learning with more opportunities that address the standard:

  • F-IF.8b: In Algebra 1 Topic 14 and Algebra II Topic 13 there are problems where students interpret expressions in exponential functions, and there could be more opportunities for students to use properties of exponents to interpret functions.
  • G-SRT.7: In Geometry Topic 15 students work with trigonometric ratios, and throughout the topic students engage with problems involving complementary angles. Students’ understanding of G-SRT.7 could be further solidified by offering more opportunities for students to use the relationship between the sine and cosine of complementary angles.
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The instructional materials reviewed for the Agile Mind Traditional series partially meet the expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The materials regularly use age-appropriate contexts and apply key takeaways from Grades 6-8, yet they do not vary the types of real numbers being used.

The materials use age appropriate and relevant contexts throughout the series. The following examples illustrate appropriate contexts for high school students:

  • Algebra I Topic 14 gives the growth of a population at a high school and requires students to solve problems based on the enrollment data.
  • In Algebra I Topic 8 students must figure out how many miles can be driven in a dune buggy while on vacation with a budget of $75.
  • In Algebra I Topic 16 Student Activity Sheets 2 and 3 use a house floor plan as the context for the problems.
  • In Geometry Topic 15 students use trigonometric ratios to find the height of the flagpole in the courtyard.
  • Algebra II Topic 15 begins with context that involves graduation money being put towards the purchase of a new car.

The following problems represent key takeaways from Grades 6-8:

  • Algebra I Topic 3 expands upon 8.F.1 as students define, evaluate, and compare functions. Students look at various situations, create functions, and move into recursively defined functions.
  • Students work with proportions and ratios as a key takeaway from grades 6-8 when working with similar figures and dilations in Geometry Topic 12. Students determine the scale factor (ratio) and a missing coordinate and examine if two figures are similar using proportions.
  • Students extend their knowledge of function concepts as students work with linear, exponential, and quadratic functions in Algebra I. In Algebra II students continue this work with polynomial, rational, logarithmic, and trigonometric functions.

Problems throughout the series provide regular practice with operations on integers and whole numbers. However, problems within the series provide limited practice with operations on fractions, decimals, and irrational numbers. The majority of the series uses whole number coefficients and values unless the context involves money, percents, or irrational constants like π or e. Examples include the following:

  • In Algebra I Topic 8 students solve linear equations and inequalities. Only one exercise in the Guided Practice involved fractions in the equations.
  • In Algebra I Topic 16 Student Activity Sheets 2 and 3, House floor plan, and Student Activity Sheet 5, Dividing Polynomials, a few of the problems have fractional answers. While some problems have fractional terms, calculated answers are generally whole numbers. Within the Guided and More Practice sections, #9 of More Practice contained fractions for coefficients or constants within the polynomials.
  • Although rational numbers are visible throughout the series, student utilization of rational numbers in practice that are not integers is missing in a multitude of topics. For example, when discussing quadratic equations in Algebra I Topic 19 and Algebra II Topic 12, students are asked to solve quadratic equations and inequalities that have coefficients that are integers.
  • In Geometry Topic 14 Pythagorean Theorem and the Distance Formula students work with decimals rounded to the nearest tenth. In the More and Guided Practice sections, triangle problems feature decimals (usually ending in .5); however, no problems involve the use of fractions. Students mostly use whole number side lengths to calculate using the Pythagorean Theorem to find the 3rd side. In Topic 24 Prisms and Cylinders, few problems feature prisms and cylinders with decimals with more than one decimal place, and no problems feature fractional side lengths or answers.
  • In Algebra II Topic 1 Arithmetic and Geometric Sequences and Series, two problems in the More and Guided Practice sections use a fractional difference in a geometric series. Few sequences and series use decimals within this topic.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for fostering coherence through meaningful connections in a single course and throughout the series. Overall, connections between and across multiple standards are made in meaningful ways. Each topic provides a Pre-requisite Skills list and an overview of the topic in Topic at a Glance. The Topic at a Glance provides generic connections within each course and throughout the series.

Examples of connections made within courses include the following:

  • Algebra I Topic 14 Exponential Functions and Equations connects a number of standards as students create and solve equations in one or two variables (A-CED.1,2) as well as recognize the difference between linear and exponential growth (F-LE) and fit an exponential model to a data set and use models to solve problems (S-ID.6a).
  • In Algebra II Topic 5 Polynomial Functions students find zeros using suitable factorizations, if possible, and graph them, connecting A-APR.3, F-IF.7c, and F-IF.8a.
  • In Geometry Topic 12 Dilations and Similarity begins by connecting the idea of transformations (G-CO.2) to deciding if two triangles are similar (G-SRT.2). This Topic uses the properties of similarity to prove AA congruence (G-SRT.2) as well as congruence and similarity criteria to solve problems (G-SRT.5). At the end of this topic, students also prove that all circles are similar (G-C.1).

Examples of connections made between the courses include the following:

  • Transformations can be found throughout the series. The materials first introduce the idea in Algebra I with translating graphs of functions using the graphing calculator in Topics 15 and 17. Students have extensive work with transformations in Geometry using all transformations (Reflect, Rotate, Translate, and Dilate) on shapes. In Algebra II Topic 3 Transforming Functions, the last lesson in this Topic is titled “Making the algebra-geometry connection,” which makes the algebra-geometry connection between transformations. Transformations are seen in a number of Topics after students extensively work on it in Topic 3 of Algebra II.
  • The F-LE standards are connected throughout the series. In Algebra I students compare linear growth and exponential growth in a number of ways and in a number of topics. Students use the idea of linear growth in Geometry to find lines that are parallel and perpendicular. This can be found in Topic 6 which has a lesson called Lines and Algebra. In Algebra II students use the idea from previous coursework to work with logarithms using prerequisite knowledge of exponential functions.
  • A-SSE.2 and A-SSE.3.a begin in Algebra I Topics 16 and 18 as students work with operations on polynomials and solving quadratic equations, and they are further developed in Algebra II Topic 6 as students work with polynomial equations.
  • In Algebra II Topic 22 Modeling Data focuses on determining an appropriate model for data, interpreting the strength of the relationship between two variables, and making predictions in the context of the problem situation. In the Advice for Instruction, a connection is made between characteristics of function families (F-IF.4) and determining the appropriate model for data in Topic 22. In the Choosing a Model subtopic of Topic 22 students are asked to look at the way data points are spread for the US Census data. Based on prerequisite skills students then determine which shape and/or model is most appropriate. When students reach the Fitting Quadratic Data subtopic in Topic 22 they connect their knowledge of Polynomial Functions Topic 15 from Algebra I and Topic 5 from Algebra II to determine that a parabola is the best fit for the data.
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The instructional materials reviewed for the Agile Mind Traditional series partially meet the expectations that the series explicitly identifies and builds on knowledge from Grades 6-8. Materials include and build on content from grades 6-8, however, the content is not clearly identified or connected to specific middle school standards. Although the provided content from Grades 6-8 supports progressions of the high school standards, the Grade 6-8 standards are not identified in either the teacher or student materials.

The following are examples of where the materials build on, but do not explicitly identify standards from Grades 6-8.

  • Algebra I Topic 3 Functions includes Prerequisite Skills listed in the Teacher Materials in Prepare Instruction. The list of skills includes “order of operations, operations with rational numbers, domain and range of a function, solving two-step equations by inspection, and plotting points on the coordinate plane.” However, there is no mention in the teacher and/or student materials of where these prerequisite skills can be found in prior content. The teacher is directed in the Deliver Instruction to review independent and dependent variables. These ideas are used to identify domain and range of functions.
  • In Algebra I Topic 1 students use prerequisite skills such as perimeter and area of polygons and volume of rectangular solids (7.G.6) in Operations with Polynomials. This topic builds on knowledge from Grades 6-8.
  • In Geometry Topic 12 students begin work on an introduction to similarity through dilations. The materials state “The topic dilations and similarity builds on what students have learned about similarity and transformations in middle school in order to generate a precise definition, make connections to transformations, and analyze ways to prove that two triangles are similar.” In the opening question students are expected to determine the distance to place a toy from a flashlight and the height at which to hang the toy such that it casts a particular size shadow on a wall. Students must use proportions (7.RP.2) to solve this problem in relation to similar triangles (G-SRT.5).
  • In Geometry Topic 25 students develop formulas for volume and surface area of pyramids and cones. In the Topics at a Glance section of the Advice for Instruction, the materials indicate that “Students should have seen formulas for computing surface area and volume of three-dimensional figures in middle school mathematics.” In this topic students work with the materials to determine the volume of chocolate needed to make a chocolate pyramid and the amount of materials needed to package the chocolate pyramid (Design problem, G-MG.3 using Surface Area, 7.G.6).
  • Themes beginning in middle school algebra continue and deepen during high school. As early as grades 6 and 7, students begin to use the properties of operations to generate equivalent expressions (6.EE.3, 7.EE.1). In grade 7, they begin to recognize that rewriting expressions in different forms could be useful in problem solving (7.EE.2). In Algebra I Topic 2 Student Activity Sheet 2 Question 5 students look at a situation where they will build borders of a garden. They look at various ways to set up the two gardens. Throughout the topic students are presented with various situations that ask them to write equivalent expressions both in words and in a numerical representation.
  • Students in Grade 8 solve linear equations (8.EE.7) and systems of linear equations (8.EE.8). This concept is built upon in Algebra I Topic 10 as students use this concept to solve real-world problems. In Student Activity Sheet 2 students are given various scenarios to apply their prior knowledge. For example, question 1 involves the context of repairing a gas-powered mower versus buying a new energy-efficient, electric-powered mower. Throughout Student Activity Sheets 3 and 4 students are presented with both systems of equations and inequalities.
  • In Algebra II Topic 20 Design and Data Collection in Statistical Studies begins with an overview introducing the idea of sampling. Once in the topic more information is given concerning surveys and sampling. Within the Subtopic Surveys and Sampling, page 12 uses several random samples to produce a dot plot to make decisions about a population. This topic is aligned with 7.SP.2, “Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.”
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The plus (+) standards, when included, are explicitly identified and coherently support the mathematics which all students should study in order to be college and career ready.

Of the 43 plus standards and 5 plus substandards included in the CCSSM, the materials address 24 of them: N-CN.8,9; N-VM.6-11; A-APR.7; A-REI.8,9; F-IF.7d; F-BF.4c; F-BF.4; F-BF.5; G-SRT.9-11; G-C.4; G-GMD.2; S-CP.8,9; and S-MD.6,7. The materials attend to the full intent of these standards. In general the materials treat these 24 standards as additional content that extends or enriches topics within the unit and do not interrupt the flow of the course. No plus standards were located within the first course of the series, Algebra I.

The following are examples of components of the materials that address the full intent of the plus standards:

  • In Algebra II Topic 6 Exploring Other Polynomial Equations, the materials address the fundamental theorem of algebra alongside finding roots of higher order polynomial equations. Students must find one root and then find additional roots using the quadratic formula to identify complex roots (N-CN.9). In the “Check” section, x^2 +1=0 is shown as (x+i)(x-i) (N-CN.8).
  • In Algebra II Topic 17 Exploring Using the Inverse Matrix, the materials provide examples of identity matrices for a 2x2 and a 3x3 matrix and ask students to identify what a 4x4 identity matrix would look like based on the provided examples. Students then are asked to describe what they notice about the relationship between the number of rows and columns of identity matrices (N-VM.10).
  • In Algebra II Topic 9 Exploring Graphing Rational Functions students use an applet to determine how different parts of rational functions change the graph of a rational function (transformations). Students are also provided opportunities to graph rational functions on the student activity sheets for the topic (F-IF.7d)
  • In Geometry Topic 16 Exploring Law of Sines and Law of Cosines students complete proofs of the laws of sines and cosines and use the laws of sines and cosines to solve problems. (G-SRT.10)
  • In Geometry Topic 26 Exploring Chocolate Hemispheres students are shown and discuss how the volume of a sphere is derived from a cone and cylinder using Cavalieri’s principle. (G-GMD.2)
  • In Algebra II Topic 18 Exploring Permutations and Combinations, students use combinations to determine the number of possible jury members for a trial. (S-CP.9)
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation that the three aspects of rigor are not always treated together and are not always treated separately. Overall, all three elements of rigor are thoroughly attended to and interwoven in a way that focuses on the needs of a specific standard as well as balancing procedural skill and fluency, application and conceptual understanding.

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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation that materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters. There are instances in the materials where students are prompted to use multiple representations to further develop conceptual understanding. In addition, throughout the materials real-world context is used in order to give “concreteness” to abstract concepts, especially when introducing a new topic.

A few examples of the development of conceptual understanding related to specific standards are shown below:

  • A-REI.A: In Algebra I Topic 8 students use algebra tiles to solve linear equations. Students check their work using tables and graphs. Algebra II Topic 13 uses graphing technology to introduce logarithmic equations as students examine graphs and tables of logarithmic functions in a real-world context before solving them analytically in the lesson Analytic Techniques.
  • A-APR.B: Algebra I Topic 18 begins by using a garden to connect x-intercepts of a graph to zeros of a function. There are a series of questions that students work through in order to connect what is happening graphically with the factored form of a function. This is found again in Algebra II Topic 5 Higher Degree Polynomials. There are graphs and functions (standard and factored form) to show the relationship between zeros and factors of polynomials. Students construct polynomials given the zeros.
  • N-RN.1: In Algebra I Topic 13 Laws of Exponents students make tables to see patterns in whole numbers raised to integer exponents, including zero and negative exponents. Students extend this idea to rational exponents by using positive integer exponents and radicals to understand rational exponents.
  • F-LE.1: Algebra I Topic 14 begins by introducing students to linear and exponential growth. Students are introduced to different scenarios using fruit flies and fire ants. Materials use the contexts, along with graphs, tables, and functions, to develop students’ conceptual understanding around linear and exponential growth.
  • G-SRT.2: Geometry Topic 12 contains applets throughout the lesson that allow students to manipulate triangles in order to further understand triangle similarity through student exploration and guided questioning in Advice for Instruction.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for providing intentional opportunities for students to develop procedural skills. Within the lessons, students are provided with opportunities to develop procedural skills for solving problems. Guided Practice and More Practice sections are included within each lesson. These practice sections are often problems with no context and provide students the opportunity to practice procedural skills when called for by the standards.

Some highlights of development of procedural skills include the following:

  • A-APR.1: In Algebra I Topic 16 students multiply binomials to determine the area of rectangles as well as simplify expressions. Students also simplify expressions using polynomial multiplication, addition, and subtraction in the More and Guided Practice sections. In Algebra II Topic 4 students multiply binomials to determine the volume of a rectangular prism as well as simplify expressions using addition, subtraction, multiplication, and division within the More Practice section.
  • G-GPE.7: In Geometry Topic 14 students use the distance formula to compute the perimeter of a triangle as well as to determine if the diagonals of a rectangle bisect each other during More Practice. In Topic 21 students use the distance formula to compute the area of a rhombus as well as find the area and perimeter of a hexagon.
  • F-BF.3: Students are given examples and applicable activities throughout both Algebra I (Topics 3, 5, 6, 12, 15, and 17) and Algebra II (Topics 3, 4, 9, 10, 12, 13, and 20). For example, Algebra II Topic 3 Making the Algebra-Geometry Connection presents several examples addressing this standard. Student Activity Sheet 4, as well as Practice and Assessment, provide opportunities for students to develop necessary skills.
  • G-GPE.4: Geometry Topic 17 Polygons and Special Quadrilaterals explores this standard. Students are given ten examples to view. Example 1 provides definitions and an overview, and Examples 2-10 provide proofs for simple geometric theorems algebraically. Students are then given Student Activity Sheet 4 Practice Problems and Assessment to practice these skills.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation of the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. Students work with mathematical concepts within a real-world context. There are Topics for which the content of the Topic is framed by a real-world context through the Overview. The context used in the Overview is expanded upon throughout the lessons in the Topic.

Examples of students utilizing mathematical concepts and skills in engaging applications include:

  • F-IF.B: The Algebra I Topic 4 Overview introduces students to a skateboarder and his motion. Students use the context of the skateboarder to match his motion to graphs and answer a variety of questions regarding the motion of the skateboarder throughout the lesson. The Topic also uses the idea of elevators and their movement to create graphs and understand various features of the graph. This Topic focuses on interpreting rates of change, and the entire Topic uses a variety of various contexts. In Topic 6 of Algebra I the Constructed Response Assessment problems are examples of applications where students are utilizing mathematical skills to answer various single- and multi-step problems. Students are asked to do things such as find the domain, find and interpret the y-intercept, find a parallel data set, and find the zero and interpret it in relation to the context of the problem. This standard is also found in Algebra II Topic 10 as students engage with the problems in the Topic using sets of data modeled by square root equations to further develop the mathematical skills for examining and identifying the features of graphs.
  • G-SRT.8: Geometry Topic 15 Indirect Measurement begins with the student council of a school finding the height of the flagpole in the school courtyard. The lesson takes students through different strategies to find the height. The last question in the lesson uses an airplane to find the angle of depression. There are a number of application problems in the practice and assessment.
  • S.ID.2: Algebra II Topic 19 has a number of application situations for students to use surveys and sampling. The Overview for the Topic uses real-world context by doing a survey to see if Americans believe life exists beyond earth. This application is used throughout in order to make sense of key terms. The questions in the Advice for Instruction provides the teacher with strategies to support students to work through the application in more meaningful ways. The entire Topic has a variety of application problems in Exploring, Practice, More Practice, Assessment, and Student Activity Sheets.
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for the three aspects of rigor being balanced with respect to the standards being addressed. The structure of the materials lends itself to balancing the three aspects of rigor.

Each Topic includes an Overview, Exploring, Practice, Assessment, and Student Activity Sheets.

  • The Topic Overviews provide a focal point for students to begin thinking about the Topic. They allow for students to relate the topic to a real-world application and/or prior knowledge. This gives students an opportunity to develop conceptual understanding through applications and/or prior knowledge. For example, in Algebra II Topic 4 the teacher is provided with the pieces of a puzzle. In the opening to the lesson teachers state “suppose you buy 100 pencils for $25.” The teacher is then presented with several framing questions “What is the cost of each pencil? How do you know? What operation did you perform to find the answer? If you bought x pencils for $25, what expression would represent the cost of each pencil?”
  • The Exploring section focuses on developing conceptual understanding, in context and/or by using applets. Students are given the tools to build their procedural skills throughout as algorithmic steps are connected to the concepts in this section.
  • Practice has Guided Practice and More Practice for students. There are a variety of types of problems (multiple choice, multiple select, true or false, etc) with a focus on conceptual understanding and procedural skills. Students can get hints and immediate feedback if their answer is correct. If it is incorrect, students receive a statement/question to help direct their thinking.
  • Assessment has two parts, Automatically Scored and Constructed Response. Automatically Scored includes Multiple Choice and Short Answer. This section has questions that require conceptual knowledge, procedural skills, and application of the Topic.
  • Student Activity Sheets follow the online instruction but include additional procedural skill and application problems.

In addition to this, there are MARS tasks throughout that focus on conceptual understanding and application.

The following are examples of balancing the three aspects of rigor in the instructional materials:

  • Algebra I Topic 5 Moving Beyond Sloping Intercept (S.ID.7) has students study data from a table of a skateboarder and their distance traveled during a set of skateboard drills. They use the data table to match the graph of the motion detector data to the path created by moving the computerized skater on the app provided. Students discuss the two parameters necessary to match the graph and develop understanding around steepness (slope) of the line and the constant (intercept). Students use this knowledge to do more procedural skills around standard form and point-slope form. Throughout the Topic, students are given real-world context to explore all concepts in this Topic.
  • Geometry Topic 17 Polygons and Special Quadrilaterals (G-GPE.4) includes examples that review key concepts from previous topics and subtopics. In the topic, the meaning of coordinate proof is given and then stepped through the idea of special quadrilaterals. Students have a series of questions posed to answer (developing conceptual understanding), examples of the process to do a coordinate proof (developing procedural skills), and more questions to check their understanding throughout the lesson on Coordinate Proofs. Students then work in small groups to write a coordinate proof showing that diagonals of a rhombus are perpendicular. After completing these things, students can do the Guided Practice and More Practice to master the skills just learned.
  • Algebra II Topic 2 Understanding Inverse Relations (F-BF.4.a) begins with a real-world context to introduce the idea of inverse relations. Throughout the topic, students are given contextual situations to bridge the idea of functions and their inverses together. The use of tables, graphs, and equations are all used throughout in order for students to understand the idea of inverse. Students are also given the opportunity to compare a function and its inverse as well as practice the skill of finding an inverse from a function. Students are also given real-world context to answer questions which require a conceptual understanding of inverse relations
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation that materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice. The instructional materials reviewed meet the expectations for making sense of problems and persevering in solving them as well as attending to precision; reasoning and explaining; and seeing structure and generalizing and partially meet the expectations for modeling and using tools.

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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation for supporting the intentional development of overarching, mathematical practices (MPs 1 and 6). The instructional materials develop both MP1 and MP6 to the full extent of the MPs. Accurate and precise mathematical language and conventions are encouraged by both students and teachers as they work with course materials. Teachers are given guidance on materials in the Advice for Instruction provided in each Topic. Topics that include a MARS Task list MPs used within the task in the Advice for Instruction. The Advice for Instruction also provides teachers with guidance to foster discussion throughout the materials. This discussion often stresses accurate vocabulary used to increase precision of mathematical language. Emphasis is placed on using units of measure and labeling axes throughout the series. Making sense of answers within the context of a problem is also emphasized. Students persevere in problem solving in lessons through the many real-world application scenarios.

  • In Geometry Topic 14 Block 5 MARS task students make sense of problems and persevere in solving them (MP1) to determine angles and lengths of pieces of wood of a wooden garden chair using similarity and Pythagorean Theorem (G-SRT.8).
  • In Algebra I Topic 1 Student Activity Sheet 1 the first question has students look at three linear graphs and asks them “How are the three graphs similar? How are the three graphs different?” The students analyze the graphs, make conjectures, and plan their strategy. As the teacher directs the lesson, the teacher is told to encourage students’ thinking and emphasize the importance of precise communication by helping students use precise language (MP6) such as slope, how steep, increasing, decreasing, and line versus segment (F-IF.6).
  • Algebra I Topic 8 Student Activity Sheet 4 Question 31 introduces a babysitting situation through Maggie who charges $15 an hour for babysitting. The students look for the possible numbers of hours she can babysit to make enough money for the bag she wants to buy without having to pay taxes on her earnings. In this problem, students are asked to first understand the meaning of the problem, analyze the information, make a conjecture, and try to find an answer. Students need to check their answers and see if their answer makes sense (MP1) (A-CED.3).
  • In Algebra II Topic 18 students are presented with an overarching scenario. In this scenario, Mr. Jones witnesses a crime and gives a witness description. In the scenario, students reason through multiple probability concepts for each portion of Mr. Jones’ account. “How likely is it that Rob is the bad guy? Mr. Jones indicated that the license plate had 6 non-repeating letters.” In this case, the students (playing the role of Rob’s attorney) will have to use the fundamental counting principle to determine how likely it is to have a license plate with non-repeating letters. Students then use basic probability concepts to determine probabilities of events occurring together and/or occurring given that another event occurred first. Lastly, students use the normal distribution to find the percentage of people in Arresta that have a height of 6 feet or more. Students persevere through multiple nuances of the problem to finally determine their solution to whether the jury comes back with a guilty or not guilty verdict (MP1) (S-ID.4).
  • In Algebra II Topic 1 Exploring Arithmetic Sequences and Series students are first presented with an auditorium seating scenario. In this scenario, students are told that the first row has 20 seats and each subsequent row has 3 seats more than the row in front of it. Students are given a table to complete in Student Activity Sheet 2 question 3. From here, the students are asked to determine the constant difference of the linear function that models this situation and also determine the domain and range. The student then writes a precise arithmetic sequence that will help them predict the 21st term in the sequence (MP6) (A-SEE.4).
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation of supporting the intentional development of reasoning and explaining (MP2 and MP3), in connection to the high school content standards, as required by the Standards for Mathematical Practice (MPs). Overall, the majority of the time MP2 and MP3 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, students are expected to reason abstractly and quantitatively, and students are expected to construct viable arguments by justifying along with a few opportunities to critique the reasoning of others in each course. Support and guidance is provided for teachers in the Advice for Instruction for each lesson to assist teacher development of these MPs although the MPs are not explicitly listed in the Advice for Instruction with the exception of the MARS Tasks. MARS Tasks identify specific MPs used within each task in the Advice for Instruction.

  • In Algebra I Topic 3 Functions, Exploring Modeling with Functions begins with the real-world application of a soccer team selling roses as a fundraiser. Students are asked at the beginning of the lesson “What a successful project looks like” and “How will the factors in the list influence the success of the project?” Students think about successful money-making ventures and discuss the factors that make the ventures successful. Students are then given amounts and pricings for two shops to determine which shop would be the best deal for them. Students are then asked to determine costs for ordering a specific numbers of roses from each shop and explaining their reasoning to the class. In this scenario students formulate their own reasoning and explain the evidence and/or prompts that led to their reasoning, illustrating both MP2 and MP3 (F-IF.1).
  • In the Geometry Topic 15 MARS Task students reason abstractly (MP2) about whether a particular triangle is a right triangle by finding the side lengths of neighboring triangles and then using the Pythagorean Theorem to draw a conclusion. Students must also explain how they decided the triangle was a right triangle (MP3) (G-SRT.8, G-SRT.5).
  • One example where students are given the opportunity to critique the reasoning of others (MP3) can be found in Algebra II Topic 21 Student Activity Sheet 5 Problem 8. Students are shown a graph of a cotangent function along with a student’s description of the function. Students are directed to critique the student’s description of the graph of the cotangent function.
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The instructional materials reviewed for the Agile Mind Traditional series partially meet the expectations for supporting the intentional development of modeling and using tools (MPs 4 and 5).

The materials fully develop MP4 as students build upon prior knowledge to solve problems, and they create and use models within many lessons. The materials pose problems connected to previous concepts and a variety of real-world contexts. Students are provided meaningful real-world problems to model using mathematics.

  • In Algebra I Topic 5 Moving Beyond Slope Intercept students try to re-create a graphical representation of the skateboarder’s data. From this data, students are then asked to view and complete a table of the skateboarder’s elapsed time and subsequent distance traveled. Students are asked to determine the change in the time per time interval and the change in distance. Once determined, the students are given a slope and an intercept that allows the student to use algebraic concepts (slope-intercept form of the equation of a line) to determine the formula for determining Teresa’s distance from the motion detector at x seconds. This allows students to apply the mathematics they know to solve problems arising in everyday life (MP4).

The materials do not fully develop MP5 as students are not given the opportunity to choose their own tools, but rather, tools are provided to them. The materials do not encourage the use of multiple tools to conduct investigations even though tools are incorporated throughout the series as students engage in mathematics. Tools are appropriately modeled throughout the series by the teacher, but limited opportunities exist for students to discuss the benefits/limitations of different tools and when to use one tool over another.

  • Within the Geometry course, students rarely have the opportunity to select the appropriate tool with which to solve a problem (MP5). For example, in Topic 1 Exploring The language of geometry students conduct an exploration on angle bisectors. Students are directed to use patty paper, rulers, and protractors. In Block 4 of Topic 21 students use patty paper to manipulate the parts of a “puzzle” within a MARS Task.
  • In Algebra II Topic 6 Polynomial Equations Student Activity Sheet 2 problem 18 students are directed to solve quadratic equations and classify the roots. The materials provide a graph next to each equation and direct students to use their graphing calculator to compare each equation to the graph of the related function. Materials direct students to use a particular method to verify the results, so they do not allow students to choose their own tools (MP5).
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The instructional materials reviewed for the Agile Mind Traditional series meet the expectation of supporting the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards, as required by the Standards for Mathematical Practice (MPs). Overall, the majority of the time MP7 and MP8 are used to enrich the mathematical content and are not treated separately from the content standards. Throughout the materials, support is present for the intentional development of seeing structure and generalizing.

  • In Geometry Topic 13 Geometric Mean students define the geometric mean. Students receive instruction on the connection between the geometric mean and right triangles. They work a puzzle to apply what they have learned and determine the difference between the arithmetic mean and the geometric mean. They are asked to make a general statement about when the arithmetic mean and the geometric mean of two numbers are the same. This provides students with the opportunity to see structure and generalize to a larger idea (MP7).
  • In Algebra II Topic 19 Conditional Probability and Independence students are asked to determine the conditional probability that a British ship is armed given that it appears armed. This question arises from British ships’ need to deter pirates. They then use their conclusions from Exploring Conditional Probability to help determine calculations from the French ship data to show that the event of selecting an armed French ship is not dependent on the event of selecting a French ship that appears to be armed. This allows the students to use requisite knowledge to make generalizations from the sample data (French Ships) to the population (British Ships) (MP7).
  • In Geometry Topic 21 students make estimates about the area of the model and then refine that estimate by using different scales for estimating the area through different geometric figures through repeated reasoning (MP8).
  • In the Algebra I Topic 4 MARS Task Differences students look for patterns and express regularity in repeated reasoning by completing a table (MP8). By completing the table, students are able to notice patterns in a sequence and determine the 7th and 8th term in the sequence. Students must also use their completed table, along with a table of expressions, to look for patterns in both tables to determine the coefficients of the expressions given in the second table.
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The instructional materials reviewed for the Agile Mind Traditional series meet expectations for alignment to the CCSSM for high school, Gateways 1 and 2. In Gateway 1, the instructional materials attend to the full intent of the non-plus standards and allow students to fully learn each non-plus standard, but they do not attend to the full intent of the modeling process when applied to the modeling standards. Although the materials regularly use age-appropriate contexts and apply key takeaways from Grades 6-8, they do not vary the types of numbers being used, and the materials do not explicitly identify and build on knowledge from Grades 6-8 although they do foster coherence through meaningful connections in a single course and throughout the series. In Gateway 2, the instructional materials meet the expectation that materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice. The instructional materials reviewed meet the expectations for making sense of problems and persevering in solving them as well as attending to precision; reasoning and explaining; and seeing structure and generalizing and partially meet the expectations for modeling and using tools.

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The instructional materials reviewed for Agile Mind Traditional series meet expectations that the materials are well designed and take into account effective lesson structure and pacing. Overall, materials are well-designed, and lessons are intentionally sequenced. Students learn new mathematics in the Exploring section of each Topic as they apply the mathematics and work toward mastery. Students produce a variety of types of answers including both verbal and written answers. The Overview for the Topic introduces the mathematical concepts, and the Summary highlights connections within and between the concepts of the Topic. Manipulatives such as algebra tiles and virtual algebra tiles are used throughout the instructional materials as mathematical representations and to build conceptual understanding.

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The instructional materials reviewed for Agile Mind Traditional series meet the expectation for having an underlying design that distinguishes between problems and exercises.

  • Each Topic includes three sections: Overview, Exploring, and Summary. The Overview section introduces the mathematical concepts that will be addressed in the Topic. The Exploring section includes two to four explorations. In these explorations, students learn the mathematical concepts of the Topic through problems that include technology-enhanced animations and full-class activities. The Summary section highlights the most important concepts from the Topic and gives students another opportunity to connect these concepts with each other.
  • Each Topic also includes three additional sections: Practice, Assessment, and Activity Sheets. The Practice section includes Guided Practice and More Practice. Guided Practice consists of exercises that students complete during class periods, providing opportunities for students to apply the concepts learned during the explorations. More Practice contains exercises that are completed as homework assignments. The Assessment section includes Automatically Scored and Constructed Response. These items are exercises to be completed during class periods or as part of homework assignments. They provide more opportunities for students to apply the concepts learned during the explorations. The Activity Sheets also contain exercises, which can be completed during class periods or as part of homework assignments, that are opportunities for students to apply the concepts learned during the explorations.
  • Some Topics also include MARS Tasks, which are exercises that present students with opportunities to apply concepts they have learned from the Topic in which the MARS Task resides or to apply and connect concepts from multiple Topics.
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The instructional materials reviewed for Agile Mind Traditional series meet the expectation for having a design of assignments that is not haphazard with problems and exercises given in intentional sequences.

The sequencing of Topics, and explorations within the Exploring section for each Topic, develops in a way that helps to build students’ mathematical foundations.

  • The Topics are comprised of similar content. For example, in Algebra I, Topic 3 Functions, the Exploring section consists of: Function Notation, Modeling with Functions, and Graphs.
  • Within the explorations for each Topic, problems generally progress from simpler to more complex, incorporating knowledge from prior problems or Topics, which offers students opportunities to make connections among mathematical concepts. For example, in Algebra I, creating linear models for data in Topic 6 incorporates and builds on rate of change from Topic 4.
  • As students progress through the Overview, Exploring, and Summary sections, the Practice (Guided and More), Assessment (Automatically Scored and Constructed Response), and Activity Sheets sections are placed intentionally in the sequencing of the materials to help students build their knowledge and understanding of the mathematical concepts addressed in the Topic.
  • The MARS Tasks are also placed intentionally in the sequencing of the materials to support the development of the students’ knowledge and understanding of the mathematical concepts that are addressed by the tasks.
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The instructional materials reviewed for Agile Mind Traditional series meet the expectation for having a variety in what students are asked to produce.

Throughout a Topic, students are asked to produce answers and solutions as well as explain their work, justify their reasoning, and use appropriate models. The Practice section and Automatically Scored items include questions in the following formats: fill-in-the-blank, multiple choice with a single correct answer, and multiple choice with more than one correct answer. Constructed Response items include a variety of ways in which students might respond, i.e. multiple representations of a situation, modeling, or explanation of a process. Also, the types of responses required vary in intentional ways. For example, concrete models or visual representations are expected when a concept is introduced, but as students progress in their knowledge, students are expected to transition to more efficient solution strategies or representations.

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The instructional materials reviewed for Agile Mind Traditional series meet the expectation for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models. The materials include a variety of virtual manipulatives, as well as, integrate hands-on activities that allow the use of physical manipulatives.

Most of the physical manipulatives used in Agile Mind are commonly available: ruler, patty paper, graph paper, algebra tiles, and graphing calculators. Due to the digital format of the materials, students also have the opportunity to represent proportional relationships virtually with a table and graph and generate random samples to draw inferences. Each Topic has a Prepare Instruction section that lists the materials needed for the Topic. Manipulatives accurately represent the related mathematics. For example, in Geometry Topic 23, Relating 2-D and 3-D objects, students use models of prisms, cones, and spheres that can be cut. In addition, they use modeling clay, fishing wire (or dental floss), and linking cubes throughout the topic.

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The instructional materials reviewed for Agile Mind Traditional series have a visual design that is not distracting or chaotic but supports students in engaging thoughtfully with the subject. The student materials are clear and consistent between Topics within a grade-level as well as across grade-levels. Each piece of a Topic is clearly labeled, and the explorations include Page numbers for easy reference. Problems and Exercises from the Practice, Assessment, and Activity Sheets are also clearly labeled and consistently numbered for easy reference by the students. There are no distracting or extraneous pictures, captions, or "facts" within the materials.

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The instructional materials reviewed for Agile Mind Traditional series meet expectations that materials support teacher learning and understanding of the standards. The instructional materials provide Framing Questions and Further Questions that support teachers in delivering quality instruction, and the teacher’s edition is easy to use and consistently organized and annotated. Different sets of interactive, print, and video essays provide teachers with adult-level explanations or examples of advanced mathematics concepts to help them improve their own knowledge of the subject. Although each Topic contains a list of Prerequisite Skills, this list does not connect any of the skills to specific standards from previous grade levels, so the instructional materials partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

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The instructional materials reviewed for Agile Mind Traditional series meet the expectation for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development. The Deliver Instruction section for each Block of a Topic includes Framing Questions for the start of each lesson. For example, in Algebra 1, Topic 8 Block 6, the Framing Questions are: “Would you use an equation or an inequality to describe this situation? How many variables will you need to describe this situation? Why?” During the lesson, the Deliver Instruction section includes multiple questions that teachers can ask while students are completing the activities. At the the end of each lesson, Deliver Instruction includes Further Questions. For example, in Geometry, Topic 7 Block 3 “Why can a triangle never have two obtuse angles? Two right angles? How could knowing the sum of the angles of a triangle help you find the sum of the angles of a quadrilateral? What about any polygon?”

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The instructional materials reviewed for Agile Mind Traditional series meet the expectation for containing a teacher’s edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The materials contain Professional Support which includes a Plan the Course section and a Scope and Sequence document. The Plan the Course section includes Suggested Lesson-planning Strategies and Planning Resources. Each Topic contains an Advice for Instruction section, and that is divided into Prepare Instruction and Deliver Instruction. For each Topic, Prepare Instruction includes Goals and Objectives, Topic at a Glance, Prerequisite Skills, Resources, and Language Support, and for each Block within a Topic, Deliver Instruction includes Agile Mind Materials, Opening the Lesson, Framing Questions, Lesson Activities, and Suggested Assignment. In Lesson Activities, teachers are given ample annotations and suggestions as to what parts of the materials should be used when and Classroom Strategies that include questions to ask, connections to mathematical practices, or statements that suggest when to introduce certain mathematical terms or concepts.

Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. For example, in Algebra II, Topic 8 Block 5 teachers are directed to, “Use the animation on page 1 to introduce the idea of area varying with more than one dimension. As you view each new panel, have students respond to the appropriate questions on their Student Activity Sheets. Then, play the panel to confirm their responses. [SAS 4, questions 1-5]”

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The instructional materials reviewed for Agile Mind Traditional series meet the expectation for containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

In Professional Support, Professional Learning, there is a group of four interactive essays in each course entitled “Developing Concepts Across Grades”, and the topics for these four essays are Functions, Volume, Rate, and Proportionality. Each essay examines the progression of the concept from Grades 6-8 through Algebra I, Algebra II, Geometry and beyond. These interactive essays give teachers the opportunity to not only make connections between the courses they are teaching and previous courses, but they also give teachers the opportunity to improve their own knowledge in regards to connections that will be made between the courses they are teaching and future courses.

In addition to “Developing Concepts Across Grades”, each course also contains a section of interactive essays entitled “Going Beyond (course name)”. In Algebra I, there are three essays in this section: Average and Instantaneous Rates of Change, The Slope of a Curve, and The Relationship Between Exponential and Logarithmic Functions. In Algebra II, there are two essays in this section: Linearizing Data Using Logarithms, and From Rates of Change to Derivatives. In Geometry, there are three essays in this section: Trigonometric Functions, Understanding Area of Irregular Shapes using Calculus, and Radians. Along with having their own section in Professional learning, each of these essays are also referenced in Deliver Instruction for the Blocks where they are appropriate under the title of Teacher Corner. For example, in Geometry, the essay Trigonometric Functions is referenced for teachers in Block 2 of Topic 15, Right Triangle and Trig Relationships, or in Algebra II, Linearizing Data using Logarithms is referenced in Block 3 of Topic 14, Logarithmic Functions.

In Professional learning, there are also sets of Video or Print essays. The Print essays are divided as either Curriculum or Course Management Topics, and there is a series of three essays in Algebra II titled “Rational Functions and Crossing Asymptotes” that addresses mathematical concepts that extend beyond the current course. The Video Essays are: Teaching with Agile Mind, More Teaching with Agile Mind, and Dimensions of Mathematics Instruction.

) [37] => stdClass Object ( [code] => 3i [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Prepare Instruction section for each Topic contains a list of Prerequisite Skills, but this list does not connect any of the skills to specific standards from previous grade levels. For example, in Geometry Topic 3 the Prerequisite Skills include “identify translations, reflections, and rotations in images; measure angles with a protractor; apply the definition of perpendicular bisector and midpoint.” In Prepare Instruction, the Topic at a Glance occasionally provides general references to how concepts will be used in future courses. For example, in Algebra II, Topic 7 “builds upon students' previous work with rational exponents. Students will move from working with numeric expressions involving rational exponents to algebraic expressions. Students will also translate between exponent form and radicals.”

) [38] => stdClass Object ( [code] => 3j [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series provide a list of lessons in the teacher's edition, cross­‐referencing the standards covered and providing an estimated instructional time for Topics and Blocks. For each course, the materials provide a Scope and Sequence document which includes the number of Blocks of instruction for the duration of the year, time in minutes that each Block should take, and the number of Blocks needed to complete each Topic. The Scope and Sequence document lists the CCSSM addressed in each Topic, but there is no part of the materials that aligns Blocks to specific content standards. The materials also provide Alignment to Standards in the Course Materials which allows users to see the alignment of Topics to the CCSSM or the alignment of the CCCSM to the Topics. The Deliver Instruction section contains the Blocks for each Topic. The Practice Standards Connections, found in Professional Support, gives examples of places in the materials where each MP is identified.

) [39] => stdClass Object ( [code] => 3k [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series do not contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

) [40] => stdClass Object ( [code] => 3l [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series do not contain explanations of the instructional approaches of the program and identification of the research-based strategies within the teaching materials. There is a Professional Essays section which addresses a broad overview of mathematics and clips of teachers using Agile Mind in Algebra I, Geometry, and Algebra II as discussed in indicator 3h.

) [41] => stdClass Object ( [code] => assessment [type] => component [report] =>

The instructional materials for Agile Mind Traditional series partially meet exceptions that materials offer teachers resources and tools to collect ongoing data about students progress on the Standards. Opportunities for ongoing review and practice, and feedback occur in various forms. Standards are identified that align to the Topic; however, there is no mapping of Standards to items. There are limited opportunities for students to monitor their own progress, and there are no assessments that explicitly identify prior knowledge within and across grade levels. The materials include few opportunities to identify common misconceptions, and strategies to address common errors and misconceptions are only found in a few Deliver Instruction topics.

) [42] => stdClass Object ( [code] => 3m3q [type] => criterion [report] => ) [43] => stdClass Object ( [code] => 3m [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series partially meet the expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels. The materials do not provide any assessments that are specifically designed for the purpose of gathering information about students’ prior knowledge, but the materials do provide indirect ways for teachers to gather information about students’ prior knowledge if teachers decide to use them that way.

In Prepare Instruction for each Topic, there is a set of Prerequisite Skills needed for the Topic, and the Overview for each Topic provides teachers with an opportunity to informally assess students prior knowledge of the Prerequisite Skills. For example, in Algebra 1, Topic 4 Block 1: Rate of Change, students should understand multiple concepts to be successful in the lesson. The prerequisite skills required for the lesson are: “Reading and constructing graphs, Domain and range and Exponential and quadratic patterns in data”. In Algebra II, there are a few places where Deliver Instruction offers teachers explicit guidance for gathering information about students’ prior knowledge within and across courses. For example, in Topic 1, Deliver Instruction for the Overview and Student Activity Sheet 1 states, “The material on these two pages is designed to activate students' prior learning from Algebra I, but keep it in the context of setting the stage for Algebra II. Do not succumb to the temptation of re-teaching everything students should have learned in Algebra I. Instead, use the material on these pages to actively engage students in recall of prior work, facilitating students’ conversations to resurface what they have learned previously about the key function families of Algebra I. This will set students up for success not only for this topic but also for work in future topics with new function families.”

) [44] => stdClass Object ( [code] => 3n [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for providing strategies for teachers to identify and address common student errors and misconceptions. Across the series, common student errors and misconceptions are identified and addressed in Deliver Instruction as parts of “Classroom Strategy”, but “Classroom Strategy” is not solely used for identifying and addressing common student errors and misconceptions.

  • In Algebra I Topic 6, Deliver Instruction for Block 3 states, “Another common mistake is for students to look at the differences in the y-values only, and not relate these changes to the differences in the x-values. This mistake sometimes comes from misconceptions students create when exploring linear data where the x-values only increase by 1 unit. Throughout all data interpretation in tables, refer to the ratio or rate of change. If the students say that y-values are increasing by some number, ask them to complete their sentence by adding a description of how the corresponding x-values are changing, even when the change in x is only 1 unit.”
  • In Geometry Topic 22, Deliver Instruction for Block 2 states, “Students often mistakenly use diameter in computations instead of radius and vice versa. Similarly, students often lose track of when to use a 1/2 or 2 in computations. This can be especially confusing when looking at something like half of a circumference.”
  • In Algebra II Topic 11, Deliver Instruction for Block 4 states, “Be sure to discuss with students the importance of isolating the radical. Show students an example of what happens if they don’t. Also remind them about how to expand a binomial. Be sure they do not distribute the exponent over addition or subtraction.”
) [45] => stdClass Object ( [code] => 3o [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series meet the expectation for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials provide opportunities for ongoing review and practice, and feedback occurs in various forms. Within interactive animations, students submit answers to questions or problems and feedback is provided by the materials. Practice problems and Automatically Scored Assessment items are submitted by the students, and immediate feedback is provided letting students know whether or not they are correct and, if incorrect, suggestions are given as to how the answer can be improved. The Lesson Activities in Deliver Instruction provide some suggestions for feedback that teachers can give while students are completing the lessons.

) [46] => stdClass Object ( [code] => 3p [type] => indicator [report] => ) [47] => stdClass Object ( [code] => 3p.i [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for assessments clearly denoting which standards are being emphasized. The items provided in the Assessment section align to the standards addressed by the Topic, but the individual items are not clearly aligned to particular standards. The set of standards being addressed by a Topic can be found in the Scope and Sequence document or in Course Materials through Alignment to Standards. The MARS Tasks also do not clearly denote which CCSSM are being emphasized.

) [48] => stdClass Object ( [code] => 3p.ii [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The MARS Tasks and selected Constructed Response items in Algebra II are accompanied by rubrics aligned to the task or item that show the total points possible for the task and exactly what students need to do in order to earn each of those points. The remainder of the Constructed Response items in Algebra II, along with all of the Constructed Response items in Algebra I and Geometry, are accompanied by complete solutions, but rubrics aligned to these Constructed Response items are not included. For both the MARS Tasks and the Constructed Response items, alternate solutions are provided when appropriate, but sufficient guidance to teachers for interpreting student performance and suggestions for follow-up are not provided with most of the MARS Tasks or the Constructed Response items. In Algebra I, there are four Constructed Response items that are accompanied by a professional essay titled “Learning from Student Work”, and Algebra I and Geometry each include a MARS Task that is accompanied by a professional essay that provides guidance to teachers for interpreting student performance and suggestions for follow-up.

) [49] => stdClass Object ( [code] => 3q [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series offer few opportunities for students to monitor their own progress. Throughout the Exploring, Practice, and Automatically Scored Assessment sections, students get feedback once they submit an answer, and in that moment, they can adjust their thinking or strategy. Goals and Objectives for each Topic are not provided directly to students, but they are given to teachers in Prepare Instruction. There is not a systematic way for students to monitor their own progress on assignments or the Goals and Objectives for each Topic.

) [50] => stdClass Object ( [code] => differentiated-instruction [type] => component [report] =>

The instructional materials for Agile Mind Traditional series partially meet expectations that materials support teachers in differentiating instruction for diverse learners within and across grades. Activities provide students with multiple entry points and a variety of solution strategies and representations. The materials also provide strategies for ELL and other special populations, but they do not provide strategies for advanced students to deepen their understanding of the mathematics. Grouping strategies are designed to ensure roles for each group member.

) [51] => stdClass Object ( [code] => 3r3y [type] => criterion [report] => ) [52] => stdClass Object ( [code] => 3r [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Each Topic consists of three main sections, Overview, Exploring, and Summary, and these three sections are divided into Blocks. Each Block contains lesson activities, materials for Practice, Assessment, and Activity Sheets, along with a MARS Tasks if applicable for the Topic. In each Topic, the Blocks and lesson activities are sequenced for the teacher. In the Advice for Instruction for each Topic, Deliver Instruction for each Block contains instructional notes and classroom strategies that provide teachers with key math concepts to develop, sample questions to ask, ways in which to share student answers, and other similar instructional supports.

) [53] => stdClass Object ( [code] => 3s [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation for providing teachers with strategies for meeting the needs of a range of learners. Overall, the instructional materials embed multiple visual representations of mathematical concepts where appropriate, include audio recordings in many explorations, and give students opportunities to engage physically with the mathematical concepts. However, the instructional notes provided to teachers do not consistently highlight strategies that can be used to meet the needs of a range of learners. When instructional notes are provided to teachers, they are general in nature and are intended for all students in the class, and they do not explicitly address the possible range of needs for learners. For example, Algebra II, Topic 6 Block 3 the Deliver Instruction states, “To save time, break the classroom into three sections. Have one section solve the first equation, another the second, and the last section the third equation. Give each section time to solve their equation and check their work with each other, as well as time to interpret their graph and number of solutions and to pick a person to present. Have a member from each section come up and present the work.”

In some explorations, teachers are provided with questions that can be used to extend the tasks students are completing, which are beneficial to excelling students. For struggling students, teachers are occasionally provided with strategies or questions they can use to help move a student’s learning forward. The Summary for each Topic does not provide any strategies or resources for either excelling or struggling students to help with their understanding of the mathematical concepts in the Topic.

) [54] => stdClass Object ( [code] => 3t [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series meet the expectation that materials embed tasks with multiple entry­-points that can be solved using a variety of solution strategies or representations. Overall, tasks that meet the expectations for this indicator are found in some of the Constructed Response Assessment items and Student Activity Sheets that are a part of all Topics. MARS Tasks embedded in some of the Topics have multiple entry-points and can be solved using a variety of solution strategies or representations. For example, Geometry, Topic 14 Mars Task: Garden Chair, students determine an angle made by the wooden construction of the chair. Students can begin the problem by using either the angle sum theorem or the exterior angle theorem. Students could find all the angles in the problem first or the minimum required for the problem.

) [55] => stdClass Object ( [code] => 3u [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series meet the expectation that the materials suggest accommodations and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The materials provide suggestions for English Language Learners and other special populations in regards to vocabulary and instructional practices throughout each course in the series. In Prepare Instruction for Topic 1 of each course, Teaching Special Populations of Students refers teachers to the Print Essay entitled “Teaching English Language Learners” in Professional Support, and that essay describes general strategies that are used across the series such as a vocabulary notebook, word walls, and concept maps. Teaching Special Populations of Students also describes general strategies that are used across the series for other special populations, and these strategies include progressing from concrete stage to representational stage to abstract stage and explicitly teaching metacognitive strategies through think alouds, graphic organizers, and other visual representations of concepts and problems.

In addition to the general strategies mentioned in Teaching Special Populations of Students, there are also many specific strategies listed across each course of the series in Deliver Instruction. In Deliver Instruction, Support for ELL/other special populations includes strategies that can be used with both English Language Learners and students from other special populations, and strategies specific to other special populations can also be found in Classroom strategy or Language strategy. An example of Support for ELL/other special populations from Geometry, Topic 1, Block 5, Pages 2-3 is “This puzzle acts much like a Cloze activity, in which key vocabulary words are removed from a paragraph, to build confidence and quickly assess fluency with the vocabulary. This type of activity can be particularly helpful to reinforce key understandings for students with a variety of learning differences, including challenges with language acquisition and processing. ELL students should add the labeled diagram to their vocabulary notebooks.” An example of a strategy for other special populations from Algebra II, Topic 13, Block 4, Page 10 is “Language strategy. You may wish to use a paired reading strategy for this page. In paired reading, one student in the pair reads the first sentence to the other student in the pair. The second student then paraphrases what was read back to the first student. Then, the students switch roles and repeat the process for the next sentence. This continues until the entire page is read and processed. This strategy can be modified if one student in the pair has a reading challenge so that only one student reads the passage, but both students take turns paraphrasing what was read.”

) [56] => stdClass Object ( [code] => 3v [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Traditional series partially meet the expectation that the materials provide opportunities for advanced students to investigate mathematics content at greater depth. Overall, all of the problems provided in the materials are on grade level, and the materials are designed so that most of the problems are assigned to all students over the course of the school year. However, there are a few problems that are on grade level and are not assigned to all students, and these problems could be used for advanced students to investigate mathematics content at greater depth. For example, in Algebra I, Topic 4 the MARS Task “Differences” does not have to be assigned to all students at the completion of the Topic, and could be assigned to advanced students.

) [57] => stdClass Object ( [code] => 3w [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series provide a balanced portrayal of various demographic and personal characteristics. The activities are diverse, meeting the interests of a demographically, diverse student population. The names, contexts, videos, and images presented display a balanced portrayal of various demographic and personal characteristics.

) [58] => stdClass Object ( [code] => 3x [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series provide opportunities for teachers to use a variety of grouping strategies. The Deliver Instruction Lesson Activities include suggestions for when students could work individually, in pairs, or in small groups. When suggestions are made for students to work in small groups, there are no specific roles suggested for group members, but teachers are given suggestions to ensure the involvement of each group member. For example, in Algebra II, Topic 10 Block 2 Deliver Instruction teachers are told to “Have student pairs solve the equation algebraically, then use the solution to determine how fast Chloe runs and how long her workout takes. [SAS 2, question 17] Ask for student volunteers to share their processes. Use page 10 as needed to verify responses.”

) [59] => stdClass Object ( [code] => 3y [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series do not encourage teachers to draw upon home language and culture to facilitate learning. Questions and contexts are provided for teachers in the materials, and there are no opportunities for teachers to adjust the questions or contexts in order to integrate the home language and culture of students into the materials to facilitate learning.

) [60] => stdClass Object ( [code] => effective-technology-use [type] => component [report] =>

The instructional materials for Agile Mind Traditional series are web-based and platform neutral but do not include the ability to view the teacher and student editions simultaneously. The materials embed technology enhanced, interactive virtual tools, and dynamic software that engage students with the mathematics. Opportunities to assess students through technology are embedded. The technology provides opportunities to personalize instruction; however, these are limited to the assignment of problems and exercises. The materials cannot be customized for local use. The technology is not used to foster communications between students, with the teacher, or for teachers to collaborate with one another.

) [61] => stdClass Object ( [code] => 3z3ad [type] => indicator [report] => ) [62] => stdClass Object ( [code] => 3z [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices. Given the digital platform of the materials, the inclusion of interactive tools and virtual manipulatives/objects helps to engage students in the MPs in all of the Topics, and the use of animations in all of the Topics provides for some examples as to how the interactive tools and virtual manipulatives can be used.

) [63] => stdClass Object ( [code] => 3aa [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series are web-based and compatible with multiple internet browsers (Chrome, Firefox, and Internet Explorer). In addition, the materials are “platform neutral” and allow the use of tablets with ChromeOS, Android, or iOS operating systems, but they do not support the use of mobile devices. However, the transition between student and teacher materials is not fluid. There are no direct links between the student and teacher materials, and the student and teacher materials cannot be viewed simultaneously.

) [64] => stdClass Object ( [code] => 3ab [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. All Practice and Automatically Scored Assessment questions are designed to be completed using technology. These items cannot be edited or customized.

) [65] => stdClass Object ( [code] => 3ac [type] => indicator [report] => ) [66] => stdClass Object ( [code] => 3ac.i [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series include few opportunities for teachers to personalize learning for all students. Within the Practice and Assessment sections, the teacher can choose which problems and exercises to assign students, but these problems and exercises cannot be modified for content or wording from the way in which they are given. Other than being able to switch between English and Spanish in My Glossary, there are no other adaptive or technological innovations that allow teachers to personalize learning for all students.

) [67] => stdClass Object ( [code] => 3ac.ii [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series cannot be easily customized for local use. Within My Courses, there are not any options for modifying the sequence or structure of the Topics or any of the sections within the Topics.

) [68] => stdClass Object ( [code] => 3ad [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Traditional series provide few opportunities for teachers and/or students to collaborate with each other. Under My Agile Mind, teachers can communicate with students through the Calendar and Score and Review. There are no opportunities for teachers to be able to collaborate with other teachers.

) ) [isbns] => Array ( ) ) 1

Agile Mind Middle School Mathematics (2016)

Agile Mind | 6-8 | 2016 Edition

Sixth Grade

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    [title] => Agile Mind Middle School Mathematics (2016)
    [url] => https://www.edreports.org/math/agile-mind-middle-school-mathematics-a/sixth-grade.html
    [grade] => Sixth Grade
    [type] => math-k-8
    [gw_1] => Array
        (
            [score] => 13
            [rating] => meets
        )

    [gw_2] => Array
        (
            [score] => 17
            [rating] => meets
        )

    [gw_3] => Array
        (
            [score] => 31
            [rating] => meets
        )

)
1                                            stdClass Object
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    [version] => 2.0.0
    [id] => 324
    [title] => Middle School Math - Grade 6
    [report_date] => 2018-03-20
    [grade_taxonomy_id] => 19
    [subject_taxonomy_id] => 5
    [notes] => 

Agile Mind made revisions that affected the scoring and reports for Rating Sheet 2, indicator 3h, and Rating Sheet 4, indicator 3u. These revisions also affected the rating for Gateway 3 so that Agile Mind Grade 6 meets expectations for Instructional Supports and Usability.

[reviewed_date] => 2017-06-08 [revised_date] => 2018-03-20 [gateway_1_points] => 13 [gateway_1_rating] => meets [gateway_1_report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet expectations for focus and coherence. The instructional materials do not assess topics beyond Grade 6, and students and teachers using the materials as designed devote the large majority of instructional time to the major work of the grade. The instructional materials meet the expectations for coherence, and they show strengths in having an amount of content that is viable for one school year and fostering coherence through connections within the grade.

[gateway_2_points] => 17 [gateway_2_rating] => meets [gateway_2_report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for rigor and the mathematical practices. The materials meets the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and applications. The materials also meet the expectations for mathematical practices. Overall, the materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialized language of mathematics.

[gateway_3_points] => 31 [gateway_3_rating] => meets [report_type] => math-k-8 [series_id] => 74 [report_url] => https://www.edreports.org/math/agile-mind-middle-school-mathematics-a/sixth-grade.html [gateway_2_no_review_copy] => Materials were not reviewed for Gateway Two because materials did not meet or partially meet expectations for Gateway One [gateway_3_no_review_copy] => This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two [meta_title] => [meta_description] => [meta_image] => [data] => Array ( [0] => stdClass Object ( [code] => focus [type] => component [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet expectations for focus. The materials can be utilized to appropriately assess grade-level content, and a majority of the time is spent on major work of the grade.

) [1] => stdClass Object ( [code] => 1a [type] => criterion [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet expectations for not assessing topics before the grade-level in which the topic should be introduced. Overall, there are not assessment items that align to topics beyond Grade 6.

) [2] => stdClass Object ( [code] => 1a [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials for Agile Mind Grade 6 meet the expectations for assessing grade-level content. The majority of the Grade 6 assessment content was appropriate for the grade, but there are some items in the assessments that align to standards above Grade 6. The assessment items that align to standards above Grade 6 could be modified/omitted without effecting the underlying structure of the materials or are mathematically reasonable in relationship to the Grade 6 standards.

The questions within the Practice and Assessment sections were reviewed for this indicator. The Practice sections within each topic contain multiple questions under the categories of Guided Practice and More Practice. The Assessment sections within each topic contain Automatically Scored questions and Constructed Response questions.

The assessment items that are above Grade 6 but are mathematically reasonable in relationship to the Grade 6 standards are as follows:

  • In Topic 2 Automatically Scored 3, Guided Practice 13 and More Practice 12 ask students to identify different ways to write a given proportion (7.RP.2).
  • In Topic 2 Constructed Response 2a has students describe how proportions can be used to show that three photos are proportional (7.RP.2).
  • In Topic 2 Guided Practice 14 asks students to determine if one figure is proportional to another one, and More Practice 8 and 9 have students analyze the outcome of specified enlargements to photographs to determine if the result is proportional to the original (7.RP.2a).
  • In Topic 2 Constructed Response 3, parts c and d ask students to determine the scale factor needed to create one geometric figure from another and to determine the area of a figure created with a given scale factor (7.G.1).

The assessment items that include content from future grades and could be modified/omitted without effecting the underlying structure of the materials are as follows:

  • In Topic 7 Automatically Scored 9 and Constructed Responses 1 and 3 have students qualitatively describe a relationship between two variables by analyzing distance-time graphs (8.F.5).
  • In Topic 11 More Practice 26, 27, and 29 have students write equations and inequalities of the form y = mx + b (7.EE.4).
  • In Topic 12 Automatically Scored 1 students use square root to find the length of one side of a square (8.EE.2).
) [3] => stdClass Object ( [code] => 1b [type] => criterion [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for students and teachers devoting the majority of class time to the major work of the grade when materials are used as designed. Overall, the materials spend approximately 65% of class time on the major work of Grade 6.

) [4] => stdClass Object ( [code] => 1b [type] => indicator [points] => 4 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for spending the majority of class time on the major clusters of the grade. Overall, the instructional materials spend approximately 65% of class time on the major clusters of Grade 6.

For this indicator, the following were examined: all Blocks of instruction within all Topics in Course Contents, Alignment to Standards in Course Materials, the Mathematics 6 Scope & Sequence with Common Core State Standards document in Professional Support, and the Block descriptions for each Topic located within Deliver Instruction under Advice for Instruction in Professional Support.

There are fifteen topics divided into the following categories: Overview, Explore, Summary, Practice, and Assessment. Each Topic contains 6 to 14 Blocks of instruction, and each Block of instruction represents a 45-minute class period.

In the Block descriptions for each Topic, individual activities are not assigned specific amounts of time, or ranges of time, for the activities to be completed. Thus, when calculating the percentage of class time spent on the major cluster of the grade, two perspectives were appropriate, Topics and Blocks. For these materials, analysis by Blocks is the most appropriate because the Topics do not have an equal number of Blocks within them, and the Blocks are not subdivided into smaller increments.

There are 141 Blocks in Grade 6 Agile Mind.

  • All Blocks in Topics 2, 3, 4, 7, 8, 9, 10, and 11 aligned to major work of the grade. In addition, three Blocks in Topic 6 were also aligned to major work of the grade. Thus, 73 of 141 Blocks, approximately 52%, aligned to major clusters of Grade 6.
  • A review of Topics and Blocks aligned to supporting clusters in Grade 6 demonstrated that in 19 of these Blocks, the supporting clusters connected to major clusters as well. Thus 19 of 141 Blocks connected supporting clusters of the grade to major clusters of Grade 6.

Overall, 92 out of 141 Blocks, approximately 65%, connected or aligned to major work of the grade.

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The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for being coherent and consistent with the Standards. The instructional materials show strengths in having an amount of content that is viable for one school year but do not always make explicit connections between prior knowledge and future learning and the major work of the grade. Therefore, the progressions in the Standards are not always evident. The materials foster coherence within grade level work.

) [6] => stdClass Object ( [code] => 1c1f [type] => criterion ) [7] => stdClass Object ( [code] => 1c [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. Overall, supporting content is found primarily in Topics 5-7 and 9-15, and the supporting content does enhance focus and coherence by engaging students in the major work of the grade. Examples of the connections between supporting work and major work include the following:

  • In Topic 1 Block 4 major standards 6.EE.1,2 are tied to supporting standard 6.NS.2 as students divide multi-digit numbers when evaluating numerical expressions.
  • In Topic 1 Block 7 major standard 6.EE.3 is tied to supporting standard 6.NS.4 as students find the area of a segmented banner using the distributive property.
  • In Topic 7 Blocks 1-8 major standard 6.RP.3 is tied to supporting standards 6.NS.2,3 as students are solving various rate and measurement problems involving the division of multi-digit numbers and the four operations with multi-digit decimals.
  • In Topic 11 major cluster 6.EE.B is tied to supporting standards 6.NS.2,3 as students are writing and solving equations and inequalities with a variety of rational numbers.
  • In Topic 12 Block 4 major standard 6.RP.3 is tied to supporting standard 6.G.1 as students find the percentage of areas in the MARS Task: Flag.
  • In Topics 12 and 13 supporting cluster 6.G.A is connected to major cluster 6.EE.B as students occasionally write the expressions and equations needed to calculate areas, surface areas, and volumes.
  • In Topic 14 Block 4 major standard 6.RP.3 is tied to supporting standard 6.SP.4 as students use percentages to display and describe numerical data in a circle graph.
  • In Topic 15 Block 2 major standard 6.EE.1 is tied to supporting standard 6.SP.3 as students write expressions that are evaluated to find the mean of a set of data.
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The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for the amount of content designated for one grade-level being viable for one school year in order to foster coherence between grades. The suggested pacing contains 15 Topics and 138-141 Blocks (days) of instruction, including assessments. According to the Agile Mind Mathematics 6 Scope and Sequence, each block is expected to last 45 minutes. Some lessons (Constructed Response, MARS tasks) may take longer than indicated.

Each Block includes the following sections: Overview, Exploring, Summary, and Assessment. The Exploring pages are categorized by math concept and can be discussed and reviewed as a class or by individuals/small groups of students. The Scope and Sequence suggests that Block 6: Metric Conversion in Topic 7: Rates and Measurement can be used as extension content. In addition, Blocks 7-8: Representing 3-Dimensional Shapes in Topic 13: Surface Area and Volume can also be used as an extension activity related to different views of 3-dimensional shapes.

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The instructional materials reviewed for Agile Mind Grade 6 partially meet the expectations for being consistent with the progressions in the Standards. Overall, the materials develop according to the grade-by-grade progressions in the Standards, and they give all students extensive work with grade-level problems. However, content from prior or future grades is not always clearly identified or related to grade-level work, and the materials do not always relate grade-level concepts explicitly to prior knowledge from earlier grades.

Examples of Grade 6 materials in which off grade-level content is present and not identified as such includes the following:

  • Topics 2 and 3 address scale factor and 3-D representations (7.G.A).
  • In Topic 5 Blocks 1, 2, 4, and 6 and Constructed Response 2, identified as 6.NS.B, focus on addition and subtraction of fractions, 5.NF.1,2.
  • In Topic 6 Blocks 1-4 and Constructed Response 1, identified as 6.NS.B, focus on multiplication of fractions, 5.NF.4-6.

The Grade 6 materials provide extensive work with grade-level standards. All students are expected to complete the same problems, and lessons or ideas presented for differentiated instruction also include grade-level problems. The MARS tasks that are included, especially the ones in Topics 9 and 12, are places where students are given the opportunity to engage with the grade-level standards to their full intent.

In lessons where prior knowledge is included, identification of content from prior grades is mentioned in four components of the materials, but the identification is general and not explicitly connected to a grade-level or standard. Examples from the four components are as follows:

  • In the first paragraph of the About the Course section, there is a brief, general overview of topics of which students acquired a foundation prior to Grade 6.
  • The first paragraph in Agile Mind Mathematics 6 Scope and Sequence, 2016-2017 briefly references prior work in numbers and operations, geometry, measurement, data, and equivalent fractions and how these concepts connect to Grade 6 work.
  • The Advice for Instruction section references prior work in different places, although specific standards are not referenced. Some examples of this include:
    • Topic 1 Block 1 states “Use page 5 to remind students that they have used these operations with whole numbers in their previous math classes. The intent of this topic is to develop fluency and extend that work to some other ways to represent numbers, including factors, multiples, and exponents.”
    • In Topic 6, Topic at at glance states, “This topic is intended to help students make sense of and formalize processes for operating with positive rational numbers. Once students understand how to operate with fractions, they can use that knowledge to learn how to operate with decimals. Students will learn through the topic that fractions and decimals are more similar than different.”
    • In Topic 6 the first Classroom Strategy in Block 3 references both prior and future work as it connects a lesson on multiplying and dividing rational numbers to the base‐10 area model of multiplying multi‐digit whole number and the later use of algebra tiles for multiplying polynomials. The connection to prior work does not explicitly connect to a previous, grade-level standard.
    • In Topic 9 the following Prerequisite skills are listed under Prepare instruction: write equivalent numerical expressions, such as 16 + 2 = 20 — 2; construct tables and graphs in the first quadrant; apply the Distributive Property in numerical contexts; and apply order of operations. The Prerequisite skills are not explicitly connected to any previous, grade-level standards.
    • In Topic 12 Block 1 a Classroom strategy states, “The formulas presented on this page were introduced in earlier grades. Depending on your students’ prior experiences, they may not recall them or how to apply them. Use this page to connect to prior learning by engaging students in these formulas.”
  • The Overview of the student material sometimes informs students what they will learn within the Topic and occasionally gives a general connection to previous learning. For example, the end of the Overview of Topic 4 states, “In this topic, you will further explore the meaning of these different forms of numbers and learn how to convert from one form to another. Once you are able to do that, you will be able to compare and order them.”
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The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards. Overall, the materials include learning objectives that are visibly shaped by CCSSM cluster headings, and they provide problems and activities that connect two or more clusters in a domain or two or more domains when these connections are natural and important.

Some examples of Topic Headings and Goals and Objectives shaped by cluster headings include the following:

  • Topic Headings:
    • In Topic 6 "Multiplying and Dividing Rational Numbers" is shaped by 6.NSA.
    • In Topic 8,"Extending the Number System" is shaped by 6.NS.C.
  • Goals and Objectives:
    • In Topic 2 “understand ratio concepts and apply ratio and proportional reasoning in problem situations” is shaped by 6.RP.A.
    • In Topics 3 and 7, “understand ratio and rate concepts…” and “apply ratio and rate reasoning…” are also shaped by 6.RP.A.
    • In Topic 5 “compute fluently with multi‐digit numbers” is shaped by 6.NS.B.
    • In Topics 9, 10, and 11, “apply and extend previous understandings of arithmetic to algebraic expressions” is shaped by 6.EE.A.
    • In Topics 12 and 13 “solve real‐world and mathematical problems involving area and surface area” are shaped by 6.G.A.
    • In Topic 14 “develop an initial understanding of statistical variability” is shaped by 6.SP.A.

The following are topics that contain problems and/or activities which connect two or more clusters in a domain or two or more domains in a grade.

  • In Topic 2, 6.NS.C and 6.RP.A are connected as students plot pairs of numbers from a ratio table on the coordinate plane, identifying the graph as a line as long as the ratios are equivalent.
  • In Topic 3, 6.NS.C and 6.RP.A are connected as students are comparing rates of two or more quantities using the position of the rate on the coordinate plane in which the x- and y-axes represent the labels of the rate.
  • In Topic 9, 6.EE.B and 6.EE.C are connected as students write and solve one-variable equations in order to represent and analyze quantitative relationships between dependent and independent variables.
  • In Topic 10, 6.EE.B and 6.NS.C are connected as students begin writing inequalities and relating them to number lines.
  • In Topic 12, 6.G.A and 6.RP.A are connected as students convert measurement units for the area of rectangles.
  • In Topic 13, 6.G.A and 6.NS.B are connected as students calculate volume of prisms with fractional/decimal edge lengths.
  • In Topic 14, 6.SP.B and 6.NS.B are connected as students perform numerical calculations to describe data.
  • In Topic 15, 6.SP.A and 6.SP.B are connected as students develop an understanding of statistical variability along with summarizing and describing distributions.
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The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance in all three.

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The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Multiple opportunities exist for students to work with standards that specifically call for conceptual understanding and include the use of visual representations, interactive examples, and different strategies.

Cluster 6.RP.A addresses understanding ratio concepts and using ratio reasoning to solve problems.

  • In Topic 2 “Understanding Ratio and Proportion” and Topic 3 “Introduction to Rates” students develop their understanding of ratios and rates through real-world, interactive examples. Students represent ratios and rates in various forms, including ratio tables, bar models, equivalent fractions, and points on a line or coordinate plane. Students are also given opportunities to solve problems using these concepts in multiple topics within the materials.
  • In Topic 4 students understand the connections between fractions, decimals, and percents, and during the topic, they are reminded that each of these forms represent ratios of rational numbers.

Cluster 6.NS.C calls for applying and extending previous understandings of numbers to the system of rational numbers.

  • In Topic 8 “Extending the Number System” students are introduced to rational numbers, integers, absolute value, and opposites. The animations allow students to develop an understanding of these terms visually, and many of the animations include the use of number lines and a coordinate grid. Through the topic, students have the opportunity to understand that the absolute value of a number is its distance from 0 on a number line. Students also have an opportunity to explore rational numbers in real-world contexts such as banking, sea level, and traveling distances. One aspect of 6.NS.C for which students are given limited opportunities to develop an understanding is that the opposite of the opposite of a number is the number itself.
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The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, there are opportunities for students to practice dividing multi-digit numbers using the standard algorithm, and students are given opportunities to develop fluency with decimal operations.

Standard 6.NS.2 addresses students being able to fluently divide multi-digit numbers using the standard algorithm.

  • In Topic 1 Blocks 1-3 there are multiple opportunities for students to engage with dividing multi-digit numbers within the Practice section, along with more problems in the Assessment section and Student Activity Sheets.

Standard 6.NS.3 addresses students being able to fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

  • In Topic 5 there are multiple opportunities for students to develop fluency with adding and subtracting decimals within the Practice section, along with more problems in the Assessment section and Student Activity Sheets.
  • In Topic 6 there are multiple opportunities for students to develop fluency with multiplying and dividing decimals within the Practice section, along with more problems in the Assessment section and Student Activity Sheets.
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The materials reviewed for Agile Mind Grade 6 meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade. Overall, students are given opportunities to solve application problems that include multiple steps, real-world contexts, and are non-routine.

Application problems allowing students to make their own assumptions in order to apply their mathematical knowledge can be found in different parts of the materials, including MARS Tasks, Constructed Response items and occasionally within the Student Activity Sheets (SAS).

Standard 6.RP.3 addresses using ratio and rate reasoning to solve real-world and mathematical problems.

  • In Topic 2 Constructed Response 1 students are expected to use rate reasoning to determine the number of candles in boxes of different sizes or the price of boxes of different sizes. This problem does not include any questions or prompts for scaffolding, and the context is unique when compared to other contexts used in the Topic.
  • In Topic 7 Constructed Response 2 students determine how long it will take for two people to paint a wall when they work together. Students calculate the area of the wall and then calculate how much of the wall each person paints per minute.

Standards 6.EE.7 and 6.EE.9 address students writing and solving linear equations in order to solve real-world and mathematical problems.

  • In Topic 9 Constructed Response 1 students write linear equations, along with tables and graphs for the equations, in order to answer questions about the type of growth for two different plants. The context is unique for this Topic.
  • In Topic 11 Constructed Response 1 students write linear equations in order to answer questions about how much time is needed to save a certain amount of money or how much money needs to be saved if there is only a certain amount of time. This context is the same as one that is used in the Topic, and there are scaffolded questions that let students know how to define the variables and how many equations they need to write.
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The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for balance. Overall, the three aspects of rigor are not always treated together and are not always treated separately. Most Topics attempt to provide opportunities through lessons and assessments for students to connect conceptual understanding, procedural skill and fluency, and application when appropriate or engage with them separately as needed.

Balance is displayed in Topics 14 and 15. In Topic 14 students begin to understand how statistical questions involve variability, and they also begin to develop procedural skills with creating different plots that represent data distributions. In Topic 15 students further their understanding of statistical variability as they summarize and describe data distributions with measures of center and measures of spread, and they also explain which measures best represent the distributions. These understandings and skills are developed in conjunction with sets of data that allow students to apply them in real-world contexts.

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The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for practice–content connections. Overall, the materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialized language of mathematics. However, the materials do not attend to the full meaning of MPs 4 and 5, and there are few opportunities for students to choose their own models or tools when solving problems.

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The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for the Standards for Mathematical Practices (MPs) being identified and used to enrich the mathematics content within and throughout the grade. The instructional materials for the teacher identify the MPs, and students using the materials as intended will engage in the MPs along with the content standards for the grade.

  • The Practice Standards Connections are found within the Professional Support section for the teacher. The eight MPs are listed with four to ten examples for each. According to the Practice Standards Connections, “each citation is intended to show how the materials provide students with ongoing opportunities to develop and demonstrate proficiency with the Standards for Mathematical Practice.”
  • Deliver Instruction is located within Advice for Instruction under Professional Support in the teacher material. Occasionally, there will be information within the Deliver Instruction section giving some guidance on how to implement the MP within the task/activity.
    • In Topic 7 Block 5 while students are working individually on Constructed Response 1, the teacher is asked to encourage the practice of modeling (MP4) and sense‐making with mathematics (MP1) by helping students make connections between what they are exploring about rates in graphs with questions such as: “How is this situation similar to the ones we have been studying?; What information do you need in order to determine a rate?; and What operations help you determine a unit rate?”
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The instructional materials reviewed for Agile Mind Grade 6 partially meet the expectations for carefully attending to the full meaning of each practice standard (MP). Overall, the materials attend to the full meaning of most of the MPs, but there are two MPs for which the full meaning is not addressed.

The instructional materials do not attend to the full meaning of MPs 4 and 5.

  • MP4: This MP is integrated several times throughout the materials, but the full meaning of the MP is not developed through these different parts of the materials. In Topic 3 during Introduction to rates and the constructed response items, students answer questions about different contexts where quantities are defined for them and models are provided for them as well. There are also no opportunities for students to revise initial assumptions or models once calculations have been made. In Topic 8 during the Overview and Constructed Response 3, students focus on how a number line or coordinate grid can be used as a model for rational numbers, but other aspects of MP4 are not attended to during this Topic. In Topic 12 during Covering the Pedestal, students engage in a problem in which they are led through some aspects of MP4- defining quantities, creating a model, and making calculations, but in this problem, the students do not get the opportunity to engage in any of these aspects on their own.
  • MP5: This MP is integrated at different points in the materials, but the full meaning of the MP is not developed through these different parts of the materials. In Topic 8 during Rational numbers in the coordinate plane, teachers are told to promote this MP “by using masking tape to create a coordinate grid on the floor,” but students are not getting to choose any tools as they use the grid to practice plotting points. In Topic 8 there is a problem where students are directed to use a coordinate plane and patty paper to help them plot points and reflect them, but the students are not given a choice as to what to tools they might use. In Topic 12 there are multiple places where this MP is referenced, but in each of these places, students are told to measure with a ruler/yardstick and what units to use when measuring or are provided pictures on pre-labeled grids.
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The instructional materials reviewed for Agile Mind Grade 6 meet the expectations for prompting students to construct viable arguments and analyze the arguments of others. Overall, the materials prompt students to construct viable arguments and present opportunities to analyze the arguments of others.

The instructional materials provide opportunities for students to construct viable arguments.

  • In Topic 3 Block 2 students are matching dimensions of photographs to the corresponding coordinate on a graphical representation. The Student Activity Sheet asks the student to explain the process used to match the name with the picture.
  • In Topic 12 Block 12 students are given the coordinates of a parallelogram and are asked to determine whether statements are true or false and record their reasoning.
  • In Topic 15 Block 5 during the MARS Task “Suzi’s Company,” students are given a table depicting jobs and annual salaries of fifteen people in a small company. Students connect their understanding of measures of center as they explain their solution to several questions.

The instructional materials provide opportunities for students to analyze the arguments of others.

  • In Topic 1 Block 8 Constructed Response 2 students engage in analyzing the arguments of fictitious students. Even though students are told in which step the error occurs, this is an opportunity for students to analyze the mathematical arguments that are presented to them and justify their suggested corrections.
  • In Topic 15 Block 5 one part of the MARS Task “Suzi’s Company” asks students to analyze the statement made by a person in the problem, identify the mistake that was made, and present the correct mode for the problem.
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The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. In Deliver Instruction, classroom strategies and question prompts are provided to assist teachers in engaging students to construct viable arguments or analyzing the arguments of others.

The following are examples of assistance provided to the teachers to promote the construction of viable arguments and analysis of other’s thinking, including prompts, sample questions to ask, and guidance for discussions.

  • In Topic 1 Block 5 students are determining all possible dimensions for a 225 square-foot school banner. The teacher is instructed as follows: “As pairs work to find all the possible dimensions for this banner, they will develop the practice of constructing viable arguments and critiquing the reasoning of the members of their group. Encourage students to justify the reasoning they used to decide when they found all possible dimensions.”
  • In Topic 1 Block 8 Constructed Response 2 teachers are told that two parts of the problem “provide an opportunity to promote the practice of constructing arguments and responding to the ideas of others.”
  • In Topic 3 Block 5 students are analyzing batting averages of two softball games at a family reunion. The teacher is instructed to “Promote the habit of communicating mathematical ideas by critiquing the reasoning of others with this activity. Allow students plenty of time to engage with each written argument and discuss their ideas with a partner.”
  • In Topic 12 Block 1 teachers are assisted with “as they share strategies, ask students to consider which strategies may be more or less accurate and have them justify their responses. This discussion promotes the mathematical practice of constructing viable arguments and critiquing the reasoning of others.”
  • In Topic 12 Block 6 students are finding the area of an irregular shape, a map of the United States. The teacher is instructed as follows: “As students share and discuss their strategies for finding the area of the unusual shape, promote the mathematical practice of constructing viable arguments and critiquing the reasoning of others. You may need to model this strategy to get the discussion started, “Who agrees with Monique’s strategy? Tell us why her strategy is effective.”
  • In Topic 15 Block 11 in the MARS Task “TV Hours” teachers are given the following assistance: “Encourage students to be active listeners during the debriefing of this task. For example, after a student has shared her strategy summarizing the results of Mrs. Campbell’s letters you can engage students by saying; “Raise your hand if you understand Sarah’s conclusion.” Then call on one or more of those students to restate the first student’s conclusion in their own words. ... As students or pairs of students are sharing their answers and strategies with the rest of the class, encourage students to critique each other’s reasoning and to compare the various strategies.”

In the Advice for Instruction there is a missed opportunity to provide support for teachers that explains and identifies where and when problems, tasks, examples, and situations lend themselves to these types of questions. Additional guidance is needed to broaden the application of these questions throughout the course so that students routinely construct viable arguments and analyze the arguments of others.

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The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for explicitly attending to the specialized language of mathematics. Overall, the materials appropriately use the specialized language of mathematics and expect students and teachers to use it appropriately as well.

Occasionally, there are suggestions within Deliver Instruction as to how teachers can reinforce mathematical language during instruction.

  • Topic 1 Block 7: "Reinforce the use of precise mathematical language as you introduce the Distributive Property. Most students are familiar with the words product, addend, factor, and sum. However, this is a new context and application for those words. Provide additional number sentences with which to identify factors, addends, products and sums (with and without common factors).”
  • Topic 9 Block 2: “Students at this grade level may have had very few experiences with using variables to write equations or expressions. Pause with each panel to clarify for students how the variables, x and y, are being used to take the place of the numbers.”

In the student materials, vocabulary terms can be found in bold print within the lesson pages, and these terms are used in context during instruction, practice, and assessment. Vocabulary terms are also available to the students at all time through My Glossary within the materials. For teachers, vocabulary terms for each Topic can be found under Language Support, which is within Advice for Instruction. Both core vocabulary and reinforced vocabulary are listed for each unit.

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The instructional materials reviewed for Agile Mind Grade 6 Agile meet the expectations for alignment to the CCSSM. The materials meet the expectations for focus and coherence in Gateway 1, and they meet the expectations for rigor and the mathematical practices in Gateway 2.

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The materials reviewed for Agile Mind Grade 6 meet the expectations for usability. The materials are well designed and take into account effective lesson structure and pacing. The instructional materials provide teachers with quality questions to help guide students' mathematical development and a teacher's edition that is easy to use and consistently organized and annotated. The materials provide adult-level explanations and examples of advanced mathematics concepts so that teachers can improve their own knowledge, but grade-level skills are not explicitly connected to standards from previous courses. The instructional materials offer teachers some resources and tools to collect ongoing data about student progress; however, there are no assessments that purposely identify prior knowledge within and across grade levels. Opportunities for ongoing review and practice, and feedback occur in various forms, but there are limited opportunities for students to monitor their own progress. The materials provide strategies to support the needs of ELLs and other special populations and offer some support with differentiating instruction for diverse learners and advanced students.

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The instructional materials reviewed for Agile Mind Grade 6 meet expectations that the materials are well designed and take into account effective lesson structure and pacing. Overall, materials are well-designed, and lessons are intentionally sequenced. Students learn new mathematics in the Exploring section of each Topic as they apply the mathematics and work toward mastery. Students produce a variety of types of answers including both verbal and written answers. The Overview for the Topic introduces the mathematical concepts, and the Summary highlights connections within and between the concepts of the Topic. Manipulatives such as algebra tiles and virtual algebra tiles are used throughout the instructional materials as mathematical representations and to build conceptual understanding.

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The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for having an underlying design that distinguishes between problems and exercises.

  • Each Topic includes three sections: Overview, Exploring, and Summary. The Overview section introduces the mathematical concepts that will be addressed in the Topic. The Exploring section includes two to four explorations where students learn the mathematical concepts of the Topic through problems that include technology-enhanced animations and full-class activities. The Summary section highlights the most important concepts from the Topic and gives students another opportunity to connect these concepts with each other.
  • Each Topic also includes three additional sections: Practice, Assessment, and Activity Sheets. The Practice section includes Guided Practice and More Practice. Guided Practice consists of exercises that students complete during class periods and give opportunities for students to apply the concepts learned during the explorations. More Practice contains exercises that are completed as homework assignments. The Assessment section includes Automatically Scored and Constructed Response items. These items are exercises to be completed during class periods or as part of homework assignments. They provide more opportunities for students to apply the concepts learned during the explorations. The Activity Sheets also contain exercises, which can be completed during class periods or as part of homework assignments, that are opportunities for students to apply the concepts learned during the explorations.
  • Some Topics also include MARS Tasks, which are exercises that present students with opportunities to apply concepts they have learned from the Topic in which the MARS Task resides or to apply and connect concepts from multiple Topics.
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The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for having a design of assignments that is not haphazard with problems and exercises given in intentional sequences.

The sequencing of Topics, and explorations within the Exploring section for each Topic, develops in a way that helps to build students’ mathematical foundations.

  • The Topics are comprised of similar content.
  • Within the explorations for each Topic, problems generally develop from more simple to more complex problems and incorporate knowledge from prior problems or Topics, which offers students opportunities to make connections among mathematical concepts. For example, using equations and inequalities to solve problems in Topic 11 builds upon writing algebraic expressions, equations, and inequalities from Topics 9 and 10.
  • As students progress through the Overview, Exploring, and Summary sections, the Practice (Guided and More), Assessment (Automatically Scored and Constructed Response), and Activity Sheets sections are placed intentionally in the sequencing of the materials to help students build their knowledge and understanding of the mathematical concepts addressed in the Topic.
  • The MARS Tasks are also placed intentionally in the sequencing of the materials to support the development of the students’ knowledge and understanding of the mathematical concepts that are addressed by the tasks.
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The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for having a variety in what students are asked to produce.

Throughout a Topic, students are asked to produce answers and solutions as well as explain their work, justify their reasoning, and use appropriate models. The Practice section and Automatically Scored items include questions in the following formats: fill-in-the-blank, multiple choice with a single correct answer, and multiple choice with more than one correct answer. Constructed Response items include a variety of ways in which students might respond, i.e. multiple representations of a situation, modeling, or explanation of a process. Also, the types of responses required vary in intentional ways. For example, concrete models or visual representations are expected when a concept is introduced, but as students progress in their knowledge, students are expected to transition to more efficient solution strategies or representations.

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The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models. The materials include a variety of virtual manipulatives, as well as integrate hands-on activities that allow the use of physical manipulatives.

Most of the physical manipulatives used in Agile Mind are commonly available: ruler, patty paper, graph paper, algebra tiles, and graphing calculators. Due to the digital format of the materials, students also have the opportunity to represent equations or functions virtually with tables and graphs. Each Topic has a Prepare Instruction section that lists the materials needed for the Topic. Manipulatives accurately represent the related mathematics. For example, Topic 13, Surface Area and Volume, students work with physical unit cubes and a cardboard box, along with having virtual representations of them, in order to expand their understanding of volume to include rectangular prisms whose edge lengths are not whole numbers.

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The instructional materials reviewed for Agile Mind Grade 6 have a visual design that is not distracting or chaotic but supports students in engaging thoughtfully with the subject. The student materials are clear and consistent between Topics within a grade-level as well as across grade-levels. Each piece of a Topic is clearly labeled, and the explorations include Page numbers for easy reference. Problems and Exercises from the Practice, Assessment, and Activity Sheets are also clearly labeled and consistently numbered for easy reference by the students. There are no distracting or extraneous pictures, captions, or "facts" within the materials.

) [33] => stdClass Object ( [code] => 3f3l [type] => criterion [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet expectations that materials support teacher learning and understanding of the standards. The instructional materials provide Framing Questions and Further Questions that support teachers in delivering quality instruction, and the teacher’s edition is easy to use and consistently organized and annotated. The materials provide full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject. Although each Topic contains a list of Prerequisite Skills, this list does not connect any of the skills to specific standards from previous grade levels, so the instructional materials partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

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The instructional materials reviewed for Grade 6 meet the expectation for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development. The Deliver Instruction section for each Block of a Topic includes Framing Questions for the start of each lesson. For example, in Topic 8 Block 4 the Framing Question is: “Have you ever used a map that had a grid system for naming points?” During the lesson the Deliver Instruction section includes multiple questions that teachers can ask while students are completing the activities. At the end of each lesson, Deliver Instruction includes Further Questions. For example, in Topic 8 Block 4 “If both coordinates are positive, what does that tell you about the location of a point? If both coordinates are negative, what does that tell you about the location of a point? If one coordinate is positive and the other is negative, what does that tell you about the location of the point?”

) [35] => stdClass Object ( [code] => 3g [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for containing a teachers edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Also, where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The materials contain Professional Support which includes a Plan the Course section and a Scope and Sequence document. The Plan the Course section includes Suggested Lesson-planning Strategies and Planning Resources. Each Topic contains an Advice for Instruction section that is divided into Prepare Instruction and Deliver Instruction. For each Topic, Prepare Instruction includes Goals and Objectives, Topic at a Glance, Prerequisite Skills, Resources, and Language Support, and for each Block within a Topic, Deliver Instruction includes Agile Mind Materials, Opening the Lesson, Framing Questions, Lesson Activities, and Suggested Assignment. In Lesson Activities, teachers are given ample annotations and suggestions as to what parts of the materials should be used when and Classroom Strategies that include questions to ask, connections to mathematical practices, or statements that suggest when to introduce certain mathematical terms or concepts.

Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. For example, in Topic 7 Block 2 teachers are directed to, “Use the interactive animation to explore a distance vs. time graph. Use these questions to guide student discussions during the activity: When you drag Terrence away from the motion detector, what does the graph look like? Why? [When you drag him away, his distance from the motion detector increases. Since the graph measures Terrence's distance from the motion detector at any given time, this causes the graph to rise, or get farther away from the x-axis, as you move from left to right on the graph.]; How does dragging him back toward the motion detector affect the graph? Why? [When you drag him toward the motion detector, his distance from the motion detector decreases. Since the graph measures Terrence's distance from the motion detector at any given time, this causes the graph to fall, or get closer to the x-axis, as you move from left to right on the graph.]; What does the graph look like when you drag him at a fairly slow, steady rate away from the motion detector? [The graph will be a line that rises from left to right, but will not be very steep.] What if you drag him away from the motion detector faster, but still at a steady rate? [The graph will still be a line that rises from left to right, but it will be steeper than the first line.]; and When Terrence stops during his skate, what does the graph do? Why? [The graph will be a horizontal line. This happens because when Terrence is not moving, his distance from the motion detector is not changing, but time continues to pass.]”

) [36] => stdClass Object ( [code] => 3h [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

In Professional Support, Professional Learning, there are four interactive essays entitled “Developing concepts across grades”. There is an Overview that explains the purpose of the four essays, and the topics for these four essays are Functions, Volume, Rate, and Proportionality. Each essay is accessible to teachers and not students, and the Overview states “these essays are available for educators to illustrate connections and deepen understanding around what students may have already learned, and where they are headed on their journey.” Each essay examines the progression of the concept from Grades 6-8 through Algebra I, Geometry, Algebra II, and beyond. By examining the progressions of the concepts beyond Algebra II, teachers have the opportunity to improve their own knowledge of more advanced mathematics concepts that build upon grade-level standards. For example, in Volume, teachers progress from packing a right, rectangular prism with unit cubes to developing the formulas for the volume of cylinders and cones to finding the volume of a figure generated by rotating a two-dimensional shape around a horizontal axis. Also, in Proportionality, teachers explore how proportional relationships are part of the following mathematical concepts: scaling images, linear functions, trigonometric ratios, rational functions, and the derivative.

In addition to “Developing concepts across grades”, the Grade 6 materials also contain a section of interactive essays entitled “Going beyond Grade 6”. There is an Overview that explains the purpose of the three essays, and the Overview states, “These essays are not intended for use with your 6th grade students; rather, they are designed to provide you with important connections and background that will support you as you help your students master 6th grade content.” The topics for these three essays are Rate of Change, Understanding Area of Irregular Shapes, and Calculating Outliers. Along with having their own section in Professional learning, each of these essays are also referenced in Deliver Instruction for the Blocks where they are appropriate under the title of Teacher Corner. For example, in Topic 3, the essay Rate of Change is referenced for teachers in Block 1 as part of the Overview, and in Topic 12, Understanding Area of Irregular Shapes is referenced in Block 5 on page 4. The three essays connect the Grade 6 content to advanced mathematical concepts through multiple grades and courses.

In Professional Support, there is a section of Professional Essays which are in either Print or Video format. The Print essays are divided as either Curriculum or Course Management Topics, and although some of the Curriculum Essays are content specific, they do not address mathematical concepts that extend beyond the current grade. The Video Essays - categorized into Teaching with Agile Mind, More Teaching with Agile Mind, and Dimensions of Mathematics Instruction - do not directly provide adult-level explanations or examples of advanced mathematics concepts.

) [37] => stdClass Object ( [code] => 3i [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Prepare Instruction section for each Topic contains a list of Prerequisite Skills, but this list does not connect any of the skills to specific standards from previous grade levels. For example, in Topic 6 the Prerequisite Skills include “Multiples of a given whole number; Rational numbers in equivalent forms; and Operations with whole numbers.” In Prepare Instruction the Topic at a Glance occasionally provides general references to how concepts will be used in future courses. For example, in Topic 6 “This topic is intended to help students make sense of and formalize processes for operating with positive rational numbers. Once students understand how to operate with fractions, they can use that knowledge to learn how to operate with decimals. Students will learn through the topic that fractions and decimals are more similar than different. Throughout the topic, students build and apply fluency with these operations. They will continue to build and demonstrate fluency in future topics. The activities in this topic are outlined in ten 45-minute instructional blocks and assessment.”

) [38] => stdClass Object ( [code] => 3j [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 provide a list of lessons in the teacher's edition, cross­‐referencing the standards covered and providing an estimated instructional time for Topics and Blocks. The materials provide a Mathematics 6 Scope and Sequence document which includes the number of Blocks of instruction for the duration of the year, time in minutes that each Block should take, and the number of Blocks needed to complete each Topic. The Scope and Sequence document lists the CCSSM addressed in each Topic, but there is no part of the materials that aligns Blocks to specific content standards. The materials also provide Alignment to Standards in the Course Materials which allows users to see the alignment of Topics to the CCSSM or the alignment of the CCCSM to the Topics. The Deliver Instruction section contains the Blocks for each Topic. The Practice Standards Connections, found in Professional Support, gives examples of places in the materials where each MP is identified.

) [39] => stdClass Object ( [code] => 3k [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 do not contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

) [40] => stdClass Object ( [code] => 3l [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 do not contain explanations of the instructional approaches of the program and identification of the research-based strategies within the teaching materials. There is a Professional Essays section which addresses a broad overview of mathematics in grades 6-8 as discussed in indicator 3h.

) [41] => stdClass Object ( [code] => 3m3q [type] => criterion [report] =>

The instructional materials for Agile Mind Grade 6 partially meet exceptions that materials offer teachers resources and tools to collect ongoing data about students progress on the Standards. Opportunities for ongoing review and practice, and feedback occur in various forms. Standards are identified that align to the Topic; however, there is no mapping of Standards to items. There are limited opportunities for students to monitor their own progress, and there are no assessments that explicitly identify prior knowledge within and across grade levels. The materials include few opportunities to identify common misconceptions, and strategies to address common errors and misconceptions are only found in a few Deliver Instruction topics.

) [42] => stdClass Object ( [code] => 3m [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 partially meet the expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels. The materials do not provide any assessments that are specifically designed for the purpose of gathering information about students’ prior knowledge, but the materials do provide indirect ways for teachers to gather information about students’ prior knowledge if teachers decide to use them that way.

In Prepare Instruction for each Topic, there is a set of Prerequisite Skills needed for the Topic, and the Overview for each Topic provides teachers with an opportunity to informally assess students prior knowledge of the Prerequisite Skills. For example, in Topic 4 the Prerequisite Skills are: “Writing equivalent fractions and Multiplication and division of whole numbers.” Then, in the Lesson Activities for Page 1 of Block 2, teachers are told, “Use this page to connect with students' prior understanding and begin developing an understanding of what a rational number is.”

) [43] => stdClass Object ( [code] => 3n [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 partially meet the expectation for providing strategies for teachers to identify and address common student errors and misconceptions. There is not an explicit way in which the materials help teachers identify and address common student errors and misconceptions, but there are a few instances in the Deliver Instruction where common errors and misconceptions are identified and suggestions are given for how to address them. For example, in Topic 6 Block 3 teachers are told, “There may be some misunderstanding when using both a linear and area model for multiplication of rational numbers. Take time to distinguish the differences by using manipulatives where needed.”

) [44] => stdClass Object ( [code] => 3o [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials provide opportunities for ongoing review and practice, and feedback occurs in various forms. Within interactive animations, students submit answers to questions or problems, and feedback is provided by the materials. Practice problems and Automatically Scored Assessment items are submitted by the students. Immediate feedback is provided letting students know whether or not they are correct, and if incorrect, suggestions are given as to how the answer can be improved. The Lesson Activities in Deliver Instruction provide some suggestions for feedback that teachers can give while students are completing the lessons.

) [45] => stdClass Object ( [code] => 3p [type] => indicator [report] => ) [46] => stdClass Object ( [code] => 3p.i [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 partially meet the expectation for assessments clearly denoting which standards are being emphasized. The items provided in the Assessment section align to the standards addressed by the Topic, but the individual items are not clearly aligned to particular standards. The set of standards being addressed by a Topic can be found in the Scope and Sequence document or in Course Materials through Alignment to Standards. The MARS Tasks also do not clearly denote which standards are being emphasized.

) [47] => stdClass Object ( [code] => 3p.ii [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 partially meet the expectation for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The MARS Tasks that are included in the materials are accompanied by rubrics aligned to the task that show the total points possible for the task and exactly what students need to do in order to earn each of those points. The Constructed Response items are accompanied by complete solutions, but rubrics aligned to the Constructed Response items are not included. For both the MARS Tasks and the Constructed Response items, alternate solutions are provided when appropriate, but sufficient guidance to teachers for interpreting student performance and suggestions for follow-up are not provided with either the MARS Tasks or the Constructed Response items.

) [48] => stdClass Object ( [code] => 3q [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 offer few opportunities for students to monitor their own progress. Throughout the Exploring, Practice, and Automatically Scored Assessment sections, students get feedback once they submit an answer, and in that moment, they can adjust their thinking or strategy. Goals and Objectives for each Topic are not provided directly to students, but they are given to teachers in Prepare Instruction. There is not a systematic way for students to monitor their own progress on assignments or the Goals and Objectives for each Topic.

) [49] => stdClass Object ( [code] => 3r3y [type] => criterion [report] =>

The instructional materials for Agile Mind Grade 6 meet expectations that materials support teachers in differentiating instruction for diverse learners within and across grades. Activities provide students with multiple entry points and a variety of solution strategies and representations. The materials provide strategies for ELLs and other special populations, but the materials do not always challenge advanced students to deepen their understanding of the mathematics. Grouping strategies are designed to ensure roles for each group member.

) [50] => stdClass Object ( [code] => 3r [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Each Topic consists of three main sections- Overview, Exploring, and Summary, and these three sections are divided into Blocks. Each Block contains lesson activities including Practice, Assessment, and Activity Sheets, along with any MARS Tasks in the Topic. In each Topic, the Blocks are sequenced for the teachers, and the lesson activities within the Blocks are sequenced for the teachers. In the Advice for Instruction for each Topic, Deliver Instruction for each Block contains instructional notes and classroom strategies that provide teachers with key math concepts to develop, sample questions to ask, ways in which to share student answers, and other similar instructional supports.

) [51] => stdClass Object ( [code] => 3s [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 partially meet the expectation for providing teachers with strategies for meeting the needs of a range of learners. Overall, the instructional materials embed multiple and/or visual representations of mathematical concepts where appropriate, include audio recordings with many explorations, and give students opportunities to engage physically with the mathematical concepts. However, the instructional notes provided to teachers do not consistently highlight these strategies that can be used to meet the needs of a range of learners. When instructional notes are provided to teachers, they are general in nature and are intended for all students in the class, and they do not explicitly address the possible range of needs for learners. For example, in Topic 3 Block 7 the Deliver Instruction states “As students are working, listen for unique strategies that students use to approach this problem that you can highlight in your debrief of the problem. Encourage students to provide strong mathematical evidence for their claims. If students are not sure, encourage them to sketch and shade tape diagrams on the Student Activity Sheet. Remind students to find and use familiar percents if needed, such as 10%, 50%, and 25% in order to label their tape diagrams.”

In some explorations, teachers are provided with questions that can be used to extend the tasks students are completing, which are beneficial to excelling students. For struggling students, teachers are occasionally provided with strategies or questions they can use to help move a student’s learning forward. The Summary for each Topic does not provide any strategies or resources for either excelling or struggling students to help with their understanding of the mathematical concepts in the Topic.

) [52] => stdClass Object ( [code] => 3t [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectation that materials embed tasks with multiple entry­-points that can be solved using a variety of solution strategies or representations. Overall, tasks that meet the expectations for this indicator are found in some of the Constructed Response Assessment items and Student Activity Sheets that are a part of all Topics. MARS Tasks embedded in some Topics have multiple entry-points and can be solved using a variety of solution strategies or representations. For example, in Topic 8 Constructed Response 1 students are instructed to "Describe four more possible events in the life of Andre or his family. For each event describe when it happened. Then graph and label the rational number representing that event. Use at least one negative rational number, at least one whole number, and at least two fractions or decimals." Another example is in Topic 12 Constructed Response 2. Students are asked to find the area and/or perimeter of different figures that are superimposed on a grid. Students are given the scale for the grid.

) [53] => stdClass Object ( [code] => 3u [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectation that the materials suggest accommodations and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The materials provide suggestions for English Language Learners and other special populations in regards to vocabulary and instructional practices. In Prepare Instruction for Topic 1, Teaching Special Populations of Students refers teachers to the Print Essay entitled “Teaching English Language Learners” in Professional Support, which describes general strategies that are used across the series such as a vocabulary notebook, word walls, and concept maps. Teaching Special Populations of Students also describes general strategies that are used across the series for other special populations, including progressing from concrete stage to representational stage to abstract stage and explicitly teaching metacognitive strategies through think alouds, graphic organizers, and other visual representations of concepts and problems.

In addition to the general strategies mentioned in Teaching Special Populations of Students, there are also many specific strategies listed across each course of the series in Deliver Instruction. In Deliver Instruction, Support for ELL/other special populations includes strategies that can be used with both English Language Learners and students from other special populations. Strategies specific to other special populations can also be found in Classroom strategy or Language strategy. An example of Support for ELL/other special populations from Topic 1, Block 4, Page 5 is “Provide students who have handwriting challenges or other language challenges with index cards that contain the 4 rules of the Order of Operations. Having the text already written will allow these students to focus on understanding the meaning of each rule instead of transcribing the words. While the rest of the class is writing the rules, encourage these students to write an example next to each rule to help them process the meaning. Have them circle each rule in a different color. Then have them write the expression from the animation and color code each step in the expression with the color of the appropriate rule.” An example of a strategy for other special populations from Topic 8, Block 1, Page 1 of Exploring Positive and Negative Rational Numbers is “Classroom strategy. Bank balances will use negative numbers, but when we talk about those balances using the word “debt” we use the absolute value. We say, “I have a debt of $15,” rather than, “I have a debt of $-15.” Or, stated another way, “I have a balance of $-15, so my debt is $15.” This is a subtle distinction, especially for students at this age, and complete mastery of this concept is not expected on this page. Throughout the topic, as students are discussing debt, continue to reinforce the notion that the greatest debt is the balance with the least value, or the negative number furthest from zero. Be aware that some students with learning differences may continue to need more explicit instructions and repetition.”

) [54] => stdClass Object ( [code] => 3v [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 partially meet the expectation that the materials provide opportunities for advanced students to investigate mathematics content at greater depth. Overall, all of the problems provided in the materials are on grade level, and the materials are designed so that all of the problems are assigned to all students over the course of the school year. There are opportunities for advanced students to investigate mathematics at a greater depth in some of the Topics through notes given to teachers in the Advice for Instruction. For example, in Topic 13 Block 2 teachers are told, "Students can also do the following activity for further exploration. Give groups a set number of cubes and ask them to build a rectangular prism. Give them some numbers that will make a cube (27), some that will make several prisms (24), and some that will make only one (17; 1 x 1 x 17). This is a nice activity for spatial visualization but also for number sense."

) [55] => stdClass Object ( [code] => 3w [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 6 meet the expectation for providing a balanced portrayal of various demographic and personal characteristics. The activities are diverse, meeting the interests of a demographically, diverse student population. The names, contexts, videos, and images presented display a balanced portrayal of various demographic and personal characteristics.

) [56] => stdClass Object ( [code] => 3x [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 provide opportunities for teachers to use a variety of grouping strategies in the Deliver Instruction Lesson Activities including when students work individually, in pairs, or in small groups. When suggestions are made for students to work in small groups, there are no specific roles suggested for group members, but teachers are given suggestions to ensure the involvement of each group member. For example, Topic 1 Block 4 teachers are told to “have students work independently or with a partner to solve the problem. Ask groups to share how they used the Order of Operations. Review the importance of the Order of Operations.”

) [57] => stdClass Object ( [code] => 3y [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 do not encourage teachers to draw upon home language and culture to facilitate learning. Questions and contexts are provided for teachers in the materials, and there are no opportunities for teachers to adjust the questions or contexts in order to integrate the home language and culture of students into the materials to facilitate learning.

) [58] => stdClass Object ( [code] => 3z3ad [type] => criterion [report] =>

The instructional materials for Agile Mind Grade 6 are web-based and platform neutral but do not include the ability to view the teacher and student editions simultaneously. The materials embed technology enhanced, interactive virtual tools, and dynamic software that engage students with the mathematics. Opportunities to assess students through technology are embedded. The technology provides opportunities to personalize instruction; however, these are limited to the assignment of problems and exercises. The materials cannot be customized for local use. The technology is not used to foster communications between students, with the teacher, or for teachers to collaborate with one another.

) [59] => stdClass Object ( [code] => 3z [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices. Given the digital platform of the materials, the inclusion of interactive tools and virtual manipulatives/objects helps to engage students in the MPs in all of the Topics, and the use of animations in all of the Topics provides examples as to how the interactive tools and virtual manipulatives can be used.

) [60] => stdClass Object ( [code] => 3aa [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 are web-based and compatible with multiple internet browsers (Chrome, Firefox, and Internet Explorer). In addition, the materials are “platform neutral” and allow the use of tablets with ChromeOS, Android, or iOS operating systems, but they do not support the use of mobile devices. However, the transition between student and teacher materials is not fluid. There are no direct links between the student and teacher materials, and the student and teacher materials cannot be viewed simultaneously.

) [61] => stdClass Object ( [code] => 3ab [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. All Practice and Automatically Scored Assessment questions are designed to be completed using technology. These items cannot be edited or customized.

) [62] => stdClass Object ( [code] => 3ac [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 include few opportunities for teachers to personalize learning for all students. Within the Practice and Assessment sections, the teacher can choose which problems and exercises to assign students, but these problems and exercises cannot be modified for content or wording from the way in which they are given. Other than being able to switch between English and Spanish in My Glossary, there are no other adaptive or technological innovations that allow teachers to personalize learning for all students.

The instructional materials reviewed for Grade 6 cannot be easily customized for local use. Within My Courses, there are not any options for modifying the sequence or structure of the Topics or any of the sections within the Topics.

) [63] => stdClass Object ( [code] => 3ad [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 6 provide few opportunities for teachers and/or students to collaborate with each other. Under My Agile Mind, teachers can communicate with students through the Calendar and Score and Review. There are no opportunities for teachers to be able to collaborate with other teachers.

) ) [isbns] => Array ( [0] => stdClass Object ( [type] => custom [number] => 978-1-943460-78-6 [custom_type] => [title] => Mathematics 6 Student Activity Book [author] => [edition] => [binding] => [publisher] => Agile Mind [year] => 2016 ) [1] => stdClass Object ( [type] => custom [number] => 978-1-943460-79-3 [custom_type] => [title] => Mathematics 6 Advice for Instruction [author] => [edition] => [binding] => [publisher] => Agile Mind [year] => 2016 ) ) ) 1

Seventh Grade

                                            Array
(
    [title] => Agile Mind Middle School Mathematics (2016)
    [url] => https://www.edreports.org/math/agile-mind-middle-school-mathematics-a/seventh-grade.html
    [grade] => Seventh Grade
    [type] => math-k-8
    [gw_1] => Array
        (
            [score] => 13
            [rating] => meets
        )

    [gw_2] => Array
        (
            [score] => 17
            [rating] => meets
        )

    [gw_3] => Array
        (
            [score] => 31
            [rating] => meets
        )

)
1                                            stdClass Object
(
    [version] => 2.0.0
    [id] => 325
    [title] => Middle School Math - Grade 7
    [report_date] => 2018-03-20
    [grade_taxonomy_id] => 21
    [subject_taxonomy_id] => 5
    [notes] => 

Agile Mind made revisions that affected the scoring and reports for Rating Sheet 2, indicator 3h, and Rating Sheet 4, indicator 3u. These revisions also affected the rating for Gateway 3 so that Agile Mind Grade 7 meets expectations for Instructional Supports and Usability.

[reviewed_date] => 2017-06-08 [revised_date] => 2018-03-20 [gateway_1_points] => 13 [gateway_1_rating] => meets [gateway_1_report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet expectations for focus and coherence. The instructional materials do not assess topics beyond Grade 7, and students and teachers using the materials as designed would devote the large majority of instructional time to the major work of the grade. The instructional materials meet expectations for coherence, and they show strength in having an amount of content that is viable for one school year and fostering coherence through connections within the grade.

[gateway_2_points] => 17 [gateway_2_rating] => meets [gateway_2_report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for rigor and the mathematical practices. The materials meets the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and applications. The materials also meet the expectations for mathematical practices. Overall, the materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialized language of mathematics.

[gateway_3_points] => 31 [gateway_3_rating] => meets [report_type] => math-k-8 [series_id] => 74 [report_url] => https://www.edreports.org/math/agile-mind-middle-school-mathematics-a/seventh-grade.html [gateway_2_no_review_copy] => Materials were not reviewed for Gateway Two because materials did not meet or partially meet expectations for Gateway One [gateway_3_no_review_copy] => This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two [meta_title] => [meta_description] => [meta_image] => [data] => Array ( [0] => stdClass Object ( [code] => focus [type] => component [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for focus on the major work of the grade. The materials do not assess topics before the grade-level in which they should be introduced, and they spend the majority of class time on the major work of the grade when they are used as designed.

) [1] => stdClass Object ( [code] => 1a [type] => criterion [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet expectations for not assessing topics before the grade-level in which the topic should be introduced. Overall, there are not assessment items that align to topics beyond Grade 7.

) [2] => stdClass Object ( [code] => 1a [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for assessing grade-level content. The majority of the Grade 7 assessment content was appropriate for the grade. There are some items in the assessments that align to standards above Grade 7, but omitting or modifying these assessment items would not significantly impact the underlying structure of the Grade 7 materials.

The questions within the Practice and Assessment sections were reviewed for this indicator. The Practice sections within each topic contain multiple questions under the categories of Guided Practice and More Practice. The Assessment sections within each topic contain Automatically Scored questions and Constructed Response questions.

The following questions that include content from future grades are from Topic 13:

  • Guided Practice 8, More Practice 1-2, and Automatically Scored 1-2 expect students to work with the central and interior angles of polygons, which aligns to 8.G.5.
  • Automatically Scored 10 requires the use of the Pythagorean Theorem to find the perimeter of a right triangle (8.G.7).
) [3] => stdClass Object ( [code] => 1b [type] => criterion [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for students and teachers devoting the majority of class time to the major work of the grade when materials are used as designed. Overall, the materials spend approximately 72% of class time on the major work of Grade 7.

) [4] => stdClass Object ( [code] => 1b [type] => indicator [points] => 4 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for spending the majority of class time on the major clusters of the grade. Overall, the instructional materials spend approximately 72% of class time on the major clusters of Grade 7.

For this indicator, the following were examined: all Blocks of instruction within all Topics in Course Contents, Alignment to Standards in Course Materials, the Mathematics 7 Scope & Sequence with Common Core State Standards document in Professional Support, and the Block descriptions for each Topic located within Deliver instruction under Advice for Instruction in Professional Support. There are fifteen topics divided into the following categories: Overview, Explore, Summary, Practice, and Assessment. Each Topic contains 6 to 13 Blocks of instruction, and each Block of instruction represents a 45-minute class period.

In the Block descriptions for each Topic, individual activities are not assigned specific amounts of time, or ranges of time, for the activities to be completed. Thus, when calculating the percentage of class time spent on the major cluster of the grade, two perspectives were appropriate, Topics and Blocks. For these materials, analysis by Blocks is the most appropriate because the Topics do not have an equal number of Blocks within them and the Blocks are not subdivided into smaller increments.

In addition to the Blocks directly aligned to major clusters of the grade, all Blocks aligned to supporting clusters of the grade were also examined. Those Blocks aligned to supporting clusters that were found to incorporate major work of the grade were included in the calculations below:

  • Blocks: 102 of the 142 Blocks, approximately 72%, are spent on the major clusters of the grade.
  • Topics: 11.5 of the 15, Topics , approximately 77%, are spent on the major clusters of the grade.
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The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for being coherent and consistent with the Standards. The instructional materials show strengths in having an amount of content that is viable for one school year but do not always make explicit connections between prior knowledge and future learning and the major work of the grade. Therefore, the progressions in the Standards are not always evident. The materials foster coherence within grade level work.

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The instructional materials reviewed for Agile Mind Grade 7 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. Overall, supporting content is found primarily in Topics 1-2 and 9-15, and the supporting content does enhance focus and coherence by engaging students in the major work of the grade. Examples of the connections between supporting work and major work include the following:

  • In Topic 1 Blocks 2, 3, 6, and 11 connect major standard 7.RP.3 to supporting standard 7.G.1. Students are using scale factor in solving a variety of problems with proportional relationships.
  • In Topic 2 Blocks 8 and 9 connect major standard 7.RP.3 to supporting standard 7.G.1. Students are using scale factor in solving direct variation problems involving the effects of different sized scopes.
  • In Topic 9 Blocks 2, 6, 9, and 10 connect major standard 7.NS.3 to supporting standard 7.SP.8. Students are calculating simple and compound probabilities using rational numbers in various forms.
  • In Topic 9 Blocks 4, 5, and 11 connect major standard 7.RP.3 to supporting standard 7.SP.7. Students are using ratios and percentages to solve problems within probability models.
  • Topic 11 connects supporting cluster 7.SP.C and major standards 7.RP.3, 7.NS.3, and 7.EE.3 as students are designing an appropriate simulation using proportional reasoning and testing its validity.
  • In Topic 12 Blocks 1, 2, and 4 connect major standards 7.EE.3,4 to supporting standard 7.G.5. As students find measures of vertical, adjacent, complementary, and supplementary angles, they are using variables and writing equations that can be solved to find the missing angle measures.
  • Topic 13 connects supporting cluster 7.G.B and major standards 7.NS.3 and 7.EE.4 as students are solving a variety of real-life and mathematical problems involving area and circumference of circles and areas of two-dimensional objects.
  • In Topic 14 Blocks 2, 3, 5, and 6 connect major standard 7.NS.3 to supporting standard 7.G.6. Students are solving rational number expressions to find the surface area and volume of various figures. Major standard 7.EE.3 is also incorporated into Topic 14 as students are solving a variety of real-life and mathematical problems with positive rational numbers.
  • In Topic 15 Blocks 2-6 connect major standard 7.NS.3 to supporting standards 7.G.1,4. Students are using scale factor to calculate differences in area and perimeter of 2-dimensional shapes. Major standards 7.EE.3 and 7.RP.1,3 are also incorporated into Topic 15 as students are solving a variety of real-life and mathematical problems involving proportional reasoning as dimensions are changed and the patterns of change in perimeter and area are examined.
) [8] => stdClass Object ( [code] => 1d [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for the amount of content designated for one grade-level being viable for one school year in order to foster coherence between grades. The suggested pacing contains 15 Topics and 142 Blocks (days) of instruction, including assessments. According to the Agile Mind Mathematics 7 Scope and Sequence, each block is expected to last 45 minutes. Some lessons (Constructed Response, MARS tasks) may take longer than indicated.

Each Block includes the following sections: Overview, Exploring, Summary, and Assessment. The Exploring pages are categorized by math concept and can be discussed and reviewed as a class or by individuals/small groups of students.

) [9] => stdClass Object ( [code] => 1e [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 partially meet the expectations for being consistent with the progressions in the Standards. Overall, the materials develop according to the grade-by-grade progressions in the Standards, and they give all students extensive work with grade-level problems. However, content from prior or future grades is not always clearly identified or related to grade-level work, and the materials do not always relate grade-level concepts explicitly to prior knowledge from earlier grades.

Examples of Grade 7 materials in which off grade-level content is present and not identified as such includes the following:

  • Topic 4 Block 11, aligned to 7.RP.3, extends above grade-level when compound interest is introduced.
  • In Topic 5 there are items that have students place rational numbers on a number line (6.NS.6) and order rational numbers (6.NS.7).
  • In Topic 7 there are items that have students solving unit rate problems (6.RP.3b) and placing rational numbers on a number line (6.NS.6).
  • Topic 8 Block 5, aligned to 7.EE.4, extends above grade-level when students are expected to solve an equation that requires collecting like terms (8.EE.7b).
  • In Topic 10 there are multiple items that have students analyze individual sets of data (6.SP.B) as opposed to analyzing and comparing multiple sets of data (7.SP.B).
  • Topic 12 Block 6, aligned to 7.G.2, extends above grade-level when “congruence” and “congruent triangles” are used within the explanation (8.G.2).

One example where off grade-level material is identified is in the teacher materials for Topic 10. "Although the Exploring 'Misleading graphs' is not focused on primary standards of the grade, it has strong connections to the ideas of ratio and area. Additionally, it allows for students to reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, and attend to precision.”

The Grade 7 materials provide extensive work with grade-level standards. All students are expected to complete the same problems, and lessons or ideas presented for differentiated instruction also include grade-level problems. The MARS tasks that are included, especially the ones in Topics 1, 3, 4, 9, 10, and 15, are places where students are given the opportunity to engage with the grade-level standards to their full intent.

In lessons where prior knowledge is included, identification of content from prior grades is mentioned in four components of the materials, but the identification is general and not explicitly connected to a grade-level or standard. Examples from the four components are as follows:

  • In the first paragraph of the About the Course section, there is a brief, general overview of topics of which students acquired a foundation prior to Grade 7.
  • The first paragraph in Agile Mind Mathematics 7 Scope and Sequence, 2016-2017 briefly references prior work in variables, properties of operations, equations, proportional reasoning, rational number operations, 2-D figures, area, and volume and how these concepts connect to Grade 7 work and beyond.
  • The Advice for Instruction section references prior work in different places, although specific standards are not referenced. Some examples of this include:
    • The Topic 3 Topic at a glance states, “This topic, Patterns in proportional relationships, is designed to build on students' prior knowledge involving ratios, rates, and proportional reasoning developed in previous topics.”
    • The Topic 5 Topic at a glance states, “The topic makes connections to the fractions and decimals students have already studied. Students complete the number line by placing fractions and decimals on number lines along with integers.”
    • In Topic 6 the following Prerequisite skills are listed under Prepare instruction: Multiplication and division, Integers as the set of whole numbers and their opposites, Adding and subtracting integers, and Raising a number to a power. The Prerequisite skills are not explicitly connected to any previous, grade-level standards.
    • In the Deliver Instruction of Topic 8, page 5 of the Overview states, “This page reminds students about their previous work with inequalities.”
    • In Topic 13 a classroom strategy on page 2 of the Overview states, “The formulas presented on this page were introduced in earlier grades. Depending on your students’ prior experiences, they may not recall them or how to apply them. Use this page to connect to prior learning by engaging students in these formulas.”
  • The Overview of the student material sometimes informs students what they will learn within the Topic and occasionally gives a general connection to previous learning. For example, the Overview of Topic 1 states, “Ratios and proportions are very useful in everyday life. You may have explored some of these uses in your previous math courses. Do you remember how to represent ratios? Use the animation for some important reminders.”
) [10] => stdClass Object ( [code] => 1f [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards. Overall, the materials include learning objectives that are visibly shaped by CCSSM cluster headings, and they provide problems and activities that connect two or more clusters in a domain or two or more domains when these connections are natural and important.

Some examples of Topic Headings and Goals and Objectives shaped by cluster headings include the following:

  • Topic Headings:
    • In Topic 3, “Patterns in Proportional Relationships” is shaped by 7.RP.A.
    • In Topics 5 and 6, "Multiplying and Dividing Rational Integers" is shaped by 7.NS.A.
    • In Topics 9 and 11, “Probability” and “Designing Experiments,” respectively, are shaped by 7.SP.C.
    • In Topic 13, “Solving problems with 2-D shapes” is shaped by 7.G.B.
  • Goals and Objectives:
    • In Topic 1, “apply proportional reasoning in a range of contexts” is shaped by 7.RP.A.
    • In Topic 5, “apply previous understandings of rational numbers to add and subtract integers represent real-world situations using integers” is shaped by 7.NS.A.
    • In Topic 8, “solve real-life and mathematical problems using linear expressions and equations using concrete models, tables, graphs, and the properties of equality and operations” is shaped by 7.EE.B.
    • In Topic 12, “solve real-life and mathematical problems involving angle measures” is shaped by 7.G.B.

The following are topics that contain problems and/or activities which connect two or more clusters in a domain or two or more domains in a grade.

  • In Topic 2, 7.RP.A and 7.EE.B are connected as students solve real-life problems where understanding of the unit rate is central to solving the problems.
  • In Topic 3, 7.RP.A and 7.EE.B are connected as students explore proportional and nonproportional relationships in multiple representations and write equations that will describe the proportional relationships.
  • In Topic 4, 7.RP.A, 7.NS.A, and 7.EE.B are connected as students solve a variety of multi-step percent problems using written expressions and multiple operations with rational numbers.
  • In Topic 7, 7.NS.A and 7.EE.B are connected as students solve a variety of problems involving rational numbers and all operations. As students solve, they are responsible for writing expressions/equations that represent the problems before solving.
  • In Topic 7, 7.RP.A and 7.EE.B are connected as students solve a variety of problems based on a given rate or unit price. Students must apply proportional reasoning as they solve.
  • In Topic 8, 7.EE.A and 7.EE.B are connected as students use the properties of operations to generate equivalent expressions as a part of solving mathematical problems using algebraic expressions and equations.
  • In Topic 10, 7.SP.A and 7.SP.B are connected as students make inferences about a single population in order to draw comparative inferences about two or more populations.
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The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance in all three.

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The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Multiple opportunities exist for students to work with standards that specifically call for conceptual understanding and include the use of visual representations, interactive examples, and different strategies.

Cluster 7.NS.A addresses applying and extending previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

  • In Topics 5, 6, and 7 students develop their understanding of rational number operations, including integers and fractions, through several real-world, interactive examples. Students use algebra tiles, number lines, and other manipulatives to represent various problems and their solutions. Fact families and patterns are also used to build understanding of rational number operations. Real world situations include change of elevation, debt, and traveling across time zones.

Cluster 7.EE.A addresses understanding how the properties of operations can be used to generate equivalent expressions.

  • In Topic 8 students begin by looking at growing patterns in “virtual creatures.” Their understanding develops as they transition from the interactive applet in the materials to modeling equivalent expressions with algebra tiles. Furthermore, equivalent expressions leads to solving equations and inequalities, which are modeled with different representations as students transition from a concrete to an abstract understanding of equivalent expressions.
) [14] => stdClass Object ( [code] => 2b [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skills. Students are given multiple opportunities to develop procedural skills with rational numbers, expressions, equations, and inequalities.

Cluster 7.NS.A addresses students developing procedural skills with adding, subtracting, multiplying, and dividing rational numbers.

  • In Topic 5 students have multiple opportunities to develop procedural skills with 7.NS.1, including combining quantities to make zero and addition/subtraction of integers. As students generalize patterns through basic practice and verbal description, they make connections between conceptual understanding and procedural skills.
  • In Topic 6 students have multiple opportunities to multiply and divide integers through Practice problems and Student Activity Sheets. Some problems are strictly procedural while others are built into constructed response assessment items.

Standard 7.EE.1 addresses students being able to add, subtract, factor, and expand linear expressions with rational coefficients, and 7.EE.4 expects students to develop procedural skills with solving linear equations and inequalities.

  • Topics 7 and 8 give students ample opportunities to combine linear expressions with rational coefficients along with solving equations and inequalities. In both topics, various rational numbers (fractions and integers, both positive and negative) are embedded into the expressions, equations, and inequalities.
) [15] => stdClass Object ( [code] => 2c [type] => indicator [points] => 2 [rating] => meets [report] =>

The materials reviewed for Agile Mind Grade 7 meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade. Overall, students are given opportunities to solve application problems that include multiple steps, real-world contexts, and are non-routine.

Application problems allowing students to make their own assumptions in order to apply their mathematical knowledge can be found in different parts of the materials, including MARS Tasks, Constructed Response items and occasionally within the Student Activity Sheets (SAS).

Cluster 7.RP.A addresses students using proportional relationships to solve real-world and mathematical problems.

  • In Topic 1 Constructed Response 1 students use proportional reasoning to modify a vegetable soup recipe based on a given number of people or a specific amount of one of the ingredients. This problem does not include any questions or prompts for scaffolding, and the context is unique when compared to other contexts used in the Topic.
  • In Topic 2 Constructed Response 1 students use proportional reasoning to determine the best buy among different packages of sports drinks. This problem provides some scaffolding for the students as the first part of the problem directs students to find the unit price for each of the three different packages.
  • In Topic 4 Constructed Response 2 students use proportional reasoning to determine the new price of an item after a sale and an additional reduction in price. Students must create a model to represent the situation and determine the full discount without any scaffolding questions or prompts.

Standard 7.NS.3 addresses solving real-world and mathematical problems involving the four operations with rational numbers.

  • In Topic 7 Constructed Response 1 students apply their knowledge of operations with rational numbers when modifying a recipe for trail mix. However, the context for this problem is almost identical to a context that is used during the Topic, and the students are provided with partially completed tables of values that help them determine which operations to use when modifying the recipe.
  • In Topic 7 Constructed Response 2, students answer multiple questions about the costs incurred when different numbers of people participate in different activities, and they are provided a table that lists the activities and their associated costs. In this problem, students are not provided with questions or prompts that lead them toward answers, and the context is unique from others used in the Topic.
) [16] => stdClass Object ( [code] => 2d [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for balance. Overall, the three aspects of rigor are not always treated together and are not always treated separately. Most Topics provide opportunities through lessons and assessments for students to connect conceptual understanding, procedural skill and fluency, and application when appropriate or engage with them separately as needed.

Balance is displayed in Topic 5 in which students apply and extend previous understanding of addition/subtraction of integers using several models. Balance is further evidenced in Topic 13 in which students apply and extend previous understanding of area and volume in a series of animations using number cubes to fill a rectangular prism and cutting the box open to determine its surface area.

) [17] => stdClass Object ( [code] => mathematical-practice-content-connections [type] => component [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for practice–content connections. Overall, the materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialized language of mathematics. However, the materials do not attend to the full meaning of MPs 4 and 5, and there are few opportunities for students to choose their own models or tools when solving problems.

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The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for the Standards for Mathematical Practices (MPs) being identified and used to enrich the mathematics content within and throughout the grade. The instructional materials for the teacher identify the MPs, and students using the materials as intended will engage in the MPs along with the content standards for the grade.

  • The Practice Standards Connections are found within the Professional Support section for the teacher. The eight MPs are listed with three to eight examples for each. According to the Practice Standards Connections “each citation is intended to show how the materials provide students with ongoing opportunities to develop and demonstrate proficiency with the Standards for Mathematical Practice.” Several opportunities exist within the Grade 7 materials where the MPs enrich student learning.
  • Deliver Instruction is located within Advice for Instruction under Professional Support in the teacher material. Occasionally, there will be information within the Deliver Instruction section giving some guidance on how to implement the MP within the task/activity.
    • In Topic 3 Block 8 the teacher guides the student to complete the MARS task “Tiling Squares.” In Deliver Instruction, the questions and activities help the teacher to guide students to identify and generalize patterns verbally and with an algebraic rule. The investigation allows students to engage in MPs 1, 4, and 8.
    • In Topic 9 Block 3 provides an opportunity to reason abstractly and quantitatively (MP2) as students relate experimental probability to theoretical probability through probability experiments with flipping coins.
) [20] => stdClass Object ( [code] => 2f [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 partially meet the expectations for carefully attending to the full meaning of each practice standard (MP). Overall, the materials attend to the full meaning of most of the MPs, but there are two MPs for which the full meaning is not addressed.

The instructional materials do not attend to the full meaning of MPs 4 and 5.

  • MP4: This MP is integrated several times throughout the materials, but the full meaning of the MP is not developed through these different parts of the materials. In Topic 5 during Using the number line and Using the vertical number line, students are given the number lines to model addition/subtraction of integers and numbers and their opposites, but there are no other aspects of MP4 addressed in this Topic. In Topic 12 Block 2, students write equations that model angle relationships and use them to find the measure of different angles, but there are no other aspects of MP4 addressed during this Block. In Topic 13 Block 3 the animation provides a specific example of how students could use square grids to estimate the area of a circle rather than have students determine their own grid. This animation does not allow students to make any assumptions about the problem or identify important quantities as these things are done for them. Also, students do not get to reflect upon their work and make any adjustments that might be necessary.
  • MP5: This MP is integrated at different points in the materials, but the full meaning of the MP is not developed through these different parts of the materials. In Topic 8 during Block 9 students are guided through using a graph to solve an inequality manually and with technology. In this example, the teacher is the person predominantly using the tools, and the students are not able to choose the tools. In Topic 10 during Block 3 the teacher is provided with questions that has students discuss the advantages and disadvantages of using different types of plots with a set of data, but in this example, the types of plots are provided to the students and teachers. In the rest of the Topic, either plots are provided to students in problems or students are directed which plot to make to represent a set of data. In Topic 12 Block 7 students are directed to use a ruler and a protractor to draw triangles that have given characteristics.
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The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for prompting students to construct viable arguments and analyze the arguments of others. Overall, the materials prompt students to construct viable arguments and to analyze the arguments of others.

The instructional materials provide opportunities for students to construct viable arguments.

  • In Topic 1 Block 5 during the MARS task “Mixing paints” students use proportional reasoning to answer several questions in the context of paint mixtures. They are prompted to explain their answer.
  • In Topic 1 Block 7 during the MARS task “Cereal” students explain their reasoning when determining which cereal has the higher ratio of protein.
  • In Topic 10 Block 8 during the MARS task “Best Guess” students determine which student is the best at guessing how long 30 seconds is with a given scenario and data set. They are then asked to justify their own reasoning.
  • In Topic 12 Block 7 students construct the triangles with given angle measurements and side lengths. Students then compare their answers with a partner. There is a missed opportunity for students to defend their own answers and critique the answers of their partners.

The instructional materials provide opportunities for students to analyze the arguments of others.

  • In Topic 4 Block 8 during the MARS task “25% Sale” students are presented with a statement made by a person in the problem, requiring them to identify, analyze, and explain why the person is wrong.
  • In Topic 8 Block 6 Constructed Response 1 students are presented a situation where a teacher is taking his students to a pizza parlor and asks them to to write an equation to represent the total cost for taking any number of students to BB’s Park and Pizza. Then, the students are prompted to evaluate four possible equations with the following two questions. “Which equation(s) correctly represent the cost for BB’s Park and Pizza? Explain why each equation is or is not correct. Which equations are equivalent? Provide evidence that the equations are equivalent.”
) [23] => stdClass Object ( [code] => 2g.ii [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. In Deliver Instruction, classroom strategies and question prompts are provided to assist teachers in engaging students to construct viable arguments or analyzing the arguments of others.

The following are examples of assistance provided to the teachers to promote the construction of viable arguments and analysis of other’s thinking, including prompts, sample questions to ask, and guidance for discussions.

  • In Topic 1 Block 5 the MARS task “Mixing Paints” assists teachers with “Encourage students to be active listeners during the debrief of this task. For example, after a student has shared his or her answer or strategy you can engage students by saying; “Raise your hand if you understand Michael’s strategy.” Then call on one or more of those students to restate the strategy in their own words. ... encourage students to critique each other’s reasoning and to compare the various strategies.”
  • InTopic 2 Block 4 students are asked to solve a punch recipe problem and then play an animation to compare their answer with Briana’s. The teacher is directed to “have them critique Briana’s strategy and discuss how effective it is and how it compares to other strategies used by classmates. Emphasize that there are many different ways that students can use a rate table to reason about the situation.”
  • In Topic 3 Block 3 students work to determine what the cost will be for having medium pizzas delivered. The assistance that teachers are provided with is as follows: “This presents a great opportunity to reinforce the mathematical practice of constructing viable arguments and critiquing the reasoning of others.”
  • In Topic 8 Block 6 as students complete Constructed Response 1, they get the opportunity to analyze the solutions of fictitious students that are presented to them. The assistance teachers are given for engaging students in analyzing each other’s critiques of the fictitious students is as follows: “After students have had 10-12 minutes to work on the task, have them pair up and share the solutions and strategies. Encourage students to critique others' arguments for which equations are equivalent.”
  • In Topic 10 Block 8 the MARS Task “Best Guess” assists teachers with “Students will also need to provide strong evidence for the arguments, so their written responses should include a description of how they applied statistical reasoning and measures of center and spread to make their decisions. ... After students have formulated their arguments, have students pair up and share their responses and rationale. Encourage them to critique each other’s reasoning. ... During the class debrief, continue to encourage students' careful analysis of the various arguments. Some students may be uncomfortable with the fact that there is not a single correct answer. Be sure that measures of center and variability are both discussed, especially when analyzing the guesses of Ben.”
  • In Topic 12 Block 7 students construct the triangles with given angle measurements and side lengths. Students are then asked to compare their answers to a partner. The teacher is instructed to ask students to present their work to the class and given the following assistance: “Doing so will get the students to think about how to articulate their reasoning and defend their work in front of their peers. It will also allow students the opportunity to ask questions of one another so that the student presenting the ideas can justify their thinking.”

The Advice for Instruction misses the opportunity to assist teachers to understand the construct of problems and situations that lend themselves to these prompts in any lesson throughout the course, instilling the habit for communicating mathematical ideas that construct viable arguments and analyze the arguments of others.

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The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for explicitly attending to the specialized language of mathematics. Overall, the materials appropriately use the specialized language of mathematics and expect students and teachers to use it appropriately as well.

Occasionally, there are suggestions within Deliver Instruction as to how teachers can reinforce mathematical language during instruction.

  • Topic 4 Block 1: "This animation reinforces that equivalencies among ratios, percents, and decimals exist because each can be written as an equivalent ratio. Its purpose is not to demonstrate the most efficient process for translating among the representations of ratios, decimals, and percents. The animation builds on the idea that any rational number, including compound ratios, can be expressed as a ratio with a denominator of 1.”
  • Topic 7 Block 1: "Review the different types of numbers: natural, whole, integer, and rational…. Ask students to share their own versions of each definition, correcting any misconceptions as needed.”
  • Topic 12 Block 3: “The animation develops the Angle Sum Theorem for triangles. For this course, the theorem is neither formalized nor proven, but students should be fluent with the fact that the measures of the interior angles in a triangle sum to 180°, and be able to explain why.”

In the student materials, vocabulary terms can be found in bold print within the lesson pages, and these terms are used in context during instruction, practice, and assessment. Vocabulary terms are also available to the students at all time through My Glossary within the materials. For teachers, vocabulary terms for each Topic can be found under Language Support, which is within Advice for Instruction. Both core vocabulary and reinforced vocabulary are listed for each unit.

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The instructional materials reviewed for Agile Mind Grade 7 meet the expectations for alignment to the CCSSM. The materials meet the expectations for focus and coherence in Gateway 1, and they meet the expectations for rigor and the mathematical practices in Gateway 2.

[rating] => meets ) [26] => stdClass Object ( [code] => usability [type] => component [report] =>

The materials reviewed for Agile Mind Grade 7 meet the expectations for usability. The materials are well designed and take into account effective lesson structure and pacing. The instructional materials provide teachers with quality questions to help guide students' mathematical development and a teacher's edition that is easy to use and consistently organized and annotated. The materials provide adult-level explanations and examples of advanced mathematics concepts so that teachers can improve their own knowledge, but grade-level skills are not explicitly connected to standards from previous courses. The instructional materials offer teachers some resources and tools to collect ongoing data about student progress; however, there are no assessments that purposely identify prior knowledge within and across grade levels. Opportunities for ongoing review and practice, and feedback occur in various forms, but there are limited opportunities for students to monitor their own progress. The materials provide strategies to support the needs of ELLs and other special populations and offer some support with differentiating instruction for diverse learners and advanced students.

) [27] => stdClass Object ( [code] => 3a3e [type] => criterion [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet expectations that the materials are well designed and take into account effective lesson structure and pacing. Overall, materials are well-designed, and lessons are intentionally sequenced. Students learn new mathematics in the Exploring section of each Topic as they apply the mathematics and work toward mastery. Students produce a variety of types of answers including both verbal and written answers. The Overview for the Topic introduces the mathematical concepts, and the Summary highlights connections within and between the concepts of the Topic. Manipulatives such as algebra tiles and virtual algebra tiles are used throughout the instructional materials as mathematical representations and to build conceptual understanding.

) [28] => stdClass Object ( [code] => 3a [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for having an underlying design that distinguishes between problems and exercises.

  • Each Topic includes three sections: Overview, Exploring, and Summary. The Overview section introduces the mathematical concepts that will be addressed in the Topic. The Exploring section includes two to four explorations. In these explorations students learn the mathematical concepts of the Topic through problems that include technology-enhanced animations and full-class activities. The Summary section highlights the most important concepts from the Topic and gives students another opportunity to connect these concepts with each other.
  • Each Topic also includes three additional sections: Practice, Assessment, and Activity Sheets. The Practice section includes Guided Practice and More Practice. Guided Practice consists of exercises that students complete during class periods, providing opportunities for students to apply the concepts learned during the explorations. More Practice contains exercises that are completed as homework assignments. The Assessment section includes Automatically Scored and Constructed Response items. These items are exercises to be completed during class periods or as part of homework assignments. They provide more opportunities for students to apply the concepts learned during the explorations. The Activity Sheets also contain exercises, which can be completed during class periods or as part of homework assignments, that are opportunities for students to apply the concepts learned during the explorations.
  • Some Topics also include MARS Tasks, which are exercises that present students with opportunities to apply concepts they have learned from the Topic in which the MARS Task resides or to apply and connect concepts from multiple Topics.
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The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for having a design of assignments that is not haphazard with problems and exercises given in intentional sequences.

The sequencing of Topics, and explorations within the Exploring section for each Topic, develops in a way that helps to build students’ mathematical foundations.

  • The Topics are comprised of similar content. For example, in Topic 5 Adding and Subtracting Integers the Exploring section includes Modeling with Algebra Tiles, Using the Number Line, and Using the Vertical Number Line.
  • Within the explorations for each Topic, problems generally progress from more simple to more complex problems and incorporate knowledge from prior problems or Topics, which offers students opportunities to make connections among mathematical concepts. For example, solving problems related to relationships for different pairs of angles in Topic 12 incorporates and builds upon writing and solving equations from Topic 8.
  • As students progress through the Overview, Exploring, and Summary sections, the Practice (Guided and More), Assessment (Automatically Scored and Constructed Response), and Activity Sheets sections are placed intentionally in the sequencing of the materials to help students build their knowledge and understanding of the mathematical concepts addressed in the Topic.
  • The MARS Tasks are also placed intentionally in the sequencing of the materials to support the development of the students’ knowledge and understanding of the mathematical concepts that are addressed by the tasks.
) [30] => stdClass Object ( [code] => 3c [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for having a variety in what students are asked to produce.

Throughout a Topic, students are asked to produce answers and solutions as well as explain their work, justify their reasoning, and use appropriate models. The Practice section and Automatically Scored items include questions in the following formats: fill-in-the-blank, multiple choice with a single correct answer, and multiple choice with more than one correct answer. Constructed Response items include a variety of ways in which students might respond, i.e. multiple representations of a situation, modeling, or explanation of a process. Also, the types of responses required vary in intentional ways. For example, concrete models or visual representations are expected when a concept is introduced, but as students progress in their knowledge, students are expected to transition to more efficient solution strategies or representations.

) [31] => stdClass Object ( [code] => 3d [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models. The materials include a variety of virtual manipulatives, as well as integrate hands-on activities that allow the use of physical manipulatives.

Most of the physical manipulatives used in Agile Mind are commonly available: ruler, patty paper, graph paper, algebra tiles, and graphing calculators. Due to the digital format of the materials, students also have the opportunity to represent proportional relationships virtually with a table and graph and generate random samples to draw inferences. Each Topic has a Prepare Instruction section that lists the materials needed for the Topic. Manipulatives accurately represent the related mathematics. For example, Topic 8, Exploring Modeling and Solving Linear Equations, students use physical Algebra Tiles to model an equation and write a mathematical procedure along with having an interactive, virtual model of the same problem.

) [32] => stdClass Object ( [code] => 3e [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 have a visual design that is not distracting or chaotic but supports students in engaging thoughtfully with the subject. The student materials are clear and consistent between Topics within a grade-level as well as across grade-levels. Each piece of a Topic is clearly labeled, and the explorations include Page numbers for easy reference. Problems and Exercises from the Practice, Assessment, and Activity Sheets are also clearly labeled and consistently numbered for easy reference by the students. There are no distracting or extraneous pictures, captions, or "facts" within the materials.

) [33] => stdClass Object ( [code] => 3f3l [type] => criterion [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet expectations that materials support teacher learning and understanding of the standards. The instructional materials provide Framing Questions and Further Questions that support teachers in delivering quality instruction, and the teacher’s edition is easy to use and consistently organized and annotated. The materials provide full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject. Although each Topic contains a list of Prerequisite Skills, this list does not connect any of the skills to specific standards from previous grade levels, so the instructional materials partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

) [34] => stdClass Object ( [code] => 3f [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development. The Deliver Instruction section for each Block of a Topic includes Framing Questions for the start of each lesson. For example, in Topic 6 Block 2, the Framing Questions are: “Can you model the multiplication problems from yesterday? (If students struggle, model a few to get them going.) How do you think we could model division with algebra tiles?” During the lesson, the Deliver Instruction section includes multiple questions that teachers can ask while students are completing the activities. At the the end of each lesson, Deliver Instruction includes Further Questions. For example, in Topic 6 Block 6 “How are the rules for integer multiplication related to the rules for raising integers to powers?”

) [35] => stdClass Object ( [code] => 3g [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for containing a teachers edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The materials contain Professional Support which includes a Plan the Course section and a Scope and Sequence document. The Plan the Course section includes Suggested Lesson-planning Strategies and Planning Resources. Each Topic contains an Advice for Instruction section that is divided into Prepare Instruction and Deliver Instruction. For each Topic, Prepare Instruction includes Goals and Objectives, Topic at a Glance, Prerequisite Skills, Resources, and Language Support, and for each Block within a Topic, Deliver Instruction includes Agile Mind Materials, Opening the Lesson, Framing Questions, Lesson Activities, and Suggested Assignment. In Lesson Activities, teachers are given ample annotations and suggestions as to what parts of the materials should be used when and Classroom Strategies that include questions to ask, connections to mathematical practices, or statements that suggest when to introduce certain mathematical terms or concepts.

Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. For example, in Topic 9 Block 1 teachers are directed to, “Show the animation on page 3. Stop the animation on each panel and have students work in pairs to understand the definition of each bolded vocabulary word. Allow students to use previous knowledge or resources to record their definitions and examples. [SAS, question 1]. Show page 4. Have students discuss each profession. Ask for suggestions for other professions that utilize probability.”

) [36] => stdClass Object ( [code] => 3h [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

In Professional Support, Professional Learning, there are four interactive essays entitled “Developing concepts across grades”. There is an Overview that explains the purpose of the four essays, and the topics for these four essays are Functions, Volume, Rate, and Proportionality. Each essay is accessible to teachers and not students, and the Overview states “these essays are available for educators to illustrate connections and deepen understanding around what students may have already learned, and where they are headed on their journey.” Each essay examines the progression of the concept from Grades 6-8 through Algebra I, Geometry, Algebra II, and beyond. By examining the progressions of the concepts beyond Algebra II, teachers have the opportunity to improve their own knowledge of more advanced mathematics concepts that build upon grade-level standards. For example, in Volume, teachers progress from packing a right, rectangular prism with unit cubes to developing the formulas for the volume of cylinders and cones to finding the volume of a figure generated by rotating a two-dimensional shape around a horizontal axis. Also, in Proportionality, teachers explore how proportional relationships are part of the following mathematical concepts: scaling images, linear functions, trigonometric ratios, rational functions, and the derivative.

In addition to “Developing concepts across grades”, the Grade 7 materials also contain a section of interactive essays entitled “Going beyond Grade 7”. There is an Overview that explains the purpose of the three essays, and the Overview states, “These essays are not intended for use with your 7th grade students; rather, they are designed to provide you with important connections and background that will support you as you help your students master 7th grade content.” The topics for these three essays are Rate of Change, Random Variables, and Volume of Solids with Known Cross Sections. Along with having their own section in Professional learning, each of these essays are also referenced in Deliver Instruction for the Blocks where they are appropriate under the title of Teacher Corner. For example, in Topic 3, the essay Rate of Change is referenced for teachers in Block 2 on page 4, and in Topic 9, Random Variables is referenced in Block 4 on page 6. Two of the essays, Rate of Change and Random Variables, connect the Grade 7 content to advanced mathematical concepts through multiple grades and courses, but Volume of Solids with Known Cross Sections progresses quickly from Grade 7 content to content that would be part of a Calculus course. This transition provides few, if any, opportunities for Grade 7 teachers to connect the concept to advanced high school content in Calculus.

In Professional Support, there is a section of Professional Essays which are in either Print or Video format. The Print essays are divided as either Curriculum or Course Management Topics, and although some of the Curriculum Essays are content specific, they do not address mathematical concepts that extend beyond the current grade. The Video Essays - categorized into Teaching with Agile Mind, More Teaching with Agile Mind, and Dimensions of Mathematics Instruction - do not directly provide adult-level explanations or examples of advanced mathematics concepts.

) [37] => stdClass Object ( [code] => 3i [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Prepare Instruction section for each Topic contains a list of Prerequisite Skills, but this list does not connect any of the skills to specific standards from previous grade levels. For example, in Topic 3 the Prerequisite Skills include “Explain what makes a relationship proportional; Describe a relationship using ratios and rates; Simplify a ratio; Identify a unit rate; and Map a coordinate pair on a coordinate grid.” In Prepare Instruction, the Topic at a Glance occasionally provides general references to how concepts will be used in future courses. For example, in Topic 3 “This topic, Patterns in proportional relationships, is designed to build on students' prior knowledge involving ratios, rates, and proportional reasoning developed in previous topics. Students will learn how to analyze relationships using tables and graphs, and develop algebraic equations that describe the relationships. Students will also explore various patterns, developing skills to predict future iterations of a pattern by developing equations. The ability to analyze relationships will be valuable in future topics as students continue to explore relationships using multiple representations.”

) [38] => stdClass Object ( [code] => 3j [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 provide a list of lessons in the teacher's edition, cross­‐referencing the standards covered and providing an estimated instructional time for Topics and Blocks. The materials provide a Mathematics 7 Scope and Sequence document which includes the number of Blocks of instruction for the duration of the year, time in minutes that each Block should take, and the number of Blocks needed to complete each Topic. The Scope and Sequence document lists the CCSSM addressed in each Topic, but there is no part of the materials that aligns Blocks to specific content standards. The materials also provide Alignment to Standards in the Course Materials which allows users to see the alignment of Topics to the CCSSM or the alignment of the CCCSM to the Topics. The Deliver Instruction section contains the Blocks for each Topic. The Practice Standards Connections, found in Professional Support, gives examples of places in the materials where each MP is identified.

) [39] => stdClass Object ( [code] => 3k [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 do not contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

) [40] => stdClass Object ( [code] => 3l [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 do not contain explanations of the instructional approaches of the program and identification of the research-based strategies within the teaching materials. There is a Professional Essays section which addresses a broad overview of mathematics in grades 6-8 as discussed in indicator 3h.

) [41] => stdClass Object ( [code] => 3m3q [type] => criterion [report] =>

The instructional materials for Agile Mind Grade 7 partially meet exceptions that materials offer teachers resources and tools to collect ongoing data about students progress on the Standards. Opportunities for ongoing review and practice, and feedback occur in various forms. Standards are identified that align to the Topic; however, there is no mapping of Standards to items. There are limited opportunities for students to monitor their own progress, and there are no assessments that explicitly identify prior knowledge within and across grade levels. The materials include few opportunities to identify common misconceptions, and strategies to address common errors and misconceptions are only found in a few Deliver Instruction topics.

) [42] => stdClass Object ( [code] => 3m [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 partially meet the expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels. The materials do not provide any assessments that are specifically designed for the purpose of gathering information about students’ prior knowledge, but the materials do provide indirect ways for teachers to gather information about students’ prior knowledge if teachers decide to use them that way.

In Prepare Instruction for each Topic, there is a set of Prerequisite Skills needed for the Topic, and the Overview for each Topic provides teachers with an opportunity to informally assess students prior knowledge of the Prerequisite Skills. For example, in Topic 8 three of the Prerequisite Skills are: “Solve one-step equations algebraically; Create an equation in the form px + q = r from a pattern; and Use models such as a pan balance and algebra tiles to represent a one-step equation.” Then, in the Lesson Activities for the Overview, teachers are told, “This page reminds students of their earlier work writing algebraic rules and equations to represent relationships. Have students verbalize the relationship represented by each equation, paying careful attention to what each variable and each number represents in the given scenario. Show pages from the previous topic as needed. Ask students to reflect on other algebraic rules and equations that they have used before. Have students record some examples including a description of the relationship and what each variable represents.”

) [43] => stdClass Object ( [code] => 3n [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 partially meet the expectation for providing strategies for teachers to identify and address common student errors and misconceptions. There is not an explicit way in which the materials help teachers identify and address common student errors and misconceptions, but there are a few instances in the Deliver Instruction where common errors and misconceptions are identified and suggestions are given for how to address them. For example, in Topic 5 Block 4 teachers are told, “Students find subtraction of integers challenging. They often need extra practice. Don't move too quickly to the symbolic level; let students use the model of their choice until they feel comfortable. Explain that because of their familiarity so far with only positive numbers, students commonly, erroneously model subtraction with a concrete direction on the number line (e.g. that it will always mean go to the left). Emphasizing the topic's treatment of subtraction as adding the opposite can help; adding the opposite doesn't always mean going to the left.”

) [44] => stdClass Object ( [code] => 3o [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials provide opportunities for ongoing review and practice, and feedback occurs in various forms. Within interactive animations, students submit answers to questions or problems, and feedback is provided by the materials. Practice problems and Automatically Scored Assessment items are submitted by the students. Immediate feedback is provided letting students know whether or not they are correct, and if incorrect, suggestions are given as to how the answer can be improved. The Lesson Activities in Deliver Instruction provide some suggestions for feedback that teachers can give while students are completing the lessons.

) [45] => stdClass Object ( [code] => 3p [type] => indicator [report] => ) [46] => stdClass Object ( [code] => 3p.i [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 partially meet the expectation for assessments clearly denoting which standards are being emphasized. The items provided in the Assessment section align to the standards addressed by the Topic, but the individual items are not clearly aligned to particular standards. The set of standards being addressed by a Topic can be found in the Scope and Sequence document or in Course Materials through Alignment to Standards. The MARS Tasks also do not clearly denote which standards are being emphasized.

) [47] => stdClass Object ( [code] => 3p.ii [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 partially meet the expectation for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The MARS Tasks that are included in the materials are accompanied by rubrics aligned to the task that show the total points possible for the task and exactly what students need to do in order to earn each of those points. The Constructed Response items are accompanied by complete solutions, but rubrics aligned to the Constructed Response items are not included. For both the MARS Tasks and the Constructed Response items, alternate solutions are provided when appropriate, but sufficient guidance to teachers for interpreting student performance and suggestions for follow-up are not provided with either the MARS Tasks or the Constructed Response items.

) [48] => stdClass Object ( [code] => 3q [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 offer few opportunities for students to monitor their own progress. Throughout the Exploring, Practice, and Automatically Scored Assessment sections, students get feedback once they submit an answer, and in that moment, they can adjust their thinking or strategy. Goals and Objectives for each Topic are not provided directly to students, but they are given to teachers in Prepare Instruction. There is not a systematic way for students to monitor their own progress on assignments or the Goals and Objectives for each Topic.

) [49] => stdClass Object ( [code] => 3r3y [type] => criterion [report] =>

The instructional materials for Agile Mind Grade 7 meet expectations that materials support teachers in differentiating instruction for diverse learners within and across grades. Activities provide students with multiple entry points and a variety of solution strategies and representations. The materials provide strategies for ELLs and other special populations, but the materials do not always challenge advanced students to deepen their understanding of the mathematics. Grouping strategies are designed to ensure roles for each group member.

) [50] => stdClass Object ( [code] => 3r [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Each Topic consists of three main sections- Overview, Exploring, and Summary, and these three sections are divided into Blocks. Each Block contains lesson activities, materials for Practice, Assessment, and Activity Sheets, along with a MARS Tasks if applicable for the Topic. In each Topic, the Blocks and lesson activities are sequenced for the teacher. In the Advice for Instruction for each Topic, Deliver Instruction for each Block contains instructional notes and classroom strategies that provide teachers with key math concepts to develop, sample questions to ask, ways in which to share student answers, and other similar instructional supports.

) [51] => stdClass Object ( [code] => 3s [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 partially meet the expectation for providing teachers with strategies for meeting the needs of a range of learners. Overall, the instructional materials embed multiple visual representations of mathematical concepts where appropriate, include audio recordings in many explorations, and give students opportunities to engage physically with the mathematical concepts. However, the instructional notes provided to teachers do not consistently highlight strategies that can be used to meet the needs of a range of learners. When instructional notes are provided to teachers, they are general in nature and are intended for all students in the class, and they do not explicitly address the possible range of needs for learners. For example, Topic 2 Block 8 the Deliver Instruction states, “In addition to playing the animation, have students recreate the experience using a rolled up piece of paper. You can have different students or groups choose different dimensions for scope Type A and discuss what difference this makes. You may want to have students choose the same distance to back up so it will be easier to compare results, or allow this to change also for more variety. Groups can present their findings.”

In some explorations, teachers are provided with questions that can be used to extend the tasks students are completing, which are beneficial to excelling students. For struggling students, teachers are occasionally provided with strategies or questions they can use to help move a student’s learning forward. The Summary for each Topic does not provide any strategies or resources for either excelling or struggling students to help with their understanding of the mathematical concepts in the Topic.

) [52] => stdClass Object ( [code] => 3t [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation that materials embed tasks with multiple entry­-points that can be solved using a variety of solution strategies or representations. Overall, tasks that meet the expectations for this indicator are found in some of the Constructed Response Assessment items and Student Activity Sheets that are a part of all Topics. MARS Tasks embedded in some of the Topics have multiple entry-points and can be solved using a variety of solution strategies or representations. For example, in Topic 4 Constructed Response 1 students determine if two fictitious people are correct in their assumption about mixing two types of popcorn together. Students have a choice in the different representations or strategies they can use to correctly analyze the assumption. Another example is Topic 13 Student Activity Sheet 6 Problem 5; students find the area of irregular shapes which can be decomposed in different ways into shapes with which students are more familiar, such as triangles, parallelograms, and trapezoids.

) [53] => stdClass Object ( [code] => 3u [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation that the materials suggest accommodations and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The materials provide suggestions for English Language Learners and other special populations in regards to vocabulary and instructional practices. In Prepare Instruction for Topic 1, Teaching Special Populations of Students refers teachers to the Print Essay entitled “Teaching English Language Learners” in Professional Support, which describes general strategies that are used across the series such as a vocabulary notebook, word walls, and concept maps. Teaching Special Populations of Students also describes general strategies that are used across the series for other special populations, including progressing from concrete stage to representational stage to abstract stage and explicitly teaching metacognitive strategies through think alouds, graphic organizers, and other visual representations of concepts and problems.

In addition to the general strategies mentioned in Teaching Special Populations of Students, there are also many specific strategies listed across each course of the series in Deliver Instruction. In Deliver Instruction, Support for ELL/other special populations includes strategies that can be used with both English Language Learners and students from other special populations. Strategies specific to other special populations can also be found in Classroom strategy or Language strategy. An example of Support for ELL/other special populations from Topic 2, Block 1, Page 2 is “The Latin prefix uni is common in many languages; connect it to unit for ELL students. Directly proportional is a new idea and ELL students as well as students with learning differences may need a physical connection to the concept. Consider using a choral chant to help students begin to internalize what this means. Have students stand up and say “x and y are directly proportional means x times a number equals y. They are always related only through multiplication.”” An example of a strategy for other special populations from Topic 12, Block 5, Page 3 is “Classroom strategy. Students can also build the triangles with straws or other tools. The use of various tools, including technology, to explore forming triangles helps develop conceptual understanding of the conditions needed and is particularly important for students with certain learning differences.”

) [54] => stdClass Object ( [code] => 3v [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 partially meet the expectation that the materials provide opportunities for advanced students to investigate mathematics content at greater depth. Overall, all of the problems provided in the materials are on grade level, and the materials are designed so that all of the problems are assigned to all students over the course of the school year. There are opportunities for advanced students to investigate mathematics at a greater depth in some of the Topics through notes given to teachers in the Advice for Instruction. For example, in Topic 15 Block 5 teachers are given this question to ask students, "How do you think the ratio of the volumes of scaled figures would relate to the scale factor? Explore this question using some simple rectangular prisms."

) [55] => stdClass Object ( [code] => 3w [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 7 meet the expectation for providing a balanced portrayal of various demographic and personal characteristics. The activities are diverse, meeting the interests of a demographically, diverse student population. The names, contexts, videos, and images presented display a balanced portrayal of various demographic and personal characteristics.

) [56] => stdClass Object ( [code] => 3x [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 provide opportunities for teachers to use a variety of grouping strategies. The Deliver Instruction Lesson Activities include suggestions for when students could work individually, in pairs, or in small groups. When suggestions are made for students to work in small groups, there are no specific roles suggested for group members, but teachers are given suggestions to ensure the involvement of each group member. For example, in Topic 11 Block 5 teachers are told to “have groups try various methods for conducting the simulation, and then share their results with the class. Students should justify their work through explanations, posters, or a class discussion.”

) [57] => stdClass Object ( [code] => 3y [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 do not encourage teachers to draw upon home language and culture to facilitate learning. Questions and contexts are provided for teachers in the materials, and there are no opportunities for teachers to adjust the questions or contexts in order to integrate the home language and culture of students into the materials to facilitate learning.

) [58] => stdClass Object ( [code] => 3z3ad [type] => criterion [report] =>

The instructional materials for Agile Mind Grade 7 are web-based and platform neutral but do not include the ability to view the teacher and student editions simultaneously. The materials embed technology enhanced, interactive virtual tools, and dynamic software that engage students with the mathematics. Opportunities to assess students through technology are embedded. The technology provides opportunities to personalize instruction; however, these are limited to the assignment of problems and exercises. The materials cannot be customized for local use. The technology is not used to foster communications between students, with the teacher, or for teachers to collaborate with one another.

) [59] => stdClass Object ( [code] => 3z [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices. Given the digital platform of the materials, the inclusion of interactive tools and virtual manipulatives/objects helps to engage students in the MPs in all of the Topics, and the use of animations in all of the Topics provides for some examples as to how the interactive tools and virtual manipulatives can be used.

) [60] => stdClass Object ( [code] => 3aa [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 are web-based and compatible with multiple internet browsers (Chrome, Firefox, and Internet Explorer). In addition, the materials are “platform neutral” and allow the use of tablets with ChromeOS, Android, or iOS operating systems, but they do not support the use of mobile devices. However, the transition between student and teacher materials is not fluid. There are no direct links between the student and teacher materials, and the student and teacher materials cannot be viewed simultaneously.

) [61] => stdClass Object ( [code] => 3ab [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. All Practice and Automatically Scored Assessment questions are designed to be completed using technology. These items cannot be edited or customized.

) [62] => stdClass Object ( [code] => 3ac [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 include few opportunities for teachers to personalize learning for all students. Within the Practice and Assessment sections, the teacher can choose which problems and exercises to assign students, but these problems and exercises cannot be modified for content or wording from the way in which they are given. Other than being able to switch between English and Spanish in My Glossary, there are no other adaptive or technological innovations that allow teachers to personalize learning for all students.

The instructional materials reviewed for Grade 7 cannot be easily customized for local use. Within My Courses, there are not any options for modifying the sequence or structure of the Topics or any of the sections within the Topics.

) [63] => stdClass Object ( [code] => 3ad [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 7 provide few opportunities for teachers and/or students to collaborate with each other. Under My Agile Mind, teachers can communicate with students through the Calendar and Score and Review. There are no opportunities for teachers to be able to collaborate with other teachers.

) ) [isbns] => Array ( [0] => stdClass Object ( [type] => custom [number] => 978-1-943460-80-9 [custom_type] => [title] => Mathematics 7 Student Activity Book [author] => [edition] => [binding] => [publisher] => Agile Mind [year] => 2016 ) [1] => stdClass Object ( [type] => custom [number] => 978-1-943460-81-6 [custom_type] => [title] => Mathematics 7 Advice for Instruction [author] => [edition] => [binding] => [publisher] => Agile Mind [year] => 2016 ) ) ) 1

Eighth Grade

                                            Array
(
    [title] => Agile Mind Middle School Mathematics (2016)
    [url] => https://www.edreports.org/math/agile-mind-middle-school-mathematics-a/eighth-grade.html
    [grade] => Eighth Grade
    [type] => math-k-8
    [gw_1] => Array
        (
            [score] => 13
            [rating] => meets
        )

    [gw_2] => Array
        (
            [score] => 17
            [rating] => meets
        )

    [gw_3] => Array
        (
            [score] => 31
            [rating] => meets
        )

)
1                                            stdClass Object
(
    [version] => 2.0.0
    [id] => 326
    [title] => Middle School Math - Grade 8
    [report_date] => 2018-03-20
    [grade_taxonomy_id] => 23
    [subject_taxonomy_id] => 5
    [notes] => 

Agile Mind made revisions that affected the scoring and reports for Rating Sheet 2, indicator 3h, and Rating Sheet 4, indicator 3u. These revisions also affected the rating for Gateway 3 so that Agile Mind Grade 8 meets expectations for Instructional Supports and Usability.

[reviewed_date] => 2017-06-08 [revised_date] => 2018-03-20 [gateway_1_points] => 13 [gateway_1_rating] => meets [gateway_1_report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet expectations for focus and coherence. The instructional materials do not assess topics beyond Grade 8, and students and teachers using the materials as designed would devote the large majority of instructional time to the major work of the grade. The instructional materials meet expectations for coherence, and they show strength in having an amount of content that is viable for one school year and fostering coherence through connections within the grade.

[gateway_2_points] => 17 [gateway_2_rating] => meets [gateway_2_report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for rigor and the mathematical practices. The materials meets the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and applications. The materials also meet the expectations for mathematical practices. Overall, the materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialized language of mathematics.

[gateway_3_points] => 31 [gateway_3_rating] => meets [report_type] => math-k-8 [series_id] => 74 [report_url] => https://www.edreports.org/math/agile-mind-middle-school-mathematics-a/eighth-grade.html [gateway_2_no_review_copy] => Materials were not reviewed for Gateway Two because materials did not meet or partially meet expectations for Gateway One [gateway_3_no_review_copy] => This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two [meta_title] => [meta_description] => [meta_image] => [data] => Array ( [0] => stdClass Object ( [code] => focus [type] => component [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for focus on the major work of the grade. The materials do not assess topics before the grade-level in which they should be introduced, and they spend the majority of class time on the major work of the grade when they are used as designed.

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The instructional materials reviewed for Agile Mind Grade 8 meet expectations for not assessing topics before the grade-level in which the topic should be introduced. Overall, there are not assessment items that align to topics beyond Grade 8.

) [2] => stdClass Object ( [code] => 1a [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for assessing grade-level content. The majority of the Grade 8 assessment content was appropriate for the grade. There are some items in the assessments that align to standards above Grade 8 or address content not explicitly addressed in the CCSSM, but omitting or modifying these assessment items would not significantly impact the underlying structure of the Grade 8 materials.

The questions within the Practice and Assessment sections were reviewed for this indicator. The Practice sections within each topic contain multiple questions under the categories of Guided Practice and More Practice. The Assessment sections within each topic contain Automatically Scored questions and Constructed Response questions.

The questions that include content from future grades or address content not explicitly addressed in the CCSSM are as follows:

  • Topic 10 Guided Practice 14 asks students to write an exponential rule (F-LE.2): “Now, write an exponential function rule to find M, the amount of medicine left in the body t hours after the scan. Complete the process column in the table to help you find this rule.”
  • Topic 15 includes items assessing surface area of spheres, which is a topic not explicitly addressed by any standards from the CCSSM. These items are as follows:
    • Guided Practice: 3, 4, and 9
    • More Practice: 7, 8, and 9
    • Automatically scored: 2 and 4
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The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for students and teachers devoting the majority of class time to the major work of the grade when materials are used as designed. Overall, the materials spend approximately 83% of class time on the major work of Grade 8.

) [4] => stdClass Object ( [code] => 1b [type] => indicator [points] => 4 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for spending the majority of class time on the major clusters of the grade. Overall, the instructional materials spend approximately 83% of class time on the major clusters of Grade 8.

For this indicator, the following were examined: all Blocks of instruction within all Topics in Course Contents, Alignment to Standards in Course Materials, the Mathematics 8 Scope & Sequence with Common Core State Standards document in Professional Support, and the Block descriptions for each Topic located within Deliver instruction under Advice for Instruction in Professional Support. There are fifteen topics divided into the following categories: Overview, Explore, Summary, Practice, and Assessment. Each Topic contains 7 to 12 Blocks of instruction, and each Block of instruction represents a 45-minute class period.

In the Block descriptions for each Topic, individual activities are not assigned specific amounts of time, or ranges of time, for the activities to be completed. Thus, when calculating the percentage of class time spent on the major cluster of the grade, two perspectives were appropriate, Topics and Blocks. For these materials, analysis by Blocks is the most appropriate because the Topics do not have an equal number of Blocks within them and the Blocks are not subdivided into smaller increments.

In addition to the Blocks directly aligned to major clusters of the grade, all Blocks aligned to supporting clusters of the grade were also examined. Those Blocks aligned to supporting clusters that were found to incorporate major work of the grade were included in the calculations below:

  • Blocks: 109 of the 132 Blocks, approximately 83%, are spent on the major clusters of the grade.
  • Topics: 14 of the 15, Topics , approximately 93%, are spent on the major clusters of the grade.
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The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for being coherent and consistent with the Standards. The instructional materials show strengths in having an amount of content that is viable for one school year but do not always make explicit connections between prior knowledge and future learning and the major work of the grade. Therefore, the progressions in the Standards are not always evident. The materials foster coherence within grade level work.

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The instructional materials reviewed for Agile Mind Grade 8 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. Overall, supporting content is found primarily in Topics 2, 9, and 15, and the supporting content in these Topics does enhance focus and coherence by engaging students in the major work of the grade. Examples of the connections between supporting work and major work found in these topics include the following:

  • In Topic 2 Blocks 4, 8, 10, and 11 connect major standard 8.EE.2 to supporting standards 8.NS.1 and 8.NS.2. Students are identifying irrational numbers using perfect squares and writing and solving equations to find approximate values of radicals.
  • In Topic 9 Blocks 1, 2, 4, 5, and 7 connect major standards 8.F.3 and 8.F.4 to supporting standards 8.SP.1, 8.SP.2, and 8.SP.3. As students explore bivariate data they are making scatter plots, finding the line of best fit, writing the equation of the line, and using the line of best fit to interpret additional data.
  • In Topic 15 Blocks 3 and 6 connect major standards 8.F.3 and 8.F.4 to supporting standard 8.G.9. As students explore the surface area and volume formulas for cones, cylinders, and spheres they are comparing properties of functions that are represented in different ways and looking at examples of functions that are not linear.
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The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for the amount of content designated for one grade-level being viable for one school year in order to foster coherence between grades. The suggested pacing contains 15 Topics and 132 Blocks (days) of instruction, including assessments. According to the Agile Mind Mathematics 8 Scope and Sequence, each block is expected to last 45 minutes. Some lessons (Constructed Response, MARS tasks) may take longer than indicated.

Each Block includes the following sections: Overview, Exploring, Summary, and Assessment. The Exploring pages are categorized by math concept and can be discussed and reviewed as a class or by individuals/small groups of students.

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The instructional materials reviewed for Agile Mind Grade 8 partially meet the expectations for being consistent with the progressions in the Standards. Overall, the materials develop according to the grade-by-grade progressions in the Standards, and they give all students extensive work with grade-level problems. However, content from prior or future grades is not always clearly identified or related to grade-level work, and the materials do not always relate grade-level concepts explicitly to prior knowledge from earlier grades.

Examples of Grade 8 materials in which off grade-level content is present and not identified as such includes the following:

  • In Topic 10 Blocks 4, 5, and 6, aligned to 8.F.3, extend above grade-level when students must write equations for nonlinear functions, especially exponential functions (F-LE.2).
  • In Topic 11 Block 4, aligned to 8.EE.7, involves solving one-step (6.EE.7) and two-step (7.EE.4) equations algebraically.

The Grade 8 materials provide extensive work with grade-level standards. All students are expected to complete the same problems, and lessons or ideas presented for differentiated instruction also include grade-level problems. The MARS tasks that are included, especially the ones in Topics 5, 7, 8, 9, and 12, are places where students are given the opportunity to engage with the grade-level standards to their full intent.

In lessons where prior knowledge is included, identification of content from prior grades is mentioned in four components of the materials, but the identification is general and not explicitly connected to a grade-level or standard. Examples from the four components are as follows:

  • In the first paragraph of the About the Course section, there is a brief, general overview of topics of which students acquired a foundation prior to Grade 8.
  • The first paragraph in Agile Mind Mathematics 8 Scope and Sequence, 2016-2017 briefly references prior work with expressions, equations/inequalities, dependent/independent variables, area, surface area, and volume and how these concepts connect to Grade 8 work.
  • The Advice for Instruction section references prior work in different places, although specific standards are not referenced. Some examples of this include:
    • In Topic 3 Topic at a glance states, “Throughout this topic, it is assumed that students have had previous exposure to the meaning of exponents.”
    • In Topic 5 the following Prerequisite skills are listed under Prepare instruction: Labeling coordinate axes, Plotting points, Reading information from graphs, Recognizing constant rates of change given a graphical representation, Representing constant rates of change given a verbal description, and Calculating unit rates. The Prerequisite skills are not explicitly connected to any previous, grade-level standards.
    • In Topic 7 the Goals and objectives start by stating, “The topic Linear patterns and functions builds on students' work in patterning in previous grades to develop the formal notion of 'function' and to begin to recognize patterns that can be modeled with linear functions.”
    • In Topic 14 Opening the lesson for Block 4 states, “Students have seen angle relationships from two intersecting lines and from two lines cut by a transversal. They have also seen how the various angle pairs they have been working with relate to each other and how they can be used to establish the fact that two lines are parallel. Now, we want to find ways to extend these geometric relationships to other shapes and situations.”
    • In Topic 15 the first Classroom strategy for Block 1 states, “This Exploring connects to students’ prior understandings of surface area of pyramids and prisms and extends that understanding to include cylinders, cones and spheres.”
  • The Overview of the student material sometimes informs students what they will learn within the Topic and occasionally gives a general connection to previous learning. For example, the Overview of Topic 2 states, “In this topic, you will learn about a set of numbers that behave quite differently from numbers you are currently familiar with. Let’s review the types of numbers that you have learned about in previous mathematics courses.”
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The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for fostering coherence through connections at a single grade, where appropriate and when the standards require. Overall, the materials include learning objectives that are visibly shaped by CCSSM cluster headings, and they provide problems and activities that connect two or more clusters in a domain or two or more domains when these connections are natural and important.

Some examples of Topic Headings and Goals and Objectives shaped by cluster headings include the following:

  • Topic Headings:
    • In Topic 7 “ Linear patterns and functions” is shaped by 8.F.A,B.
    • In Topic 11 “Solving Linear Equations” is shaped by 8.EE.C.
  • Goals and Objectives:
    • In Topic 2 “know that there are numbers that are not rational” is shaped by 8.NS.A.
    • In Topic 3 “generate the laws of exponents and apply them in problem solving situations” is shaped by 8.EE.A.
    • In Topic 4 “apply the Pythagorean Theorem and its converse in problem situations” is shaped by 8.G.B.
    • In Topic 7 “use variables to generalize linear patterns and represent problem situations that can be modeled by linear functions” is shaped by 8.F.B.
    • In Topic 13 “solve systems of linear equations using the substitution method” is shaped by 8.EE.C.

The following are topics that contain problems and/or activities which connect two or more clusters in a domain or two or more domains in a grade.

  • In Topic 4, 8.G.B and 8.EE.A are connected as students solve problems involving the Pythagorean Theorem by applying knowledge of square roots.
  • In Topic 7, 8.F.A and 8.F.B are connected as students define, evaluate, and compare functions along with using functions to model relationships between quantities.
  • In Topic 8, 8.F.A and 8.EE.B are connected as students connect various representations of a function by writing equations, making graphs and tables, and interpreting values within these representations.
  • In Topic 10, 8.F.A and 8.F.B are connected as students explore nonlinear relationships by examining examples of functions that are not linear and by qualitatively describing the relationship between two quantities that are not linear.
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The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance in all three.

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The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Multiple opportunities exist for students to work with standards that specifically call for conceptual understanding and include the use of visual representations, interactive examples, and different strategies.

Cluster 8.EE.B addresses understanding the connections between proportional relationships, lines, and linear equations, and standards 8.F.2,3 address comparing and defining functions.

  • In Topic 8 students are given opportunities to connect verbal descriptions of situations to graphs and write situations that can be represented by a specific graph. The MARS task “Vacations” gives students the opportunity to compare slopes on two different graphs. Students also use proportional and nonproportional reasoning to derive a two-variable equation representing a situation. Overall, students are responsible for interpreting the given real-world situation and representing it in multiple ways, i.e. tables, graphs, verbal descriptions, and equations.

Standard 8.F.1 addresses understanding that a function is a rule that assigns to each input exactly one output.

  • In Topic 7 the definition of a function is developed in multiple ways. Students understand functions through the use of verbal description, input-output machines, real-world situations, graphs, and mappings.

Cluster 8.G.A addresses understanding congruence and similarity through different tools.

  • In Topic 1 students are given multiple opportunities to describe the effects of a transformation of a shape on the coordinate plane. Students demonstrate understanding by moving points on the coordinate plane given a specific transformation and by describing the movement of a point algebraically. Although there are a few opportunities to describe a sequence of transformations and to explain the effects of change on the figure as a whole, greater focus is placed on individual ordered pairs. In this topic, angle measures and side lengths are referenced in dilations.
  • In Topic 14 students further develop their understanding of how angle relationships are affected by transformations through the use of verbal descriptions, animations, real-world examples with maps, and the use of geometric tools.
) [14] => stdClass Object ( [code] => 2b [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skills. Overall, students are given opportunities to develop procedural skills within clusters 8.EE.C and 8.G.C.

Cluster 8.EE.C addresses students developing procedural skills with analyzing and solving linear equations and pairs of linear equations in one variable.

  • In Topic 11 students have opportunities to practice solving multi-step equations in one variable using the Distributive Property and combining like terms through the Practice and Assessment sections, along with the Student Activity Sheets. Also, within this topic there are opportunities to develop procedural skills with solving equations resulting in infinitely many or no solutions.
  • In Topic 12 students have opportunities to solve systems of equations both graphically and algebraically. Also, within this Topic there are opportunities to develop procedural skills with solving systems of linear equations resulting in infinitely many or no solutions.

Cluster 8.G.C addresses developing procedural skills with the formulas for the volume of cylinders, cones, and spheres.

  • In Topic 15, 3-dimensional animations are provided as assistance for students in understanding the development of the volume formulas. There is limited practice in using the formulas to find the volume of cones, cylinders, and spheres. There are problems within the Practice and Assessment questions, which are multiple choice, that involve finding the volume of cones, cylinders, and spheres, and there are three Constructed Response items that involve these formulas as well.
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The materials reviewed for Agile Mind Grade 8 meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade. Overall, students are given opportunities to solve application problems that include multiple steps, real-world contexts, and are non-routine.

Application problems allowing students to make their own assumptions in order to apply their mathematical knowledge can be found in different parts of the materials, including MARS Tasks, Constructed Response items, and occasionally within the Student Activity Sheets (SAS).

Standard 8.EE.8c addresses students solving real-world and mathematical problems leading to two linear equations in two variables.

  • In Topic 12 Constructed Response 1 students are given tables of data for two students walking in front of a motion detector. The tables include the distances in feet from the motion detector after different amounts of time in seconds. Students are ultimately asked to interpret the point of intersection for the graphs of the two sets of data. Students are provided with scaffolded steps in this problem that lead them to interpreting the point of intersection.
  • Topic 12 Constructed Response 2 is a scaffolded problem that leads students through the steps of creating a solution that contains a certain percentage of pure acid.
  • In Topic 12 Constructed Response 3 students are presented a problem that has them create a solution containing a certain percentage of pure acid, just as in Constructed Response 2 of Topic 12, but the problem does not provide them any scaffolded questions to help them obtain the answer.
  • In Topic 12 the MARS task, Pathways, allows students to write and solve a system of equations that will yield the appropriate dimensions of a paving stone based on a desired design design of the pathway. This problem does not include any questions or prompts for scaffolding, and the context is unique to the topic, which makes the problem non-routine.
  • In Topic 13 Constructed Response 1 students write and solve a system of equations that will result in the dimensions of three horse pens. This problem does not include any questions or prompts for scaffolding, and the context is unique to the topic, which makes the problem non-routine. Furthermore, students must alter the equations and recompute the dimensions based on a change in feet of fencing used.

Cluster 8.F.B addresses students being able to use functions to model relationships between quantities.

  • In Topic 5 Constructed Response 2 students are given a graph that shows the volume of four different gas tanks and how much time is needed for each gas tank to become empty. Students have to answer different questions using the graph, and the questions involve analyzing the volumes of the tanks, the time needed to empty the tanks, distance traveled, and gas mileage. There are no questions or prompts that provide scaffolding to lead students toward the answers, and although the context is similar to one students encountered during the Topic, the use of four gas tanks as opposed to one gives students the opportunity to apply their mathematical knowledge in a non-routine way.
  • In Topic 6 Constructed Response 1 students are presented with a graph that shows distance traveled from a motion detector over time. Students are expected to answer different questions about the graph, but the context used is exactly the same as the context used with many other problems throughout the Topic. Also, the questions used in this problem are the same in wording and structure as other questions posed in the Topic.
  • In Topic 7 Constructed Response 2 students are presented with the first three steps of a tile pattern and must answer questions about different steps in the pattern. Students are provided with some scaffolding during the problem as they are instructed to include a general function rule and a description of what is constant and what changes in the tile pattern as they respond. The context of tiles is unique for this problem in comparison to the other contexts used in the Topic.
) [16] => stdClass Object ( [code] => 2d [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for balance. Overall, the three aspects of rigor are not always treated together and are not always treated separately. Most Topics provide opportunities through lessons and assessments for students to connect conceptual understanding, procedural skill and fluency, and application when appropriate or engage with them separately as needed.

Balance is displayed in Topic 4 when students apply and extend previous understanding of the Pythagorean Theorem as they complete an activity. Balance is further evidenced in Topic 11 where students conceptually solve linear equations using different models.

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The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for practice–content connections. Overall, the materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialized language of mathematics. However, the materials do not attend to the full meaning of MPs 4 and 5, and there are few opportunities for students to choose their own models or tools when solving problems.

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The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for the Standards for Mathematical Practices (MPs) being identified and used to enrich the mathematics content within and throughout the grade. The instructional materials for the teacher identify the MPs, and students using the materials as intended will engage in the MPs along with the content standards for the grade.

  • The Practice Standards Connections are found within the Professional Support section for the teacher. The eight MPs are listed with six to ten examples for each. According to the Practice Standards Connections, “each citation is intended to show how the materials provide students with ongoing opportunities to develop and demonstrate proficiency with the Standards for Mathematical Practice.”
  • Deliver Instruction is located within Advice for Instruction under Professional Support in the teacher material. Occasionally, there will be information within the Deliver Instruction section giving some guidance on how to implement the MP within the task/activity.
    • In Topic 8 Block 8 the teacher leads the class through the “Connecting Representations” pages. With the use of teacher questioning and activities found in the Deliver Instruction teacher material, the teacher helps the students understand that relationships can be expressed abstractly and quantitatively. Throughout this lesson, the students are engaging in MP2.
    • Topic 12 Block 6 Deliver Instruction suggests teachers tell students to make sense of the problem by explaining it to their partner. By doing this, students are engaging in MP1; however, that is not noted within the teacher information.
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The instructional materials reviewed for Agile Mind Grade 8 partially meet the expectations for carefully attending to the full meaning of each practice standard (MP). Overall, the materials attend to the full meaning of most of the MPs, but there are two MPs for which the full meaning is not addressed.

The instructional materials do not attend to the full meaning of MPs 4 and 5.

  • MP4: This MP is integrated several times throughout the materials, but the full meaning of the MP is not developed through these different parts of the materials. In Topic 7 during the MARS Task “Squares and Circles” the teacher is directed to “observe how students are modeling with mathematics” by noting different representations that the students pick/create in order to model a context. In this task, most of the models are provided for the students, and students are not defining quantities for themselves or needing to revise their initial choices. In Topic 8 there are different opportunities for students to engage with this MP, but the problems do not allow for students to define their own quantities, and most of the models are provided for the students. There are some opportunities for students to revisit their initial calculations, but this is due to new information being introduced into the problem and not because there could be other solutions that are more optimal. In Topic 13 students create a system of equations to solve a real-world problem, but the quantities needed are defined for the students and revisions to the initial calculations are due to new information being introduced into the problem.
  • MP5: This MP is integrated at different points in the materials, but the full meaning of the MP is not developed through these different parts of the materials. In Topic 1 Block 1 teachers are given assistance for discussing the full meaning of MP5 with students, but the students are not engaged with using appropriate tools strategically at this time as the tools for the activity are given to them. Also in Topic 1, students are told to use patty paper as their tool in Block 2, and teachers are directed to tell students that they will have dilations as a new tool to use in problems in Block 9. In Topic 4 during Block 3, students are told to use patty paper, a ruler, and a pencil as they work on a proof related to the Pythagorean Theorem. In Topic 11 during Blocks 2 and 3 students are lead through using a graphing calculator to solve an equation with both tables and graphs. In Block 4 this MP is identified, but students are shown how to use algebra tiles to solve an equation.
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The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for prompting students to construct viable arguments and analyze the arguments of others. Overall, the materials prompt students to construct viable arguments and present opportunities for students to analyze the arguments of others.

The instructional materials provide opportunities for students to construct viable arguments.

  • In Topic 4 Block 4 the class is investigating the converse of the Pythagorean Theorem and must answer, “What do the conclusions you have reached so far tell you about the triangles and ∠?????”
  • In Topic 5 during the MARS task “Graphs” students are told to “Explain how you made your choices” after matching several equations and graphs.
  • In Topic 8 Block 4 students compare data in a table during a class discussion and must answer, “Which amount grows at a faster rate—the amount paid or the amount collected? How do you know?”
  • In Topic 14 Block 2 the class is introduced to parallel lines cut by a transversal and the related angles. The class is asked “Use what you have learned about parallel lines, supplementary angles and corresponding angles to explain why ????∠2 = ????∠7.”

The instructional materials provide opportunities for students to analyze the arguments of others.

  • In Topic 2 Constructed Response 3 students create a cube out of modeling clay that has a volume of 20 cubic centimeters. In the last part of the problem, students are instructed to “Compare your group's estimates and results with another group by reading your written responses to another group. Did your methods differ for finding an approximation for the edge length? Did your explanations differ? Revise your writing after reflection and feedback from the other group.”
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The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. In Deliver Instruction, classroom strategies and question prompts are provided to assist teachers in engaging students to construct viable arguments or analyzing the arguments of others.

The following are examples of assistance provided to the teachers to promote the construction of viable arguments and analysis of other’s thinking, including prompts, sample questions to ask, and guidance for discussions.

  • In Topic 2 after the students complete Constructed Response 1, there is a class discussion and the teacher is instructed, “The debrief of this task provides an opportunity for students to construct viable arguments and critique the reasoning of others. As students are explaining their solution methods, they should also attend to precision. Encourage the audience members to ask questions of the presenters if information is not clear, either in vocabulary or logic. Push students to clarify their thinking and to use precise vocabulary when explaining their solution method(s).”
  • In Topic 3 Block 1 Professional Support Deliver Instruction, teachers are guided to listen for misconceptions related to exponents, use questions as needed that specifically ask students to analyze why or how a hypothetical person thinks about an answer or gets an answer, and identify whether the person correctly interpreted the meaning of the mathematics in the problem. The intent of these questions is to help students analyze the arguments presented by others and determine how those arguments support the mathematics in this specific problem.
  • In Topic 4 Block 3 students watch animations to help them understand the Pythagorean Theorem, and the teacher is instructed to pose these questions: “Consider one of the right triangles on your Patty Paper. What is the base of the triangle? What is the height of the triangle? Do all four right triangles have the same area? How do you know?” These questions assist teachers in engaging students in constructing viable arguments.
  • In Topic 4 Block 4 students investigate the converse of the Pythagorean Theorem. Teachers are provided with the following assistance, “This may be the first time students have been pushed to make precise mathematical arguments about a geometric relationship. ... Give students the opportunity to make their own arguments before showing the final Check button. Then let students compare their arguments with the one shown online. Discuss the precision of the language and the logical reasoning used. The intent of this page is not for students in this course to develop strict mathematical proofs, but instead to expose students to the reasoning and language used in such arguments.” This assistance is specific in that teachers can draw students’ attention to specific aspects of the solutions provided, which helps in constructing an argument. Also, students can use the correct solutions as a way to analyze their own arguments and improve them as needed.
  • In Topic 5 of the MARS Task “Graphs” teachers are provided with the following assistance, “As the majority of students seem to be finishing the task, put students into pairs and assign one of the four graphs to the each pair of students by counting off student pairs by four. ... This can help students grow in their ability to construct sound arguments and provide meaningful critiques of others’ arguments.”
  • In Topic 5 as the students work on Constructed Response 1, the teacher is instructed to have “students verify their graphs and written responses with a partner. Provide students an opportunity to revise their work as needed but ask that they note any modifications and justification for changes.”
  • In Topic 5 as the students work on Constructed Response 2, the teacher is given the following Classroom Strategy: “Divide the class into two “teams.” Then have each team come to a consensus on their responses for each part. Have team leaders present their team’s answers. Allow time for the other team to ask for clarifications. This will provide students with an opportunity to practice constructing viable arguments and critiquing the reasoning of others as they justify their conclusions, communicate them to others, and respond to the arguments of others.” The assistance provided for the teacher helps create an environment where MP3 can occur.
  • In Topic 14 Block 2 students construct an argument to show that the measures of two angles are equal. The assistance provided to the teacher is as follows: “Encourage students to begin their proving process by measuring or tracing angles. They can write algebraic equations using the variables labeling each angle. Students may need to state their reasons verbally before recording their ideas on paper. These are a great opportunity to promote the mathematical practice of constructing viable arguments and critiquing the reasoning of others. Engage students in each others' arguments by asking them to restate key arguments in their own words or describe how those arguments are related to angles within each image.” This assistance gives teachers specific strategies for helping students construct a viable argument, and it also provides specific ways in which students can begin to analyze the arguments of others.
  • In Topic 14 Constructed Response 1 students construct an argument to prove that two rays are parallel. The assistance that is provided to teachers with this problem is to “Ask a few students to share their explanation. Again, encourage students to critique the reasoning of the students, in a respectful way, and come to a class consensus on a strong explanation."

In the Advice for Instruction there is a missed opportunity to provide support for teachers that explains and identifies where and when problems, tasks, examples, and situations lend themselves to these types of questions. Additional guidance is needed to broaden the application of these questions throughout the course so that students routinely construct viable arguments and analyze the arguments of others.

) [24] => stdClass Object ( [code] => 2g.iii [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for explicitly attending to the specialized language of mathematics. Overall, the materials appropriately use the specialized language of mathematics and expect students and teachers to use it appropriately as well.

Occasionally, there are suggestions within Deliver Instruction as to how teachers can reinforce mathematical language during instruction.

  • Topic 1 Block 2: “Discuss the term corresponding points. Make sure students understand this, as it is part of all three transformations.”
  • Topic 3 Block 2: “...encourage them to use technical vocabulary to describe what is happening: base, exponent, and sum.”
  • Topic 11 Block 1: “Language strategy. Students may have trouble at first telling the difference between a function and an equation, and may need to review some core vocabulary from previous topics: function rule, function, equation, input, output, domain, and range may need further review before, during, and after the lessons. Stress the relationship between a function, which describes the relationship between two varying quantities, and an equation, which represents a specific instance of the functional relationship.”

In the student materials, vocabulary terms can be found in bold print within the lesson pages, and these terms are used in context during instruction, practice, and assessment. Vocabulary terms are also available to the students at all time through My Glossary within the materials. For teachers, vocabulary terms for each Topic can be found under Language Support, which is within Advice for Instruction. Both core vocabulary and reinforced vocabulary are listed for each unit.

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The instructional materials reviewed for Agile Mind Grade 8 meet the expectations for alignment to the CCSSM. The materials meet the expectations for focus and coherence in Gateway 1, and they meet the expectations for rigor and the mathematical practices in Gateway 2.

[rating] => meets ) [26] => stdClass Object ( [code] => usability [type] => component [report] =>

The materials reviewed for Agile Mind Grade 8 meet the expectations for usability. The materials are well designed and take into account effective lesson structure and pacing. The instructional materials provide teachers with quality questions to help guide students' mathematical development and a teacher's edition that is easy to use and consistently organized and annotated. The materials provide adult-level explanations and examples of advanced mathematics concepts so that teachers can improve their own knowledge, but grade-level skills are not explicitly connected to standards from previous courses. The instructional materials offer teachers some resources and tools to collect ongoing data about student progress; however, there are no assessments that purposely identify prior knowledge within and across grade levels. Opportunities for ongoing review and practice, and feedback occur in various forms, but there are limited opportunities for students to monitor their own progress. The materials provide strategies to support the needs of ELLs and other special populations and offer some support with differentiating instruction for diverse learners and advanced students.

) [27] => stdClass Object ( [code] => 3a3e [type] => criterion [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet expectations that the materials are well designed and take into account effective lesson structure and pacing. Overall, materials are well-designed, and lessons are intentionally sequenced. Students learn new mathematics in the Exploring section of each Topic as they apply the mathematics and work toward mastery. Students produce a variety of types of answers including both verbal and written answers. The Overview for the Topic introduces the mathematical concepts, and the Summary highlights connections within and between the concepts of the Topic. Manipulatives such as algebra tiles and virtual algebra tiles are used throughout the instructional materials as mathematical representations and to build conceptual understanding.

) [28] => stdClass Object ( [code] => 3a [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for having an underlying design that distinguishes between problems and exercises.

  • Each Topic includes three sections: Overview, Exploring, and Summary. The Overview section introduces the mathematical concepts that will be addressed in the Topic. The Exploring section includes two to four explorations where students learn the mathematical concepts of the Topic through problems that include technology-enhanced animations and full-class activities. The Summary section highlights the most important concepts from the Topic and gives students another opportunity to connect these concepts with each other.
  • Each Topic also includes three additional sections: Practice, Assessment, and Activity Sheets. The Practice section includes Guided Practice and More Practice. Guided Practice consists of exercises that students complete during class periods and give opportunities for students to apply the concepts learned during the explorations. More Practice contains exercises that are completed as homework assignments. The Assessment section includes Automatically Scored and Constructed Response items. These items are exercises to be completed during class periods or as part of homework assignments. They provide more opportunities for students to apply the concepts learned during the explorations. The Activity Sheets also contain exercises, which can be completed during class periods or as part of homework assignments, that are opportunities for students to apply the concepts learned during the explorations.
  • Some Topics also include MARS Tasks, which are exercises that present students with opportunities to apply concepts they have learned from the Topic in which the MARS Task resides or to apply and connect concepts from multiple Topics.
) [29] => stdClass Object ( [code] => 3b [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for having a design of assignments that is not haphazard with problems and exercises given in intentional sequences.

The sequencing of Topics, and explorations within the Exploring section for each Topic, develops in a way that helps to build students’ mathematical foundations.

  • The Topics are comprised of similar content.
  • Within the explorations for each Topic, problems generally develop from more simple to more complex problems and incorporate knowledge from prior problems or Topics, which offers students opportunities to make connections among mathematical concepts. For example, solving systems of linear equations in Topic 12 incorporates and builds upon solving individual linear equations from Topic 11.
  • As students progress through the Overview, Exploring, and Summary sections, the Practice (Guided and More), Assessment (Automatically Scored and Constructed Response), and Activity Sheets sections are placed intentionally in the sequencing of the materials to help students build their knowledge and understanding of the mathematical concepts addressed in the Topic.
  • The MARS Tasks are also placed intentionally in the sequencing of the materials to support the development of the students’ knowledge and understanding of the mathematical concepts that are addressed by the tasks.
) [30] => stdClass Object ( [code] => 3c [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for having a variety in what students are asked to produce.

Throughout a Topic, students are asked to produce answers and solutions as well as explain their work, justify their reasoning, and use appropriate models. The Practice section and Automatically Scored items include questions in the following formats: fill-in-the-blank, multiple choice with a single correct answer, and multiple choice with more than one correct answer. Constructed Response items include a variety of ways in which students might respond, i.e. multiple representations of a situation, modeling, or explanation of a process. Also, the types of responses required vary in intentional ways. For example, concrete models or visual representations are expected when a concept is introduced, but as students progress in their knowledge, students are expected to transition to more efficient solution strategies or representations.

) [31] => stdClass Object ( [code] => 3d [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for having manipulatives that are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models. The materials include a variety of virtual manipulatives, as well as integrate hands-on activities that allow the use of physical manipulatives.

Most of the physical manipulatives used in Agile Mind are commonly available: ruler, patty paper, graph paper, algebra tiles, and graphing calculators. Due to the digital format of the materials, students also have the opportunity to represent equations or functions virtually with tables and graphs. Each Topic has a Prepare Instruction section that lists the materials needed for the Topic. Manipulatives accurately represent the related mathematics. For example, Topic 15 Exploring Volume, students use physical geometric models to explore finding the volumes of various solids along with having interactive, virtual models of the same shapes.

) [32] => stdClass Object ( [code] => 3e [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 8 have a visual design that is not distracting or chaotic but supports students in engaging thoughtfully with the subject. The student materials are clear and consistent between Topics within a grade-level as well as across grade-levels. Each piece of a Topic is clearly labeled, and the explorations include Page numbers for easy reference. Problems and Exercises from the Practice, Assessment, and Activity Sheets are also clearly labeled and consistently numbered for easy reference by the students. There are no distracting or extraneous pictures, captions, or "facts" within the materials.

) [33] => stdClass Object ( [code] => 3f3l [type] => criterion [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet expectations that materials support teacher learning and understanding of the standards. The instructional materials provide Framing Questions and Further Questions that support teachers in delivering quality instruction, and the teacher’s edition is easy to use and consistently organized and annotated. The materials provide full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject. Although each Topic contains a list of Prerequisite Skills, this list does not connect any of the skills to specific standards from previous grade levels, so the instructional materials partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

) [34] => stdClass Object ( [code] => 3f [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Grade 8 meet the expectation for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development. The Deliver Instruction section for each Block of a Topic includes Framing Questions for the start of each lesson. For example, in Topic 12 Block 3 the Framing Questions are: “What do we mean by a 'solution' to a system of equations? How would solve these systems?” During the lesson the Deliver Instruction section includes multiple questions that teachers can ask while students are completing the activities. At the end of each lesson, Deliver Instruction includes Further Questions. For example, in Topic 12 Block 5 “Could we have solved the two equations modeling the tickets in terms of d? What would the equations have been? If so, what would the solution have looked like?”

) [35] => stdClass Object ( [code] => 3g [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for containing a teachers edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Also, where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The materials contain Professional Support which includes a Plan the Course section and a Scope and Sequence document. The Plan the Course section includes Suggested Lesson-planning Strategies and Planning Resources. Each Topic contains an Advice for Instruction section that is divided into Prepare Instruction and Deliver Instruction. For each Topic, Prepare Instruction includes Goals and Objectives, Topic at a Glance, Prerequisite Skills, Resources, and Language Support, and for each Block within a Topic, Deliver Instruction includes Agile Mind Materials, Opening the Lesson, Framing Questions, Lesson Activities, and Suggested Assignment. In Lesson Activities, teachers are given ample annotations and suggestions as to what parts of the materials should be used when and Classroom Strategies that include questions to ask, connections to mathematical practices, or statements that suggest when to introduce certain mathematical terms or concepts.

Where applicable, the materials include teacher guidance for the use of embedded technology to support and enhance student learning. For example, in Topic 3 Block 4 teachers are directed to, “Show students the animation on page 7 that combines multiplication and division with different bases in the same problem. [SAS, question 7] Next, have students work on finding the solution for the expression on page 8. [SAS, question 8]”

) [36] => stdClass Object ( [code] => 3h [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for containing full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

In Professional Support, Professional Learning, there are four interactive essays entitled “Developing concepts across grades”. There is an Overview that explains the purpose of the four essays, and the topics for these four essays are Functions, Volume, Rate, and Proportionality. Each essay is accessible to teachers and not students, and the Overview states “these essays are available for educators to illustrate connections and deepen understanding around what students may have already learned, and where they are headed on their journey.” Each essay examines the progression of the concept from Grades 6-8 through Algebra I, Geometry, Algebra II, and beyond. By examining the progressions of the concepts beyond Algebra II, teachers have the opportunity to improve their own knowledge of more advanced mathematics concepts that build upon grade-level standards. For example, in Volume, teachers progress from packing a right, rectangular prism with unit cubes to developing the formulas for the volume of cylinders and cones to finding the volume of a figure generated by rotating a two-dimensional shape around a horizontal axis. Also, in Proportionality, teachers explore how proportional relationships are part of the following mathematical concepts: scaling images, linear functions, trigonometric ratios, rational functions, and the derivative.

In addition to “Developing concepts across grades”, the Grade 8 materials also contain a section of interactive essays entitled “Going beyond Grade 8”. There is an Overview that explains the purpose of the three essays, and the Overview states, “These essays are not intended for use with your 8th grade students; rather, they are designed to provide you with important connections and background that will support you as you help your students master 8th grade content.” The topics for these three essays are are Rate of Change, Finding the Line of Best Fit, and Trigonometric Ratios. Along with having their own section in Professional learning, each of these essays are also referenced in Deliver Instruction for the Blocks where they are appropriate under the title of Teacher Corner. For example, in Topic 9, the essay Finding the Line of Best Fit is referenced for teachers in Block 1 on pages 1-2 of Exploring Trend Lines, and in Topic 1, Trigonometric Ratios is referenced in Block 9 on page 12. The three essays connect the Grade 8 content to advanced mathematical concepts through multiple grades and courses.

In Professional Support, there is a section of Professional Essays which are in either Print or Video format. The Print essays are divided as either Curriculum or Course Management Topics, and although some of the Curriculum Essays are content specific, they do not address mathematical concepts that extend beyond the current grade. The Video Essays - categorized into Teaching with Agile Mind, More Teaching with Agile Mind, and Dimensions of Mathematics Instruction - do not directly provide adult-level explanations or examples of advanced mathematics concepts.

) [37] => stdClass Object ( [code] => 3i [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 partially meet the expectation for explaining the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

The Prepare Instruction section for each Topic contains a list of Prerequisite Skills, but this list does not connect any of the skills to specific standards from previous grade levels. For example, in Topic 1 the Prerequisite Skills include “Using operations with integers; Creating ratios; Using operations with fractions; Knowledge of the coordinate plane; and Identifying lines of symmetry.” In Prepare Instruction the Topic at a Glance occasionally provides general references to how concepts will be used in future courses. For example, in Topic 1 “This topic, Transformational geometry and similarity, explores congruence transformations and dilations. These transformations are fun for students and most students have an intuitive understanding of what 'looks right' when performing the transformations. However, students need to have a deeper understanding of what's happening algebraically, as these principles will be applied in later math. These transformations are the basis for how the coordinate plane will be used.”

) [38] => stdClass Object ( [code] => 3j [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 8 provide a list of lessons in the teacher's edition, cross­‐referencing the standards covered and providing an estimated instructional time for Topics and Blocks. The materials provide a Mathematics 8 Scope and Sequence document which includes the number of Blocks of instruction for the duration of the year, time in minutes that each Block should take, and the number of Blocks needed to complete each Topic. The Scope and Sequence document lists the CCSSM addressed in each Topic, but there is no part of the materials that aligns Blocks to specific content standards. The materials also provide Alignment to Standards in the Course Materials which allows users to see the alignment of Topics to the CCSSM or the alignment of the CCCSM to the Topics. The Deliver Instruction section contains the Blocks for each Topic. The Practice Standards Connections, found in Professional Support, gives examples of places in the materials where each MP is identified.

) [39] => stdClass Object ( [code] => 3k [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 8 do not contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

) [40] => stdClass Object ( [code] => 3l [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 8 do not contain explanations of the instructional approaches of the program and identification of the research-based strategies within the teaching materials. There is a Professional Essays section which addresses a broad overview of mathematics in grades 6-8 as discussed in indicator 3h.

) [41] => stdClass Object ( [code] => 3m3q [type] => criterion [report] =>

The instructional materials for Agile Mind Grade 8 partially meet exceptions that materials offer teachers resources and tools to collect ongoing data about students progress on the Standards. Opportunities for ongoing review and practice, and feedback occur in various forms. Standards are identified that align to the Topic; however, there is no mapping of Standards to items. There are limited opportunities for students to monitor their own progress, and there are no assessments that explicitly identify prior knowledge within and across grade levels. The materials include few opportunities to identify common misconceptions, and strategies to address common errors and misconceptions are only found in a few Deliver Instruction topics.

) [42] => stdClass Object ( [code] => 3m [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 partially meet the expectations for providing strategies for gathering information about students' prior knowledge within and across grade levels. The materials do not provide any assessments that are specifically designed for the purpose of gathering information about students’ prior knowledge, but the materials do provide indirect ways for teachers to gather information about students’ prior knowledge if teachers decide to use them that way.

In Prepare Instruction for each Topic, there is a set of Prerequisite Skills needed for the Topic, and the Overview for each Topic provides teachers with an opportunity to informally assess students prior knowledge of the Prerequisite Skills. For example, in Topic 15 two of the Prerequisite Skills are: “Area and circumference of circles and Surface area and volume of prisms and pyramids.” Then, in the Lesson Activities for the Overview, teachers are told, “This Exploring connects to students’ prior understandings of surface area of pyramids and prisms and extends that understanding to include cylinders, cones and spheres. The formulas for surface area and volume are developed from physical models so that students are able to see and make strong connections.”

) [43] => stdClass Object ( [code] => 3n [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 partially meet the expectation for providing strategies for teachers to identify and address common student errors and misconceptions. There is not an explicit way in which the materials help teachers identify and address common student errors and misconceptions, but there are a few instances in the Deliver Instruction where common errors and misconceptions are identified and suggestions are given for how to address them. For example, in Topic 7 Block 5 teachers are told, “A common misunderstanding that students make when working with this task (MARS Task: Squares and CIrcles) is that they graph points incorrectly, or forget to graph the points, in questions 1 and 2. If this misunderstandings occurs, ask students to re-read both parts of each prompt or ask them to explain how they determined the coordinates or location of the point they graphed.”

) [44] => stdClass Object ( [code] => 3o [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills. The materials provide opportunities for ongoing review and practice, and feedback occurs in various forms. Within interactive animations, students submit answers to questions or problems, and feedback is provided by the materials. Practice problems and Automatically Scored Assessment items are submitted by the students. Immediate feedback is provided letting students know whether or not they are correct, and if incorrect, suggestions are given as to how the answer can be improved. The Lesson Activities in Deliver Instruction provide some suggestions for feedback that teachers can give while students are completing the lessons.

) [45] => stdClass Object ( [code] => 3p [type] => indicator [report] => ) [46] => stdClass Object ( [code] => 3p.i [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 partially meet the expectation for assessments clearly denoting which standards are being emphasized. The items provided in the Assessment section align to the standards addressed by the Topic, but the individual items are not clearly aligned to particular standards. The set of standards being addressed by a Topic can be found in the Scope and Sequence document or in Course Materials through Alignment to Standards. The MARS Tasks also do not clearly denote which standards are being emphasized.

) [47] => stdClass Object ( [code] => 3p.ii [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 partially meet the expectation for assessments including aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The MARS Tasks that are included in the materials are accompanied by rubrics aligned to the task that show the total points possible for the task and exactly what students need to do in order to earn each of those points. The Constructed Response items are accompanied by complete solutions, but rubrics aligned to the Constructed Response items are not included. For both the MARS Tasks and the Constructed Response items, alternate solutions are provided when appropriate, but sufficient guidance to teachers for interpreting student performance and suggestions for follow-up are not provided with either the MARS Tasks or the Constructed Response items.

) [48] => stdClass Object ( [code] => 3q [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 8 offer few opportunities for students to monitor their own progress. Throughout the Exploring, Practice, and Automatically Scored Assessment sections, students get feedback once they submit an answer, and in that moment, they can adjust their thinking or strategy. Goals and Objectives for each Topic are not provided directly to students, but they are given to teachers in Prepare Instruction. There is not a systematic way for students to monitor their own progress on assignments or the Goals and Objectives for each Topic.

) [49] => stdClass Object ( [code] => 3r3y [type] => criterion [report] =>

The instructional materials for Agile Mind Grade 8 meet expectations that materials support teachers in differentiating instruction for diverse learners within and across grades. Activities provide students with multiple entry points and a variety of solution strategies and representations. The materials provide strategies for ELLs and other special populations, but the materials do not always challenge advanced students to deepen their understanding of the mathematics. Grouping strategies are designed to ensure roles for each group member.

) [50] => stdClass Object ( [code] => 3r [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Each Topic consists of three main sections- Overview, Exploring, and Summary, and these three sections are divided into Blocks. Each Block contains lesson activities including Practice, Assessment, and Activity Sheets, along with any MARS Tasks in the Topic. In each Topic, the Blocks are sequenced for the teachers, and the lesson activities within the Blocks are sequenced for the teachers. In the Advice for Instruction for each Topic, Deliver Instruction for each Block contains instructional notes and classroom strategies that provide teachers with key math concepts to develop, sample questions to ask, ways in which to share student answers, and other similar instructional supports.

) [51] => stdClass Object ( [code] => 3s [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 partially meet the expectation for providing teachers with strategies for meeting the needs of a range of learners. Overall, the instructional materials embed multiple and/or visual representations of mathematical concepts where appropriate, include audio recordings with many explorations, and give students opportunities to engage physically with the mathematical concepts. However, the instructional notes provided to teachers do not consistently highlight these strategies that can be used to meet the needs of a range of learners. When instructional notes are provided to teachers, they are general in nature and are intended for all students in the class, and they do not explicitly address the possible range of needs for learners. For example, in Topic 5 Block 1 the Deliver Instruction states “Give students the situation, either the verbal description or the graph, and give them time to see what information they can pull from it. Then, give them additional time to present that information in the opposing format. Students can usually catch on to analyzing graphs quickly, when given sufficient time initially to process important data from the graph. Ask students smaller, more specific questions about parts of a graph or situation before tackling the entire problem.”

In some explorations, teachers are provided with questions that can be used to extend the tasks students are completing, which are beneficial to excelling students. For struggling students, teachers are occasionally provided with strategies or questions they can use to help move a student’s learning forward. The Summary for each Topic does not provide any strategies or resources for either excelling or struggling students to help with their understanding of the mathematical concepts in the Topic.

) [52] => stdClass Object ( [code] => 3t [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation that materials embed tasks with multiple entry­-points that can be solved using a variety of solution strategies or representations. Overall, tasks that meet the expectations for this indicator are found in some of the Constructed Response Assessment items and Student Activity Sheets that are a part of all Topics. MARS Tasks embedded in some Topics have multiple entry-points and can be solved using a variety of solution strategies or representations. For example, in Topic 14 Constructed Response 2 students complete a geometry puzzle called an angle network. Students are given information about a picture containing three parallel lines intersected by two transversals, and they have choices as to which information to use first and how to proceed in determining the angles measures for the angles in the puzzle. Another example is in Topic 13 Student Activity Sheet 6 Problem 11. Students create one system of linear equations for each of the following three conditions: a system with one solution, a system with many solutions, and a system with no solutions.

) [53] => stdClass Object ( [code] => 3u [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation that the materials suggest accommodations and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

The materials provide suggestions for English Language Learners and other special populations in regards to vocabulary and instructional practices. In Prepare Instruction for Topic 1, Teaching Special Populations of Students refers teachers to the Print Essay entitled “Teaching English Language Learners” in Professional Support, which describes general strategies that are used across the series such as a vocabulary notebook, word walls, and concept maps. Teaching Special Populations of Students also describes general strategies that are used across the series for other special populations, including progressing from concrete stage to representational stage to abstract stage and explicitly teaching metacognitive strategies through think alouds, graphic organizers, and other visual representations of concepts and problems.

In addition to the general strategies mentioned in Teaching Special Populations of Students, there are also many specific strategies listed across each course of the series in Deliver Instruction. In Deliver Instruction, Support for ELL/other special populations includes strategies that can be used with both English Language Learners and students from other special populations. Strategies specific to other special populations can also be found in Classroom strategy or Language strategy. An example of Support for ELL/other special populations from Topic 6, Block 3 is “Use a think-write-pair-share strategy to help ELL and those with other learning differences process the given information and the framing questions. Give time for students to individually think about the questions and time to write a response. Then pair a non-native English speaker with a native English speaker and have each student share his or her responses.” An example of a strategy for other special populations from Topic 2, Block 2, Pages 3-4 is “Classroom strategy. Students may not be familiar with bar notation used to represent repeating decimals. Take time now to review this notation. The use of technology in this case is especially important for those students with certain learning differences; it can remove barriers and allow them to access the content they need to learn.”

) [54] => stdClass Object ( [code] => 3v [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 partially meet the expectation that the materials provide opportunities for advanced students to investigate mathematics content at greater depth. Overall, all of the problems provided in the materials are on grade level, and the materials are designed so that all of the problems are assigned to all students over the course of the school year. There are opportunities for advanced students to investigate mathematics at a greater depth in some of the Topics through notes given to teachers in the Advice for Instruction. For example, in Topic 14 Block 2 teachers are given the following assistance, "The animation shows only one case, so you may prefer to ask the students to conduct a further exploration in the following way: Give each group of students three pieces of spaghetti and a different angle measure to explore. Ask each group to use spaghetti to construct lines cut by a transversal in such a way that one pair of corresponding angles has the measure assigned to the group. Ask the group to measure the distance between the two parallel‐line candidates. Then, ask the group to try to keep the angles congruent but make the parallel lines not be parallel. Then ask the group to shift the position of the transversal and redo the process."

) [55] => stdClass Object ( [code] => 3w [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Agile Mind Grade 8 meet the expectation for providing a balanced portrayal of various demographic and personal characteristics. The activities are diverse, meeting the interests of a demographically, diverse student population. The names, contexts, videos, and images presented display a balanced portrayal of various demographic and personal characteristics.

) [56] => stdClass Object ( [code] => 3x [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 8 provide opportunities for teachers to use a variety of grouping strategies in the Deliver Instruction Lesson Activities including when students work individually, in pairs, or in small groups. When suggestions are made for students to work in small groups, there are no specific roles suggested for group members, but teachers are given suggestions to ensure the involvement of each group member. For example, Topic 7 Block 7 teachers are told to “have each student in a group present one part of the problem.”

) [57] => stdClass Object ( [code] => 3y [type] => indicator [report] =>

The instructional materials reviewed for Agile Mind Grade 8 do not encourage teachers to draw upon home language and culture to facilitate learning. Questions and contexts are provided for teachers in the materials, and there are no opportunities for teachers to adjust the questions or contexts in order to integrate the home language and culture of students into the materials to facilitate learning.

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The instructional materials for Agile Mind Grade 8 are web-based and platform neutral but do not include the ability to view the teacher and student editions simultaneously. The materials embed technology enhanced, interactive virtual tools, and dynamic software that engage students with the mathematics. Opportunities to assess students through technology are embedded. The technology provides opportunities to personalize instruction; however, these are limited to the assignment of problems and exercises. The materials cannot be customized for local use. The technology is not used to foster communications between students, with the teacher, or for teachers to collaborate with one another.

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The instructional materials reviewed for Agile Mind Grade 8 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices. Given the digital platform of the materials, the inclusion of interactive tools and virtual manipulatives/objects helps to engage students in the MPs in all of the Topics, and the use of animations in all of the Topics provides examples as to how the interactive tools and virtual manipulatives can be used.

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The instructional materials reviewed for Agile Mind Grade 8 are web-based and compatible with multiple internet browsers (Chrome, Firefox, and Internet Explorer). In addition, the materials are “platform neutral” and allow the use of tablets with ChromeOS, Android, or iOS operating systems, but they do not support the use of mobile devices. However, the transition between student and teacher materials is not fluid. There are no direct links between the student and teacher materials, and the student and teacher materials cannot be viewed simultaneously.

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The instructional materials reviewed for Agile Mind Grade 8 include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. All Practice and Automatically Scored Assessment questions are designed to be completed using technology. These items cannot be edited or customized.

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The instructional materials reviewed for Agile Mind Grade 8 include few opportunities for teachers to personalize learning for all students. Within the Practice and Assessment sections, the teacher can choose which problems and exercises to assign students, but these problems and exercises cannot be modified for content or wording from the way in which they are given. Other than being able to switch between English and Spanish in My Glossary, there are no other adaptive or technological innovations that allow teachers to personalize learning for all students.

The instructional materials reviewed for Grade 8 cannot be easily customized for local use. Within My Courses, there are not any options for modifying the sequence or structure of the Topics or any of the sections within the Topics.

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The instructional materials reviewed for Agile Mind Grade 8 provide few opportunities for teachers and/or students to collaborate with each other. Under My Agile Mind, teachers can communicate with students through the Calendar and Score and Review. There are no opportunities for teachers to be able to collaborate with other teachers.

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AMSCO Math: Algebra 1, Geometry, Algebra 2 (2016)

Perfection Learning | High School | 2016 Edition

High School

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The instructional materials reviewed for the AMSCO Traditional Series do not meet expectations for focusing on the non-plus standards of the CCSSM and exhibiting coherence within and across courses that is consistent with a logical structure of mathematics. The instructional materials attend to the full intent of the high school standards and spend a majority of time on the widely applicable prerequisites from the CCSSM. The instructional materials partially attend to engaging students in mathematics at a level of sophistication appropriate to high school, making connections within courses and across the series, and explicitly identifying standards from Grades 6-8 and building on them to the High School Standards. The materials do not attend to the full intent of the modeling process when applied to the modeling standards and allowing students to fully learn each non-plus standard.

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The instructional materials reviewed for the AMSCO Traditional series meet expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. There are few instances where all the aspects of the standards are not addressed, and there is one standard that is not addressed by the materials. Overall, most non-plus standards are addressed to the full intent of the mathematical content by the instructional materials.

The following are standards that have been addressed fully in the instructional materials:

  • A-CED: The standards from A-CED are addressed throughout the Algebra 1 and Algebra 2 courses. In Algebra 1, students create linear equations and inequalities in one and two variables to represent relationships in problems. Students extend their understanding of creating inequalities to linear programming problems in Algebra 2 when they consider viable solutions within given constraints. Additionally, students create absolute value (Algebra 1, Algebra 2), quadratic (Algebra 1, Algebra 2), exponential (Algebra 1, Algebra 2), and rational (Algebra 2) equations to solve problems.
  • F-BF.3: Transformations of functions are emphasized throughout the Algebra 1 and Algebra 2 courses. In Algebra 1, students transform absolute value functions in Lesson 4.4 (pages 164-170), quadratic functions in Lesson 8.7 (pages 288-290), and exponential functions in Lesson 9.2 (pages 326-334). In Algebra 2, students transform quadratic and absolute value functions in Lesson 1.1 (pages 49-54), square root and cube root functions in Lesson 5.5 (pages 251-255), exponential functions in Lesson 6.1 (pages 264-265), and logarithmic functions in Lesson 7.2 (pages 302-305).
  • G-CO.9: Students prove theorems about lines and angles. Proofs include: vertical angles are congruent; alternate interior angles are congruent when a transversal intersects parallel lines; alternate exterior angles are congruent when a transversal intersects parallel lines; corresponding angles are congruent when a transversal intersects parallel lines; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints; segment addition postulate; if two angles are complementary to the same angle they are congruent to each other; if two angles are supplementary to the same angle they are congruent to each other; and two angles that form a linear pair have measures that sum to 180 degrees. Through these proofs, students enhance their understanding of the relationship between lines and angles in geometric figures.

The following standards are partially addressed in the instructional materials:

  • A-REI.11: In Algebra 1, Lesson 5.1, the materials state, “The coordinates of the point of intersection, or points of intersection, of the lines define the solution of the system” (page 185). However, an explanation as to why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equations f(x)=g(x) is not provided. The materials do provide examples where f(x) and g(x) are linear, absolute value, polynomials, rationals, exponential, and logarithmic functions.
  • G-CO.4: In Geometry, Lesson 1.6, materials define reflections in terms of line of symmetry, but not in terms of angles, circles, perpendicular lines, parallel lines, and/or line segments.
  • G-CO.13: Students construct an equilateral triangle, square, and equilateral hexagon; however, not all of these constructions are inscribed in a circle. Students construct a hexagon inscribed in a circle in Geometry, Lesson 8.3, pages 372-373.
  • S-IC.5: Students compare two treatments in an experiment in an online activity paired with Algebra 2, Lesson 10.7. However, students do not compare differences between two parameters or a statistic and a parameter to determine if the data is statistically significant. The concept of significance is not defined or discussed in materials.
  • S-CP.4: In Algebra 2, Lesson 10.4, pages 490-493, students interpret two-way tables to calculate probabilities and determine if two categorical variables are independent. However, students do not construct a two-way table to represent categorical data.

The following standard is not addressed in the instructional materials:

  • A-REI.5: In Algebra 1, Lesson 5.3, students solve systems of equations using elimination by replacing one equation with the sum of that equation and a multiple of another equation in the system, but a proof by comparison of methods or how this method works is not provided in the materials.
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The instructional materials reviewed for the AMSCO Traditional series do not meet expectations for attending to the full intent of the modeling process when applied to the modeling standards. The instructional materials do not include all aspects of the modeling process, and students do not engage in parts of the modeling process.

Students do not have an opportunity to work through the entire modeling cycle independently due to extensive scaffolding. Students do not have opportunities to make assumptions about problems, develop their own solution strategies, validate their solutions, and either improve upon their model or report their conclusions. Students apply and analyze given problems and scenarios, but the variables, parameters, units, equations, or problem-solving methods have been identified.

While aspects of the modeling process are attended to, there are components of the modeling process that are altogether missing from the series. Examples from the instructional materials include:

  • In Algebra 1, Lesson 3.8, Problem 9, page 135, students compare prices for two car repair services. Students are provided the variables and equations, yet they explain the equations in words. Students are directed to graph the equations (as to which variable goes on the x-axis and which goes on the y-axis) in order to find the difference in cost for service that would take 2, 3, or 5 hours before making a conclusion about which service to choose. Students do not formulate the equations nor do they have the opportunity to validate their conclusion. (A-SSE.1, F-IF.7a)
  • In Algebra 1, Chapters 1-6, Cumulative Review, Problem 23, page 238, students use information about hourly wages earned working two different jobs as well as the maximum number of hours that can be worked and the minimum amount of money needed to be earned weekly. Students write a system of linear inequalities to represent the situation and graph the system in order to describe the range of possible combinations of hours worked at each job to make at least $240 per week. Students formulate the inequalities representing the information provided, but they do not decide on a pathway for solving the problem or validate or report their conclusion. (A-CED.3)
  • In Geometry, Chapters 1-10, Cumulative Review, Problem 30, page 525, students determine a new height for a platform on a zip-line course in order to reduce the speed when participants approach the landing platform. Students are given certain parameters, change the height of the landing platform (not the take-off platform) by making the angle of depression between the two platforms smaller, and this removes assumptions students can make as they identify variables in the problem. In addition, students justify their answer but are not given an opportunity to validate their solution. (G-SRT.8)
  • In Geometry, Lesson 9.6, Multi-Part Problem Practice, page 56, students use coordinates representing street intersections that bound a park. Students apply properties of quadrilaterals to find the area of the park, determine how much fencing would be needed to enclose the park, and determine the coordinate for the center of the park where a tree will be planted. Students do not develop a plan to approach this multi-part problem, rather they are offered a step-by-step outline. Students compute area, perimeter, and midpoint and interpret their calculations in the context of the problem, but students do not validate or report their solution. (G-GPE.7)
  • In Algebra 2, Lesson 9.5, Problem 31, page 431, students write an equation to model the oscillating motion of a puck attached to a spring. The instructional materials identify the variables d (displacement in inches from its equilibrium position) and t (time in seconds) and that the motion is modeled by the sine or cosine function. Students formulate the equation, but they do not make assumptions about the variables and parameters involved in this scenario in order to develop the equation. Students do not interpret their equation in terms of the original scenario nor do they validate that their equation accurately represents the scenario. (F-TF.5)
  • In Algebra 2, Lesson 6.2, Problem 29, students write an equation to model the dilution of a solution of alcohol with water. The instructional materials identify the variables x (number of dilutions) and y (portion of alcohol that remains), so students do not make assumptions about the variables and parameters in order to develop the equation. Students graph the equation and use the equation or graph for two computations. Students do not interpret their equation in terms of the original scenario nor do they validate their equation to ensure it accurately represents the scenario. (F-LE.1c)
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The instructional materials reviewed for the AMSCO Traditional series meet expectations, when used as designed, for spending the majority of time on the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs and careers (WAPs). Some examples of how the materials spend the majority of time on the WAPs include:

  • N-RN.2: In Algebra 1, Lessons 1.7 and 9.1 and Algebra 2, Lessons 5.1 and 5.3, students rewrite expressions involving rational exponents and radicals to create equivalent expressions.
  • A-APR.3: In Algebra 1, Lesson 8.5, students use zeros of quadratics to sketch a graph. This knowledge is extended in Algebra 2, Lesson 3.5 when students find zeros given a graph of a polynomial and write the function of the graphed polynomial.
  • F-IF.6: In Algebra 1, Lesson 3.3, students calculate and interpret the average rate of change of a linear function presented symbolically and as a table. Students later calculate and interpret the rate of change for quadratic functions in Algebra 1, Lesson 8.5. In Algebra 2, the concept of rate of change is extended to exponential functions in Lesson 6.1 and logarithmic functions in Lesson 7.5.
  • G-SRT.B, C: In Geometry, Chapter 7, students build upon knowledge of similarity from Geometry, Chapter 2. In Chapter 7, the materials address proving triangle similarity in order to develop the Pythagorean Theorem. This extends to solving problems with special right triangles and trigonometric ratios.
  • S-ID.7: In Algebra 1, Lesson 10.4 and Algebra 2, Lesson 1.2, students find a linear regression equation to represent a data set and describe the meaning of the slope and y-intercept within the context of the problem for linear models.

There is little evidence of where plus standards or non-CCSSM standards distract from the WAPs. An example of distracting topics include:

  • Algebra 2, Lesson 9.6 addresses reciprocal trigonometric functions. This topic is not included in the CCSSM high school standards. This lesson is labeled “Optional” in the instructional materials.
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The instructional materials reviewed for the AMSCO Traditional series do not meet expectations, when used as designed, for letting students fully learn each non-plus standard. Overall, where the standards expect students to prove, derive, or develop understanding of a concept, the materials often provide students the information. For some standards, the materials provide limited opportunities for students to learn the non-plus standard fully.

The following are examples of how the materials, when used as designed, do not enable students to learn the non-plus standards fully:

  • N-RN.3: In Algebra 1, Lesson 9.1, the materials state, “The sum or product of two rational numbers is always rational;” however, students do not explain why these two properties are true (page 322). Furthermore, students identify whether the following two properties are always, sometimes, or never true: (1) Multiplying a rational number by an irrational number results in a rational number, and (2) Adding a rational number and an irrational number results in a rational number (page 363). Students do not explain their reasoning.
  • A-SSE.3: Students regularly produce equivalent forms of expressions, yet do not use these equivalent expressions to explain properties of the quantity represented by the expression. For example, in Algebra 1, Chapter 8, students utilize procedures to produce equivalent quadratic expressions but do not make connections between factored form and x-intercepts of the graphed quadratic or vertex form and whether the vertex is a maximum or minimum.
  • A-SSE.3c: In Algebra 1, Lesson 9.3, Practice Problem 8b, students determine an equivalent way to express $$N(t)=50(2^{2t})$$ (page 339). In Algebra 2, Lesson 6.2, Practice Problems 26-28, students transform expressions for exponential functions to change the time measurement for the independent variable (page 277). These practice problems provide limited opportunities for students to learn the standard fully.
  • A-SSE.4: In Algebra 2, Lesson 8.4, the materials derive the formula for the sum of a finite geometric series (page 361). Students do not derive the formula but do use the formula to solve problems.
  • A-APR.4: The materials prove several polynomial identities - square of a binomial sum (Algebra 2, Lesson 2.1, page 81), product of the sum and difference of two terms (Algebra 2, Lesson 2.1, page 81), identity to generate Pythagorean triples (Algebra 2, Lesson 2.3, page 93), but students do not prove these identities.
  • A-REI.1: In Algebra 1, Lesson 2.1, students explain each step in solving a linear equation as following from the equality of numbers asserted at the previous step. Students do not explain each step in solving a non-linear equation.
  • A-REI.4a: In Algebra 1, Lesson 8.8, the materials derive the quadratic formula by completing the square on a general quadratic equation in standard form (page 291), but students do not derive the quadratic formula.
  • F-IF.8b: In Algebra 1, Lesson 9.2, Practice Problem 11, students identify six functions as exponential growth or exponential decay (page 332); however, students do not interpret parts of the exponential equations. In Algebra 1, Lesson 9.3, Practice Problem 21, students identify the decay rate for a given exponential function (page 340), yet properties of exponents are not needed to answer the question. In Algebra 2, Lesson 6.1, Practice Problem 24, students use properties of exponents to rewrite the function $$g(x)=(\frac{5}{2})^{3-x}$$ into an exponential function of the form $$a^{x+b}$$ (page 268), however, students do not need to interpret this function. In Algebra 2, Chapter 6, Review Problems 12 and 13, students use properties of exponents to interpret expressions for exponential functions involving rates of inflation (page 291).
  • F-BF.4a: Students write equations for the inverses of linear functions in Algebra 1, Lesson 3.7 and Algebra 2, Lesson 6.4, Opportunities, when students write equations for the inverses of non-linear functions, are limited to Algebra 1, Chapter 3, Review Problem 44, page 139, when students determine if $$f(x)^{-1}=\sqrt{x}$$ is the inverse of $$f(x)=x^2$$, Algebra 2, Lesson 6.4, Practice Problem 2, page 285, when students identify the graph of the inverse for a quadratic function with a restricted domain in a multiple choice question and Algebra 2, Cumulative Review Chapters 1-9, Problem 1 (page 460), when students identify the inverse of a rational function in a multiple choice question.
  • F-LE.1: In Algebra 1 and Algebra 2, students model situations with linear functions and exponential functions. However, students have limited opportunities to distinguish between situations that could be modeled with a linear or an exponential function. Limited opportunities for students to distinguish between linear and exponential models are provided in Algebra 2, Lesson 9.7.
  • F-LE.1a: In Algebra 2, Lesson 6.1, the materials show how linear functions grow by equal differences over equal intervals, and exponential functions grow by equal factors over equal intervals in an example (page 265). Students do not prove this relationship.
  • F-LE.3: In Algebra 1, Lesson 9.2, students observe the relationship that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically in Model Problem 5 (page 329) when comparing quadratic and exponential functions using a table and graph and Model Problem 6 (page 330) when comparing linear, quadratic, and exponential functions using a graph. These examples are completed for the students.
  • F-TF.2: In Algebra 2, Lesson 9.4, students apply understandings of the unit circle; however, they do not explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers.
  • G-CO.3: In Geometry, Lesson 1.6, students find the smallest angle of rotation that matches an image to the original figure. There was no evidence found where students take a given rectangle, parallelogram, trapezoid, or regular polygon and rotate it onto itself.
  • G-CO.8: In Geometry, Lessons 5.3 and 5.4, students use SSS, SAS, and ASA criteria to show that two triangles are congruent. Students do not explain how these criteria stem from the definition of congruence in terms of rigid motions.
  • G-GPE.6: In Geometry, Lesson 1.2, students use ratios to find the length of a partitioned segment or find the coordinate of an endpoint of a line segment. In Practice Problem 5, students find point B that partitions line segment AC in a ratio of 1:1, so students find the midpoint of segment AC (page 50). There is no evidence of students finding the point on a directed line segment between two points that partition the segment in a ratio other than 1:1.
  • G-GMD.4: In Geometry, Lesson 10, students identify the shape of two-dimensional cross-sections of three-dimensional objects (pages 475, 481-482). However, students' identification of three-dimensional objects generated by rotations of two-dimensional objects is limited to Problem 17 in the Practice Problem portion of the lesson.
  • G-MG.1: Throughout the Geometry course, Model Problems use geometric shapes, their measures, and their properties to describe objects include representing the shape of a yard using trapezoids (Lesson 9.6, page 450-451), a soup can as a cylinder (Lesson 10.2, page 486), and the shape of a hand using cylinders and prisms. Since these examples are completed for the students, students do not learn the standard fully. In Lesson 10.3, Practice Problem 36, page 507, students select a complex object and sketch the object reducing it to four or more solids that have a known formula for volume.
  • G-MG.3: Throughout the Geometry course, Model Problems apply geometric methods to solve design problems include building a new basketball court using specified ratios and limited gym space (Lesson 2.3, page 104) and deciding on the shape of an ornament to minimize the surface area (Lesson 10.3, page 503). Students apply geometric methods to solve design problems in Multi-Part Problem practice (Lesson 9.8, page 464) when designing a sculpture out of aluminum and steel with budget constraints. These problems combined provide limited opportunities for students to learn the standard fully.
  • S-ID.6b: In Algebra 1, Lesson 10.4, Practice Problems 1e, 2e, 3e, 4e, and 9d, students create residual plots (pages 388-389); however, students do not analyze the residuals to assess the fit of a function.
  • S-ID.9: In Algebra 1, Lesson 10.4, the materials state, “Correlation does not always mean causation,” (page 387) and provide two examples and Practice Problems 5 and 6 for students to distinguish the difference between causation and correlation. These problems combined provide limited opportunities for students to fully learn the standard.
  • S-IC.2: In Algebra 2, Lesson 10.1, Model Problem 1, page 468 and Practice Problem 16, page 471, students analyze whether results from a spinner are fair. These problems provide limited opportunities for students to learn the standard fully.
  • S-IC.6: In Algebra 2, Lesson 10.6, Practice Problems 9 and 10, page 512, students determine the validity of a decision based on reported data. In Algebra 2, Lesson 10.7, Practice Problem 22, page 522, students determine whether a polling company was accurate in their report given that the actual results fell outside their reported margin of error. These problems provide limited opportunities to learn the standard fully.
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The instructional materials reviewed for the AMSCO Traditional series partially meet expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The instructional materials regularly use age-appropriate contexts and apply key takeaways from grades 6-8, yet do not vary the types of numbers being used.

The materials provide opportunities to solve problems in real-world contexts that are relevant to high school students. Examples include:

  • In Algebra 1, Lesson 2.4, students determine how many miles a car can drive on 24 gallons of gas (Practice Problem 5, page 71).
  • In Algebra 1, Lesson 9.3, students determine the value of an autographed baseball card given that it grew 10% in value every year (Practice Problem 7, page 339).
  • In Geometry, Lesson 7.2, students use similar triangles and the Law of Reflection from physics to determine how to tag a friend in a game of laser tag (Practice Problem 34, page 296).
  • In Geometry, Lesson 7.8, students use the Law of Cosines to determine the length of a slide at an amusement park (Practice Problem 30, page 344).
  • In Algebra 2, Lesson 2.4, students determine whether a field goal kicker kicked the ball high enough to clear the goalpost (Practice Problem 21, page 102).
  • In Algebra 2, Lesson 6.2, students compare how the amount of money they invest changes depending on whether the account compounds interest annually, quarterly, monthly, or daily (Multi-Part Problem Practice, page 277).

The instructional materials offer opportunities for students to apply/extend key takeaways from Grades 6-8. Examples include:

  • In Algebra 1, Lesson 10.4, students apply statistical concepts by graphing scatterplots, using a calculator to find the correlation coefficient, and plotting residuals.
  • In Algebra 1, Lesson 9.2 and Algebra 2, Lesson 6.1, students apply the rate of change for a linear function to the average rate of change for an exponential function.
  • In Geometry, Lesson 7.6, students extend their understanding of ratios as they define the trigonometric ratios of sine, cosine, and tangent.
  • In Geometry, Chapters 2 and 7, students extend their understanding of proportional relationships by dilating geometric figures, proving triangles similar, and using trigonometric functions to solve problems.

The types of real numbers being used are not varied throughout any single course or the entire series. Examples include:

  • In Algebra 1, Lesson 2.1, students solve two-sided equations in Practice Problems 5-20, page 55. While four of the equations consist of non-integer values, all but one of the equations results in a positive whole-number answer.
  • In Algebra 1, Lesson 8.4, students solve quadratic equations by graphing. Of the quadratic equations graphed in Practice Problems 1-20, Practice Problem 4 contains a decimal: $$x^2+7x+12.25=0$$.
  • In Geometry, Lesson 1.2, problems involving the distance formula and midpoint formula include integer coordinates.
  • In Geometry, Lessons 4.1 and 4.2, students learn about angle relationships resulting from a transversal intersecting a pair of parallel lines. Model Problems and Practice Problems include positive integer values for all angle measures.
  • In Geometry, Lesson 7.8, integer values are used for given side lengths and angle measures of triangles in all but three problems (Problems 7, 8, and 14 provide the measure of a side length that is 6.2, 9.1, and 4.5 units in length, respectively).
  • In Algebra 2, Chapter R, students review topics from Algebra 1. Most Model Problems and Practice Problems use integer values in the given problems and result in positive whole number solutions.
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The instructional materials reviewed for the AMSCO Traditional series partially meet expectations for being mathematically coherent and making meaningful connections in a single course and throughout the series. Connections within a course or between courses are often not made explicit for teachers and students.

Examples of where the materials foster coherence through meaningful mathematical connections in a single course and between courses include:

  • In Algebra 1, Lesson 3.4, students graph linear equations by creating a table of values or identifying the slope and y-intercept from an equation in slope-intercept form or point-slope form. In Algebra 1, Lesson 4.1, students make a table or use the slope and y-intercept to graph linear inequalities.
  • Students identify the effect on the graph of f(x) when it is replaced by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k when transforming functions throughout the series (F-BF.3). In Algebra 1, students work with transformations of absolute value functions (Lesson 4.4), quadratic functions (Lesson 8.7), and exponential functions (Lesson 9.2). In Algebra 2, students connect to their work from Algebra 1 when they transform polynomial functions (Lesson 3.7), exponential functions (Lesson 6.1), logarithmic functions (Lesson 7.2), and rational functions (Lesson 4.4). In Geometry, students translate parabolas (Lesson 11.4) from the origin.
  • Students work with the slope criteria for parallel and perpendicular lines to solve problems in Algebra 1, Lesson 3.3 and Geometry, Lesson 4.4. In Lesson 4.4, the materials explicitly state, “We recall from Algebra how to determine if two lines are parallel” and then prove why perpendicular lines have negative reciprocal slopes using rotations to find parallel lines.
  • Students solve quadratic equations using a variety of methods in Algebra 1, Chapter 8. In Algebra 2, Chapter 2, students review those methods for solving quadratic equations in Lessons 2.1 and 2.4, and in Lessons 2.5 and 2.6, students connect those methods to solving quadratic equations with complex number solutions.

Examples of where the materials do not foster coherence through meaningful mathematical connections in a single course and between courses include:

  • In Algebra 1, Lesson 4.3, students graph a piecewise function in which a portion of the graph is an absolute value function. The materials state, “The absolute value function we will study later can also be defined algebraically as a piecewise function,” (page 156). However, when students graph absolute value functions in Lesson 4.4, no connection is made to defining an absolute value function as a piecewise function.
  • In Algebra 1, Chapter 8, students solve quadratic equations algebraically by factoring and taking the square root (Lesson 8.2), completing the square (Lesson 8.3) and the quadratic formula (Lesson 8.8). Students solve quadratic equations graphically in Lesson 8.4. In Lesson 8.4. There are connections made for students in the Model Problem between the factored form of a quadratic equation and the x-intercepts of the graphed equation. However, students do not make these connections in the Practice Problem portion of the lesson.
  • In Geometry, Chapter R, Algebra Review, students review concepts and skills from Algebra 1 including solving equations, inequalities, and systems, the slope intercept form of a line, multiplying and factoring polynomials, simplifying square roots, completing the square, graphing parabolas, area and perimeter fundamentals, and the Pythagorean Theorem. There are no specific connections made between these topics and specific Geometry concepts or skills to be taught later in the course. For example, Geometry, Lesson R.8, pages 24-25 and Algebra 2, Lesson 2.4, pages 94-96 address completing the square, which was addressed in Algebra 1, Lesson 8.3. These two lessons are duplicate lessons with the same Model Problems and Practice Problems.
  • In Algebra 2, Chapter R, Review, students review concepts and skills from Algebra 1 including solving equations, the rate of change for linear functions, graphing functions, solving systems of linear equations and inequalities, operations with polynomials, and translating parabolas in vertex form. There are no explicit connections made between these topics and specific concepts or skills to be taught later in the course. For example, in Algebra 2, Lesson R.4, students solve systems of linear equations and inequalities by graphing or using the algebraic methods of substitution and elimination. However, students do not solve problems related to systems of linear equations or inequalities later in the course.
  • Statistics and Probability standards are addressed in Algebra 1, Chapter 10 (interpreting quantitative and categorical data), Geometry, Chapter 12 (probability), and Algebra 2, Chapter 10 (probability). These chapters are taught in isolation and have limited connections to other content in the series.
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The instructional materials reviewed for the AMSCO Traditional Series partially meet expectations for explicitly identifying and building on knowledge from Grades 6-8 to the high school standards. The instructional materials do not explicitly identify content from Grades 6-8, but they do support the progressions of the high school standards. Connections between the non-plus standards and standards from Grades 6-8 are not clearly articulated in the student or teacher materials. Certain lessons or parts of lessons are labeled “Review,” but these review sections do not identify standards from Grades 6-8.

Although the materials do not explicitly identify Grade 6-8 standards when addressed in the materials, evidence that the materials build on knowledge from Grades 6-8 standards and connect to the high school standards include:

  • Students extend their knowledge of properties of operations to add, subtract, factor, and expand linear expressions (7.EE.1) when adding, subtracting, and multiplying polynomials in Algebra 1, Lessons 6.1 - 6.5 and Algebra 2, Lesson 3.1 (A-APR.1). Connections between the multiplication of polynomials and factoring are made in Algebra 1, Lesson 7.1-7.3 as students determine the factored form of a quadratic expression, with an emphasis on the structure of the quadratic expression (A-SSE.2 connects to 7.EE.2).
  • Students build upon their knowledge of the formulas for the circumference and area of a circle (7.G.4) and proportional reasoning (7.RP.3) as they calculate the length of an arc and the area of a sector in Geometry, Lesson 8.5 (G-C.5).
  • Students extend their knowledge of the relationships between angles that are created when parallel lines are cut by a transversal (8.G.5) by proving postulates and theorems about angle measures in Geometry, Chapter 4 (G-CO.A).
  • Students review the properties of integer exponents used to generate equivalent numerical expressions (8.EE.1) in Algebra 2, Lesson 5.1 and extend their knowledge to generate equivalent numerical expressions for numeric and algebraic expressions in Lessons 5.1 and 5.2 (N-RN.2).
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The instructional materials reviewed for the AMSCO Traditional series use the plus standards to coherently support the mathematics which all students should study in order to be college and career ready. Plus standards are integrated into the chapters in such a way that omitting the lessons or portions of lessons aligned to plus standards would not disrupt the coherence of the remainder of the mathematical content in the series.

Generally, plus standards are explicitly identified as “Optional” in both teacher and student materials. For example, Geometry, Lesson 11.5 is titled, “Optional: Ellipses at the Origin” and aligns to G-GPE.3. If a portion of a lesson addresses a plus standard, then the portion is identified as optional, for example, Algebra 2, Lesson 6.4, “Optional: Domain Restrictions of Composite Functions.” There are instances when plus standards are not explicitly identified, such as Algebra 2, Lesson 4.4, “Graphing Rational Functions” that aligns to F-IF.7d.

The following plus standards are addressed fully in the instructional materials:

  • N-CN.8: In Algebra 2, Lesson 2.5, the materials derive the identity for the sum of two squares, and students use the identity in Practice Problems 1-18, pages 107 and 108.
  • N-CN.9: In Algebra 2, Lesson 3.5, the materials state the Fundamental Theorem of Algebra and explain why it is true for quadratic polynomials.
  • A-APR.5: In Algebra 2, Lesson 8.5, the materials state the Binomial Theorem, and students apply the Binomial Theorem in the Practice Problems for the lesson.
  • A-APR.7: In Algebra 2, Lesson 4.1, the materials state that operations with rational expressions are part of a closed system. Students multiply and divide rational expressions in Algebra 2, Lesson 4.1 and add and subtract rational expressions in Algebra 2, Lesson 4.2.
  • F-BF.1c: In Algebra 2, Lesson 6.4, students compose functions in problems on pages 286-289.
  • F-BF.4b: In Algebra 2, Lesson 6.4, Multi-Part Problem Practice, students verify that a function is the inverse of a given function using composition.
  • F-BF.4d: In Algebra 2, Lesson 6.4, the materials restrict the domain of a non-invertible function to produce an invertible function.
  • G-SRT.9: In Geometry, Lesson 7.8, the materials derive the formula A=$$\frac{1}{2}$$ bc sin A on page 338.
  • G-SRT.10: In Geometry, Lesson 7.8, the materials prove the Law of Cosines and Law of Sines. Students use these laws in the Practice Problems for the lesson.
  • G-SRT.11: In Geometry, Lesson 7.8, students apply the Law of Cosines and Law of Sines to find unknown measurements in right and non-right triangles.
  • G-C.4: In Geometry, Lesson 8.1, the materials outline the steps for constructing a tangent line from a point outside a given circle to the circle and prove the construction on pages 357 and 358.
  • G-GPE.3: In Geometry, Lesson 11.5, the materials derive the equation for an ellipse given the foci, and in Geometry, Lesson 11.6, the materials derive the equation for a hyperbola given the foci.
  • G-GMD.2: In Geometry, Lesson 10.4, the materials state Cavalieri’s Principle, and students use it to find the volume of solids.
  • S-CP.9: In Geometry, Lesson 12.2, students use permutations and combinations to compute probabilities of compound events and solve problems.
  • S-MD.5a: In Geometry, Lesson 12.1, students have opportunities in the Practice Problem set and an online activity to find the expected payoff for a game of chance.
  • S-MD.5b: In Geometry, Lesson 12.1, Multi-Part Problem Practice 3, students evaluate whether to plead guilty or go to trial when given expected chances of going to prison for each outcome.
  • S-MD.6: In Geometry, Lesson 12.1, Model Problem 4, the probability to make a decision based on randomly flipping a coin versus using a deck of cards is considered when determining which high school in a town gets a new football field.
  • S-MD.7: In Practice problems in Geometry, Lesson 12.1 and Algebra 2, Lessons 10.6 and 10.7, students analyze decisions and strategies using probability concepts.

The following plus standards are partially addressed in the instructional materials:

  • N-CN.3: In Algebra 2, Lesson 2.5, students find the conjugate of a complex number. However, students do not use conjugates to find moduli and quotients of complex numbers.
  • F-IF.7d: In Algebra 2, Lesson 4.4, students graph rational functions as they transform reciprocal functions. While the materials have students identify asymptotes of a rational function, students do not factor in order to identify zeros and asymptotes of a given function. Furthermore, the end behavior for the parent function $$f(x)=\frac{1}{x}$$ is provided (page 205), and students do not determine end behavior for any other rational functions.
  • F-BF.5: In Algebra 2, Lesson 7.4, students solve exponential equations with logarithms on pages 316 and 317. However, neither the materials nor students explain the inverse relationship between exponential and logarithmic functions.
  • F-TF.3: In Algebra 2, Lesson 9.4, the materials use the unit circle to express the values of sine, cosine, and tangent for $$z$$, $$\pi-z$$, and $$\pi+z$$ in terms of given values for $$z$$, where $$z$$ is any real number. However, the materials do not express the values of sine, cosine, and tangent for $$2\pi-z$$.

The following plus standards are not addressed in the instructional materials:

  • N-CN.4
  • N-CN.5
  • N-CN.6
  • N-VM.1
  • N-VM.2
  • N-VM.3
  • N-VM.4a
  • N-VM.4b
  • N-VM.4c
  • N-VM.5a
  • N-VM.5b
  • N-VM.6
  • N-VM.7
  • N-VM.8
  • N-VM.9
  • N-VM.10
  • N-VM.11
  • N-VM.12
  • A-REI.8
  • A-REI.9
  • F-BF.4c
  • F-TF.4
  • F-TF.6
  • F-TF.7
  • F-TF.9
  • S-CP.8
  • S-MD.1
  • S-MD.2
  • S-MD.3
  • S-MD.4
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The instructional materials reviewed for the AMSCO Traditional Series do not meet expectations for alignment to the CCSSM for high school. The instructional materials attend to the full intent of the high school standards and spend a majority of time on the widely applicable prerequisites from the CCSSM. However, the instructional materials partially attend to engaging students in mathematics at a level of sophistication appropriate to high school, making connections within courses and across the series, and explicitly identifying standards from Grades 6-8 and building on them to the High School Standards. Since the materials do not meet the expectations for focus and coherence, evidence for rigor and the mathematical practices in Gateway 2 was not collected.

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Big Ideas Integrated (2013)

Big Ideas Learning, LLC | High School | 2013 Edition

High School

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The instructional materials reviewed for Big Ideas Integrated do not meet the expectation for Focus and Coherence within the CCSSM. For focus, even though students have the opportunity to spend the majority of time on the WAPs, students do not have the opportunity to fully learn all aspects of the non-plus standards. The contexts of problems are appropriate for high school students, but the numbers used in the exercises are often integers or lead to integer solutions. Also, the full intent of the modeling process is not applied to the modeling standards. For coherence, there are partial connections within and between courses, and explicit and purposeful connections to the standards from Grades 6-8 are also partially present.

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The materials partially meet the expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. Although most non-plus standards were addressed, multiple standards were only partially addressed.

For some standards, the materials only addressed certain aspects of the standard. For example:

  • N-Q.1,2: Throughout the series, students are always provided the units to use in their problems. When contexts are used, students are given the appropriate units, and in most problems, they are given the units to use in their solution process. Examples of this include:
    • In exercises involving area and volume, the unit of measure is consistent so that the student does not have to choose to convert units and decide which unit of measure to use for reporting solutions.
    • In other real-life problems, units could be used to gain understanding, as in problem 38 of Lesson 1.5 or problem 22 in Lesson 3.3 (both in Integrated Mathematics I), but students are not given opportunities to use units to solve problems that would require a series of conversions.
  • For G-CO.3, the materials do not address the reflection of an object onto itself. In Integrated Mathematics I, 11.2, the materials ask students about lines of symmetry but do not ask students to describe reflections that carry a polygon onto itself. Lesson 11.3 does ask students to describe rotations that map a figure onto itself in problem 20 and to select angles of rotation symmetry for a given regular polygon in problems 21-24, which addresses the portion of the standard concerning rotations.
  • S-IC.4 requires students to develop a margin of error through the use of simulation models for random sampling. In Integrated Mathematics III, Lesson 10.5, students calculate a sample mean and sample proportion and use those to estimate the population parameters. Students learn a formula for calculating a margin of error but do not “develop a margin of error through the use of simulation models for random sampling.”
  • S-IC.5: In Integrated Mathematics III, Lesson 10.6, problems 3-4 and 7-9 do "use data from a randomized experiment to compare two treatments," and although problems 7-9 have students use simulations to calculate the differences between the means of two groups and draw a conclusion, the conclusion that is drawn does not include whether or not the differences are significant.
  • For S-CP.5, students are given the formula for conditional probability, and then they are directed to use and apply the formula. When asked to explain the concept of conditional probability, they are asked to explain in terms of dependence and not independence. Students are not prompted to explain or make the connection between conditional probability and independence.
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The materials do not meet the expectations for attending to the full intent of the modeling process when applied to the modeling standards. Many of the modeling standards have not been completely addressed with the full intent of the modeling process by the instructional materials of the series, and some aspects of the modeling process are missing within the materials.

According to the CCSSM, modeling has attributes such as choice, decision-making, creativity, estimation, drawing and validating conclusions, design and re-design, as well as reasoning and communicating. Scaffolding within the lessons, practice problems, extension resources, and the performance assessment tasks prevents students from engaging in the full modeling process, and opportunities for students to engage in validation, reporting of conclusions, and the reasoning behind them were routinely omitted from problems. Also, there were many problems in the materials labeled as Modeling with Mathematics that attended to either MP4 or were application problems because they were missing at least one part of the modeling process. Examples of incomplete opportunities to engage in the full modeling process include hte following:

  • In Integrated Mathematics I, the exploration for Section 6.6 focuses on a context of rabbits reproducing. Students determine the number of pairs of rabbits after a given number of months. The reproduction pattern is described, and the exploration states that the numerical pattern is exponential and gives a picture. Numbers are organized in a table with places for students to finish by filling in blank spaces. This guides the student solution strategy and inhibits the modeling process from unfolding.
  • In the student resources for Integrated Mathematics I, the enrichment and extension exercise for 3.1 gives a quadratic rule in the context of jumping off a diving board. Within this exercise in relation to F-IF.4, students are given the opportunity to interpret key features of graphs and tables in terms of the context. However, rather than asking students to model, analyze and interpret the function, the problem specifically tells them how and what to interpret. Students are instructed to graph the function with t on the horizontal axis and construct a table with time increments of tenths of a second. These specifications take away the first part of the modeling process in which students identify variables in the situation and select those that represent essential features. There is also no request for the final part of the modeling process in which students report their conclusions and the reasoning behind them.
  • In Integrated Mathematics II, the performance task for Chapter 2, "Flight Path of a Bird," includes scaffolding that prevents the full modeling process from taking place. A quadratic model is given, and then students are guided to write an equation, graph it, interpret the graph and answer questions about the model using the graph. Student decisions about how to analyze and interpret the quadratic model for the bird path is overly guided by instructions, diagrams, and guidance about solution strategies and the completion of calculations.
  • In Integrated Mathematics II Lesson 9.1, problems 13 and 14 use the Pythagorean theorem to solve a right triangle application problem, but students are not engaged in the entire modeling process. The questions are prescriptive as they refer to an example to follow, and students are told to use the Pythagorean theorem. The right angle is drawn and labeled for the students.
  • In Integrated Mathematics II Lesson 3.7, problems 19, 20 and 30 have the variables already defined, and students are told which variable is independent. Students are asked to determine if the given data is linear, exponential, or quadratic and to explain their reason, but there is no opportunity to define variables, interpret results in context or consider the need for reevaluation.
  • In Integrated Mathematics III, the Performance Task for Chapter 1, entitled Population Density, gives students the opportunity to formulate a model/plan for attendance boundaries given a list of constraints, and then students can compute and interpret their results based on the formation of the problem. This task does not, however, indicate how students would be expected to validate their results and, then, either report those results or complete any re-formulations that might be needed.
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The materials, when used as designed, meet the expectations for allowing students to spend the majority of their time on the content from the CCSSM widely applicable as prerequisites for a range of college majors, postsecondary programs, and careers. The Widely Applicable Prerequisite Standards (WAPs) are a focus across the series.

  • In Integrated Mathematics I, the first half of the course (Chapters 1-6) is focused on the WAPs from Algebra and Functions. The Algebra WAPs continue to be supported through integration with geometric concepts in Chapters 8, 9, 10, and 12.
  • Chapters 1-4 of Integrated Mathematics II are focused on WAPs as students work with a broader range of equations and functions. Chapters 6-9 focus on WAPs through geometric investigations of triangle relationships, polygons, similarity and right triangle trigonometry.
  • The only chapters within Integrated Mathematics III that do not have a heavy focus on the WAPs are 1, 9, and 10. Chapter 1 focuses on G-MG.1-3 and G-GMD.4, and Chapters 9 and 10 address trigonometric identities and formulas, data analysis, and statistics. Otherwise, Integrated Mathematics III is focused on the WAPs throughout Chapters 2-8.

This relatively balanced distribution of WAPs across all three courses of the Big Ideas Integrated series for high school is a strength of the materials.

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The instructional materials reviewed for Big Ideas Integrated Series partially meet the expectations that the materials, when used as designed, provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics. The materials for the series, when used as designed, would not enable students to fully learn some of the non-plus standards.

There were a number of examples where the materials would not enable students to fully learn a particular standard. Specific examples are shown below:

  • G-CO.1: This standard asks students to know geometric definitions based on "undefined notions of point, line, distance along a line, and distance around a circular arc." Across the series, this standard was addressed by telling students there are undefined notions in geometry (with the exception of distance around a circular arc), and the geometric definitions were not developed based on the undefined notions.
  • N-RN.3: The standard asks students to "explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational." In Algebra 1, Lesson 4.1, problem 99 has students perform a limited number of calculations with pre-selected numbers, and then, students use that information in problem 100 to answer if this is always, sometimes, or never true. These are the two problems that offer opportunities to address N-RN.3.
  • Additionally, it is important to note that there were a number of standards, for example, A-REI.5 (Integrated Mathematics I, 5.3), G-SRT.7 (Integrated Mathematics II, 9.5) and S-ID.5 (Integrated Mathematics I, 7.5), which were addressed in one lesson throughout the series.

For standards that require students to derive, prove or explain, the materials often provide a derivation, proof or explanation rather than providing students with the opportunity to show their own understanding. Examples are shown below:

  • N-RN.1 requires students to explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponent to those values. In Integrated Mathematics II, Lesson 1.5, an explanation was provided to students, with no opportunity for student explanations.
  • G-C.5 calls for students to derive the fact that the length of the arc intercepted by an angle is proportional to the radius. In Integrated Mathematics II, Lessons 11.1-11.2, the derivation of arc length and the formula for area of sectors is provided for students. Students are not required to engage in constructing a derivation.
  • G-GPE.2 requires students to derive the equation of a parabola given a focus and directrix. In Integrated Mathematics II, Lesson 3.6, the derivation is given to students, with no opportunity for active engagement or input by the student.

For some standards, the materials do provide sufficient opportunities for students to fully learn the standards. Examples where the materials provide sufficient opportunities for students to fully learn a standard include the following:

  • F-IF.2: The materials introduce function notation to students in Integrated Mathematics I during Lesson 3.3, providing a description of function notation and how to use it to evaluate functions for specific input values and practice in using function notation when solving familiar problems in mathematical contexts and real-world contexts. Skill in utilizing function notation is continually promoted throughout the remainder of the series with the regularity of using function notation increasing from Integrated Mathematics I through Integrated Mathematics III. For example, by the end of Integrated Mathematics I, students build fluency in seeing f(4) = -2 as (4, -2) . In Integrated Mathematics II and III, basic skills extend to more complex uses of function notation, such as seeing Δy/Δx as or describing function transformations using f(x) notation.
  • F-BF.3: In Integrated Mathematics I, students develop understanding of transformations with linear and exponential functions. In Integrated Mathematics II, students begin working with absolute value and quadratic function transformations. In section 3.4 of this course, students learn how to identify even and odd functions. There is a specific example to address identifying even and odd functions from the rules, and there is a study tip to expose students to identifying even and odd functions from a graph. There are practice problems to address both. Students continue to practice and develop understanding of transformations of functions in Integrated Mathematics III. In earlier chapters of this course, students work with transformations on function types previously learned in the first two courses. Students then develop understanding of transformations with new functions (polynomials, radical, exponential and logarithm, rational, and trigonometric) as they learn them.
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The instructional materials reviewed partially meet the expectations for engaging students in mathematics at a level of sophistication appropriate to high school. The materials regularly use age-appropriate contexts and apply key takeaways from Grades 6-8, yet do not vary the types of real numbers being used.

Throughout the series, many examples, exercises, and problems include real numbers that, in many instances, provide limited opportunities for students to use decimal and fractional constants, coefficients and solutions.

  • In Integrated Mathematics I, the chapters that address linear equations, linear inequalities and geometric concepts avoid fractions, decimals and irrational numbers. Chapter 1 focuses on solving linear equations, and in Lesson 1.1, students solve one-step equations in examples and practice problems that use values integer coefficients, numbers that divide or multiply easily to give integer results, or fractions that have the same denominator.
  • In Chapter 11 of Integrated Mathematics II, the figures in the exercises addressing area, surface area, and volume typically have dimensions that are whole numbers. When calculating area, surface area, or volume, the solutions are typically whole numbers unless the figure involves a circle or finding the area of a regular polygon. In those exercises where a missing dimension needs to be found, the solution is typically a whole number except for figures that involve a circle or finding the area of a regular polygon.
  • In Lesson 6.5 of Integrated Mathematics III, students solve rational equations, and for most of the equations, the solutions are integers or simple rational numbers. This also occurs when students solve radical equation and inequalities in Lesson 4.4 and exponential and logarithmic equations in lesson 5.5.

Contexts include a variety of topics that help engage learners with different interests at a level appropriate to high school. Contexts include, but are not limited to: sports, animals, income, profit, investments, driving cars, and population density. Opportunities for experience with key take-aways from Grades 6-8 were prevalent, with the following seen more strongly than others:

  • Applying ratios and proportional relationships
  • Applying basic function concepts; e.g., by interpreting the features of a graph in the context of an applied problem
  • Applying concepts and skills of geometric measurement; e.g., when analyzing a diagram or schematic
  • Applying concepts and skills of basic statistics and probability (see 6-8.SP)

There were limited opportunities to apply the following key-takeaways:

  • Performing rational number arithmetic fluently
  • Applying percentages and unit conversions; e.g., in the context of complicated measurement problems involving quantities with derived or compound units (such as mg/mL, kg/m3, acre-feet, etc.)
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The instructional materials reviewed partially meet the expectations for fostering coherence through meaningful mathematical connections in a single course and throughout the series, where appropriate and where required by the Standards. While there is some evidence of mathematical connections within courses and across the series, overall the connections among standards within and between courses are not clearly shown for teachers. Even when solid connections exist within the mathematics, those connections may not be utilized to enhance the students' learning and understanding of the mathematics.

Some of the lessons and chapters are not connected to other lessons and chapters within the course/series where connections would be appropriate. For example:

  • In Integrated Mathematics I, Chapter 7 primarily addresses analyzing and displaying univariate data, but there are no connections made to the analysis and display of bivariate data presented in lessons 4.4 and 4.5 of the same course. There was also no indication for teachers as to if or how the content of Chapter 7 would connect to future courses in the series.
  • In Integrated Mathematics II, there is no connection made between geometric probability addressed in Lesson 11.2 and Chapter 5, Probability.
  • In Integrated Mathematics III, the summary for Chapter 1, Geometric Modeling, indicates general concepts from the first two courses in the series that students will use in the chapter, but there are no other direct references to either Chapter 11, Circumference, Area, and Volume, of Integrated Mathematics II or Lesson 8.4, Perimeter and Area in the Coordinate Plane, of the first course.
  • In Integrated Mathematics III, the summary for Chapter 10, Data Analysis and Statistics, indicates general concepts from the first two courses in the series that students will use in the chapter, but there are no other direct references to either Chapter 5, Probability, of Integrated Mathematics II or Chapter 7, Data Analysis and Displays, of Integrated Mathematics I.

The following examples include characteristics of the instructional materials reviewed that may promote mathematical connections within and between courses but do not clearly articulate those connections.

  • The Teacher Edition includes overviews of the sections in each chapter as well as a chapter summary. Both the overviews and the chapter summaries indicate general concepts and skills teachers can expect their students to know from middle school and/or previous courses, but neither of these sections provide direct connections to previous courses.
  • The student materials occasionally include “remember” arrows (as on pages 254 and 255 of Integrated Mathematics II) or phrases like “Previously, you…” that contain information about a previously learned concept. For example, in Lesson 2.6 of Integrated Mathematics III, the materials state, "Previously, you used transformations to graph quadratic functions in vertex form. You can also use the axis of symmetry and the vertex to graph quadratic functions written in vertex form". However, these aids do not foster direct connections within and among courses of the series.
  • The connections between standards are listed in the Correlation to the Common Core State Standards Table of Contents documents for each course. However, the connections are not explained or emphasized for the teachers. These documents are not included in the Teacher Editions.
  • There are "Connections to Algebra" and "Connections to Geometry" symbols and notes included in the lesson examples that make students and teachers aware that this is a concept that can be connected to other concepts in the course or series; however, these are used only occasionally. Some of these connections are very specific, as on page 390 of Integrated Mathematics I that states "In this step, you are applying the Substitution Property of Equality that you learned about in Section 1.1." Others are general or vague, as on page 405 of Integrated Mathematics I, where it states, "In this exploration, you expand your work on perimeter and area into the coordinate plane".

The following are examples within the series materials that do promote and foster coherence:

  • At the beginning of each chapter across the series, there is a "Maintaining Mathematical Proficiency" activity that reviews important concepts and skills from previous grades or courses. These activities help students connect prior learning to new learning in the chapter they are beginning.
  • Students work with area, surface area, and volume in middle school, and these geometric concepts are reviewed in the last chapter of Integrated Mathematics II. The chapter extends this review work to include the area of regular polygons and surface area and volume of a variety of 3D figures not included in the middle school standards.
  • In Integrated Mathematics I, students develop linear and exponential equations and functions. Then in Integrated Mathematics II, they extend their algebraic reasoning and understanding of functions with quadratic, absolute value, and polynomial functions. Finally, in Integrated Mathematics III, students study polynomials, radicals, logarithmic, rational, and trigonometric functions. In each of the three courses, students' ability to interpret the structure (A-SSE), perform arithmetic (A-APR), create equations (A-CED), reason with equations and inequalities (A-REI), and interpret functions (F-IF) builds upon the previous course. Students are given the opportunities to augment their work with many function types in these domains across the series.
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The materials partially meet the expectations for explicitly identifying and building on knowledge from Grades 6-8 to the High School Standards. Overall, content from Grades 6-8 is present but is not explicitly identified and does not always fully support the progressions of the high school standards. There is some support for making connections between standards from Grades 6-8 and high school as seen in Laurie's Notes and Maintaining Mathematical Proficiency.

The following are examples of where the materials do not explicitly identify and/or build on standards from Grades 6-8:

  • The Chapter Summary pages for all courses have a What Your Students Have Learned…Will Learn section, and throughout the series, this section includes references to topics or concepts students learned in middle school. However, when referencing these concepts from Grades 6-8, specific standards are not explicitly identified, and there are not clear connections between standards from Grades 6-8 and high school.
  • Lessons 1.1-1.3 and 2.1-2.4 of Integrated Mathematics I are identified in the CCSSM document as addressing A-CED.1 and A-REI.3. These lessons address solving linear equations and inequalities, which aligns to standards from 8.EE.C, but there are no standards identified from Grades 6-8.
  • Lessons 5.1-5.4 of Integrated Mathematics I address solving systems of linear equations graphically and symbolically for systems having one solution, no solutions and infinitely many solutions, which aligns to standards from 8.EE.C. The teacher materials do indicate that the concepts are being repeated from Grades 6-8, but no specific standards are identified. Also, the teacher materials do provide some pacing suggestions for these lessons, but there is no guidance for how to build on these concepts in order to enhance the learning for high school students.
  • The Chapter 7 Summary of Integrated Mathematics I indicates the construction and interpretation of box-and-whisker plots as new learning rather than prior learning from middle school, but the overview for Lesson 7.2 states that “students should be familiar with representing data using box-and-whisker plots from middle school.” There are no standards from Grades 6-8 identified.
  • Chapter 11 of Integrated Mathematics II mainly addresses area, volume, and surface area, which are concepts that have their origins in standards from Grades 6-8. However, there are no standards from Grades 6-8 identified, and the connections between the high school concepts and the concepts for Grades 6-8 are only described in the teacher materials.

The Maintaining Mathematical Proficiency lessons at the beginning of each chapter connect to concepts from Grades 6-8 as this section reviews concepts or skills that will be needed throughout the chapter. Most of the work with middle school concepts or skills in these lessons is intended to reinforce course-level standards, but identification of the concepts or skills from Grades 6-8 is inconsistent. For example, in Chapter 1 of Integrated Mathematics 1, adding, subtracting, multiplying, and dividing integers is specifically noted as Grade 7 work, but in Chapter 2 of the same course, graphing numbers on a number line and comparing real numbers are not identified as work from a previous grade.

An example of the materials attending to the progressions of standards that are referenced in the progression documents is in Chapter 11 of Integrated Mathematics 1. The publisher describes concepts from middle school that relate to what students will be learning, and the middle school concepts include the types of transformations students are familiar with and their understanding of congruency. In the introduction to Lesson 11.4, teachers are told, "In Grade 8, the concept of congruency was introduced. Students should understand that a two-dimensional figure is congruent to another when the second can be obtained from the first by a sequence of rotations reflections and translations." In Lesson 11.4, the understanding of congruency through rigid motions is continued, and then, students develop triangle congruency theorems through rigid motions in Chapter 12.

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The plus standards, when included, are not explicitly identified in the teacher or student editions of the materials. The correlation charts of lessons-to-standards and standards-to-lessons found in the front matter of the Teacher Editions do not denote which standards are plus standards. Alignment of plus standards was in a separate set of course alignment documents titled, Correlation to the CCSSM (I, II, and III) Table of Contents.

The plus standards that are included in the materials typically support the mathematics which all students should study in order to be college- and career-ready in a coherent manner, and the plus standards typically could be omitted without interfering with the flow of the content within the series.

In Integrated Mathematics II, the following plus standards are addressed in the given lessons: S-CP.8 (Lesson 5.2); S-CP.9 (Lesson 5.5); G-C.4 (Lesson 10.1); and G-GMD.2 (Lessons 11.4 and 11.7). In Integrated Mathematics III, the following plus standards are addressed in the given lessons: A-APR.5 (Lesson 3.2); N-CN.8,9 (Lesson 3.6); A-APR.7 (Lessons 6.3 and 6.4); F-TF.9 (Lesson 9.2); G-SRT.9-11 (Lessons 9.3 and 9.4); and S-MD.6,7 (Lessons 10.2 and 10.5).

Some lessons do not reach the full intent of the plus standards.

  • In Integrated Mathematics III, Lesson 3.6 does not give students the opportunity to show that the Fundamental Theorem of Algebra is true for all quadratic polynomials, which is an aspect of N-CN.9, but it does give students the opportunity to fully address N-CN.8.
  • In Integrated Mathematics III, Lessons 6.3 and 6.4 do not give students the opportunity to "understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression", which is an aspect of A-APR.7.
  • In Integrated Mathematics III, Lesson 9.2 does not give students the opportunity to prove the addition and subtraction formulas for the tangent ratio, which is part of F-TF.9.
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The instructional materials reviewed for the Big Ideas Integrated series do not meet expectations for alignment to the CCSSM for high school. The materials do meet the expectations for allowing students to spend the majority of their time on the content from the CCSSM widely applicable as prerequisites, but they do not meet the expectations for attending to the full intent of the modeling process when applied to the modeling standards. The materials partially meet the expectations for the remainder of the indicators within Gateway 1, and since the materials did not meet the expectations for focus and coherence, evidence for rigor and the mathematical practices in Gateway 2 was not collected.

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Big Ideas Math (2013)

Big Ideas Learning, LLC | Grades 6-8 | 2013 Edition

Sixth Grade

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The instructional materials reviewed for Grade 6 Big Ideas do not meet the expectations for focus and coherence. Future grade-level standards are rarely assessed and could be easily modified or omitted. The materials do not devote a majority of the time to the major work of the grade. The instructional materials do not connect supporting work with the major work of the grade. Although the materials provide a full program of study that is viable for a school year, students are not always given extensive work with grade-level problems. Connections between grade levels and domains are missing. Overall, the instructional materials do not meet the expectations for focusing on the major work of the grade, and the materials are not always consistent and coherent with the standards.

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The Grade 6 Big Ideas materials do not meet the expectations for focus. Future, grade-level standards are rarely assessed and could be easily modified or removed. However, the materials do not devote a majority of the time to the major work of the grade.

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The materials meet the expectation for not assessing topics before the grade-level in which they should be introduced. The majority of the assessments are on grade-level with a few items that could be easily modified or removed to remain on grade-level.

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The instructional materials reviewed for Grade 6 meet the expectations for assessing the grade-level content and, if applicable, content from earlier grades. Above grade-level assessment items could be modified or omitted without a significant impact on the underlying structure of the instructional materials. Overall, summative assessments focus on the Grade 6 standards with minimal occurrences of above grade-level work.

The following assessments were reviewed for this indicator from the print and digital materials: Forms A and B of the Chapter Tests, Chapter Quizzes, Standards Assessments, and Alternate Assessments.

On the Chapter 7 Assessments, all equations are of the form x + p = q or px = q required in 6.EE.7 with the exception of the following above grade-level items:

  • On Test A, problem 27/Test B, problem 26, students must write an equation that represents a px + q = r situation, which most closely aligns with 7.EE.4a.
  • On item 7 on the Standards Assessments in both Chapters 7 and 8, students must choose the appropriate equation in the form px + q = r which most closely represents the given real world context.
  • These items assess the problems in Lesson 7-4 in which students encounter the px + q = r structure in multiple examples and exercises.
  • In all of these cases, the context of the item can be adapted to a proportional relationship instead of a linear one without substantially altering the material.

It is also noted that students are asked to calculate a least common multiple above what is expected in 6.NS.4, which requires finding the LCM of two whole numbers less than or equal to 12. The following items on Chapter 1 assessments include values outside of the expected range and can be removed without drastically changing the material:

  • Items 9 and 10 on Chapter 1 Quiz
  • Item 21 on Test A
  • Item 22 on Test B

Overall, items on the assessments reflect Grade 6 standards as well as some below grade-level items in Chapters 1 and 2.

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The Grade 6 Big Ideas materials do not meet expectations for devoting the large majority of class time to the major work of the grade-level. The materials engage students in the major work of the grade less than 65 percent of the time.

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The instructional materials reviewed for Grade 6 do not meet the expectations for focus on major clusters. The Grade 6 instructional materials do not spend the majority of class time on the major clusters of the grade.

The Common Core State Standards to Book Correlation (pages xx-xxvi) and the Book to Common Core State Standards Correlation (page xxvii) were used to identify major work, as well as the first page in each chapter which includes Common Core progression information, a chapter summary, and a pacing guide (and related online pages). The pacing guide provides the number of days to spend on each chapter opener, activity, lesson, any extensions, and review/assessment days. This guide was used to determine the number of instructional days allotted by the publisher for each standard found in the material. Lessons identified by the publisher as addressing major work were reviewed. Lessons with standards identified by the publisher as non-major work were also examined to ensure that these lessons did not contain enough material to strengthen major work.

All percentages are below 65 percent and were calculated to reflect the chapters, lessons, and instructional time spent on major work:

  • The material devoted approximately 50 percent of chapters to major work of the grade (Chapters 1, 3, 5, 6 and 7). If over 50 percent of a chapter addressed major work, then the chapter was counted as major work.
  • 54 percent of lessons (28 out of 52) were dedicated to major work. Lessons 4.1, 4.2 and 4.3 were not identified as major work by the material but were found to have enough examples and problems connected to major clusters 6.EE.A and 6.EE.B. Some lessons identified as major work were not counted as major work even if identified by the publisher. These lessons include 1.1, 2.1, and 2.4, as they only address work below grade-level, and 1.4 where prime factorization, a singular strategy for finding a GCF or LCM, is taught and not aligned to a particular CCSSM standard.
  • Of the instructional days, 56 percent (or 87 out of 154) were spent on lessons aligned to major work.

Days were counted based on the recommendation of the pacing guide in the beginning of each chapter for all lessons reviewers found aligned to major work.

) [5] => stdClass Object ( [code] => coherence [type] => component [report] =>

The instructional materials reviewed for Grade 6 do not meet the expectations for coherence. The instructional materials do not use supporting content as a way to continue working with the major work of the grade. The materials include a full program of study that is viable content for a school year. Content from prior grades is not clearly identified or connected to grade-level work, and not all students are given extensive work with grade-level problems. Material related to prior grade-level content is not clearly identified or related to grade-level work. These instructional materials are shaped by the cluster headings in the standards. Overall, the Grade 6 materials do not support coherence and are not consistent with the progressions in the standards.

) [6] => stdClass Object ( [code] => 1c1f [type] => criterion ) [7] => stdClass Object ( [code] => 1c [type] => indicator [points] => 0 [rating] => does-not-meet [report] =>

The instructional materials reviewed do not meet the expectations for having supporting content that enhances focus and coherence simultaneously by engaging students in the major work of the grade. Overall, the structure of the chapters and lessons in the Grade 6 material rarely engage students in both supporting and major work to allow natural connections.

Most connections between supporting work and major clusters are related to computation, and connections between supporting work and major clusters were either not supported by the lessons when identified or not stated for teachers or students when they did occur.

Supporting standard 6.NS.4 is isolated into several different lessons in two different chapters in this material.

  • In Lesson 1-5, 6.NS.4 and 6.EE.2b are both identified by the material; however, the examples explore factors as specific parts of a product within a multiplication equation in the absence of variables (4.OA.4) and then identify the common factors to find the greatest common factor. While this partially satisfies 6.NS.4 because students are finding the GCF and LCM of two whole numbers, at no point are they having to “use the distributive property to express a sum of two whole numbers with no common factor” in either lesson 1-5 or 1-6. Since there is no occurrence of this part of the standard, these particular lessons do not fully allow students to “view one or more parts of the expression as a single entity.”
  • Standard 6.NS.4 is most closely aligned with Extension 3-4, Factoring Expressions. The lesson extension does support 6.EE.3 and 6.EE.4 when students apply properties to generate equivalent expressions and identify when they are equivalent; however, the lesson does not discuss using a LCM and to generate equivalent expressions.

6.G is addressed in Chapters 4 and 8. In these chapters, students explore area, surface area, and volume in real world contexts. In order for these lessons to connect to the 6.EE domain, students should use variables to represent numbers when solving a real-world or mathematical problem and understand that a variable can represent an unknown number (6.EE.6).

  • Lessons 4.1 through the lesson 4.3 extension involve applying area formulas to different geometric shapes. The majority of the questions do not explicitly ask students to write the formulas in algebraic form in the exercises. Although, the presentation of problems and some application problems (e.g., 4.1: problems 9, 17, 20; 4-2: problems 9, 20; and 4-3: problems 10, 19, 21, 22) encourage the application of standards 6.EE.2, 6.EE.5, and 6.EE.6.
  • There are partial connections in lessons 8-2 (examples 1 and 2) and 8-3 (examples 1 and 2) when an equation is used to find the surface area of prisms using a given net, but the only variable used is the “S” for surface area, not the labeled parts of the equivalent expression representing the faces. In Laurie’s notes, teachers are encouraged to have students “write a verbal model and substitute the areas of the faces as they are computed,” but equations are not mentioned nor are these connections explicitly noted for teachers or students. Students are not asked to use an equation in the exercise directions.
  • Lesson 8-4 does use volume formulas in worked examples, so a partial connection to 6.EE.2c is present; however, much like in Lessons 4.1-4.3 students are not encouraged to write the expressions.

Even though there are some natural connections, the material does not explicitly express these connections. Lessons are often treated as disconnected topics, and there are few visible connections.

) [8] => stdClass Object ( [code] => 1d [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed meet the expectation for having an amount of content designated for one grade-level that is viable for one school year in order to foster coherence between grade-levels. Overall, the instructional materials reviewed for Grade 6 provide a year’s worth of content as written.

In Big Ideas, the length of each class period is 45 minutes, so 154 45-minute class periods would be needed to cover Chapters 1-10. The 154 days of instruction are outlined in the pacing guide on pages xxxiv and xxxv. This pacing includes one day for a scavenger hunt, one day in each chapter for study help and review before the mid-chapter quizzes, and two days for review and assessment at the end of each chapter. In total, ten days are devoted to Study Help/Quizzes and 20 instructional days on chapter review assignments and chapter assessments, which leaves 123 days for instructional lessons, activities, and extension lessons. Before each chapter, information is provided for the teacher on how much time to spend on each section including activities, lessons, and any extensions.

It should be noted that the following extension activities found in Chapters 1, 3, 4 and 10 are of particular importance and should not be skipped, as this is where the following standards are fully addressed.

  • Chapter 1, Extension 1.6 - Finding a common denominator versus using the LCM to add and subtract fractions (6.NS.B).
  • Chapter 3, Extension 3.4 - Using the distributive property to factor expressions as well as identify and generate equivalent expressions (6.EE.3,4).
  • Chapter 4, Extension 4.3 - Decomposing composite figures into triangles and other shapes to find area. However, there are no examples or items that require students to compose shapes into rectangles, which is also not addressed in any other lesson (6.G.1).
  • Chapter 10, Extension 10.3 - Choosing appropriate measures of center and variability to describe a set of data (6.SP.5d).

The online lesson plans provided in Chapters at a Glance also include detailed information about when to use the supplemental activities, such as extra examples, as well as performance tasks for each standard. Any additional days of instruction can be spent implementing these tasks or the additional skills practice found in the online resources.

) [9] => stdClass Object ( [code] => 1e [type] => indicator [points] => 0 [rating] => does-not-meet [report] =>

The instructional materials reviewed for Grade 6 do not meet the expectation for having materials that are consistent with the progressions in the Standards. Materials are not intentionally written to follow the progressions of the grade level as few lessons are identified as work from prior grade levels, and there are no lessons identified to connect grade 6 work to the work of future grades. General explanations for how lessons are related to prior knowledge are present. Materials do not give all students extensive work with grade-level problems.

The materials do not develop according to the grade-by-grade progressions in the standards. Content from prior or future grades is not clearly identified and related to grade-level work.

  • Explanations of Common Core Progressions are given at the beginning of each chapter connecting both Grade 4 and Grade 5 level work to the Grade 6 work students will encounter in each of the chapters. These connections to below grade-level work are presented as bulleted lists of skills and are not aligned to specific standards.
  • Math Background Notes include vocabulary review as well as a general explanation of the most important skills and understandings from the prior grade level(s). For example, in Chapter 6, the notes mention that students should convert one of the numbers so both numbers are in the same form. It does not explain for the teacher the representations of rational numbers, such as tape diagrams or double number lines, used in Grade 4 and Grade 5.
  • The first page of each chapter is “What You Learned Before.” The teacher page adjacent to this page identifies the CCSSM addressed, which is usually from a previous grade level, but no explanation of what connects this previous material to current concepts is included.
  • Once into the chapter, teachers can see previous skills being reviewed, but they are not identified. Examples of this include:
    • In Section 1.1, addition and subtraction of whole numbers is reviewed. This is stated as a Grade 4 standard in the Common Core progression chart. It also reviews multiplying whole numbers, which is stated as a Grade 5 standard in the progressions chart but not in the section. Although the material does address dividing fluently, Section 1.1 is identified as aligned to standard 6.NS.2 without reference to the lower grade content present in the section,.
    • In Section 1.3, Activity 3 Reviewing Fractions and Decimals, the Teacher Edition states this is an opportunity for students to review prior work. It does not identify from which grade-level the prior work is most closely aligned.
    • Section 2.1 is identified as preparing for 6.NS.1, but the entire chapter is about multiplying fractions and not identified as 5.NF.B.
    • Section 2.4 Activity 4, Using a Place Value Chart, is an unidentified Grade 4 skill.
  • Content of progressions beyond the current grade-level is neither visible nor identified in the material.

Extensive grade-level problems are provided for all students if they are given the opportunity to access all of them. There is an assignment guide in each lesson that levels students into basic, average, or advanced. These charts exclude the “basic” learner from the reasoning and critical thinking problems. These problems are critical for all students in order for them to reach the depth of the standard in many of the lessons. Specific examples follow:

  • In lesson 1-6, the critical thinking problems 29-31 support students in making generalizations about the LCM and GCF of two values.
  • Many lessons contain explanations in “Laurie’s Notes” of a specific homework problem and how “Taking Math Deeper” can apply to that problem. Usually, it is a item that can get to the depth of a standard; however, it is in part of the Basic Assignment in 6 out of 52 lessons in the material. If students are only assigned Basic or Average Level Assignments, they will often not engage with the problems reaching the full meaning of the standard.
  • Occasionally, “Laurie’s Notes” contains a basic idea for differentiating instruction for low-level or average students.
  • Each lesson also contains “Reteaching and Enrichment Strategies” that can be found within the chapter or online resources.

The materials do not relate grade-level concepts explicitly to prior knowledge from earlier grades.

  • Each chapter begins with a What You Learned Before page just before the first lesson. These pages contain problems for students from prior grade-levels and/or chapters found earlier in the material. Connections to specific grade levels or standards are not identified.
  • Laurie’s Notes are found in each lesson. In the margin of these notes for instruction, specific Grade 6 standards that will be addressed are identified. Most of them contain a Previous Learning section that describes prior knowledge students should possess before engaging in the lesson, but again, they are not explicit about the particular grade level or standard tied to the skills or understanding needed. For example, in section 7.1 the Previous Learning states, “Students should be familiar with the vocabulary specific to the four operations.” Neither the specific CCSSM standards nor the grade level is stated.


Overall, explicit connections to prior knowledge are made at a general level through the chapter and lesson features in this series. Connections are not clearly articulated for teachers and are merely lists of skills without indication of standards, clusters, or domains. There is not a clearly defined progression for teachers to demonstrate how prior knowledge is being extended or developed.

) [10] => stdClass Object ( [code] => 1f [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Grade 6 partially meet the expectation for fostering coherence through connections at a single grade, where appropriate and required by the Standards. Overall, the materials do not include learning objectives that are visibly shaped by CCSSM cluster headings, but there are some opportunities to connect clusters and domains.

Examples of the materials not including learning objectives that are visibly shaped by CCSSM cluster headings include:

  • Cluster headings were explicitly addressed in the materials on page xxxvii, where it appears that a chapter is dedicated to each. There is no explanation as to how the lessons are tied together under the cluster heading besides the information found on this page. In most cases, standards are addressed in isolated lessons with very little overlap of CCSSM across chapters, and the language used in the cluster heading was not found.
  • The lesson “Goal” appears in Laurie’s Notes before the lessons in each section and most closely aligns to an objective. These are descriptions of the parts of the standard that are addressed in the lesson and were not found to describe cluster headings.
    • For example, “Today’s lesson is making and using ratio tables” in Chapter 5, section 2, partially captures what is expected in 6.RP.A, but the "Goal" does not reach the underlying connections necessary to understand ratio concepts and use ratio reasoning to solve problems.

Examples of the materials providing some problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important include:

  • In general, lessons focus on one standard, and lessons in each chapter address standards from the same cluster. Chapters are taught as disconnected units.
  • An identified connection is found in Activity 7.4 on page 314 when both 6.RP.A and 6.EE.C are identified. Students explore how independent and dependent variables are related in the given tabular and graphic representations and are required to write an equation.
  • An unidentified connection occurs when 6.SP.A,B are identified in Chapters 9 and 10. The problems involving decimals that are designated for advanced students are opportunities to make connections with 6.NS.B when the mean of the data set is calculated. Other data sets in the chapters involve whole numbers.
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The instructional materials reviewed for Grade 6 Big Ideas do not meet the expectations for Gateway One. Future grade-level standards are rarely assessed and could be easily modified or omitted. The materials do not devote a majority of the time to the major work of the grade. The instructional materials do not connect supporting work with the major work of the grade. Although the materials provide a full program of study that is viable for a school year, students are not always given extensive work with grade-level problems, and connections between grade levels and domains are missing. Since the materials do not meet expectations for Gateway One, evidence for Gateways Two and Three was not collected.

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Seventh Grade

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The instructional materials reviewed for Grade 7 Big Ideas do not meet the expectations for focus and coherence. Future grade-level standards are rarely assessed and could be easily modified or omitted. The materials do not devote a majority of the time to the major work of the grade. The instructional materials infrequently connect supporting work with the major work of the grade. Although the materials provide a full program of study that is viable for a school year, students are not always given extensive work with grade-level problems. Connections between grade-levels are missing. Overall, the instructional materials do not meet the expectations for focusing on the major work of the grade, and the materials are not always consistent and coherent with the standards.

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The Grade 7 Big Ideas materials do not meet the expectations for focus. Future, grade-level standards are rarely assessed and could be easily modified or removed. However, the materials do not devote a majority of the time to the major work of the grade.

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The materials meet the expectation for not assessing topics before the grade-level in which they should be introduced. The majority of the assessments are on grade-level with a few items that could be easily modified or removed to remain on grade-level.

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The instructional materials reviewed for Grade 7 meet the expectations for assessing the grade-level content and, if applicable, content from earlier grades. Summative assessments focus on the Grade 7 standards with minimal occurrences of above grade-level work, which include finding slope and identifying direct variation. The above grade-level items in which students are asked to find slope can be modified by adding a real world context and changing the term “slope” to “unit rate.” Additionally, the specific language “direct variation” can be changed to “proportional relationship” to reflect Grade 7 standards without altering the structure of the materials.

The following assessments were reviewed for this indicator from the print and digital materials: forms A and B of the Chapter Tests, Chapter Quizzes, Standards Assessments, and Alternate Assessments.

On the Chapter 5 assessments, students are asked to calculate slope and identify direct variations on the following above grade-level items:

  • On Form A items 17-18 and Form B items 19-20, students are asked to find slope from 2 points on a given graph, which most closely aligns with 8.EE.5.
  • On item 28 on Form B in Chapter 5, students must use slope to determine a missing y value in a linear relationship and explain, which more closely aligns to 8.F.4. This item can be omitted without affecting the intent of the chapter.
  • On item 5 on the Chapter 6 Standards Assessment, students are asked to find the slope of a line from two given points on the line. All items can be omitted or edited to find the unit rate instead of the slope to align questions to grade-level content.
  • On the Quiz after Lesson 5-6 items 8-9, Test Form A, items 19-21, and Form B. items 23-25 in Chapter 5, several questions require students to identify a direct variation from a table or equation. These questions can be omitted or edited to find/explain the “constant of proportionality” in the presented table or equation to reflect the Grade 7 standard, 7.RP.2.

In addition to these items, students are asked to calculate the volume of pyramids on the Chapter 9 assessments. Students are not required to find volume or use the formula for pyramids specifically, as this particular three dimensional figure is linked to 8.G.9, which calls for students to give an informal argument for the volume of a pyramid. In the materials there is one lesson dedicated to this topic in which students are allowed to experiment with the conceptual nature of volume using beans and cubes in the embedded activity. For this reason, the inclusion of the activity is appropriate, and the score was not affected.

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The Grade 7 Big Ideas materials do meet expectations for devoting the large majority of class time to the major work of the grade-level. The materials engage students in the major work of the grade less than 65 percent of the time.

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The instructional materials reviewed for Grade 7 do not meet the expectations for focus on major clusters. The Grade 7 instructional materials do not spend the majority of class time on the major clusters of the grade.

The Common Core State Standards to Book Correlation (pages xx-xxiv) and the Book to Common Core State Standards Correlation (page xxv) were used to identify major work, as well as the first page in each chapter which includes Common Core progression information, a chapter summary, and a pacing guide (and related online pages). The pacing guide provides the number of days to spend on each section in the chapter as well as days for review and assessment. This guide was used to determine the number of instructional days allotted by the publisher for each standard found in the major work of the grade. The lessons containing major work were reviewed for alignment to each of the identified standards. Reviewers also examined all lessons with standards identified by the publisher as non-major work to ensure that these lessons did not contain enough material to strengthen major work.

All percentages are below 65 percent and were calculated to reflect the chapters, lessons, and instructional time spent on major work:

  • The materials devoted approximately 60 percent of chapters to major work of the grade (Chapters 1 through 6). If over 50 percent of a chapter addressed major work, then the chapter was counted as major work.
  • 56 percent of lessons (29 out of 52) were dedicated to major work. Lessons 7.5, 10.2, and 10.3 were not identified as major work by the materials but were found to have enough examples and problems connected to major clusters 7.RP and 7.NS. Some lessons identified as major work were below grade-level and were not counted as major work even if identified by the publisher. These lessons include 1.1, which is identified by the publisher as “Preparing For” grade-level standards, and 2.1, 6.1, and 6.2 for addressing work with rational numbers which align to 6.NS and 6.RP standards.
  • 59 percent of instructional days (90 out of 154) were spent on lessons aligned to major work. Days were counted based on the recommendation of the pacing guide in the beginning of each chapter for all lessons reviewers found aligned to major work.
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The instructional materials reviewed for Grade 7 do not meet the expectations for coherence. The instructional materials infrequently use supporting content as a way to continue working with the major work of the grade. The materials include a full program of study that is viable content for a school year. Content from prior grades is not clearly identified or connected to grade-level work, and not all students are given extensive work with grade-level problems. Material related to prior, grade-level content is not clearly identified or related to grade-level work. These instructional materials are not shaped by the cluster headings in the standards. Overall, the Grade 7 materials do not support coherence and are not consistent with the progressions in the standards.

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The instructional materials reviewed for Grade 7 partially meet the expectations for having supporting content that enhances focus and coherence simultaneously by engaging students in the major work of the grade. The content designed to address the supporting work was isolated from major work in most cases. Some unidentified connections between supporting and major work were found.

Supporting work is identified by the publisher in Chapters 7, 8, 9 and 10 on page xxv of the Teacher Edition. Overall, the chapters occur in isolation of major work with each domain addressed in separate chapters spanning the last four chapters in the book. Limited connections to major work were found in the following lessons:

  • In Chapter 7 Constructions and Scale Drawings, Lesson 7.5 connects scale drawings (7.G.1) to ratios and proportional reasoning (7.RP.A), and the vocabulary is noted in the lesson. It is also modeled in the online tutorial video located in the dynamic classroom. The textbook does not identify this connection. The other lessons in the chapter were not found to have connections to major work.
  • In Chapter 8, Lesson 8.1 has an unidentified connection to 7.EE in Example 3. Students are asked to use the formula for circumference to find the diameter, and it is practiced in problem 13. In the other lessons, students substitute values into the given formula and use order of operations to simplify the expressions. With the exception of lesson 8.1, there is not a connection to 7.EE as the standard requires students to work with positive and negative rational numbers and with multi-step equations. Inverse operations are not needed to find the area or the volume.
  • The lessons in Chapter 10, which address supporting Clusters 7.SP.A,C, do provide natural connections when working with creating probabilities as rational numbers and converting them to decimals and percents. In addition, each lesson, with the exception of lesson 10.1, also involves using proportional relationships to draw conclusions in both sampling and experimental probability situations. However, standards 7.RP.3 and 7.NS.2 are not explicitly identified as being part of the lessons, and neither the teacher script nor the student materials highlight this connection for the teacher or the students
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The instructional materials reviewed meet the expectation for having an amount of content designated for one grade-level that is viable for one school year in order to foster coherence between grade-levels. Overall, the instructional materials reviewed for Grade 7 provide a year’s worth of content as written.

154 days of instruction are outlined in the pacing guide on page xxxii and xxxiii with each lesson designed to fill a 45-minute instructional period. This pacing includes days for study help and review before the mid-chapter quizzes and two days for review and assessment at the end of each chapter. Remaining days can be used for review and the cumulative assessments found at the conclusion of each chapter and the assessment book. Before each chapter, information is provided for the teacher on how much time to spend on each section. Unlike the other textbooks in the series, this one includes regular and accelerated pacing instead of detailing the number of days allotted for activities, lessons, and extensions. Since the material does not explicitly define the amount of time to spend on each extension lesson, these lessons may not be given adequate attention. The following extensions allow students to encounter specific parts of the following standards found in the work of the grade-level:

  • 7.EE.2 - In 3.2 Factoring Expressions, students add and subtract expressions in the lesson but work to factor and expand them only in the 3.2 Extension.
  • 7.RP.2a,b,d - Extension 5.2 Graphing Proportional Relationships connects all of these standards in the same lesson while the rest of the chapter separates the tabular, graphic, and verbal representations of proportional relationships in different lessons.
  • 7.G.3 - In Extension 9-5, students “slice” three-dimensional figures and examine the two dimensional figure that is formed as a result.

The online lesson plans also include extra examples if teachers need them. If students need more practice on skills, teachers would have access to resources without having to find other supplemental practice. The Online section titled “Chapters at a Glance” displays all of these practice opportunities for each chapter.

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The instructional materials reviewed for Grade 7 do not meet the expectation for having materials that are consistent with the progressions in the Standards. Materials are not intentionally written to follow the progressions of the grade-level as few lessons are identified as work from prior grade-levels, and there are no lessons identified to connect Grade 7 work to the work of future grades. General explanations for how lessons are related to prior knowledge are present. Materials do not give all students extensive work with grade-level problems.

The materials do not develop according to the grade-by-grade progressions in the standards. Content from prior or future grades is not clearly identified and related to grade-level work.

  • Explanations of Common Core Progressions are given at the beginning of each chapter connecting both Grade 5 and 6 level work to the Grade 7 work students will encounter in each of the chapters. These connections to below grade-level work are presented as bulleted lists of skills and are not aligned to specific standards.
  • Math Background Notes include vocabulary review as well as a general explanation of the most important skills and understandings from the prior grade-level(s). For example, in Chapter 3 the notes instruct teachers to review key words and order of operations to evaluate expressions, but the prior grade-level standards require a flexible view of simplifying and generating equivalent expressions. Other examples include:
    • Chapter 2: LCM review before performing operations with rational numbers (page T-43)
    • Chapter 4: Specific work with Grade 6 level inequalities and comparing rational numbers is explained before writing, graphing and solving inequalities.
  • The first page of each chapter is “What You Learned Before.” The teacher page adjacent to this page identifies the CCSS addressed, which is usually from a previous grade-level, but no explanation of what connects this previous material to the upcoming lessons is included.
  • Once into the chapter, teachers can see previous skills being reviewed, but there is only one identification by the textbook.
    • In section 1.1, a review of Grade 6 work with absolute value is presented and identified as preparing for 7.NS.1, 7.NS.2, 7.NS.3
    • A review of Grade 6 work with rational numbers occurs in section 2.1, but it is identified as learning 7.NS.2b, 7.NS.2d.
  • Content of future progressions beyond the current grade-level are not identified in the materials nor are these lessons accompanied by an explanation of the progressions.
    • Students find slope of a line in 5.5, but the material does not identify this lesson as Grade 8 content.

The materials do not give all students extensive work with grade-level problems. The majority of the problems in the exercises require students to produce an answer or solution. There are open-ended, reasoning, and critical thinking items which allow students to engage in grade-level work that meets the depth of the standard in most cases. These opportunities to engage in extensive grade-level problems are provided for all students only if they are given the opportunity to access all of them.

  • An assignment guide is provided in each lesson that levels students into basic, average, or advanced. These charts exclude the “basic” learner from the reasoning and critical thinking problems. These problems are critical for all students in order for them to reach the depth of the standard in many of the lessons.
    • For example in section 2.3, both average and basic learners are excluded from item 27, an open-ended problem that requires students to fill in two different negative values to make an equation true. Item 28, which requires students to make generalizations about the structure of rational numbers, is also not listed as an opportunity for average or basic learners (7.NS.1).
    • Many lessons contain explanations in “Laurie’s Notes” of a specific homework problem and how “Taking Math Deeper” can apply to that problem. Usually it is a simple task that can get to the depth of a standard; however, it is rarely part of the Basic Assignment. If students are only assigned Basic or Average Level Assignments, they will often not engage with the problems reaching the full depth of the standard.

The materials do not relate grade-level concepts explicitly to prior knowledge from earlier grades.

  • Each chapter begins with a What You Learned Before page just before the first lesson. These pages contain problems for students from prior grade levels and/or chapters found earlier in the materials. Connections to specific grade-levels or standards are not identified.
  • Laurie’s Notes are found in each lesson. In the margin of these notes for instruction, specific Grade 7 standards that will be addressed are identified. Most of them contain a Previous Learning section that describes prior knowledge students should possess before engaging in the lesson, but again they are not explicit about the particular grade-level or standard tied to the skills or understanding needed. For example, in section 5.2 the Previous Learning states, “Students have written and simplified ratios.” Neither the specific CCSS standards nor the grade-level is stated.

Overall, explicit connections to prior knowledge are made at a very general level through the chapter and lesson features in this series. Connections are not clearly articulated for teachers and are merely lists of skills without indication of standards, clusters, or domains. There is not a clearly defined progression for teachers to demonstrate how prior knowledge is being extended or developed.

) [10] => stdClass Object ( [code] => 1f [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Grade 7 partially meet the expectation for fostering coherence through connections at a single grade, where appropriate and required by the Standards. Overall, the materials do not include learning objectives that are visibly shaped by CCSSM cluster headings, but there are some opportunities to connect clusters and domains.

Examples of the materials not including learning objectives that are visibly shaped by CCSSM cluster headings include:

  • Cluster headings were explicitly addressed in the materials on page xxxvii, where it appears that a chapter is dedicated to each. There is no explanation as to how the lessons are tied together under the cluster heading besides the information found on this page. In most cases, standards are addressed in isolated lessons with very little overlap of CCSSM across chapters, and the language used in the cluster heading was not found.
  • The lesson “Goal” appears in Laurie’s Notes before the lessons in each section and most closely aligns to an objective. These are descriptions of the parts of the standard that are addressed in the lesson and were not found to describe cluster headings.
    • Examples from Chapter 5 include, “Today’s lesson is comparing ratios using proportions and the Cross Products Property” in Section 2 and "Today's lesson is solving proportions using a variety of strategies" in Section 4. Neither lesson goal capture what is required in 7.RP.A because the students do not analyze proportional relationships; in both lessons, students use multiplication or the Cross Products strategy to produce an answer.

Examples of the materials providing some problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important include:

  • All major work is addressed before the supporting work occurs limiting the textbook’s ability to foster connections between clusters.
  • An identified connection occurs in Chapter 6, Lesson 4, which connects 7.RP.3 to 7.EE.3, when students write an equation before solving problems involving percentages on page 236. Students also work with rational numbers in this lesson, but 7.NS.3 is not identified.
  • The performance tasks could make connections between cluster headings. These tasks present open-ended problems with varying ways to represent solutions, but they address one standard at a time. Although not explicitly mentioned, the tasks aligned to the 7.RP standards included natural connections to 7.EE and 7.NS. Students may choose to represent solutions as equations, but they are not required to do so as the prompts are written.
  • In Chapter 8, Lessons 2 through 4 address 7.G as well as parts of 7.EE.4 in the limited number of items which require students to use properties of equality when engaging in work that includes semicircles. While the examples are accompanied by explicit steps to calculate the perimeter and area of various two-dimensional shapes using equations, the directions do not require students to use equations or find missing dimensions (length, width, height, radii or diameters) when completing the problems. Most items do not require students to use inverse operations or explain equality.
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The instructional materials reviewed for Grade 7 Big Ideas do not meet the expectations for Gateway One. Future grade-level standards are rarely assessed and could be easily modified or omitted. The materials do not devote a majority of the time to the major work of the grade. The instructional materials infrequently connect supporting work with the major work of the grade. Although the materials provide a full program of study that is viable for a school year, students are not always given extensive work with grade-level problems, and connections between grade levels and domains are missing. Since the materials do not meet expectations for Gateway One, evidence for Gateways Two and Three was not collected.

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Eighth Grade

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            [score] => 11
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The instructional materials reviewed for Grade 8 Big Ideas partially meet the expectations for Gateway One. Future grade-level standards are rarely assessed and could be easily modified or omitted. The materials devote a majority of the time to the major work of the grade. The instructional materials infrequently connect supporting work with the major work of the grade. Although the materials provide a full program of study that is viable for a school year, students are not always given extensive work with grade-level problems. Connections between grade-levels and domains are missing. Overall, the instructional materials meet the expectations for focusing on the major work of the grade, but the materials are not always consistent and coherent with the standards.

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The materials reviewed for Grade 8 partially meet the expectations for Gateway 2, Rigor and Mathematical Practices. All three of the aspects of rigor are present, but procedural skill and fluency are focused on in the materials. There is not a balance of the three aspects of rigor within the grade, specifically where the Standards set explicit expectations for conceptual understanding, procedural skill and fluency, and application. The MPs are not always identified correctly, and the full meaning of the MPs is sometimes missed. The materials set up opportunities for students to engage in mathematical reasoning and partially support teachers in assisting students in reasoning. The materials attend to the specialized language of mathematics.

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The Grade 8 Big Ideas materials meet the expectations for focus. Future, grade-level standards are rarely assessed and could be easily modified or removed, and the materials devote a majority of the time to the major work of the grade.

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The materials meet the expectation for not assessing topics before the grade-level in which they should be introduced. The majority of the assessments are on grade-level with a few items that could be easily modified or removed to remain on grade-level.

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The instructional materials reviewed for Grade 8 meet the expectations for assessing the grade-level content and, if applicable, content from earlier grades. Summative assessments focus on the Grade 8 standards with few occurrences of above grade-level work.

The following assessments were reviewed for this indicator from the print and digital materials: forms A and B of the Chapter Tests, Chapter Quizzes, Standards Assessments, and Alternate Assessments.

The majority of items are within Grade 8, and few above-grade level items were found. There are some items that assess two domains and call for students to explain the meaning behind mathematical concepts and/or reasoning behind their solutions.

  • On all Chapter 3 assessments, students are required to explain angle relationships and why triangles are or are not similar. This meets the depth of standards 8.G.5 and 8.EE.5. Similarly, in Chapter 4, several questions have students work with slope and y-intercepts (8.EE.5 and 8.F.4). For example, on the Chapter 4 Alternative Assessment, students must calculate slope, write an equation, explain the meaning of the slope, and use the equation to solve related real-world problems. Similar questions are included on forms A and B.
  • On form A (items 25 and 26) and B (items 24 and 25) of the Chapter 7 test, students are required to find the distance between two points. These questions do not prompt students to use the Pythagorean theorem nor are they provided a graph. In Grade 8, students are expected to use the Pythagorean theorem to find distance (8.G.8). In lesson 7.5, examples 2 and 3, the procedure for using the distance formula is explicitly taught after directed exploration found in Activity 3 on page 319.
  • On the Chapter 7 Standards Based Assessment, students must identify and use the Pythagorean theorem, and they are asked to find a missing hypotenuse on three different triangles in a composite figure (8.G.7). They must also “show [their] work clearly” and explain how they can identify any irrational numbers (8.NS.A).
  • On the alternative assessment in Chapter 7, students must use the Pythagorean theorem to find the missing sides of the given right triangles, and they also identify if the two are similar. In 8.G.4, students are expected to find similarity through transformations and dilations, not by finding proportionality in corresponding sides as noted in the answers on page A13.

It should also be noted that there are items on the Chapter 9 assessments, problems 35-36, and end-of-course tests that do not have connections to any Grade 8 standard (e.g., choosing proper display, explain why the data is misleading).

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The Grade 8 Big Ideas materials meet expectations for devoting the large majority of class time to the major work of the grade-level. The materials engage students in the major work of the grade more than 65 percent of the time.

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The instructional materials reviewed for Grade 8 meet the expectations for focus on major clusters. The Grade 8 instructional materials do spend the majority of class time on the major clusters of the grade.

The Common Core State Standards to Book Correlation (pages xx-xxvi) and the Book to Common Core State Standards Correlation (page xxvii) were used to identify major work, as well as the first page in each chapter which includes common core progression information, a chapter summary, and a pacing guide (and related online pages). The pacing guide provides the number of days to spend on each chapter opener, activity, lesson, any extensions, and review/assessment days. This guide was used to determine the number of instructional days allotted by the publisher for each standard found in the major work of the grade. Reviewers also examined all lessons with standards identified by the publisher as non-major work to ensure that these lessons did not contain enough material to strengthen major work.

All percentages are greater than 65 percent and were calculated to reflect the chapters, lessons, and instructional time spent on major work:

  • The material devoted approximately 80 percent of chapters to major work of the grade. Chapters 8 and 9 were excluded because they only address supporting standards. If over 50 percent of a chapter addressed major work, then the chapter was counted as major work.
  • Of the lessons, 86 percent (44 out of 51) were dedicated to major work. One lesson, 7.4, identified as major work did not reach the depth of the standard and was not counted as major work. Lesson 7.4, Approximating Cube Roots, is aligned to 8.EE.2, but students are not required to use square and cube roots to represent solutions to equations.
  • Of the instructional days, 85 percent (125 out of 148 ) in the Grade 8 materials are spent focusing on the major clusters of Grade 8. Days were counted based on the recommendation of the pacing guide in the beginning of each chapter for all lessons reviewers found aligned to major work.
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The instructional materials reviewed for Grade 8 do not meet the expectations for coherence. The instructional materials infrequently use supporting content as a way to continue working with the major work of the grade. The materials include a full program of study that is viable content for a school year. Content from prior grades is not clearly identified or connected to grade-level work, and not all students are given extensive work with grade-level problems. Material related to prior, grade-level content is not clearly identified or related to grade-level work. These instructional materials are not shaped by the cluster headings in the standards. Overall, the Grade 8 materials do not support coherence and are not consistent with the progressions in the standards.

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The instructional materials reviewed for Grade 8 partially meet the expectations for having supporting content that enhances focus and coherence simultaneously by engaging students in the major work of the grade. The structure of the chapters and the lessons in the Grade 8 materials do not fully engage students in both supporting and major work to allow natural connections.

Supporting work is identified in Chapters 7, 8, and 9 and shows some connection to major work.

Connections to major work are only identified in Chapter 7 as students move from finding square and cube roots (8.EE.2) in lessons 1 and 2 to using the Pythagorean theorem to find the missing sides of right triangles (8.G.7) in lesson 3, and finally, working with irrational numbers in order to categorize real numbers (8.NS.2) in lesson 4. While an attempt was made to connect these standards by sequencing them in one chapter, they are isolated and addressed in different lessons with one activity and three items found to connect them.

  • In Approximating Square Roots, Activity 2 on page 308, students must use the Pythagorean theorem to find the diagonal of a square situated on a number line between 0 and 1 and then estimate the length of the diagonal using the number line in the graphic. Lesson 7.4 has students estimate square roots, compare real numbers, and approximate the value of expressions.
  • Items 37-39 in Lesson 7.4 connect 8.G.7 to 8.NS.2 by having students approximate the length of a diagonal of a square or rectangle.
  • An opportunity to have students connect approximating square roots and the Pythagorean theorem is missed in Lesson 7.5 when students are not required to estimate an answer while engaging in the practice and problem solving set, and all solutions are expressed in radical form. This connection could be noted in the margin of Laurie’s Notes on page 318 under Previous Learning, and explicit direction requiring students to find approximations could be included in the exercises to strengthen major work.

Connections between 8.G.C and 8.EE.A are found in Chapter 8, Volume and Surface Area of Similar Solids, due to the formulas containing both rational and irrational numbers as well as exponents. 8.EE.2 could be mentioned in the Previous Learning section found in the margin of Laurie’s Notes in each of the following lessons:

  • Lessons 8.1, 8.2 and 8.3 are aligned to 8.G.9 and 8.EE.7 where examples use equations to solve problems involving volume.
  • In Lesson 8.4, 8.G.9 has connections to 8.G.4 when students use the volumes and surface areas of similar solids to develop understanding of the related formula. While this particular connection is explored in the lesson, this concept regarding three dimensional figures is not addressed until the high school Geometry standards.
  • Also, 8.EE.2 is found in Chapters 7 and 8, but then it is isolated from Chapter 10 where the major work with exponents occurs.

Unidentified connections were also found in Chapter 9 in Lesson 2 in which students draw lines of best fit to model a set of data and then use the slope and y-intercept to make predictions about future events. Laurie’s Notes does mention that students “should know how to make scatter plots and write equations in slope-intercept form,” but 8.EE.6 is not explicitly identified.

) [8] => stdClass Object ( [code] => 1d [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed meet the expectation for having an amount of content designated for one grade level that is viable for one school year in order to foster coherence between grade levels. Overall, the instructional materials reviewed for Grade 8 provide a year’s worth of content as written.

In Big Ideas, the length of each class period is 45 minutes, so 149 45-minute class periods would be needed to cover Chapters 1-10. The 149 days of instruction are outlined in the pacing guide on pages xxxii and xxxiii. This pacing includes one day for a scavenger hunt, one day in each chapter for study help and review before the mid-chapter quizzes, and two days for review and assessment at the end of each chapter. In total, ten days are devoted to Study Help/Quizzes and 20 instructional days spent on chapter review assignments and chapter assessments, which leaves 119 days for instructional lessons, activities, and extension lessons. Before each chapter, information is provided for the teacher on how much time to spend on each section including activities, lessons, and any extensions.

The following extension activity found in Chapter 7 is of particular importance and should not be skipped, as this is the place in the material where 8.NS.1 is fully addressed.

  • Chapter 7, Extension 7.4 - Writing a repeating decimal in rational form (8.NS.1).

The online lesson plans provided in Chapters at a Glance also include detailed information about when to use the supplemental activities such as extra examples as well as performance tasks for each standard. Any additional days of instruction can be spent implementing these tasks or the additional skills practice found in the online resources.

) [9] => stdClass Object ( [code] => 1e [type] => indicator [points] => 0 [rating] => does-not-meet [report] =>

The instructional materials reviewed for Grade 8 do not meet the expectation for having materials that are consistent with the progressions in the Standards. Materials are not intentionally written to follow the progressions of the grade-level as few lessons are identified as work from prior grade-levels, and there are no lessons identified to connect Grade 8 work to the work of future grades. Materials do not give all students extensive work with grade-level problems although general explanations for how lessons are related to prior knowledge are present.

The materials do not develop according to the grade-by-grade progressions in the standards. Content from prior or future grades is not clearly identified and related to grade-level work.

  • Explanations of Common Core Progressions are given at the beginning of each chapter connecting both Grade 6 and Grade 7 work to the Grade 8 work students will encounter in each of the chapters. These connections to below grade-level work are presented as bulleted lists of skills and are not aligned to specific standards.
  • Math Background Notes include vocabulary review as well as a general explanation of the most important skills and understandings from the prior grade-level(s) found in the “What You Learned Before” activities on the following page. For the most part, these are procedural in nature and do not add connections or meaning to the mathematics which occurred in prior grade-levels. For example, in Chapter 5, the notes instruct teachers to remind students to use inverse operations to isolate a variable, but there is no mention of the underlying properties of equality that make this possible. Other examples include:
    • Chapter 7: Order of operations review cites the “Please Excuse My Dear Aunt Sally” pneumonic before engaging in evaluating expressions with square and cube roots (page T-287).
    • Chapter 10: The steps to multiplying decimals are reviewed, and teachers are encouraged to “remind students to count the number of digits in both factors that appear to the right of the decimal point, and then put that many digits to the right of the decimal point in the answer” instead of using estimation or place value, which would further develop the structure of numbers for struggling learners.
  • The first page of each chapter is What You Learned Before. The teacher page adjacent to this page identifies the CCSS addressed, which is usually from a previous grade-level, but no explanation of what connects this previous material to the upcoming lessons is included.
  • Chapter 1, Section 1 serves as review lesson as students solve one-step equations with rational numbers, but this section is marked as “Learning” 8.EE.7a and 8.EE.7b instead of 6.EE.5 and 7.EE.3.
  • Content of future progressions beyond the current grade-level are not identified in the material nor are these lessons accompanied by an explanation of the progressions.
    • Students find the distance between two points on the coordinate plane using the distance formula in Chapter 7, but the material does not identify this lesson as high school content.

The materials do not give all students extensive work with grade-level problems. The majority of the problems in the exercises require students to produce an answer or solution. There are open ended, reasoning, and critical thinking items which allow students to engage in grade-level work that meets the depth of the standard in most cases. These opportunities to engage in extensive grade-level problems are provided for all students only if they are given the opportunity to access all of them.

  • An assignment guide is provided in each lesson that levels students into basic, average, or advanced. These charts exclude the “basic” learner from the reasoning and critical thinking problems. These problems are critical for all students in order for them to reach the depth of the standard in many of the lessons.
    • For example, in section 4.5, basic learners are excluded from item 23, a critical thinking problem which asks students to reason about the values in the equation in context. Item 22, which requires students to make generalizations about the the graphic representation of all linear equations, is not listed as an opportunity for average or basic learners (8.EE.8).
    • Many lessons contain explanations in Laurie’s Notes of a specific homework problem and how Taking Math Deeper can apply to that problem. Usually it is a simple task that can reach the depth of a standard; however, it is rarely part of the Basic Assignment. If students are assigned Basic or Average Level Assignments, they will often not engage with the problems reaching the full depth of the standard.

The materials do not relate grade-level concepts explicitly to prior knowledge from earlier grades.

  • Each chapter begins with a What You Learned Before page just before the first lesson. These pages contain problems for students from prior grade-levels and/or chapters found earlier in the material. Connections to specific grade-levels or standards are not identified.
  • Laurie’s Notes are found in each lesson. In the margin of these notes for instruction, specific Grade 8 standards that will be addressed are identified. Most of them contain a Previous Learning section that describes prior knowledge students should possess before engaging in the lesson, but again they are not explicit about the particular grade-level or standard tied to the skills or understanding needed. For example, in section 2.6 the Previous Learning states, “Students should know how to plot ordered pairs. Students also need to remember how to solve a proportion.” Neither the specific CCSS standards nor the grade-level is stated.

Overall, explicit connections to prior knowledge are made at a very general level through the chapter and lesson features in this series. Connections are not clearly articulated for teachers and are merely lists of skills without indication of standards, clusters, or domains. There is not a clearly defined progression for teachers to demonstrate how prior knowledge is being extended or developed.

) [10] => stdClass Object ( [code] => 1f [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Grade 8 partially meet the expectation for fostering coherence through connections at a single grade, where appropriate and required by the Standards. Overall, the materials do not include learning objectives that are visibly shaped by CCSSM cluster headings, but there are some opportunities to connect clusters and domains.

Examples of the materials not including learning objectives that are visibly shaped by CCSSM cluster headings include:

  • Cluster headings were explicitly addressed in the textbook on page xxxv. There is no explanation as to how the lessons are tied together under the cluster heading besides the information found on this page. The language used in the cluster heading was not found.
  • Chapter and lesson titles are connected to, but do not appear to be influenced by, cluster headings. They are often descriptive of specific skills or topics but not the overarching idea of the cluster heading. For example, the “Expressions and Equations” domain has the following headings, which are connected to the these chapter titles:
    • “Work with radicals and integer exponents” is connected to Chapter 1, “Equations.”
    • “Understand the connections between proportional relationships, lines, and linear equations” is connected to Chapter 4, “Graphing and Writing Linear Equations.”
    • “Analyze and solve linear equations and pairs of simultaneous equations” is connected to Chapter 5, “Systems of Linear Equations.”
  • The lesson goal appears in Laurie’s Notes before the lessons in each section and most closely aligns to an objective. These are descriptions of the parts of the standard that are tackled in the lesson and were not found to describe cluster headings.
    • For example, “Today’s lesson is graphing linear equations” is found in section 4.1 and “Today’s lesson is graphing proportional relationships” is found in section 4.3, but neither is shaped by the full meaning of 8.EE.B, which calls for students to understand the connections between proportional relationships, lines, and linear equations.

Examples of the materials providing some opportunities of problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important include the following, but these are not identified for the teacher except in the Chapter 6 example below.

  • In Chapter 6, 8.G.A and 8.F.B are connected in the Activity for Lesson 6.4 when students use the perimeters and areas of similar rectangles (8.G.4) to compare linear and nonlinear functions (8.F.3).
  • In Chapter 9, 8.SP.A and 8.EE.C are connected in Lesson 9.2 when students draw lines of best fit using slope intercept form to investigate patterns of association in bivariate data.
  • The performance tasks could make connections between cluster headings. These tasks present open ended problems with varying ways to represent solutions, but they address one standard at a time with the exception of performance task 8.G.6. Natural connections between 8.G.6 and 8.EE.A occur in this task as students grapple with making connections between the area of the trapezoid and the areas of the embedded right triangles using equivalent expressions and the Pythagorean theorem.

Chapter 7 is the place where the material identifies multiple domains in the same chapter, but the standards are addressed in isolated lessons with few connections across chapters.

Chapter 7 is aligned to the number system, expressions and equations, and geometry domains on page xxxv. An explanation for why these domains were connected was not found in the material. An explanation could be provided in Laurie’s Notes in the “Connect” section where teachers are given a one sentence summary of “yesterday’s learning” and “today’s learning.” It appears in 7.1 and 7.2 when students work with exponents (8.NS) and roots (8.NS), but not in 7.3, which would connect square roots (8.NS) with the Pythagorean Theorem (8.G). ) [11] => stdClass Object ( [code] => rigor-and-balance [type] => component [report] =>

The materials reviewed for Grade 8 partially meet the expectations for Gateway 2, Rigor and Mathematical Practices. All three of the aspects of rigor are present, but procedural skill and fluency are focused on in the materials. There is not a balance of the three aspects of rigor within the grade, specifically where the Standards set explicit expectations for conceptual understanding, procedural skill and fluency, and application. The MPs are not always identified correctly, and the full meaning of the MPs is sometimes missed. The materials set up opportunities for students to engage in mathematical reasoning and partially support teachers in assisting students in reasoning. The materials attend to the specialized language of mathematics.

) [12] => stdClass Object ( [code] => 2a2d [type] => criterion ) [13] => stdClass Object ( [code] => 2a [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed partially meet the expectation for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Overall, the development of conceptual understanding is presented in a directed way so that students would not fully develop and refine their ability to reason mathematically.

All sections in the material begin with a one-day activity intended to build understanding of the concept within each section. Most of the activities are hands-on, but they are presented in a step-by-step manner, which leads all students to solve them in the same way and, in turn, produce the same results. This limits opportunities to explore and make connections between various solution paths. The following activities address conceptual development in an explicit way:

  • The initial activity for Chapter 2, Lesson 6 has students find the perimeters and areas of various similar figures. After completing the chart, students have to describe the pattern and find the relationship between perimeter, area, and the original figure. The conceptual understanding of 8.G.4 is the primary goal.
  • Activities in Chapter 4, Lesson 1 address 8.EE.5 by requiring students to find multiple ordered pairs that satisfy the equation, graph them, and then find and check additional ordered pairs. Students understand that in order to be a solution of an equation, a point has to lie on the line.
  • Lessons attending to 8.F are found in Chapter 6. The 6.1 activities have students generate the outputs, or areas of the given two- and three-dimensional figures, using a given input for the length, and then students map these values in activity 1. In activity 2 students generate rules. In the activities for section 6.2, students write equations in problems 1 and 2, and students graph points from a given table to see if the given statements are true in problem 3. These connections are built in a procedural way until students are shown all three representations in Example 3 and the summary of section 6.2. Students are often expected to complete the representation rather than make connections between them.
  • The activities found in Chapter 10, Lessons 1-4, align to 8.EE.1, and all use the expanded form of exponents to help students understand operations with powers and develop a rule using patterns.

On the second day, the lesson is presented through multiple examples that can be reviewed as a class and On Your Own examples that allow students to practice the lesson concept. There is usually one reasoning or logic problem per lesson in the exercises of each section. One problem in each of the lessons is explained at length in the Taking Math Deeper. Lesson notes for the teacher mainly focus on procedures and the steps necessary to solve the problems.

Additional features included in the material show an emphasis on conceptual development:

  • The beginning of each exercise has a section called Vocabulary and Concept Check, and the end of each activity has a section called What is Your Answer? Both of these sections often expect students to explain and demonstrate conceptual understanding through reasoning and writing about the concepts.
  • The online lesson plans also account for sections titled Start Thinking, and the student’s copy is located in the Resources by Chapter book. These questions assist in connecting to previous knowledge that students need in order to engage in a new task and also require explanations, justifications, or comparisons, which are important to conceptual understanding.

In the overall structure of the material, concepts are proceduralized in each section because the activities are accompanied by directed steps. Students are asked to make generalizations in the reasoning and logic items before engaging with the exercises. Communications between the students and teacher addressing conceptual understanding are not always stressed.

) [14] => stdClass Object ( [code] => 2b [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Grade 8 meet the expectation for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, there are many problems provided that help develop procedures and fluencies expected by the grade-level standards.

Many of the assigned problems focus on using procedures to compute an answer. The majority of the other problems in the lessons are word problems that require the use of the procedures found in the examples to solve the problem. Specific examples that reflect procedural development include:

  • In Chapter 1, lessons 1-3, students have many opportunities to engage in 8.EE.7 by following specific steps to solve equations with variables on one or both sides.
  • In Chapter 2, lessons 2 and 3 address 8.G.4, and students are encouraged to use the “rule” when transforming objects. For example, on page 50, the Key Idea states that the translation of (x, y) is (x + a, y + b).
  • In Chapter 4, lesson 1 gives students an equation and has them graph it by creating a table of values, graphing the points, and graphing the line. The equations are all in slope-intercept form, but the table of values is the only suggested solution method (8.EE.5).
  • In Chapter 8, lessons 1 through 3, students complete activities in order to understand the parts of each formula before they are given the steps to follow. Examples give explicit instructions on how to calculate the volumes of cylinders (8.1), cones (8.2), and spheres (8.3). The majority of each section focuses on finding the volume of the shapes with a few opportunities to find missing dimensions.
  • In Chapter 9, lesson 2 has students solve problems involving lines of best fit (8.SP.1, 8.SP.2, 8.SP.3). While there is a real–world context, each problem is written in the same way by prompting students to use a table, graph, write an equation, and use the equation to answer a question.

Additional opportunities to build procedural skill and fluency can be found in the “Fair Game Review” at the end of each lesson, which build fluency by repeating skills found earlier in the material as well as previous grades. Each lesson has extra practice problems in the Record and Practice Journal. Teachers also have access to supplemental worksheet Forms A and B in the Resources by Chapter workbook.

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The materials reviewed partially meet the expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of the grade. Overall, there were few opportunities for students to engage in non-routine problems, and in many cases, the items direct students in such a way that they do not have a chance to decide on alternate ways to solve them.

The following structures and resources sometimes attend to application, but many times are scaffolded for students with steps and given paths to the solution:

  • Performance Tasks for each section are provided online for each grade-level standard. These provide opportunities for students to engage in the mathematics from the section in a new way, but these can sometimes be directive and rarely address more than one standard.
  • Enrichment and Extension is found in Resources by Chapter, and there is one for each lesson. These are not included in the lesson plans provided online, but they are suggested as extra practice. There are questions in these activities that would meet the expectations of application, but a direct path for the answer is provided. For example, in lesson 1.4 students are asked to rewrite the formula for volume of a cylinder. They are given two methods for rewriting the formula with explicit steps for each and then asked which method they prefer.
  • In Appendix A, there are additional opportunities for students to apply what they have learned over the course of the year. This section has four Big Idea Projects. Students look at examples of math in the real world and in a cross-curricular way. The projects contain scaffolding and guide students in how to complete the work.
  • Each lesson exercise has one problem that applies the skill of the lesson, and it is titled Taking Math Deeper in the teacher edition. In the student materials, these problems do not have any special identification. This section provides an extended solution for the teacher for one specific item in the exercises and then provides a related project that the teacher can assign. For example, in Lesson 10.2, “Taking Math Deeper” explains the steps to solving exercise 31, where students show the number of pieces of mail sent by the United States Postal Service in 6 days as a product of powers expression. The related project involves researching the price of a postage stamp and finding the range in the cost.

There are some examples of items that allow students to engage in applying mathematics to the given situations.

  • In Chapter 1 Lesson 2, Solving Multi-Step Equations, there are two activities that address 8.EE.7 on page 11. These problems require students to apply past and current concepts to solve a multistep problem. In both, students may work with a partner to find multiple unknowns.
  • In Chapter 1 Lesson 3, Solving Equations with Variables on Both Sides, item 39 is an opportunity to apply 8.EE.7 in a real world context. Students must find the price of mailing DVDs by setting the two different companies’ rates equal to each other. Students are given a chart with the needed information to solve the problem.
  • As students work with functions (8.F) in Chapter 6 (pages 254, 262), they often work with real-world problems requiring them to write an equation, graph the relationship, and then answer various questions using both the equation and graph. These questions vary as to content and encourage students to think and analyze the relationships.
  • Activity 2 in Chapter 8, lesson 1 has students engage with 8.G.9 by asking students to use a defined amount of wax purchased for $20 to make eight candles of three different sizes and then decide on candle size and price using properties of cylinders. Students are not given guidance on this item, making this a very rigorous problem for students.

While opportunities for application are seen in several features and sections found in the material, evidence of proceduralizing opportunities for application was also found.

  • In Chapter 4, lesson 3, Graphing and Comparing Proportional Relationships (8.EE.5,6), students are presented with two 2-variable relationships, one is in algebraic form and the other in graphical form. Students must write both equations and graph as well. Nearly all problems are application, i.e., cost in dollars/hour, growth in mm/year; however, all are presented in the same format so that students actually follow the procedures more than applying the mathematics.
  • The initial activity for Chapter 5, Lesson 1 begins by asking students to write two equations for a real-world situation involving starting a bed-and-breakfast and finding when it would actually make money. Instead of leaving the problem open for students to solve, the students are walked through writing the equations and given a table to complete using the given equations. In the 5.1 lesson, students solve systems of equations by graphing and are asked to use the graphical representation to compare the relationships (8.EE.8a). Four out of 24 problems include a real world scenario. These are all routine problems that follow the steps given in example 2, Real Life Application, on page 205.

Overall, students are given multiple opportunities to solve mathematics in real world contexts, but given the way the material is structured, students can easily use the examples provided in the section to figure out which types of relationships or structures to apply to given situations. Problems are routine and occur in the labeled lesson. For example, all Pythagorean theorem problems appear in Chapter 7 in lessons where students are given worked examples and already know what procedures to apply.

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The materials reviewed for Grade 8 partially meet the expectations for not always treating the three aspects of rigor together and not always treating them separately. Overall, there is not a balance of the three aspects of rigor within the grade.

  • Most of the items require calculating a solution, and students are given step-by-step procedures to use when solving them. The activities sometimes offer opportunities for students to engage in procedures with connections to explore concepts, but they are given targeted and scaffolded paths to reach the desired understanding limiting opportunities for students to apply mathematics previously learned.
  • According to the given online lesson plans, many of the application opportunities are not part of the regular program, but are offered as other opportunities.
) [17] => stdClass Object ( [code] => mathematical-practice-content-connections [type] => component [report] =>

The materials reviewed for Grade 8 partially meet the expectations for practice-content connections. The MPs are identified and sometimes used to enrich mathematics content. The materials rarely attend to the full meaning of each practice. The materials reviewed for Grade 8 partially attend to an emphasis on mathematical reasoning. Overall, students are prompted to construct viable arguments, but there are not sufficient opportunities for students to analyze the arguments of others or for teachers to assist students in analyzing the arguments of others. The materials attend to the specialized language of mathematics.

) [18] => stdClass Object ( [code] => 2e2g [type] => criterion ) [19] => stdClass Object ( [code] => 2e [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The materials reviewed for Grade 8 partially meet the expectations for identifying the mathematical practices (MPs) and using them to enrich the mathematics content within and throughout the Grade 8 materials.

The MPs are clearly labeled in the teacher edition in the activity portion of each section. They can also be found in the introduction within Laurie’s Notes, where there is also an explanation of how they are connected to the lesson and what the teacher should expect out of the students.

  • A Math Practice box is found in the student edition. It is not labeled with the specific MP for the student, but an explanation can be found in the teacher’s edition explaining the MP in which students are engaging.
  • Sometimes MP1 and MP3 are split and labeled “MPa” and “MPb”, when the full MPs are not reached. For example, in section 4.2, MP1a is listed with a note that “drawing arrow diagrams will help students visualize the slope triangle.” There is no explanation of what the “a” means. In section 4.3, MP3a is listed, and the teacher is asked to use “volunteers to justify their procedures and explain why their procedure shows a proportional relationship.” An explanation of what the “a” means is not provided.
  • The MPs are also identified in the online lesson plans. Each lesson has the specific MP stated in a box in the upper right hand corner, and within the lesson, there is a section that states the focus of the MP. This is generally an explanation of what to look for while students are engaging in the problems, but sometimes it offers a question to ask the students.

The MPs are identified for the student on page iv in the beginning of the student textbook as well as in a Math Practice box located in the activities. The Math Practice boxes cause students to think about the habits of mind to be used while solving problems, but sometimes it is not clear which MP is connected to the activity since the numbers 1 through 8, used to identify the MPs in the standards, are not used in the student textbook within the series.

  • On page 19 in the student textbook, the Math Practice box is labeled “Use Operations” and asks students, “What properties of operations do you need to use in order to find the value of x?” This appears to be a prompt more than the identification of a math practice.
  • On page 49, the box is labeled “Justify Conclusions,” connecting it to Math Practice 3, and asks students what information they need to make a conclusion.
  • The box on page 203 is labeled Use Technology to Explore, and students are asked how they decided on the values they used in setting the calculator window.
  • On page 355, students are asked how repeating calculations assist them in describing the pattern in the activity. The box is labeled Repeat Calculations.

It should also be noted that the Math Practices are only identified within lesson activities and class examples. They are not identified within the problem set. For example, teachers could be aware that part of MP3 is being addressed in item 30 on page 24 as the problem is labeled Error Analysis and MP6 is being addressed in item 35 on page 24 as the problem is labeled Precision, but these connections are not explicitly made in the materials.

) [20] => stdClass Object ( [code] => 2f [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The materials reviewed for Grade 8 partially meet the expectations for attending to the full meaning of each MP. The MPs are most frequently identified in notes where they are aligned to a particular practice activity or question item. Many times the note is guidance on what the teacher does or says rather than engaging students in the practice. Little evidence was found to show MPs used to enhance understanding of a standard in an intentional way, with the exception on MP8.

The MPs are often applied to problems where they could be beneficial. However, the depth of the MPs is often not met since teachers, not students, engage in the MPs as they show students how to solve the problems. Examples of how MPs are presented in the materials follow:

  • On page 77 in Activity 4 of section 2.6, MP1 is identified. Students are given step-by-step directions on reaching the answer. The end of the problem then tells students that there are three other similar rectangles.
  • In section 5.1, Activities 1 and 2, the materials identifies MP1 when students are “investigating” a given system of equations. The example provides all steps of representing the language of the problem with equations, moving to a table, and then finally a graph. Students would not have to formulate a plan in order to find “the break even point.”
  • MP2 is used quantitatively on page T-148 in the teacher edition in Activity 1 as students work with a partner to find the slope using slope triangles in multiple places on a line. MP2 is identified as being used abstractly on page T-167 in Activity 2 when students arrive at slope-intercept form of the given graph of the relationship. In both instances, students are given explicit directions and explanations of how to engage with the mathematics of the problems.
  • MP4 is rarely identified in situations where students are modeling a mathematical problem and making choices about that process. This MP is frequently identified in situations where a particular form of modeling is already chosen for students. In many situations, it is labeled with directions for how the teacher should “model” rather than guidance on experiences where students engage with mathematical modeling. Examples can be found in the following lessons:
    • Lesson 1.3 - The teacher is prompted to use algebra tiles “if students are familiar with algebra tiles.”
    • Lesson 2.3 “Set up a table …”
    • Lesson 3.3 “This table helps to organize data.”
    • Lesson 8.3 - The teacher is given guidance on different ways to measure a sphere and use it to make a net for a cylinder. Teacher/students are “modeling” the conceptual connection, but they are not mathematically modeling a problem situation. All activities are guided step-by-step.
  • MP5 is rarely identified in a problem solving situation or in a situation where students must choose a tool. MP5 is frequently labeled when the materials suggest a specific tool for teachers to give to students. However, guiding questions provided to the student may help develop some aspects of MP5 (i.e., page 143 asks “What are some advantages and disadvantages of using a graphing calculator to graph a linear equation?”) Language from the teacher edition that illustrates how MP5 is typically presented in the materials follows:
    • Lesson 6.2 - “If available, provide square tiles to students.”
    • Lesson 10.5 - “In the first two activities, students will use calculators to multiply very large and very small numbers.”
    • Lesson 10.7 - “You may wish to give students access to calculators for this activity.”
    • Lesson 9.2 - “It is helpful to model this with a piece of spaghetti.”
  • MP7 is identified on page T-27 in Activity 2. In this example, structure is used as the materials identifies the area of the base as B instead of using each specific area formula. This is explained in the materials for the teacher, but the students are not discussing or arriving at this use of structure.
  • MP8 is used appropriately in many activities as students are completing tables of repeated examples to arrive at a conclusion. On page 76 in Activities 2 and 3, students experiment with the perimeters and areas of similar figures to discover how they change when the dimensions of the figures change.
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The materials reviewed for Grade 8 partially meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards. While the material prompts students to construct their own arguments, there are few times when they are asked to consider and critique the reasoning of their peers.

In many cases, students are asked to construct arguments and justify them, but there are limited opportunities for them to critique and provide feedback to another student or student groups. Prompts in the “Math Practice” box found in the activities in the student textbook include questions requiring explanations and/or justifications.

  • In section 3.3, after completing Activity 2 with a partner, students are asked to analyze a given conjecture about the sum of the interior angles of any polygon, including convex polygons. They are told to “explain,” but there is no instruction or prompt aimed at considering the reasoning of others in order to build on knowledge or refine thinking.
  • On pages T-2 and T-3, MP3 is identified when the teacher asks “What rule did you write for the sum of the angle measures of a triangle?” A critique of the problem is not required, and the students are not asked to communicate with other students. This also occurs on page 49 in activity 4 as the teacher asks “What conjectures can you make about the two rectangles?” This could lead to an in-depth mathematical discussion, but there is no guidance for teachers that could ensure students would discuss and critique each other appropriately.

There are many explain “how” prompts, but students are asked to explain “why” they are able to use a certain procedure in many cases, which limits the arguments students may write or verbalize.

  • For example, on page 119, students are asked how they can find the sum of both interior and exterior angle measures of given polygons, but they are not asked to explain why it is possible to calculate them.
  • In Section 3.2, Angles of Triangles, students are working to prove that the sum of all three interior angles in a triangle measures 180 degrees in Activity 1 using their prior knowledge of parallel lines and transversals. Both MP3 and MP6 are identified by the publisher for these connected activities. Both MPs are correctly identified and met if students are given the opportunity to share their arguments with other groups in order to critique the reasoning, but the tmaterial does not prompt this conversation. An explanation for the teacher is provided on page T-110.
  • MP6 is often used to remind students about vocabulary and units. For example, on page T-56 discussion of appropriately saying (x, -y) is found. Students should say “the opposite of y” and not “negative y.” On page T-79 in example 3, the material reminds teachers to “make sure students include the correct units in their answers.”

Students are asked to engage in an Error Analysis in many of the lessons. While these problems provide an opportunity for students to “describe and correct the error,” most of the errors are based on procedures that are completed incorrectly instead of requiring students to use mathematical reasoning and conceptual knowledge to strengthen an argument. For example, in item 17 of section 7.5, students must identify the error in using the distance formula to calculate the distance between two points. In the given solution, the error was in subtracting, instead of adding, the area of the squares before finding the square root. Another example is problem 13 of section 8.2; the diameter was used instead of the radius when calculating the volume of a cone with the given formula. An exception is found in item 14 of Lesson 5.4 when students must find the error in the given response, which also includes a statement about the system of equations having infinitely many solutions.

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The materials reviewed for Grade 8 partially meet the expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The materials, especially the activities, encourage student collaboration and discussion. Students are engaged in constructing arguments in terms of explaining why and justifying their answers, but students are rarely engaged in analyzing the work and arguments of others and critiquing that specific work. Many of the activities would lend themselves to this kind of class discussion and debate, but the materials do not provide much support for teachers to use the given materials to engage students in meaningful discourse.

  • In section 2.4, MP3 is often identified in Laurie’s Notes for the teacher on pages T-61 through T-64 in reference to certain activities or examples. On page T-62 in example 1, the teacher is told to “note that the relationships between the coordinates of the vertices of a figure before and after rotation are not given” and then asked to explore the relationship if students are ready. Based on this note, students may or may not explore the relationship, and if they do, the teacher is not provided specific support, such as possible student responses, to help guide the discussion.
  • In extension 4.2, “students are asked to make a conjecture and then justify their answers.” The publisher does include a note about the importance of constructing arguments, but there is no guidance for the teacher on having them critique the reasoning of other students.
  • In the Activities for Section 4.7, within the Laurie’s Notes, teachers are told to “take time for discussions and explanations so that students’ reasoning is revealed,” but there are no prompts or possible responses to help the teacher facilitate this discussion.
  • MP3 is misidentified on page T-2 and T-3 in Lesson 1.1 when the teacher asks “What rule did you write for the sum of the angle measures of a triangle?” Students do not have to construct a viable argument to support their rule or critique the reasoning of others that supports their rules.

Few directions are provided for the teacher other than “have students share out,” “listen for _____ methods,” etc. A few places within the teacher notes include MP3 outlined with relevant questioning and prompting for students to make conjectures about completed work within lesson activities. (For example, pages T-49, T-55, T-61, T-85, and T-111). Overall, teachers are given opportunity to engage students with MP3, but are not provided much assistance.

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The materials reviewed for Grade 8 meet the expectations for explicitly attending to the specialized language of mathematics.

  • At the beginning of each exercise there is a section called Vocabulary and Concept Check. This section requires students to write about mathematics using precise language.
  • Vocabulary is taught through a Key Vocabulary box found in most lessons, and vocabulary words are identified throughout the textbook. These boxes list the word and the page on which it is located. Once on the identified page, the word will be bold, highlighted in yellow, and defined.
  • At the end of each activity, there is a section called What Is Your Answer? Students are required to describe their thoughts with precise language.
  • The teacher and student edition use consistent specialized language that does include visual examples where appropriate. Notes throughout the teacher edition give guidance on how to address potential language misconceptions. Each Topic’s Math Background Notes includes the vocabulary that may need to be reviewed and guidance on potential misconceptions that often include language misconceptions.
  • Examples of misused vocabulary were not found within the student or teacher materials.
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The instructional materials reviewed partially meet the expectations for alignment to the CCSSM. In Gateway One, the instructional materials meet the expectations for focus, but they do not meet the expectations for coherence. This leads to the materials partially meeting the expectations for focus and coherence. In Gateway 2, the materials partially meet the expectations for both rigor and balance and practice-content connections, which means the materials partially meet the expectations for rigor and the mathematical practices.

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Big Ideas Traditional (2013)

Big Ideas Learning, LLC | | 2013 Edition

High School

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The instructional materials reviewed for Big Ideas Traditional do not meet the expectation for Focus and Coherence within the CCSSM. For focus, even though the students spend the majority of time on the WAP standards, not all of the high school, non-plus standards are taught to the depth expected in order to give students the opportunity to fully learn each standard. The context of problems are appropriate for high school students; however, the numbers used to model situations are often integers, and the full intent of the modeling process is minimally applied to the modeling standards. For coherence, there is a partial connection between and among standards and courses. Explicit and purposeful connections to the middle school standards are limited.

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The instructional materials reviewed for the Big Ideas Traditional series partially meet the expectation that materials attend to the full intent of the mathematical content contained in the high school standards for all students. For this Indicator, the materials were examined, guided by included correlation documents: Common Core State Standards for Mathematical Content Correlated to Algebra 1, Geometry, and Algebra 2 and a course specific table at the beginning of each book which listed the standards addressed in each lesson. Overall, most of the non-plus standards are included in the materials; however, some aspects of the non-plus standards have not been completely addressed by the instructional materials. Additionally at least two standards are completely omitted.

  • G-CO.3: In Geometry, Lesson 4.2, students are asked about lines of symmetry but are not directly asked to describe the reflections that carry a specific polygon onto itself as called for in the standard. Lesson 4.3 does ask students to describe rotations that map a figure onto itself, page 195, problem 20, and to select angles of rotational symmetry for a given regular polygon, page 195, problems 21 - 24.
  • S-IC.4: In Algebra 2, Lesson 11.5, students are given a version of the Margin of Error formula and use it, but they do not develop the concept using simulations as required by the standard.
  • S-IC.5: In Algebra 2, Lesson 11.6, problems 3-4 and 7-9 do "use data from a randomized experiment to compare two treatments," yet no evidence was found requiring students to "use simulations to decide if differences between parameters are significant."
  • S-CP.5: In Geometry, Lesson 12.2, students are asked about a variety of everyday situations and whether they are independent (problems 3 - 10). However, there is no evidence of students connecting the concepts of conditional probability and independence.
  • N-Q.2 and N-Q.3: No evidence was found where students had to define their own quantities or determine the appropriate level of accuracy of quantities. These standards are indicated to be present throughout sections of Algebra 1 and Algebra 2 but are not noted as a primary focus in any lesson as stated on page xxxii in both the Algebra 1 and Algebra 2 Teaching Editions. Upon examining the identified sections in Algebra 1 and Algebra 2, no evidence of N-Q.2 or N-Q.3 was found to be incorporated into the lessons.
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The instructional materials reviewed for Big Ideas Traditional Series do not meet the expectations that the materials attend to the full intent of the modeling process when applied to the modeling standards. For this indicator, materials were examined for the extent to which the modeling process is incorporated. The series shows the intention to incorporate the modeling process within each chapter; however, the majority of the problems lack the incorporation of the full modeling process as described in the CCSSM. Overall, very few problems throughout the materials integrate the entire modeling process.

Many of the modeling tasks include heavy scaffolding, and the following are some examples of this:

  • Geometry, Chapter 11: Page 655 contains a performance task which asks students how much it would cost to reopen a water park if some of the structures have to be repainted, pools filled with water, and some flat surfaces have to be resurfaced. The prompt, however, takes away the modeling intent by directing students towards specific calculations and steps.
  • Algebra 1, Chapter 6: The performance task asks students, given a map, to find the best place to locate bicycle rental stations. The question has the potential to incorporate the modeling process, but by providing leading questions the “formulate” and “validate” aspects of the modeling process are lost.
  • Geometry: On page 469, problems 13 and 14 are denoted as Modeling with Mathematics problems, yet the right triangles needed to solve the problems are superimposed on the real life pictures and labeled, students are instructed to use the Pythagorean theorem in order to solve and students are directed to look back at a previous example for support in solving. Being provided the physical model as well as the computations needed, the students are unable to experience the modeling process.

The following are two of the few tasks that incorporate the full intent of the modeling process.

  • Algebra 1, Chapter 3: The performance task asks students to analyze and compare t-shirt ordering proposals from four different companies, then determine the best company to order from and create a proposal for the class officers as to why that company is the best deal.
  • Geometry, Chapter 12: The performance task asks students to put themselves in the shoes of a graphic designer and design a new dartboard. Students are placed into design teams, and each member of the team is given a different scenario for what the dartboard should look like and the probability of hitting certain colors. Students are asked to get creative and design various dartboards keeping probability in mind. The task does include a significant amount of questioning; however, the questions asked push student thinking and do not hinder the implementation of the full modeling process.
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The instructional materials reviewed for the Big Ideas Traditional Series meet the expectation that the materials, when used as designed, allow students to spend the majority of their time on the content from CCSSM widely applicable as prerequisites (WAPs) for a range of college majors, post-secondary programs, and careers.

  • The materials provided for teachers suggest a timeline and show an overall focus on WAPs.
    • In Algebra 1, the majority of the 160 days focus on the widely applicable prerequisites.
    • In Geometry, of the 160-162 days, there is not a majority of days that focus on the widely applicable prerequisites.
    • In Algebra 2, the majority of the 160 days focus on the widely applicable prerequisites.
    • Viewing the series as a whole, the majority of days focus on the widely applicable prerequisites.
  • In Algebra 2, students build upon their basic function concepts from grades 6-8 while exploring various functions: Linear Functions (Chapter 1), Quadratic Functions (Chapter 2), Polynomial Functions (Chapter 4), Radical Functions (Chapter 5), Exponential and Logarithmic Functions (Chapter 6), Rational Functions (Chapter 7), and Trigonometric Functions (Chapter 9).
  • Geometry lessons 11.1 and 11.2 review and extend measures of center and variation as well as box-and-whisker plots. Lesson 11.3 builds on these concepts from grades 6-8, addressing standard S-ID.2 in the WAPs.
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The instructional materials reviewed for Big Ideas Traditional Series partially meet the expectations that the materials, when used as designed, provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics. Overall, the lessons are presented in a way that will allow students to fully learn many of the standards. Throughout the series, students are not spending time on content previously learned but are constantly moving forward. Students, however, are frequently not given the opportunity to develop their own definitions, and where the standards expect students to prove or develop a concept, the materials often provide students the information.

The following standards were included in the materials, yet were not presented in a way that would allow students to fully learn that standard.

  • S-ID.4: In Algebra 2, Lesson 11.1 includes practice on using "the mean and standard deviation of a set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate." The materials make use of calculators and tables to estimate areas under the normal curve; however, they do not make use of spreadsheets as called for in the standard.
  • N-RN.3: In Algebra 1, Lesson 4.1, problems 99 and 100, students do a limited number of calculations with pre-selected numbers then use that information in problem 100 to answer if this is always, sometimes, or never true. This is the only place where the standard is located. With this limited amount of exposure to the standard, students may not reach the full depth of understanding.
  • G-MG.1: Students are frequently provided or told the exact geometric shape to use to solve a problem; the material does not offer students a chance to select for themselves the appropriate shape to model a situation. Throughout the series only one problem was located that asked students to “draw an object (or part of an object) that can be modeled” by a certain shape (problem 40 on page 237 in the Geometry materials).
  • G-GPE.2: In Algebra 2, Lesson 2.3 the derivation of a parabola is given to students on pages 68-69. Students are not given the opportunity to derive the equation of a parabola given a focus and directrix for themselves.
  • G-C.5: In Geometry, lesson 11.1, page 597, students are provided with the derivation of the fact that the length of the arc intercepted by an angle is proportional to the radius. Students are not given the opportunity to complete this derivation for themselves.
  • G-CO.1: Throughout the geometry materials, students are not provided the opportunity to develop their own definitions as required by the standard; instead, the definitions were given to them. The textbook does tell students that there are undefined terms in geometry (with the exception of distance around a circular arc) and point, line, and plane are mentioned to be “undefined terms;” however, the textbook does not build the definitions through this notion.
  • A-APR.1: Students add, subtract and multiply polynomials, but the “understanding” aspect of this standard is not developed. In Algebra 2, Lesson 4.2 the closed nature of polynomials is explicitly stated; however, students are not required to demonstrate understanding of the concept.
  • A-SSE.1a: Problem 49 of lesson 3.2 in Algebra 1 is the only problem that addresses this standard. The materials list various other lessons in both Algebra 1 and 2 as addressing this standard; however, evidence of meeting the full depth of this standard was not found.
  • A-REI.4a: In Algebra 1, page 515, students are given the derivation of the quadratic formula and then work with a partner to provide the justification for the given steps. Students are never given the opportunity to derive the formula as stated in the standard.
  • G-CO.10: The standard calls for students to “prove theorems about triangles;” however, many proofs are simply provided by the materials throughout the geometry materials. For example, proof that the measures of interior angles of a triangle add to 180 degrees is provided as an example on page 234, proof that the base angles of isosceles triangles are congruent is provided as an example on page 252, and proof that the medians of a triangle meet at a point is offered as a theorem on page 320 with the proof provided online.
  • G-C.1: Proof that all circles are similar is provided for the students on page 541 of the Geometry textbook.
  • S-CP.2: There were four problems—Geometry, lesson 12.2, problems 3-6 with an identical chapter in Algebra 2—throughout the series that fully addressed this standard Geometry.
  • S-CP.6: Problem 23 in lesson 12.2 of the Geometry textbook is the one problem which addresses the standard.

The following standards provide students thorough exposure through multiple experiences to fully learn each standard.

  • G-C0.4: Students are given several explorations throughout Chapter 4 of the Geometry materials through which they can develop the ideas of transformations before the mathematical definition is provided for them.
  • G-CO.12: Students are given opportunity to explore constructions using “a variety of tools” (i.e., compass, straightedge, paper folding, and dynamic software) as called for in the standard.
  • A-CED: The entire Creating Equations cluster is thoroughly developed throughout the algebra 1 and algebra 2 materials.

The materials offer additional resources to help all students fully learn each standard.

  • Each section begins with two or three exploration activities that offer students an opportunity to engage with the content before formal presentation of the terms, definitions, facts, theorems, or procedures. These explorations help students with content mastery by allowing them to "play" with the mathematics and familiarize themselves with the concepts in a seemingly informal way. Many of the explorations present content from different perspectives. For example, in Algebra 1, Lesson 1.4, students explore the concept of absolute value first by considering it as an equation and looking for values that make it true, then as a number line, and finally numerically using a spreadsheet.
  • The end of each section offers suggestions for students who may need extra help as well as for students who may need additional challenge problems. The Resources by Chapter book provides on grade-level, additional problems for struggling students and extension problems for students who need a challenge. The extension problems focus on moving the learner forward. For example, in Algebra 1, section 5.2, the lesson dealt with solving systems of equations algebraically. The extension asks students to consider how to solve a system of three equations.
  • "Laurie's Notes" in every chapter and lesson provide guidance to teachers in presenting lessons, which in turn could help all students better learn all aspects of standards. Furthermore, the beginning of each chapter includes "Scaffolding in the Classroom" notes in the margin. Each lesson also includes "Differentiated Instruction" and "English Language Learner" boxes providing strategies for teachers to use in order to reach all learners. "Assignment Guide and Homework Check" boxes included before student exercises break down what problems teachers could use for Basic/Average/Advanced assignments and homework checks. After each Chapter Test in the book, teachers are given ideas on what materials to use if students need help or if students got the info. Students are given opportunities to use paper and pencil, graphing calculators, and dynamic software.
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The Instructional materials reviewed for Big Ideas Traditional Series partially meet the expectation that all students engage in mathematics at a level of sophistication appropriate to high school. Overall, the materials offer real-life and relevant situations to high school students; however, problems often involve integers and avoid more complex solutions.

  • The materials present a majority of problems with integer values. In Algebra 1, Chapters 3 and 4 deal with linear functions, both graphing them and then writing equations. Students do not see non-integer y-intercepts until they are introduced to the linear regression feature of the calculator in Lesson 4.5. The majority of y-intercepts are integer values. Likewise, Chapter 5 (solving systems of equations) primarily contains solutions that consist of integer values. Algebra 2, Lesson 1.4 extends the ideas of solving a system of equations to three variables but still has a large majority of solutions that are integer values and rarely involves non-integer values. Similarly, the Geometry textbook remains focused on integer values much like that of the Algebra 1 and Algebra 2 textbooks.
  • The materials offer students problems frequently based on real-life and relevant situations.
    • Geometry, page 198, problem 19 connects the Tetris game to rigid transformations.
    • Geometry, page 608, problem 37 connects pizza with circle/sector area and reasoning.
    • Algebra 1, in the Chapter 6 Performance Task "The New Car," students weigh the many costs of a car when deciding which is the best buy.
    • In the Algebra 2, Chapter 2 Performance Task "Accident Reconstruction," students analyze car speed and braking distance.
  • Daily journal entries, either online or utilizing the Student Journal book, are age appropriate for high school students.
  • One worksheet per lesson provides problems for students working below course level as well as students working above level. These enrichment and extension worksheets are similar to that of the chapter.
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The instructional materials reviewed for Big Ideas Traditional Series partially meet the expectations that the materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the standards. Overall, Chapter Summaries are provided at the beginning of each chapter that indicate what content the students should already be familiar with from previous grades and courses. The Common Core Progression offered at the beginning of each chapter indicates skills learned in previous courses, however does not make connections to specific standards.

  • Notes for teachers are abundant in the margins and before each lesson; however, they focus mainly on identifying common errors, teacher’s actions, and ideas for next steps for students. Connections within a course and across a series are not made explicit for the teacher.
  • Each course is presented in a traditional progression. Chapters are presented as isolated concepts or skills with little to no connection between mathematical concepts made explicit.
    • For example, Algebra 1: Lessons 8.1-8.4 each focus on graphing various forms of quadratic functions (8.1 - "Graphing f(x) = ax^2", 8.2 - "Graphing f(x) = ax^2 + c", 8.3 - "Graphing f(x) = ax^2+bx+c", and 8.4 - "Graphing f(x) = a(x-h)^2 +k"), however not until Lesson 9.2 do students discuss solving a quadratic function by graphing. Lesson 9.2 makes no reference to concepts learned in Lessons 8.1 - 8.4.
  • “Common Core Progression” boxes located at the beginning of each chapter indicate general connections to previous courses; however, no standards are cited, simply skills and concepts.
    • For example, the "Common Core Progression" box at the beginning of Chapter 3 in Algebra 1 lists multiple skill focused statements not rooted in the standards in order to show connection among standards within and across courses.
      • "Identify linear functions, using graphs, tables, and equations."
      • "Use function notation to evaluate, interpret, and graph functions."
      • "Find the slope of a line and use it to write a linear equation in slope-intercept form."
      • "Solve real-life problems using function notation, linear equations, slopes, and y-intercepts."
      • "Translate, reflect, stretch, and shrink graphs of linear and absolute functions, and combine transformations of graphs of linear and absolute functions."
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The instructional materials reviewed for Big Ideas Traditional Series partially meet the expectations that the materials explicitly identify and build on knowledge from grades 6-8 to the high school standards. Chapter Summaries and the Common Core Progression boxes indicate skills and procedures students should already be familiar with from previous mathematics classes; however, they do not explicitly identify the middle grades standards which they are referencing. Occasionally throughout the materials, explicit connections to middle grade standards are made in the Teacher Edition, specifically in “Laurie’s Notes” and the “Maintaining Mathematical Proficiency” sections. In some lessons, topics that are middle school standards are presented as if they are high school standards being presented for the first time.

  • “Maintaining Mathematical Proficiency” sections are included throughout the materials and clearly state that these are below grade-level problems that have been included to help students retain previously learned skills necessary for further growth in mathematics. The Teacher Edition explicitly references these middle school standards.
  • Connections to middle grades are typically mentioned for teachers in notes for the “Maintaining Mathematical Proficiency” sections at the beginning of each chapter. Explicit middle school standards are not referenced in these sections, but rather basic skills such as “Finding x-intercepts” and “The Distance Formula” (Algebra 2, page T-45). These notes are not included for the students.
  • At the beginning of each chapter, the Teacher Edition does identify skills, not standards, from grades 6-8 that students should already be proficient at performing. Individual lessons, however, do not reference these connections.
  • Multiple lessons present middle grades standards as if for the first time without identifying the standards are middle school standards but rather labeling them as high school standards. For example:
    • Algebra 1, Lesson 1.1 is about solving one-step, one variable equations, which is 8.EE.7, but is labeled as A-CED.1, A-REI.1, and A-REI.3. There is no mention in Lesson 1.1 that this is a concept that was previously learned and will be built upon.
    • Algebra 1, Lesson 6.1 is a re-teach of the rules of exponents, 8.EE.1, but does not indicate this. The rules of exponents are extended to rational exponents in Algebra 2, Lesson 5.2.
    • Algebra 1, Chapter 5 presents systems of equations to students as though the material is new, not a review of standards introduced in Grade 8, 8.EE.8.
    • The Overview of Algebra 1, lesson 3.1 explicitly states “Students have prior knowledge of functions from Grade 8 (8.F.1 - 8.F.5). Their understanding may be limited to discrete functions.” However, the lesson then proceeds as though the students have no understanding of the material.

Middle grades standards are rarely explicitly noted throughout the materials. Specific reference to middle grade standards are only found in the Teacher Edition in "Laurie’s Notes" and the Maintaining Mathematical Proficiency sections.

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The plus standards, when included, are not clearly identified, and though some of the included plus standards do not unduly interfere with the course, there is no indication that they are optional or extensions.

  • Plus standards are not explicitly identified throughout the materials. The correlation charts of lessons to standards and standards to lessons found in the front matter of the Teacher Editions does not denote which standards are plus standards. Furthermore, the plus standards are not identified at the beginning of the lessons when they are present.
  • Plus standards are present in both the Geometry and Algebra 2 textbooks.
    • Algebra 2 materials include plus standards N-CN.8,9; A-APR.5,7; F-TF.9; S-CP.8,9; and S-MD.6,7 throughout Lessons 1.4, 4.2, 4.6, 7.3, 7.4, 9.8, 10.2, and 10.5.
    • Geometry materials include plus standards G-SRT.9-11; G-C.4; and G-GMD.2 throughout Lessons 9.7, 10.1, 11.5, and 11.8.
  • Materials do not identify plus standards as optional or additional extension opportunities, rather they are presented in a way as being an expectation for all learners.
    • Algebra 2, Lesson 4.6 is the Fundamental Theorem of Algebra and proposes to cover two plus standards (N-CN.8,9). While an appropriate extension into higher level mathematics, the textbook does not indicate that this should be considered optional or as an extension. It would not unduly interfere with the course for all students to complete it, but it is not necessary and could take time from other required standards.
  • Some plus standards are presented in a way that is distracting to the learning of the non-plus standards.
    • The Binomial Theorem is presented on page 574 of the Algebra 2 materials, and it is part of the lesson on probability. This standard is not necessary for the understanding of probability as called for in the non-plus standards and could detract from the focus of the lesson.
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The instructional materials reviewed for the Big Ideas Traditional series do not meet expectations for alignment to the CCSSM for high school. The materials do meet the expectations for allowing students to spend the majority of their time on the content from the CCSSM widely applicable as prerequisites, but they do not meet the expectations for attending to the full intent of the modeling process when applied to the modeling standards. The materials partially meet the expectations for the remainder of the indicators within Gateway 1, and since the materials did not meet the expectations for focus and coherence, evidence for rigor and the mathematical practices in Gateway 2 was not collected.

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Bridges In Mathematics (2015)

The Math Learning Center | Grades K-5 | 2015 Edition

Kindergarten

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The materials reviewed for Kindergarten meet the expectations for Gateway 1. These materials do not assess above-grade level content, and they spend the majority of the time on the major clusters of each grade level. Teachers using these materials as designed will use supporting clusters to enhance the major work of the grade. These materials are partially consistent with the mathematical progression in the standards, and students are offered extensive work with grade-level problems. Connections are made between clusters and domains where appropriate. Overall, the Kindergarten materials are focused and follow a coherent plan.

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The materials reviewed for Kindergarten meet the expectations for Gateway 2. The materials include each aspect of rigor: conceptual understanding, fluency and application. These three aspects are balanced within the lessons. The materials partially meet the expectations for the connections between the MP and the mathematical content. There are missed opportunities for attending to the full meaning of the MPs. More teacher guidance about how to support students in analyzing the arguments of others is needed.

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The instructional materials reviewed for Kindergarten meet the expectations for focus. Content from future grades was found to be introduced; however, above grade-level assessment items and their accompanying lessons could be modified or omitted without significantly impacting the underlying structure of the instructional materials. Although some Unit Assessment Checkpoints contain above grade-level content or content not specifically required by the standards, the Comprehensive Growth Assessment (CGA) and all of the Number Corner Quarterly Checkups are fully aligned to the Kindergarten CCSSM. The instructional materials spend the majority of the time on major clusters of the grade. This includes all standards in K.CC, K.OA, and K.NBT. Overall, the materials meet the expectations for focus.

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The instructional materials reviewed for Kindergarten meet the expectations for assessing grade-level content. Overall, the instructional materials can be modified without substantially affecting the integrity of the materials so that they do not assess content from future grades within the summative assessments provided. Summative assessments considered during the review for this indicator include unit post-assessments and Number Corner assessments that require mastery of a skill.

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The assessment materials reviewed for Kindergarten meet expectations focus within assessment. Content from future grades was found to be introduced; however, above grade-level assessment items, and their accompanying lessons, could be modified or omitted without significantly impacting the underlying structure of the instructional materials.

For this indicator, the Kindergarten Assessment Map found on pages 12 – 14 in the Assessment Overview section was used to identify “summative” assessments. The Assessment Map indicates when mastery of each standard is expected and where the mastery standard is assessed. Based on the Assessment Map, the following were considered to be the summative assessments and were reviewed for Indicator 1a:

  • Number Corner Checkups 1 – 4
  • the Comprehensive Growth Assessment
  • Select Unit Checkpoints
    • Unit 2 Module 1, Session 5, Count and Compare Checkpoint
    • Unit 3 M1, Session 4, Beat You to Ten Checkpoint
    • Unit 4 M3, Session 3, Counting and Writing Numbers Checkpoint
    • Unit 5 M1, Session 4, Sort and Count Checkpoint
    • Unit 5 M3, Session 4, 2-D Shapes and Their Attributes Checkpoint
    • Unit 6 M1, Session 4, Cylinder Tens and Ones Checkpoint
    • Unit 6 M2, Session 4, 3-D Shapes and Their Attributes Checkpoint

Assessments are student observation/interview or written in nature. The Comprehensive Growth Assessment (CGA) and all of the Number Corner Quarterly Checkups are fully aligned to the Kindergarten CCSSM. In the Number Corner Quarterly Checkups, several skills/concepts in the K.CC cluster are benchmarked and assessed throughout the year. For example, K.CC.1 (Count to 100 by 1s) is assessed to 20 on NCCU1, to 60 on NCCU3, and to 100 on NCCU4.

The Unit Assessment Checkpoints that contain above grade-level or content not specifically required by the standards are noted in the following list:

  • In the Unit 4 Module 3 Session 3 Counting and Writing Numbers Checkpoint, Prompt 4, students are asked to count backward from a number (4 – 9) until they reach zero. Counting backwards is not an explicit K CCSSM expectation; however, it makes mathematical sense to address it as a precursor to subtraction where counting backward is a necessary skill. This skill is identified in the assessment scoring guide as “Supports K.CC.”
  • In the Unit 5 Module 3 Session 4 Two-Dimensional Shapes & Their Attributes Checkpoint, students are expected to identify a rhombus and a trapezoid. Those shapes are not specifically identified in K.G.2; however, it makes sense to include them since they are shapes in the pattern block set students use throughout the unit.
  • In the Unit 6 Module 1 Session 4 Cylinder Tens & Ones Checkpoint, students are asked to create a cylinder using a strip of paper. Question 2 asks students to estimate how many unifix cubes they think it will hold and then to fill the cylinder with unifix cubes without counting them. After dumping out the cubes and arranging them into tens and ones, they are asked to count them (Question 5) and then compare the actual number of cubes to their estimate to determine if their estimate was more or less than the actual number of cubes (Question 6). Since teachers are instructed to provide each student with 40 cubes, it is reasonable to believe that the number of cubes students will be expected to count will exceed the limit of 20 designated in K.CC.5 and subsequently, students may be comparing numbers greater than 20. Additionally, the estimation of quantities is not a Kindergarten expectation. Adjusting the size of the cylinder to ensure the counting of smaller quantities and eliminating the estimation portion of the assessment would be an easy fix and would not affect the integrity of the unit.
  • In the Unit 6 Module 2 Session 4 3-Dimensional Shapes and Their Attributes Checkpoint, the majority of the student observations are aligned to the K Geometry Standards: K.G.1, K.G.2, K.G.3, and K.G.4. However, in the observational task in which students use polydrons to build 3-D shapes, the teacher is to document if a student has successfully built rectangular prisms, triangular prisms and pyramids. This expectation is more appropriately aligned to 1.G.2 and not to K.G.5 (model shapes in the world by building shapes from components, e.g., sticks and clay balls, and drawing shapes) because the 3-D shapes identified in the K Geometry standards are limited to cube, cone, sphere and cylinder. However, as long as the focus is on building shapes and not naming them, this would be acceptable as most of these shapes are introduced within the K-2 grade band.
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The instructional materials reviewed for Kindergarten meet the expectations for focus on the major clusters of each grade. Students and teachers using the materials as designated will devote the majority of class time to major clusters of the grade.

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The instructional materials reviewed for Kindergarten meet the expectations for focus by spending the majority of class time on the major clusters of the grade. All sessions (lessons), except summative and pre-assessment sessions, were counted as 60 minutes of time. Number Corner activities were counted and assigned 20 minutes of time. When sessions or Number Corner activities focused on supporting clusters and clearly supported major clusters of the grade, they were counted. Reviewers looked individually at each session and Number Corner in order to determine alignment with major clusters and supporting clusters. Optional Daily Practice pages and Home Connection pages were not considered for this indicator because they did not appear to be a required component of the sessions.

When looking at the modules (chapters) and instructional time, when considering both sessions and Number Corners together, approximately 90 percent of the time is spent on major work of the grade.

  • Units – 8 out of 8 units spend the majority of the unit on major clusters of the grade, which equals 100 percent. Each unit devotes most of the instructional time to major clusters of the grade.
  • Modules (chapters) – 28 out of 32 modules spend the majority of the time on major clusters of the grade, which equals approximately 88 percent. Units 2, 5, 7 and 8 had three Modules that focused on major work of the grade, and all other units had all four Modules focused on major work of the grade.
  • Bridges Sessions (lessons) – 143 out of 160 sessions focus on major clusters of the grade, which equals approximately 89 percent. Major work is not the focus of the following sessions:
    • Unit 1, Module 4, Sessions 1, 2, 3 and 4
    • Unit 2, Module 4, Sessions 1, 2, 3 and 4
    • Unit 4, Module 3, Session 2
    • Unit 5, Module 1, Session 1
    • Unit 5, Module 2, Session 5
    • Unit 5, Module 4, Sessions 2, 3 and 4
    • Unit 6, Module 2, Session 4
    • Unit 8, Module 4, Sessions 4 and 5
  • Bridges sessions require 60 minutes. A total of 143 sessions are focused on major work of the grade. Bridges sessions devote 8,580 minutes of 9,600 minutes to major work of the grade. A total of 155 days of Number Corner activities address major work of the grade. Number Corner activities are 20 minutes each adding another 3,100 minutes to this total. In all 11,680 of 13,000 minutes, approximately 90 percent, is devoted to major work of the grade.
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The instructional materials reviewed for Kindergarten meet the expectations for coherence. The materials use supporting content as a way to continue working with the major work of the grade. For example, students count shapes in categories and then compare the quantities. The materials include a full program of study that is viable content for a school year, including 160 days of lessons and assessment. All students are given extensive work on grade-level problems, even students who are struggling, and this work progresses mathematically. However, future grade-level content is not consistently identified. These instructional materials are visibly shaped by the cluster headings in the standards; for example, one session is called "Classify Objects Into Categories." Connections are made between domains and clusters within the grade level. For instance, materials make connections between counting and cardinality and measurement and data. Overall, the Kindergarten materials support coherence and are consistent with the progressions in the standards.

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The instructional materials reviewed for Kindergarten meet expectations that supporting content enhances focus and coherence by engaging students in the major work of the grade.

Supporting standard K.MD.1 is connected to K.CC and K.OA — major work of the grade — throughout the instructional materials. For example, in Unit 4, Module 3, Session 1, standard K.MD.1 supports major work of K.CC.6 by measuring and comparing lengths and then correlating the measurement into a number unit and comparing the quantities.

Supporting standard K.MD.2 is connected to the major work of K.CC.6 throughout the instructional materials. For example, in Unit 4, Module 3, Session 1 standard K.MD.2 supports major work of K.CC.6 by measuring and comparing lengths and then correlating the measurement into a number unit and comparing the quantities.

Supporting standard K.G.2 is connected to K.CC and K.OA, major work of the grade, throughout the instructional materials. For example, in Unit 5, Module 2, Sessions 1-3, after sorting, students are asked to count the number of shape cards in each category and then compare the quantities to find which group has the most or least, and combining amounts. This supports K.CC and cluster K.OA. Another example is found in Unit 6, Module 1, Session 2. In this session work in three-dimensional geometry is used as a vehicle to represent and solve addition situations. This supports K.OA. Also, in Unit 6, Module 2, Session 5, students use three-dimensional shapes as a vehicle to practice decomposing numbers and fluently adding within 5 (K.OA).

Supporting standard K.MD.3 is connected to K.CC and K.OA, major work of the grade, throughout the instructional materials. For example, in Unit 5, Module 2, Sessions 1-3, while sorting shape cards in various ways, students are asked to count the number in each category and then compare the quantities to find which group has the most or least, and in some sessions combining amounts. This supports standards in K.CC and K.OA. Also, in the March and May Calendar Collector, students examine data and connect it to clusters K.CC and K.OA through questions about counting, comparing and combining.

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The instructional materials reviewed for Kindergarten meet the expectations for this indicator by providing a viable level of content for one school year. Overall, the materials have expectations for teachers and students that are reasonable.

  • Materials provide for 160 days of instruction. Each Unit has 20 sessions = 20 days. There are 8 Units. (20x8=160)
  • The prescribed daily instruction includes both unit session instruction and a Number Corners session. (170 days). There are no additional days built in for re-teaching.
  • Assessments are incorporated into sessions and do not require an additional amount of time. Instead, they are embedded into module sessions one-on-one as a formative assessment.
  • The Number Corner Assessments/Checkups (a total of 10 assessments, 1 interview and 1 written, in each of the following months: September, October, January, March and May) would require additional time to conduct a 7-10 minute interview with each student.
  • A Comprehensive Growth Assessment is completed at the end of the year and will require additional number of days to administer.
  • There are no additional time/days built in for additional Support, Intervention or Enrichment in the pacing guide. The Publisher recommends re-teaching of strategies, facts and skills take place in small groups while the rest of the class is at Work Places (math stations) or doing some other independent task. There is a concern that if a particular session’s activities take up most of the 60 minutes allotted, there will be no time for the remediation and enrichment to take place.
  • Based on the Bridges Publisher Orientation Video and Guide provided to the reviewers, unit sessions are approximately 60 minutes of each instructional day.
    • Each unit session contains: Problems & Investigations (whole group), Work Places (math stations), Assessments (not found in each session), and Home Connections (homework assignments not found in each session).
  • Based on the introduction section in the Number Corners Teacher Guide, as well as the Bridges Publisher Orientation Video, Number Corners sessions are approximately 20 to 25 minutes of each instructional day.
  • Approximately 80-85 minutes is spent on the Bridges and Number Corner activities daily.
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The instructional materials reviewed for Kindergarten are partially consistent with the progressions in the standards. Although students are given extensive grade-level problems and connections to future work are made, future grade level content is not always clearly identified to the teacher or student.

At times, the session materials do not concentrate on the mathematics of the grade. Some of the sessions within each module focus on above grade-level concepts. Examples of this include addition and subtraction beyond 10, counting quantities beyond 20, the use of greater than and less than symbols, patterns, volume, and identifying and counting money amounts using coins. The inclusion of off-grade level concepts takes away from the number of sessions that could be spent more fully developing the work on the mathematics of the grade.

In some cases, the above-grade level content is identified as such by the publishers, and in other cases it is not. On the first page of every session, the Skills & Concepts are listed along with the standard to which it has been aligned by the publisher. In some cases, this alerts the user to the inclusion of off-grade level concepts. Examples include:

  • Unit 5, Module 1, Exploring Shapes Overview, page 1, the publisher describes the work in Module 1 as “extending the range of their counting and comparing skills” which somewhat signifies that students will be moving beyond the expected range of Kindergarten standards.
  • Unit 5, Module 1, Session 3 warm-ups include counting backward from 20 which is not a Kindergarten standard, but the publisher alerts teachers to this by aligning it to “supports K.CC.”
  • Unit 5, Module 4, Session 5- students make a quilt following an AB pattern which publisher identifies as “create and extend simple repetitive patterns with up to 3 elements. The publisher alerts teachers by aligning it to “supports K.OA.”
  • In the Unit 6 Introduction, page ii, the publishers state that “a mastery of the forward and backward counting sequences, one-to-one correspondence, and cardinality helps students correctly determine sums and differences as they begin to solve addition and subtraction tasks,” thereby explaining an inclusion of counting backwards throughout the unit even though it is not a Kindergarten expectation.

In other cases, the above-grade level concepts are not identified as such within the sessions in the "Skills and Concepts" listing or at the beginning of the Units in the "Skills Across the Grade Levels" sections. Examples of unidentified above-grade level content include:

  • Unit 3, Module 1, Session 2 and Session : Counting by 2’s is a skip counting strategy/skill that is not introduced until Grade 2 (2.NBT.2).
  • Unit 4, Module 3 Sessions 2 -5: These focus on above-grade level content using standard units of measurement. The lessons are worded with the language of the standard for measurement in first grade (1.MD.1).
  • Unit 4, Module 4, Session 2 and Session 5: Both sessions involve counting by 5’s, which is a skip counting strategy/skill that is not introduced until Grade 2 (2.NBT.2).
  • Unit 5, Module 1, Session 5: This focuses on addition to 20, number combinations, and comparing numbers. This goes beyond K.OA expectations of adding to 10 as it continues on to 20, asking questions such as “how many more to 20?”
  • Unit 6, Module 1, Sessions 3 and 4: The comparison of the cylinders as to which one holds more is a volume activity which is more appropriate for Grade 4.
  • Unit 6, Module 2: Activities involve three-dimensional shapes that go beyond the shape expectations as outlined in the K.G cluster. The activities in this module include drawing and identifying rectangular prisms, triangular prisms and pyramids.
  • Unit 6, Module 3, Session 4: Money is a Grade 2 standard, but is not specifically identified as such by the publisher. In this session, students are determining the total value of a collection of coins.
  • Unit, 8, Module 4, Sessions 4 and 5: Students use repetitive patterns while completing a double Irish chain frog quilt. This moves beyond the grade-level work.

Materials provide students opportunities to work with grade-level problems. The majority of differentiation/support provided is on grade-level. Extension activities are embedded within sessions and allow students to engage more deeply with grade-level work. Additional extension activities are also provided online.

) [10] => stdClass Object ( [code] => 1f [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Kindergarten meet the expectations for fostering coherence through connections at a single grade, where appropriate and when the standards require. The standards are referred to throughout the materials. Overall, materials include learning objectives that are visibly shaped by CCSSM cluster headings and include problems and activities that connect two or more clusters in a domain or two or more domains when these connections are natural and important.

Instructional materials shaped by cluster headings include the following examples:

  • Unit 5, Module 2, Session 3, "Sorting Shapes by Sides, and Corners," is shaped by K.G.B.
  • Unit 6, Module 3, "Exploring the Teen Numbers," is shaped by the K.NBT cluster heading.
  • Unit 7, Module 1, Session 1, “Compare Weights,” is shaped by the K.MD.A cluster heading.
  • Unit 8, Module 1, “Catching, Counting, and Comparing,” is shaped by the K.OA.A cluster heading.
  • The Unit 5, Module 1, Session 1 learning objectives include "Classify objects into categories," which is visibly shaped by K.MD.B.
  • The Unit 5, Module 3, Session 1 learning objectives include "Classify objects into given categories" and "Count the number of objects in each category," which are visibly shaped by K.MD.B cluster heading.

Units, modules, and sessions that connect two or more clusters in a domain or two or more domains include the following examples:

  • Unit 1 Module 1, Session 1: "One Shoe" connects cluster K.CC.A to K.CC.B as students are counting the number of shoes up to 10, saying the number in the standard form and pairing each shoe with only one number name.
  • Unit 1 Module 1, Session 2: "Two Shoes" connects clusters K.CC.A and K.CC.B to K.CC.C as students count the number of shoes to by ones, say the number in the standard form, pair each shoe with only one number name, and finally compare which group is greater than, less than, or equal.
  • Unit 1 Module 1, Session 3: "Five Shoes" connects clusters K.CC.A and K.CC.B to K.CC.C as students count the number of shoes, saying the numbers in standard order and pairing each shoe with only one number name, and identifying whether the number of shoes in one group is greater than, less than, or equal to the number of shoes in the other group.
  • Unit 1 Module 2, Session 1: "Shoes to Toes" connects cluster K.CC.B with K.OA.A as students are counting dots on five-frame cards, and then decomposing the sums in more than one way.
  • Unit 1 Module 2, Session 2: "Fabulous Fives" connects cluster K.CC.B with K.OA.A as students are counting dots on five-frame cards, and then decomposing the sums in more than one way with Unifix cubes and five-frame cards.
  • Unit 1 Module 2, Session 3: "Fives with Fingers" connects cluster K.CC.B with K.OA.A as students are counting dots on five-frame cards, and then decomposing the sums in more than one way with fingers.
  • Unit 1 Module 2, Session 4: "Numerals 1 to 5" connects K.CC.A and K.CC.B with K.OA.A and K.MD.3 as students shake two-colored beans, count each color, find the sum and record numeral on sheet.
  • Unit 1 Module 2, Session 5: "Filling Five-Frames" connects K.CC.B with K.OA.A as students are flashed five-frames they show the number of dots on one hand and the number of blank spaces on the other.
  • Unit 1 Module 3, Session 4: "Beat You to Five" connects K.CC.A and K.CC.B with K.OA.A as students play the game of spinning a number and working in teams to see if they can cover five cubes on their frames before the teacher's side is covered. Students determine which group is greater than, less than, or equal to the number on the other side of the five-frame card.
  • Unit 2 Module 1, Session 1: "Two Red, Three Blue" connects clusters K.CC.B to K.OA.A as students count, compare and answer how many more as they are looking at 5-frame cards.
  • Unit 2 Module 1, Session 2: "Funny Five-Frame Flash" connects clusters K.CC.B to K.OA.A as students compare irregular 5-frame cards with regular 5-frame cards.
  • Activities in Unit 5, Modules 1 and 2 connect K.G.4 with K.MD.3 when sorting pattern blocks, shapes and shape cards and then counting and recording the number of each.
  • Activities in Unit 5, Module 3 connect K.G to K.CC.6 and K.MD.3 when sorting and recording the number in each category and then compare the quantities to find which is greater.
  • Activities in Unit 6, Module 1 connect K.G to K.MD.3, K.CC.3, and K.CC.6 and K.CC.7 when sorting three-dimensional shapes into categories and counting and recording the number in each category and then comparing the numbers and quantities.
  • Activities in Unit 6 Module 2 connect K.G to K.CC and K.OA.3 by playing a game “Make it Five” and recording combinations of five shapes.
  • In Unit 3, Module 3, Session 3, students are counting objects (K.CC.2), comparing amounts (K.CC.6), describing the attributes of the objects (K.MD.1), and comparing more and less of the objects (K.MD.2).
  • In Unit 4, Module 3, Session1 students work on counting and cardinality domain and all three clusters while comparing measurable attributes from the measurement and data domain.
  • Activities in Unit 8, Module 1 connect K.CC.A to K.OA.A when counting objects and writing equations based on information from story problems.
) [11] => stdClass Object ( [code] => rigor-and-balance [type] => component [report] =>

The materials reviewed for Kindergarten meet the expectations for this criterion by providing a balance of all three aspects of rigor throughout the lessons. To build conceptual understanding, the instructional materials include concrete materials, visual models, and open-ended questions. In the instructional materials students have many opportunities to build fluency with adding and subtracting within five. Application problems occur throughout the materials. The three aspects are balanced within the instructional materials.

) [12] => stdClass Object ( [code] => 2a2d [type] => criterion ) [13] => stdClass Object ( [code] => 2a [type] => indicator [points] => 2 [rating] => meets [report] =>

The materials reviewed in Kindergarten for this indicator meet the expectations by attending to conceptual understanding within the instructional materials.

The instructional materials often develop a deeper understanding of clusters and standards by requiring students to use concrete materials and multiple visual models that correspond to the connections made between mathematical representations. The materials encourage students to communicate and support understanding through open-ended questions that require evidence to show their thinking and reasoning.

The following are examples of attention to conceptual understanding of K.CC.B:

  • Unit 1, Module 1, Session 2: addresses and supports the developing understanding of cardinality and the conceptual understanding of K.CC.4 and K.CC.6 by sorting shoes in two lines then counting to identify which group is larger. The investigation uses concrete visual and verbal cues; there is correspondence across the mathematical representations as students are using verbal descriptions, concrete (actual shoes lined up on two different lines), and written value of each line.
  • Unit 2, Module 1, Session 3: students reinforce their conceptual understanding of one-to-one correspondence (K.CC.4.A) as they are counting the number of boxes in the 10-frame and/or counting dots arranged on a ten-frame (K.CC.5). Students then use Unifix cubes to build a concrete representation of the 10-frame card, connecting the visual, verbal, and concrete representations.

The following are examples of attention to conceptual understanding of K.CC.6:

  • Unit 2, Module 1, Sessions 4 and 5: conceptual understanding of comparing numbers is developed with a ten-frame. Strategies for determining which number has more or less are shared through discussion. In Session 5, the students play the game independently as the teacher observes and documents how students determine value of greater and less than.
  • October Number Corner Calendar Collector: conceptual understanding is built with cubes and ten-frame representations. Discussion elicits evidence for which number is greater/less/equal by using multiple representations, including a simple array for comparison.

The following are examples of attention to conceptual understanding of K.OA.1:

  • Unit 3, Module 2, Session 2: students develop conceptual understanding of addition and subtraction by acting out situations, using Unifix cubes, giving verbal explanations, and reading equations.
  • Unit 6, Module 3, Session 3: students play the Work Place 6D Roll, Add & Compare game, roll 0-5 dice, build quantities to 10 with Unifix cubes, record the addition facts on a recording sheet, and then compare their total amount to their partners by snapping all their cubes together. Students are asked to justify their answer to who has more. Students connect the mathematical representations of dice, Unifix cubes/10-frames, written equations and Unifix trains to validate their comparison of who has more.
  • April Number Corner Calendar Collector: writing addition equations is represented through direct modeling of frogs and represented as unit squares. Conceptual understanding of the addition equation sequence can be determined in multiple ways (example: 2 +1 + 1 + 1 is the same as 2 + 3).

The following are examples of attention to conceptual understanding of K.OA.3:

  • Unit 1, Module 3, Session 1: students move from the 5-frame to the 10-frame in Terrific Tens. The 10-frame model helps develop students' understanding of part-part-whole relationship of 10 (K.OA.3). As students explore the 10-frame, they use their fingers to show the amount on various 10-frames.
  • Unit 8, Module 4, Session 1: students compose and decompose numbers less than or equal to 10 and explore how they might see equations in the ten-frame. Students record their way of seeing various quantities within 10: 5 = 4 + 1, 2 + 3, etc. Students are then asked to think about what subtraction equation they can write or the same 10-frames: 5 - 1 = 4, 5 - 3 = 2. Students are asked to "show where they see the equation on the 10-frame" and "who has a different equation?"

The following are examples of attention to conceptual understanding of K.NBT.1:

  • Unit 7 Module 2 Session 1 and Session 2: conceptual understanding of teen numbers is elicited from building numbers on a double 10-frame to see the unit of 10 as a whole with some more (10 and 3 is 13). Number line representations are also used to guide the counting sequence of more than 10.
  • Unit 8, Module 3, Session 1: students develop conceptual understanding using place value mats of ones/tens to build numbers in the 10-20 range in Place Value Build and Win. They build the quantity with cubes on the mat, compare the numbers, and write inequality statements using the greater than and less than symbols. Cubes are pre-grouped into trains of 10. As students build the numbers, they are asked to explain how they used their cubes to build the number. Emphasis is placed on the "10 and some more" concept.
  • February Number Corner Number Line: conceptual understanding of “ten and some more” is reinforced through multiple and concrete representations (double 10-frame and a manipulative number line). Connections are made between the concrete visual representation of the teen number and the written numeral representation.

The following are examples of attention to conceptual understanding of K.G:

  • Unit 2, Module 4, Session 3, Pattern Block Puzzles: students observe and explore pattern blocks, identify the shape using characteristics and correct mathematical name (K.G.2), describe the positions (above, below, beside) (K.G.1) and develop understanding that shapes are the same regardless of orientation or size. Students also compose simple shapes to form larger shapes (K.G.6), as they cover various shapes with smaller pattern blocks. There is correspondence across mathematical representations as student give verbal descriptions of a shape's characteristics, practice using the correct word for the shape, use concrete pattern blocks to compose a larger shape. Conceptual discussions with high level questions occur (students quietly observe the shapes of various pattern blocks and then are asked, "Can you tell me about these shapes?") Students pair-share ways to build designs, have the opportunity to build, and then are invited to share design and finally asked, "Can you show me a different way to cover the shapes?"
) [14] => stdClass Object ( [code] => 2b [type] => indicator [points] => 2 [rating] => meets [report] =>

The Kindergarten materials meet the expectations for procedural skill and fluency by giving attention throughout the year to individual standards which set an expectation of procedural skill and fluency.

  • Throughout the materials, computational fluency is elicited with both addition and subtraction equations. Students are able to use counters, 10-frames or drawings to assist them. Evidence is gathered to note if a student has moved to procedural fluency and no longer needs concrete materials to add and subtract within five or within ten by using a unit of five. The expectation within the last two units is that students will be able to decompose five into parts fluently without the support of concrete materials to show procedural understanding.
  • Students spend a significant amount of time and have a variety of opportunities to fluently add and subtract throughout number corners activities. K.OA.5 is addressed in two areas of Number Corner. Although the publisher does not list the K.OA.5 standard in any of their Computational Fluency workouts, instead most often listing K.OA.4 in relation to adding and subtracting within five, Computational Fluency workouts use finger patterns, 5-frames, and the number line to help students develop fluency with addition and subtraction facts to the number five. Calendar Collector workouts have students collecting various items to count throughout the month.
    • In the March Calendar procedural fluency is guided by using subitizing images to state how many more to make a unit of 10. This builds from the conceptual understanding within an organized structure to see the parts of ten fluently without having to count (perceptual subitizing).
    • In the May Computational Fluency workout fact fluency to 5 is investigated by using multiple representations (number cards, 10-frames). Routines focus on looking at decomposing 5 into 2 or even 3 addends to build number flexibility.
  • Fluency is developed throughout the sessions of the Kindergarten instructional materials.
    • In Unit 1, Module 2, Session 3 in “Fives with Fingers,” frames are flashed and students show number of dots with fingers of one hand and use their other hand to show how many empty boxes there are in the 5-frame and then add to find the total in all.
    • In the Unit 6, Module 4, Session 1 Work Place “Shake Those Beans,” students count how many red and how many white beans and how many in all to determine all combinations of five.
    • In Unit 7, Module 3, Session 5 in “Cubes in My Hand,” the teacher divides five cubes between her two hands. The teacher opens one hand to reveal cubes while keeping the other cubes hidden in other hand. Students determine how many cubes are hiding and then write the equations that represent the investigation.
    • In Unit 8, Module 1 students fluently subtract with minuends to 5 by using spinners and drawings to represent minuends and subtrahends.
) [15] => stdClass Object ( [code] => 2c [type] => indicator [points] => 2 [rating] => meets [report] =>

Materials meet the expectations for having engaging applications of mathematics as they are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade.

Materials include multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. The materials provide single step contextual problems that revolve around real world applications. Major work of the grade level is addressed within most of these contextual problems. The majority of the application problems are done with guiding questions elicited from the teacher through whole group discussions that build conceptual understanding and show multiple representations of strategies. Materials could be supplemented to allow students more independent practice for application and real world contextual problems that are not teacher guided within discussions. This would provide students opportunities to show more evidence of their mathematical reasoning through common addition and subtraction situations as outlined in the CCSSM Glossary, Table 1.

The instructional materials include problems and activities aligned to K.OA.2 that provide multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. Examples of these applications include the following:

  • In Unit 3, Module 3, Session 2, "Bicycle story problems," students are using their 10-frames to solve story problems given to them orally.
  • In Unit 6, Module 4, students engage in application of addition skills to solve story problems.
  • In Unit 7, Module 3, Sessions 1, 2 and 3, students solve frog addition/subtraction word problems using pictures. Students share out their strategies for solving. Students use Unifix cubes to model story problems and solve.
  • In Unit 8, Module 1, Sessions 1-4, students use manipulatives, pictures, and 10-frame counting mats to demonstrate application of addition and subtraction skills for solving story problems.
  • This module contains story problems set in the context of addition and subtraction. Student strategies are shared to elicit more sophisticated strategies over time within the unit. The unit also contains a checkpoint small group formative assessment to gather data to evaluate student strategies and misconceptions.
  • In the Number Corner February Computational Fluency, students began to add to 10 in the context of themed story problems and application within the number corner computational fluency routine. Thinking within these contextual situations is extended toward building conceptual understanding of subtraction as a missing addend problem.
) [16] => stdClass Object ( [code] => 2d [type] => indicator [points] => 2 [rating] => meets [report] =>

The materials reviewed in Kindergarten meet the expectations for providing a balance of rigor. The three aspects are not always combined nor are they always separate.

In the Kindergarten materials all there aspects of rigor are present in the instructional materials. All three aspects of rigor are used both in combination and individually throughout the Unit Sessions and in Number Corner activities. Application problems are seen to utilize procedural skills and require fluency of numbers. Conceptual understanding is enhanced through application of previously explored clusters. Procedural skills and fluency learned in early units are applied in later concepts to improve understanding and conceptual understanding.

) [17] => stdClass Object ( [code] => mathematical-practice-content-connections [type] => component [report] =>

The materials reviewed for Kindergarten partially meet this criterion. The MPs are often identified and often used to enrich mathematics content. There are, however, several sessions that are aligned to MPs with no alignment to Standards of Mathematical Content. The materials often attend to the full meaning of each practice. However, there are instances where the standards are superficially attended to. The materials reviewed for Kindergarten attend to the standards' emphasis on mathematical reasoning. Students are prompted to explain their thinking, listen to and verify the thinking of others, and justify their own reasoning. Although the materials often assist teachers in engaging students in constructing viable arguments, more guidance about how to guide students in analyzing the arguments of others is needed.

) [18] => stdClass Object ( [code] => 2e2g [type] => criterion ) [19] => stdClass Object ( [code] => 2e [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Kindergarten meet the expectations for identifying the MPs and using them to enrich the mathematical content. Although a few entire sessions are aligned to MPs without alignment to grade-level standards, the instructional materials do not over-identify or under-identify the MPs and the MPs are used within and throughout the grade.

The Kindergarten Assessment Guide provides teachers with a Math Practices Observation Chart to record notes about students' use of MPs during Sessions. The Chart is broken down into four categories: Habits of Mind, Reasoning and Explaining, Modeling and Using Tools, and Seeing Structure and Generalizing. The publishers also provide a detailed, "What Do the Math Practices Look Like in Kindergarten?" guide for teachers (AG, page 16).

Each Session clearly identifies the MPs used in the Skills & Concept section of the Session. Some Sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher that shows the connection between the indicated MP and the content standard. Examples of the MPs in the instructional materials include the following:

  • In Unit 1, Module 3 each of the six sessions list the same two Math Practices: MP6 and MP7. There is a "Math Practices In Action" reference in two of the six Sessions.
  • In Unit 2, Module 3 in the Skills & Concepts section, four sessions (1, 2, 3, 6) list MP6, four sessions (1, 2, 5, 6) list MP7, three sessions list MP8 (3, 4, 5) and one session lists MP3 (4).
  • In Unit 2, Module 3, Sessions 1, 4 and 6 reference the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
  • In Unit 4, Module 2 all five sessions list in the Skills & Concepts section two MPs: MP6 and MP7.
  • In Unit 7, Module 1, sessions 2 and 5 reference the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
  • In the September Number Corner MP2 is referenced in the Calendar Collector; MP4 is referenced in Days in School; MP7 is addressed in Calendar Grid, Computational Fluency, and Number Line; and MP8 is addressed in Calendar Grid, Computational Fluency, Number Line, Days in School and Calendar Collector.

Lessons are aligned to MPs with no alignment to Standards of Mathematical Content. These lessons occur at the beginning and the end of the year. These sessions that focus entirely on MPs include the following:

  • Unit 1, Module 4, Session 1
  • Unit 1, Module 4, Session 2
  • Unit 1, Module 4, Session 3
  • Unit 1, Module 4, Session 4
  • Unit 8, Module 4, Session 4
  • Unit 8, Module 4, Session 5
) [20] => stdClass Object ( [code] => 2f [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The materials partially meet the expectations for attending to the full meaning of each practice standard. Although the instructional materials often attend to the full meaning of each practice standard, there are instances where the MPs are only attended to superficially. There is limited discussion or practice standards within Sessions, Number Corner, and Assessments.

Each Session clearly identifies the MPs used in the Skills & Concept section of the Session. Typically there are two MPs listed for each session, so there is not an overabundance of identification. Some Sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. Although the MPs are listed at the session level, they are not discussed or listed in unit overviews or introductions (Major Skills/Concepts Addressed); however, they are listed in Section 3 of the Assessment Overview. With limited reference in these sections, overarching connections were not explicitly addressed.

In Number Corners, the MPs are listed in the Introduction in the Target Skills section with specific reference to which area of Number Corner in which the MP is addressed (Calendar Grid, Calendar Collector, Days in School, Computational Fluency, Number Line). The MP are also listed in the Assessment section of the Introduction as well. Although the MPs are listed in these sections, there is no further reference to or discussion of them within Number Corner.

At times, the instructional materials fully attend to a specific MP. The following are examples:

  • In Unit 1, Module 3, Session 2, the Skills & Concepts section lists MP6 and MP7. The session also references the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action." This section states that "(w)hen students pair the numerals and quantities and then arrange them in order, they are looking for and making use of structure..." Students were provided with several opportunities to "communicate precisely to others" their counting strategies as they are asked to explain how they counted their 10-frame cards, asked to explain other ways to count, and asked if there an easy way to count the dots? Students paired numerals with 10 frame cards and then arranged them in order; they are using the structure of the 10-frame cards to recognize patterns and describe the structure through repeated reasoning.
  • In Unit 1, Module 3, Session 4, the Skills & Concepts section lists MP6 and MP7. The session also references the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action." This sections states that "(w)hen you help young students keep track of their counting, you are helping them attend to precision..." Students are playing the game, "Beat you to Five." They are counting accurately attending to precision using strategies so that they include each object once without losing track. As students spin, they are using the Unifix cubes to show the number needed to reach five.
  • Unit 4, Module 2, Session 5 attends to MP7. In the "Beat You to Twenty" Work Place, grouping the cubes by 10 and having students count on from 10 helps them recognize the structure of our number system.

At times, the instructional materials only attend superficially to MPs. The following are examples:

  • Standard MP3 is addressed in Unit 2, Module 3, Session 4. Students play the game, "Which Bug Will Win?" by spinning a spinner with two different bugs. Students mark an "x" on the column according to which bug the spinner landed on. The first student to fill a column wins. Students are asked, "Who won?" and "Why?" This session does not attend to the full meaning of constructing mathematical arguments and/or critiquing the reasoning of others.
  • MP8 is addressed in Unit 2, Module 3, Session 3. As students play the game, "Which Bug Will Win?," they make predictions about the two different spinners (one has an equal amount of two different bugs and the other spinner has four of one bug and two of the other). Students then play the game multiple times using the two different spinners and then adjust their predictions based on their outcomes. Students are looking for regularities as they spin multiple times during the game. There is a missed opportunity to revisit students' predictions during the final discussion. Students could identify the differences in the spinners and then describe why one spinner results in a different outcome than the other spinner.
  • The materials partially attend to the meaning of MP4. The intent of this practice standard is to apply mathematics to contextual situations in which the math arises in everyday life. Often when MP4 is labeled within the instructional materials students are simply selecting a model to represent a situation. For example, in Unit 3, Module 1, Session 1, MP 4 is indicated, but students are simply representing a number on a ten frame. The Math Practices in Action note states that "Students will use drawings, numbers, expressions, and equations to model with mathematics."
) [21] => stdClass Object ( [code] => 2g [type] => indicator [report] => ) [22] => stdClass Object ( [code] => 2g.i [type] => indicator [points] => 2 [rating] => meets [report] =>

The materials reviewed for Kindergarten meet the expectations of this indicator by attending to the standards' emphasis on mathematical reasoning.

Students are asked to explain their thinking, listen to and verify other's thinking, and justify their reasoning. This is done in interviews, whole group teacher lead conversations, and in student pairs. For the most part, MP3 is addressed in classroom activities and not in Home Connection activities.

  • In Unit 2, Module 1, Session 2, within the Problems and Investigations portion, students are introduced to the "think-pair-share" routine. They are asked to listen and explain their partners' thinking.
  • In Unit 5, Module 4, Session 2 students are introduced to "There's a Shape in My Pocket." In this activity, students present arguments and critique the reasoning of their classmates to come to an agreement about which cards to remove.
  • In Unit 7, Module 4, Session 1 students engage in a "think-pair-share" routine. As in other Sessions in the instructional materials, this activity allows students to share their thoughts, listen to the thoughts of classmates, and justify their own reasoning.
  • In the March Calendar Grid and Number Line students share their thinking and justify their reasoning in developing their combinations of numbers to construct a ten.
) [23] => stdClass Object ( [code] => 2g.ii [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Kindergarten partially meet the expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. Although the instructional materials often assist teachers in engaging students in constructing viable arguments, there is minimal assistance to teachers in how to guide their students in analyzing the arguments of others.

There are Sessions containing the "Math Practice In Action" sidebars that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. A few of the sessions contain direction to the teacher for prompts and sample questions and problems to pose to students.

Many lessons give examples of teacher/student discourse by providing teachers a snapshot of what questions could be used to generate conjectures and possible student thinking samples. The following are examples of sample discourse:

  • Unit 4, Module 1, Session 2
  • Unit 7, Module 2, Session 5
  • Unit 8, Module 1, Session 5

Although teachers are provided guidance to help students construct arguments and students are provided many opportunities to share their arguments, more guidance is need to support teachers in guiding their students through the analysis of arguments once they are shared. For example, in Unit 5, Module 4, Session 2, students are asked to think-pair-share about their observations of a Shape Card pocket chart. Students are invited to report to the group what they heard their partner say. Students continue to engage as they turn and talk to their partner about what problem they are trying to solve and asking questions about the shape cards, and finally coming to an agreement about which cards should be removed. When students come to an agreement about which cards to remove, they are presenting arguments and critiquing the reasoning of their classmates, engaging in logical reasoning. Although this activity allows students to analyze the arguments of classmates, the teacher is not provided enough support to help students with this analysis.

) [24] => stdClass Object ( [code] => 2g.iii [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Kindergarten meet the expectations for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of mathematics that is present throughout the materials.

The instructional materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Students have opportunities to explain their thinking while using mathematical terminology, graphics, and symbols to justify their answers and arguments in small group, whole group teacher directed, and teacher one-to-one settings.

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. Examples of this include using geometry terminology such as rhombus, hexagon, and trapezoid and using operations and algebraic thinking terminology such as equation, difference, and ten-frame.

  • Many sessions include a list of mathematical vocabulary that will be utilized by students in the session.
  • The online Teacher Materials component of Bridges provides teachers with "Word Resources Cards" which are also included in the Number Corner Kit. The Word Resources Cards document includes directions to teachers regarding the use of the mathematics word cards. This includes research and suggestions on how to place the cards in the room. There is also a "Developing Understanding of Mathematics Terminology" included within this document which provides guidance on the following: providing time for students to solve problems and ask students to communicate verbally about how they solved, modeling how students can express their ideas using mathematically precise language, providing adequate explanation of words and symbols in context, and using graphic organizers to illustrate relationships among vocabulary words
  • At the beginning of each section of Number Corner, teachers are provided with "Vocabulary Lists" which lists the vocabulary words for each section.
  • In Unit 6, Module 1, Section 5, in the Problems & Investigation section, the teacher is reminded to use the vocabulary for three-dimensional shapes, such as edge, face, vertex, surface.
) [25] => stdClass Object ( [code] => alignment-to-common-core [type] => component [report] =>

The instructional materials reviewed for Kindergarten are aligned to the CCSSM. Most of the assessments are focused on grade-level standards, and the materials spend the majority of the time on the major work of the grade. The materials are also coherent. The materials follow the progression of the standards and connect the mathematics within the grade level although at times off-grade level content is not identified. There is also coherence within units of each grade. The Kindergarten materials include all three aspects of rigor, and there is a balance of the aspects of rigor. The MPs are used to enrich the learning, but the materials do not always attend to the full meaning of each MP and additional teacher assistance in engaging students in constructing viable arguments and analyzing the arguments of others is needed. Overall, the materials are aligned to the CCSSM.

[rating] => meets ) [26] => stdClass Object ( [code] => usability [type] => component [report] =>

The materials reviewed meet the expectations for usability. In reviews for use and design, the problems and exercises are developed sequentially and each activity has a mathematical purpose. Students are asked to produce a variety of assignments. Manipulatives and models are used to enhance learning, and the purpose of each is explained well. The visual design is not distracting or chaotic and supports learning. The materials support teachers in learning and understanding the standards. All materials include support for teachers in using questions to guide mathematical development. Teacher editions have many annotations and examples on how to present the content and an explanation of the math of each unit and the program as a whole.

A baseline assessment allows teachers to gather information on student's prior knowledge, and teachers are offered support in identifying and addressing common student errors and misconceptions. Materials include opportunities for ongoing review and practice. All assessments include information on standards alignment and scoring rubrics. There are no systems or suggestions for students to monitor their own progress. Activities provide ELL strategies, support strategies, challenge strategies, and grouping strategies to assist with differentiating instruction. A chart at the beginning of each unit indicates places in the instructional materials where suggestions for differentiating instruction can be found. Most activities allow opportunities for differentiation. The materials provide many grouping strategies and opportunities. Support and intervention materials are also available online.

All of the instructional materials available in print are also available online. Additionally, the Bridges website offers additional resources such as Whiteboard files, interactive tools, virtual manipulatives, and teacher blogs. Digital resources, however, do not provide additional, technology-based assessment opportunities, and the digital resources are not easily customized for individual learners. Overall, the materials meet the expectations for usability. ) [27] => stdClass Object ( [code] => 3a3e [type] => criterion [report] =>

Materials are well-designed, and lessons are intentionally sequenced. Typically students learn new mathematics in the Problems & Investigations portion of Sessions while they apply the mathematics and work towards mastery during the Work Station portion of Sessions and during Number Corner. Students produce a variety of types of answers including both verbal and written answers. Manipulatives such as 10-frames, craft sticks and tiles are used throughout the instructional materials as mathematical representations and to build conceptual understanding.

) [28] => stdClass Object ( [code] => 3a [type] => indicator [points] => 2 [rating] => meets [report] =>

The sessions within the units distinguish the problems and exercises clearly. In general, students are learning new mathematics in the Problems & Investigations portion of each session. Students are provided the opportunity to apply the math and work toward mastery during the Work Station portion of the session as well as in daily Number Corners.

For example, in Unit 2, Module 2 of Session 4, students are learning the new mathematics and in Session 5, students are applying that learning in the Work Station. In the Problem & Investigations, students are learning the new mathematics concept of identifying if the number of objects in one group is greater than, less than, or equal to the number of objects in another group. They initially observe the Count and Compare game board and discuss the meaning of the words "greater than," "less than," and "equal to." Students are shown and then demonstrate hand movements that represent greater than, less than, and equal to. Students and teacher each choose a 10-frame dot card and share strategies for determining the amount. Students then spin the greater than/less than spinner to determine who wins the two cards. In another Problems & Investigations, students get another opportunity to play the game Count and Compare Dots. The teacher observes students at play, checking for understanding of the greater than/less than concept as well as the directions of the game, and clarifies any questions. In the Work Place, students engage in the game Count and Compare Dots where they apply their understanding of identifying if the number of objects in one group is greater than, less than, or equal to the number of objects in another group (K.CC.6).

In the October Calendar Collector, students are given another opportunity to apply their understanding of identifying if the number of objects in one group is greater than, less than, or equal to the number of objects in another group. Students are observing the weekly Pattern Blocks Data Collection Graph. Students use the Word Resource Cards in the pocket chart; the cards show greater than, less than, most, least, and equal. Each card contains a base ten model that represents the word on the card. Students share their observations of the graph and are encouraged to use the mathematical terms on the Word Resource Cards.

) [29] => stdClass Object ( [code] => 3b [type] => indicator [points] => 2 [rating] => meets [report] =>

The assignments are intentionally sequenced, moving from introducing a skill to developing that skill and finally mastering the skill. After mastery, the skill is continuously reviewed, practiced and extended throughout the year.

The "Skills Across Grade Level" table is present at the beginning of each Unit. This table shows the major skills and concepts addressed in the Unit. The table also provides information about how these skills are addressed elsewhere in the Grade, including Number Corner, and in the grade that follows. Finally, the table indicates if the skill is introduced (I), developed (D), expected to be mastered (M), or reviewed, practiced or extended to higher levels (R/E).

For example, K.CC.6 is found in Units 1, 2, 3, 4, 5, 6, 7, 8 and in Number Corner in October, December, January, February, March, April, and May. In Unit 1 this standard is introduced. In Units 2, 3, 4 and 5 it is developed, and in Unit 6 the standard is mastered. The standard is again Reviewed/Practiced/Extended in Units 7 and 8. Another example is K.CC.4.B found in Units 1, 2, 3, 4, 6 and in all Number Corners. In Unit 1 this standard is introduced. It is developed in Units 2 and 3, and it is mastered in Unit 4. The standard is once again Reviewed/Practiced/Extended in Unit 6.

Concepts are developed and investigated in daily lessons and are reinforced through independent and guided activities in Work Places. Number Corner, which incorporates the same daily routines each month (not all on the same day) has a spiraling component that reinforces and builds on previous learning. Assignments, both in class and for homework, directly correlate to the lesson being investigated within the unit.

The sequence of the assignments is placed in an intentional manner. First, students complete tasks whole group in a teacher directed setting. Then students are given opportunities to share their strategies used in the tasks completed in the Problems & Investigations. The Work Place activities are done in small groups or partners to complete tasks that are based on the problems done as a whole group in the Problems & Investigations. The students then are given tasks that build on the session skills learned for the home connections. For example, in Unit 7, Module 2, Session 1 the focus in the Problems & Investigations is using double 10-frames to identify numbers between 10 and 20 with sight and equations. Then, students use a number line to determine how far from 20 the number is so they can determine the winner. Then, in the Work Place, students are given the same tasks of identifying numbers between 10 and 20 with sight and equations and determining, on a number line, how far from 20 the number line is with partners.

) [30] => stdClass Object ( [code] => 3c [type] => indicator [points] => 2 [rating] => meets [report] =>

There is variety in what students are asked to produce. Throughout the grade, students are asked to respond and produce in various manners. Often, working with concrete and moving to more abstract models as well as verbally explaining their strategies. Students are asked to produce written evidence using drawings, representations of tools or equations along with a verbal explanation to defend and make their thinking visible.

For example, in Unit 2, Module 2, Session 5 in the Problems & Investigation section of the lesson, students are working with three different models to show combinations of 5: 5-frames, finger patterns, and number racks. First, students are flashed 5-frame cards and asked to show the number of red dots with their fingers on one hand and the number of blue dots with their fingers on their other hand. Students are asked to determine the total number of dots. Various 5-frame cards are flashed as students are working to support their development of cardinality. Next, students transition to the number racks, moving their beads to represent the 5-frame cards and are guided to verbally explain the process: "I pushed 3 red beads and then I added 2 white beads. Now I have 5 beads in all." Students continue to represent the amounts on various five-frame cards and turn and talk to their partners to describe what they did using the sentence frame. The lesson is wrapped up with students using the think-pair-share routine to discuss the various ways they built combinations of five.

) [31] => stdClass Object ( [code] => 3d [type] => indicator [points] => 2 [rating] => meets [report] =>

Manipulatives are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written methods. Manipulatives are used and provided to represent mathematical representations and provide opportunities to build conceptual understanding. Some examples are the 10-frames, number lines, Unifix cubes, number racks, coins, craft sticks and tiles. When appropriate, they are connected to written representations.

For example, in Unit 8, Module 2, Session 4, students are working with Unifix cubes to measure various items around the room. After recording their estimates and actual measurements, they write actual measurements in expanded form. Also, in the Number Corner February Number Line, students are playing a game called Roll & Count On From Ten. They roll a die to determine how many hops forward the frog will make on the number line. They connect the number line to an equation to represent the frog's hops forward. Another example is Unit 4, Module 2, Session 4. Students take turns rolling the numbered 0-5 die and covering the indicated number of pictured cubes with Unifix cubes. Students work together to see if they can be the first to collect 20 Unifix cubes on their side of the game board.

Also, in Unit 7, Module 4, Session 1, students use double 10-frames and craft stick bundles to demonstrate counting by 10's and 1's and organizing the counting process. The 10-frame numbers are compared to the task of bundling the sticks into groups of 10's and 1's.

) [32] => stdClass Object ( [code] => 3e [type] => indicator [report] =>

The material is not distracting and does support the students in engaging thoughtfully with the mathematical concepts presented. The visual design of the materials is organized and enables students to make sense of the task at hand. The font, size of print, amount of written directions and language used on student pages is appropriate for kindergarten. The visual design is used to enhance the math problems and skills demonstrated on each page. The pictures match the concepts addressed such as having the characters that are in the story problems placed in picture format on the page as well. Some problems may even require students to use the pictures to solve the story problems.

For example, in the Number Corner March Calendar Grid, the grid is visually appealing, with easy to read/interpret diagrams of a sequence of numbers that represent the specific day of the month. There is also a corresponding 20-frame that represents the number as well. Kindergarteners would easily be able to see that the diagrams and 20-frame representations are increasing by one each day.

) [33] => stdClass Object ( [code] => 3f3l [type] => criterion [report] =>

The instructional materials support teacher learning and understanding of the standards. The instructional materials provide questions and discourse that support teachers in providing quality instruction. The teacher's edition is easy to use and consistently organized and annotated. The teacher's edition explains the mathematics in each unit as well as the role of the grade-level mathematics within the program as a whole. The instructional materials are all aligned to the standards, and the instructional approaches and philosophy of the program are clearly explained.

) [34] => stdClass Object ( [code] => 3f [type] => indicator [points] => 2 [rating] => meets [report] =>

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. Lessons provide teachers with guiding questions to elicit student understanding and opportunities for discourse to allow student thinking to be visible. Discussion questions provide a context for students to communicate generalizations, find patterns, and draw conclusions.

Each unit has a sessions page, which is the daily lesson plan. The materials have quality questions throughout most lessons. Most questions are open-ended and prompt students to higher level thinking.

In Unit 1, Module 1, Session 2, teachers are prompted to ask the following questions:

  • What did you notice is the same about these two shoes?
  • How are these two shoes alike?
  • How do you know there are two?

In Unit 1, Module 4, Session 4, students are working with patterns, and teachers are prompted to ask the following questions:

  • So, you're saying that these cards all show a pattern? How do you know?
  • Is that true for all the cards?
  • Can you show us what you mean with cubes?
  • What should come next? How do you know?

In Unit 7, Module 4, Session 2, students are working to count dots on a double 10-frame, and teachers are prompted to ask the following questions:

  • What do you think is the same and what is different about these cards?
  • What else do you notice?
  • What do you mean they look different?
  • Can you tell me a bit more? How do they look different?

In Number Corners, there are are sidebars labeled "Key Questions" throughout the sections. For example, in the Number Corner December Calendar Grid, the"Key Questions" sidebar includes the following examples:

  • Where do you think the teddy bear will be on the next marker? Why?
  • Where is the bear on the 3rd marker?
  • I see a teddy bear behind a box, which marker am I looking at?
  • Can you use the patterns we've discovered to predict what the marker for the day after tomorrow will look like?
) [35] => stdClass Object ( [code] => 3g [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials; however, additional teacher guidance for the use of embedded technology to support and enhance student learning is needed.

There is ample support within the Bridges material to assist teachers in presenting the materials. Teacher editions provide directions and sample scripts to guide conversations. Annotations in the margins offer connections to the mathematics practices and additional information to build teacher understanding of the mathematical relevance of the lesson.

Each of the eight units also have an Introductory section that describes the mathematical content of the unit and includes charts for teacher planning. Teachers are given an overview of mathematical background, instructional sequence, and the ways that the materials relate to what the students have already learned and what they will learn in the future units and grade levels. There is a Unit Planner, Skills Across the Grade Levels Chart, Assessment Chart, Differentiation Chart, Module Planner, Materials Preparation Chart. Each unit has a Sessions page, which is the Daily Lesson Plan.

The Sessions contain:

  • Sample Teacher/Student dialogue;
  • Math Practices In Action icons as a sidebar within the sessions - These sidebars provide information on what MP is connected to the activity;
  • A Literature Connection sidebar - These sidebars list suggested read-alouds that go with each session;
  • ELL/Challenge/Support notations where applicable throughout the sessions; and
  • A Vocabulary section within each session - This section contains vocabulary that is pertinent to the lesson and indicators showing which words have available vocabulary cards online.

Technology is referenced in the margin notes within lessons and suggests teachers go to the online resource. Although there are no embedded technology links within the lessons, there are technology resources available on the Bridges Online Resource page such as videos, whiteboard files, apps, blogs, and online resource links (virtual manipulatives, images, teacher tip articles, games, references). However, teacher guidance on how to incorporate these resource is lacking within the materials. It would be very beneficial if the technology links were embedded within each session, where applicable, instead of only in the online teacher resource. For instance, the teacher materials would be enhanced if a teacher could click on the embedded link, (if using the online teacher manual) and get to the Whiteboard flipchart and/or the virtual manipulatives.

) [36] => stdClass Object ( [code] => 3h [type] => indicator [points] => 2 [rating] => meets [report] =>

Materials contain adult-level explanations of the mathematics concepts contained in each unit. The introduction to each unit provides the mathematical background for the unit concepts, the relevance of the models and representations within the unit, and teaching tips. When applicable to the unit content, the introduction will describe the algebra connection within the unit.

At the beginning of each Unit, the teacher's edition contains a "Mathematical Background" section. This includes the mathematics concepts addressed in the unit. For example, Unit 1 states, "This unit addresses three major concepts... First, students must master the number word sequence, that is they must be able to say the number words in the correct order... Students must also understand one-to-one correspondence, the idea that when counting to find the total number of objects in a collection, they must count each object once and only once... Finally, students must have a full grasp of cardinality, that is, that the last number they say when counting a group of objects indicates the total number in the collection."

The Mathematical Background also includes sample models with diagrams and explanations, strategies, and algebra connections. There is also a Teaching Tips section following the Mathematical Background that gives explanations of routines within the sessions such as think-pair-share, craft sticks, and choral counting. There are also explanations and samples of the various models used within the unit such as frames, number racks, tallies/bundles/sticks, and number lines.

In the Implementation section of the Online Resources, there is a "Math Coach" tab that provides the Implementation Guide, Scope & Sequence, Unpacked Content, and CCSS Focus for Kindergarten Mathematics.

) [37] => stdClass Object ( [code] => 3i [type] => indicator [points] => 2 [rating] => meets [report] =>

Materials contain a teacher’s edition (in print or clearly distinguished/accessible as a teacher’s edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

In the Unit 1 binder there is a section called "Introducing Bridges in Mathematics." In this section there is an overview of the components in a day (Problems & Investigations, Work Places, Assessments, Number Corner). Then there is an explanation of the Mathematical Emphasis in the program. Content, Practices, and Models are explained with pictures, examples and explanations. There is a chart that breaks down the mathematical practices and the characteristics of children in that grade level for each of the math practices. There is an explanation of the skills across the grade levels chart, the assessments chart, and the differentiation chart to assist teachers with the use of these resources. The same explanations are available on the website. There are explanations in the Assessment Guide that goes into they Types of Assessments in Bridges sessions and Number Corner.

The CCSS Where to Focus Kindergarten Mathematics document is provided in the Implementation section of the Online Resources. This document lists the progression of the major work in grades K-8.

Each unit introduction outlines the standards within the unit. A “Skills Across the Grade Level” table provides information about the coherence of the mathematics standards that are addressed in other units in Kindergarten and in Grade 1. The "Skills Across the Grade Level" document at the beginning of each Unit is a table that shows the major skills and concepts addressed in the Unit and where that skill and concept is addressed in the curriculum in the previous grade as well as in the following grade.

) [38] => stdClass Object ( [code] => 3j [type] => indicator [report] =>

The materials provide a list of lessons in the teacher's edition cross-referencing the standards covered and providing an estimated instructional time for each lesson and unit. The "Scope and Sequence" chart lists all modules and units, the CCSSM standards covered in each unit, and a time frame for each unit. There is a separate "Scope and Sequence" chart for Number Corners.

) [39] => stdClass Object ( [code] => 3k [type] => indicator [report] =>

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

Home connection materials and games sometimes include a “Note to Families” to inform them of the mathematics being learned within the unit of study.

Additional Family Resources are found at the Bridges Educator's Site.

  • Grade K Family Welcome letter in English and Spanish- This letter introduces families to Bridges in Mathematics, welcomes them back to school, and contains a broad overview of the year's mathematical study.
  • Grade K Unit Overviews for Units 1-8, in English and Spanish.
) [40] => stdClass Object ( [code] => 3l [type] => indicator [report] =>

Materials contain explanations of the instructional approaches of the program. In the beginning of the Unit 1 binder, there is an overview of the philosophy of this curriculum and the components included in the curriculum. There is a correlation of the CCSSM and MPs as the core of the curriculum in the Mathematical Emphasis section. The assessment philosophy is given in the beginning of the assessment binder. The types of assessments and their purpose is laid out for teachers. For example, informal observation is explained as "one of the best but perhaps undervalued methods of assessing students...Teachers develop intuitive understandings of students through careful observation, but not the sort where they carry a clipboard and sticky notes. These understandings develop over a period of months and involve many layers of relaxed attention and interaction."

) [41] => stdClass Object ( [code] => 3m3q [type] => criterion [report] =>

The instructional materials offer teachers resources and tools to collect ongoing data about student progress. The September Number Corner Baseline Assessment allows teachers to gather information on student's prior knowledge, and the Comprehensive Growth Assessment can be used as a baseline, quarterly, and summative assessment. Checkpoint interviews and informal observation are included throughout the instructional materials. Throughout the materials, Support sections provide common misconceptions and strategies for addressing common errors and misconceptions. Opportunities to review and practice are provided in both the Sessions and Number Corner routines. Checkpoints, Check-ups, Comprehensive Growth Assessment, and Baseline Assessments clearly indicate the standards being assessed and include rubrics and scoring guidelines. There are, however, limited opportunities for students to monitor their own progress.

) [42] => stdClass Object ( [code] => 3m [type] => indicator [points] => 2 [rating] => meets [report] =>

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

The September Number Corner Baseline Assessment is designed to gauge incoming students' numeracy skills. Also, the Comprehensive Growth Assessment contains 22 interview items and 8 written items and addresses every Common Core standard for Kindergarten. This can be administered as a baseline assessment as well as an end of the year summative or quarterly to monitor students' progress. Each unit contains at least two interview checkpoints within small groups to gather data for progress monitoring within the unit.

Informal observation is used to gather information. Many of the sessions and Number Corner workouts open with a question prompt: a chart, visual display, a problem, or even a new game board. Students are asked to share comments and observations, first in pairs and then as a whole class. This gives the teacher an opportunity to check for prior knowledge, address misconceptions, as well as review and practice with teacher feedback. There are daily opportunities for observation of students during whole group and small group work as well as independent work as they work in Work Places.

) [43] => stdClass Object ( [code] => 3n [type] => indicator [points] => 2 [rating] => meets [report] =>

Materials provide strategies for teachers to identify and address common student errors and misconceptions.

Most Sessions have a Support section and ELL section that suggests common misconceptions and strategies for re-mediating these misconceptions that students may have with the skill being taught.

Materials provide sample dialogues to identify and address misconceptions. For example, the Unit 2 Module 2 Session 5 “Support” section gives suggestions for students struggling with one-to-one correspondence and cardinality. Each unit assessment also lists reteaching suggestions for students who did not master the learning targets for the unit.

) [44] => stdClass Object ( [code] => 3o [type] => indicator [points] => 2 [rating] => meets [report] =>

Materials provide opportunities for ongoing review and practice, with feedback for students in learning both concepts and skills.

The scope and sequence document identifies the CCSS that will be addressed in the Sessions and in the Number Corner activities. Sessions build toward practicing the concepts and skills within independent Work Places. Opportunities to review and practice are provided throughout the materials. Ongoing review and practice is often provided through Number Corner routines. Each routine builds upon the previous month’s skills and concepts. For example, K.CC.2 is reviewed and practiced in Bridges Units 4, 6 and 8, and this standard is reviewed and practiced in all Number Corner months.

) [45] => stdClass Object ( [code] => 3p [type] => indicator [report] => ) [46] => stdClass Object ( [code] => 3p.i [type] => indicator [points] => 2 [rating] => meets [report] =>

All assessments, both formative and summative, clearly outline the standards that are being assessed. In the assessment guide binder, the assessment map denotes the standards that are emphasized in each assessment throughout the year. Each assessment chart notes which CCSS is addressed.

For example, in Unit 1, Module 2, Session 5, the “Elements of Early Number Sense Checkpoint” includes four prompts targeting standard K.CC.4.B. There is a Checkpoint Scoring Guide that lists each prompt and each standard. Another example is Number Corner Checkup 2; the Interview Response Sheet has a CCSS Correlation for each of the questions at the top of the Response Sheet as well as a Number Corner Checkup 2 Scoring Guide. Also, each item on the Comprehensive Growth Assessment lists the standard being emphasized listed on the Skills & Concepts Addressed sheet as well as on the Interview Materials List and the Interview and Written Scoring Guides.

) [47] => stdClass Object ( [code] => 3p.ii [type] => indicator [points] => 2 [rating] => meets [report] =>

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting students' performance and suggestions for follow-up.

All Checkpoints, Check-ups, Comprehensive Growth Assessment, and Baseline Assessments are accompanied by a detailed rubric and scoring guideline that provide sufficient guidance to teachers for interpreting student performance. There is a percentage breakdown to indicate Meeting, Approaching, Strategic, and Intensive scores. Section 5 of the Assessments Guide is titled "Using the Results of Assessments to Inform Differentiation and Intervention.” This section provides detailed information on how Bridges supports RTI through teachers' continual use of assessments throughout the school year to guide their decisions about the level of intervention required to ensure success for each student. There are cut scores and designations assigned to each range to help teachers identify students in need of Tier 2 and Tier 3 instruction. There is also a breakdown of Tier 1, 2 and 3 instruction suggestions.

) [48] => stdClass Object ( [code] => 3q [type] => indicator [report] =>

There is limited evidence in the instructional materials that students are self-monitoring their own progress.

Section 4 of the Assessment Guide is titled, "Assessment as a Learning Opportunity." This section provides information to teachers guiding them in: setting learning targets, communicating learning targets, encouraging student reflection, exit cards and comparing work samples from earlier and later in the school year.

) [49] => stdClass Object ( [code] => 3r3y [type] => criterion [report] =>

Session and Number Corner activities provide ELL strategies, support strategies, challenge strategies, and grouping strategies to assist with differentiating instruction. A chart at the beginning of each unit indicates places in the instructional materials where suggestions for differentiating instruction can be found. Most activities allow opportunities for differentiation. The Bridges and Number Corner materials provide many grouping strategies and opportunities. Support and intervention materials are also available online.

) [50] => stdClass Object ( [code] => 3r [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

Units and modules are sequenced to support student understanding. Sessions build conceptual understanding with multiple representations that are connected. Procedural skills and fluency are grounded in reasoning that was introduced conceptually, when appropriate. An overview of each unit defines the progression of the four modules within each unit and how they are scaffolded and connected to a big idea. For example, in Unit 2 “Numbers to Ten” (K.CC) Module 1 compares five with ten frames, Module 2 compares five and ten units using the number rack (rekenrek), and Module 3 compares numbers within 10 using multiple visual models (10-frames, number rack, tally cards, craft sticks).

In the Sessions and Number Corner activities there are ELL strategies, support strategies, and challenge strategies to assist with scaffolding lessons and making content accessible to all learners.

The Assessment & Differentiation portion of Unit 1, Session 2, Module 4 in the “Spill 5 Beans” Work Place Guide provides suggestions for teachers on how to scaffold the Work Place. Guidance includes “(i)f you see that...(a) student is struggling with one-to-one correspondence then... support the student by pairing the student with someone with solid one-to-one correspondence. Together they can pull off and count beans as they organize them on 5-frame counting mat.”

In the Unit 7, Module 3, Session 1 Problem & Investigation, students are solving word problems by counting the number of eyes on the frogs. The following is "Support" and "ELL" suggestions are provided:

  • "Support" - Some students may be completely stumped and not know how to start. Have them look at the picture and ask again, "How many frogs could there be?" Continue with, "Can you think of something to use to help you?"
  • "ELL"- As you discuss and read the problem, be sure to point to the parts (eyes, log, pond) as you say them, circle all the eyes as you say, "8 eyes," point to the eyes as they're counted.

In the January Number Corners Number Line, as students are working on the number line to determine which number is greater and less than another number, the following "Support" suggestion is provided:

  • If students are having a difficult time telling which number is greater than the other using numeral cards, show your class two small groups of cubes or other small objects, count the items into 10-frames, and ask which group has the greater number - reminding them that the word "greater" in mathematics means "more."
) [51] => stdClass Object ( [code] => 3s [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials provide teachers with strategies for meeting the needs of a range of learners.

A chart at the beginning of each unit indicates which sessions contain explicit suggestions for differentiating instruction to support or challenge students. Suggestions to make instruction accessible to ELL students is also included in the chart. The same information is included within each session as it occurs within the teacher guided part of the lesson. Each Work Place Guide offers suggestions for differentiating the game or activity. The majority of activities are open-ended to allow opportunities for differentiation. Support and intervention materials are provided online and include practice pages, small-group activities and partner games.

In Unit 2, Module 2, Session 1, as students are working with two-color 10-frames, the teacher is provided with ELL, Support, and Challenge strategies to meet the needs of a range of learners.

  • ELL - "When asking students about the top row and bottom row be sure to point to that row on the card, and when asking about how many there are in all, sweep your hand in a circular motion to indicate what you mean."
  • Support - "Seat students who are not yet solid with one-to-one correspondence and numeral counting sequences to ten, close to or right in front of you. Once you have flashed the 10-frame to the rest of the students, continue to show the 10-frame to these students to view while they build what they see. Show the 10-frame again and give students the opportunity to check their work as you leave the card displayed."
  • Challenge - "Provide students who are already facile with subitizing and building quantities to ten with a student whiteboard and dry erase marker and ask them to record an equation that describes the 10-frame.”

In Unit 6, Module 3, Session 2, as students are working on guessing and writing the mystery numbers, the teacher is provided with Support and Challenge strategies to meet the needs of a range of learners.

  • Support - "If students have difficulty writing the numerals, say aloud what you are doing as you write them. For example, when writing 17, say, "For the 1, I'll start at the top and make a straight line down. For the 7, I'll start at the top and make a short line straight across and then make another slanted line down to the bottom."
  • Challenge - "Ask students to explain the "10 and some more" property of each number (14 means there are 10 and 4 more).
) [52] => stdClass Object ( [code] => 3t [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations. Tasks are typically open ended and allow for multiple entry-points in which students are representing their thinking with various strategies and representations (concrete tools as well as equations).

In the Problems and Investigations section, students are often given the opportunities to share strategies they used in solving problems that were presented by the teacher. Students are given multiple strategies for solving problems throughout a module. They are then given opportunities to use the strategies they are successful with to solve problems in Work Places, Number Corner and homework.

For example, in Unit 1, Module 3, Session 2, students are using the 10-frame, counting how many dots they see, and discussing various ways they counted. As students share their strategies, the following sample dialogue is provided: T - "How did you count the dots?" S - "I knew that there were 5 on the top and then I said 6, 7, 8." T - "Does someone have a different way that they counted?" S - "I just counted them all 1, 2, 3, 4, 5, 6, 7, 8." S - "There's 10 boxes and 2 are empty, so that makes 8 with dots.”

Another example is found in Unit 2, Module 2, Session 5, as students are working with the number rack, 5-frames, fingers, and equations to build combinations of 5. Students practice building these combinations and, then, share out the various ways they can build combinations of 5 using the various models.

In Unit 6, Module 4, Session 1, students are identifying the attributes of the group of students selected by the teacher. Students can enter the discussion with a wide variety of attributes they noticed and then provide various strategies to represent the attributes. For example, they can draw a picture of the short-sleeve shirts and the long-sleeve shirts, write 3 short-sleeve and 2 long-sleeve, write 3 + 2, write 2 + 3, or write 2 + 3 = 5. Students are able to respond with sketches, words and numbers, just numbers, expressions and equations.

) [53] => stdClass Object ( [code] => 3u [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials suggest supports, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

Online materials support students whose primary language is Spanish. The student book, home connections and component masters are all available online in Spanish. Materials have built in support in some of the lessons in which suggestions are given to make the content accessible to ELL students of any language.

There are ELL, Support, and Challenge accommodations throughout the Sessions and Number Corner activities to assist teachers with scaffolding instructions. Examples of these supports, accommodations and modifications include the following:

  • In Unit 6, Module 4, Session 3, students are introduced to a new game called "Fill It Up Five +." Students are working on filling a 10-frame with 5 and some more. The ELL support provided for this session suggests "When using the terms, ‘top row’ and ‘bottom row’ be sure to point to that row on the display card, and run your hand in a circular motion around the card when you say ‘in all.’”
  • For ELL support, in Unit 7, Module 2, Session 2, the materials suggest stressing the "-teen" ending of the numbers to differentiate from numbers ending in "-ty."
  • For ELL and Support, Unit 7, Module 4, Session 2 suggests that teachers “(r)emind students what less means by demonstrating with a large pile of cubes and a small pile of cubes.”
  • The Number Corner December Computational Fluency "Five and More" page in Activity 3 suggests “asking students to work together at the same pace while you read each prompt aloud. Students who are able to work ahead may do so, but providing this kind of scaffolding may help the students who are still learning to read and write numbers.”
) [54] => stdClass Object ( [code] => 3v [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials provide opportunities for advanced students to investigate mathematics content at greater depth. The Sessions, Work Places, and Number Corners include "Challenge" activities for students who are ready to engage deeper in the content.

Challenge activities found throughout the instructional materials include the following:

  • In Unit 2, Module 2, Session 3, the challenge part of this session encourages students to write equations related to the number rack investigation of counting two sets of beads.
  • In Unit 4, Module 2, Session 5, as students are working in the Work Place, "Beat You to Twenty." the Work Place Guide offers the following differentiation to challenge students: “Have students record an equation to describe their turn. Invite students to play Game Variation A or B.”
  • In Unit 5, Module 4, Session 4, students are working on sorting 2-D shapes using Shape Sorting Cards. Students use characteristics from the cards such as "curved sides"and "three corners to eliminate all shapes that do not have the card's characteristic. The "Challenge" suggestion is to have students explain why there are squares in the "blue group" instead of in the "square group.”
  • In the May Number Line Number Corners, students are playing "Cross out Fifty," a game requiring naming and crossing out all of the numbers to 50 on the One Hundred Grid. There are three "Challenge" suggestions: 1) Each time a team rolls, before the numbers are crossed out, ask students to figure out the last number that will be crossed out on that turn and explain their thinking. 2) Ask students to figure out how many more squares need to be crossed out to reach 50. How do they know? Can they prove it? 3) With input from the class, write an inequality statement about the two color amounts or write an addition equation about the two color amounts and the total number of squares.
) [55] => stdClass Object ( [code] => 3w [type] => indicator [points] => 2 [rating] => meets [report] =>

The materials provide a balanced portrayal of demographic and personal characteristics. Most of the contexts of problem solving involve objects and animals, such as frogs and penguins. When students are shown performing tasks, there are cartoons that appear to show a balance of demographic and personal characteristics.

) [56] => stdClass Object ( [code] => 3x [type] => indicator [report] =>

The instructional materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials offer flexible grouping and pairing options. Throughout the Units, Work Places, and Number Corners there are opportunities to group students in various ways such as whole group on the carpet, partners during pair-share, and small groups during Problem & Investigations and Work Places.

In Unit 1, Mocule 2, Session 4, students are playing the game "Spill Five Beans" with a partner. In Unit 4, Module 2, Session 5, students begin the session sitting whole group in a discussion circle. In this Session, they move into Work Places where they get to choose what Work Place they would like to participate in. Grouping can be individual, pairs, or small groups, depending on the Work Place chosen.

) [57] => stdClass Object ( [code] => 3y [type] => indicator [report] =>

There is limited evidence of the instructional materials encouraging teachers to draw upon home language and culture to facilitate learning. The materials provide parent welcome letters and unit overview letters that are available in English and Spanish.

) [58] => stdClass Object ( [code] => 3z3ad [type] => criterion [report] =>

All of the instructional materials available in print are also available online. Additionally, the Bridges website offers additional resources such as Whiteboard files, interactive tools, virtual manipulatives, and teacher blogs. Digital resources, however, do not provide additional, technology-based assessment opportunities, and the digital resources are not easily customized for individual learners.

) [59] => stdClass Object ( [code] => 3z [type] => indicator [report] =>

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the MPs.

Each session within a module offers online resources that are in alignment with the session learning goals. Online materials offer an interactive whiteboard file as a tool for group discussion to facilitate discourse in the MPs. Resources online also include virtual manipulatives and games to reinforce skills that can be used at school and home. In the Bridges Online Resources there are links to the following:

  • Virtual Manipulatives - a link to virtual manipulatives such as number lines, geoboard, number pieces, number racks, number frames, and math vocabulary
  • Interactive Whiteboard Files - Whiteboard files that go with each Bridges Session and Number Corner
  • Online Games- online games such as 100 Hunt using the hundreds grid, 2-D Shape Pictures, Interactive math dictionary, Addition With Manipulatives, and Balloon Pop Comparisons (greater than/less than)
  • Images - for example, 1,000 M&M candies arranged on hundred grids by students

Within the Teacher's Edition, there is no direct reference to online resources. If embedded within the Teacher's Edition, the resources would be more explicit and readily available to the teacher.

) [60] => stdClass Object ( [code] => 3aa [type] => indicator [report] =>

The digital materials are web-based and compatible with multiple internet browsers. They appear to be platform neutral and can be accessed on tablets and mobile devices.

All grade level Teacher Editions are available online at bridges.mathlearningcenter.org. Within the Resources link (bridges.mathlearningcenter.org/resources), there is a sidebar that links teachers to the MLC, Math Learning Center Virtual Manipulatives. These include games, Geoboards, Number Line, Number Pieces, Number Rack, Number Frames and Math Vocabulary. The resources are all free and available in platform neutral formats: Apple iOS, Microsoft and Apps from Apple App Store, Window Store, and Chrome Store. The apps can be used on iPhones and iPads. The Interactive Whiteboard files come in two different formats: SMART Notebook Files and IWB-Common Format. From the Resource page there are also many links to external sites such as ABCYA, Sheppard Software, Illuminations, Topmarks, and Youtube.

) [61] => stdClass Object ( [code] => 3ab [type] => indicator [report] =>

The instructional materials do not include opportunities to assess student mathematical understanding and knowledge of procedural skills using technology.

) [62] => stdClass Object ( [code] => 3ac [type] => indicator [report] =>

The instructional materials are not easily customizable for individual learners or users. Suggestions and methods of customization are not provided.

) [63] => stdClass Object ( [code] => 3ad [type] => indicator [report] =>

The instructional materials provide opportunities for teachers to collaborate with other teachers and with students, but opportunities for students to collaborate with each other are not provided. For example, a Bridges Blog offers teacher resources and tools to develop and facilitate classroom implementation.


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First Grade

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    [title] => Bridges In Mathematics (2015)
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    [grade] => First Grade
    [type] => math-k-8
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            [rating] => meets
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            [score] => 17
            [rating] => meets
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            [score] => 37
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    [title] => Bridges in Mathematics
    [report_date] => 2016-04-29
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    [reviewed_date] => 2016-05-05
    [gateway_1_points] => 13
    [gateway_1_rating] => meets
    [gateway_1_report] => 

The materials reviewed for Grade 1 meet the expectations for Gateway 1. These materials do not assess above-grade level content, and they spend the majority of the time on the major clusters of each grade level. Teachers using these materials as designed will use supporting clusters to enhance the major work of the grade. These materials are partially consistent with the mathematical progression in the standards, and students are offered extensive work with grade-level problems. Connections are made between clusters and domains where appropriate. Overall, the Grade 1 materials are focused and follow a coherent plan.

[gateway_2_points] => 17 [gateway_2_rating] => meets [gateway_2_report] =>

The materials reviewed for Grade 1 meet the expectations for Gateway 2. The materials include each aspect of rigor: conceptual understanding, fluency and application. These three aspects are balanced within the lessons. The materials meet the expectations for the connections between the MPs and the mathematical content. More teacher guidance about how to support students in analyzing the arguments of others is needed.

[gateway_3_points] => 37 [gateway_3_rating] => meets [report_type] => math-k-8 [series_id] => 33 [report_url] => https://www.edreports.org/math/bridges-in-mathematics/first-grade.html [gateway_2_no_review_copy] => Materials were not reviewed for Gateway Two because materials did not meet or partially meet expectations for Gateway One [gateway_3_no_review_copy] => This material was not reviewed for Gateway Three because it did not meet expectations for Gateways One and Two [meta_title] => [meta_description] => [meta_image] => [data] => Array ( [0] => stdClass Object ( [code] => focus [type] => component [report] =>

The instructional materials reviewed for Grade 1 meet the expectations for focus. Content from future grades was found to be introduced; however, above grade-level assessment items, and their accompanying lessons, could be modified or omitted without significantly impacting the underlying structure of the instructional materials. Although the Comprehensive Growth Assessment (CGA) and some Unit Assessment Checkpoints contain above grade-level content or content not specifically required by the standards, all of the Number Corner Quarterly Checkups are fully aligned to the Grade 1 CCSS. The instructional materials spend the majority of the time on major clusters of the grade. This includes all standards in 1.OA and 1.NBT and cluster 1.MD.A. Overall, the materials meet the expectations for focus.

) [1] => stdClass Object ( [code] => 1a [type] => criterion [report] =>

The instructional materials reviewed for Grade 1 meet the expectations for assessing grade-level content. Overall, the instructional materials can be modified without substantially affecting the integrity of the materials so that they do not assess content from future grades within the summative assessments provided. Summative assessments considered during the review for this indicator include unit post-assessments and Number Corner assessments that require mastery of a skill.

) [2] => stdClass Object ( [code] => 1a [type] => indicator [points] => 2 [rating] => meets [report] =>

The assessment materials reviewed for Grade 1 meet expectations for focus within assessment. Content from future grades was found to be introduced; however, above grade-level assessment items, and their accompanying lessons, could be modified or omitted without significantly impacting the underlying structure of the instructional materials.

For this indicator, several pieces in the Assessment Overview section of the Assessment Guide were used to identify summative assessments. On page 4 of the assessment overview, the authors note that "The unit assessments are generally longer, more comprehensive in terms of the material covered in the unit, and more summative in nature," and on page 5,"Unit assessments are generally longer, more comprehensive, and summative with respect to the goals of the instruction in the unit." Additionally, on pages 3 and 4, the authors identify the Number Corner Checkups as having a "focus on critical numeracy skills and concepts," and they explain that they are meant to document growth quarterly as compared to a "Baseline" Checkup at the beginning of the year and based on skills and concepts taught in that quarter. Lastly, the Grade 1 Assessment Map found on pages 12-14 in the Assessment Overview section indicates when mastery of each standard is expected and where the mastery standard is assessed. It was found that some of the mastery concepts were assessed on unit checkpoints. Based on these criteria, the following were considered to be the summative assessments and were reviewed for Indicator 1a:

  • All Unit Assessments
  • Number Corner Checkups 1 – 4
  • The Comprehensive Growth Assessment
  • Select Unit Checkpoints where mastery is indicated on Assessment Map:
    • Unit 5 M2 S5: Shapes Checkpoint
    • Unit 6 M2 S5: Combinations and Stories Checkpoint
    • Unit 7 M2 S5: Numbers to 120 Checkpoint
    • Unit 8 M2 S4: Time and Change Checkpoint

Assessments are student observation/interview or written in nature. Most Comprehensive Growth Assessment (CGA) questions are fully aligned to the Grade 1 CCSS. All of the Number Corner Quarterly Checkups are fully aligned to the Grade 1 CCSSM. There are some questions in the Unit Checkpoints that go above Grade 1 assessment expectations.

The Unit Assessment Checkpoints that contain above grade-level or content not specifically required by the standards are noted in the following list:

  • In the Comprehensive Growth Assessment (CGA) (written portion, page 18)
    • question 18 asks the student to create a composite shape. One of the suggestions is to use a rhombus, which is not part of 1.G.1 but is mathematically reasonable to have included.
  • In the Unit 5 Assessment (Module 3, Session 6, p. 54):
    • In Questions 5 and 7, fractional parts are written using symbols (½ and ¾), for example: Draw a line to divide this rectangle in half. Color in one-half (1/2) of the rectangle. Symbolic notation for fractions is a Grade 3 expectation (3.NF.1). However, since the word form is included in the question, it is not necessary for students to have mastered symbolic fraction notation in order to answer the questions. It should be noted that when looking at the sessions leading into this assessment, symbolic notation is included. For example, students play games with fraction cards which show fractional models of shapes and sets with the symbolic fraction form included (Grade 3 expectations).
    • In Question 6, students are required to color in three-fourths of the circle. Coloring in three-fourths of a circle is not mathematically reasonable in Grade 1, and this item would need to be revised or removed from the assessment. The related session, Unit 5, Module 3, Session 5, also includes fractions that are not appropriate for Grade 1 (for example, 2/3).
  • In the Unit 6 Assessment (Module 3, Session 5, p. 66):
    • In Question 2, students are given 14 addition problems (sums up to and including 20). The author aligned the task to 1.OA.6 (add and subtract to 20, demonstrating fluency for addition and subtraction within 10). However, the answer key provided indicates that students score one point for each correct answer recorded during the 3-minute timing. Timing indicates an expectation of fluency with those addition combinations, and six of the problems have sums greater than the limit of 10 as indicated in the standard. This item could be easily modified to be appropriate for Grade 1 by not timing the students.
  • In the Unit 8 Time and Change Checkpoint (Module 2 Session 4) and the Unit 8 Assessment (Module 3 Session 6):
    • Question 2 on both assessments goes beyond the intended “tell time to the hour” to assess elapsed time: soccer practice began at 4:00 and lasted 1 hour; students must identify the clock with shows what it looks like when soccer practice is over.
) [3] => stdClass Object ( [code] => 1b [type] => criterion [report] =>

The instructional materials reviewed for Grade 1 meet the expectations for focus on the major clusters of each grade. Students and teachers using the materials as designated will devote the majority of class time to major clusters of the grade.

) [4] => stdClass Object ( [code] => 1b [type] => indicator [points] => 4 [rating] => meets [report] =>

The instructional materials reviewed for Grade 1 meet the expectations for focus by spending the majority of class time on the major clusters of the grade. All sessions (lessons), except summative and pre-assessment sessions, were counted as 60 minutes of time. Number Corner activities were counted and assigned 20 minutes of time. When sessions or Number Corner activities focused on supporting clusters and clearly supported major clusters of the grade, they were counted. Reviewers looked individually at each session and Number Corner in order to determine alignment with major clusters and supporting clusters. Optional Daily Practice pages and Home Connection pages were not considered for this indicator because they did not appear to be a required component of the sessions.

When looking at the modules (chapters) and instructional time, when considering both sessions and Number Corners together, approximately 86 percent of the time is spent on major work of the grade.

  • Units – Approximately seven out of eight units spend the majority of the unit on major clusters of the grade, which is approximately 88 percent. Much of Unit 5 is not focused on major work of the grade. The other units devote most of the instructional time to major clusters of the grade.
  • Modules (chapters) – Approximately 27 out of 32 modules spend the majority of the time on major clusters of the grade, which equals approximately 84 percent. Units 2, 5, and 8 had modules that were not focused on major work of the grade.
  • Bridges Sessions (lessons) – 137 out of 160 sessions focus on major clusters of the grade, which equals approximately 86 percent. Major work is not the focus of the following sessions:
    • Unit 1, Module 2, Session 1
    • Unit 2, Module 4, Session 1
    • Unit 2, Module 4, Sessions 2 and 3
    • Unit 5, Module 1, Sessions 1-5
    • Unit 5, Module 2, Sessions 1-5
    • Unit 5, Module 3, Sessions 2-5
    • Unit 8, Module 1, Sessions 2 and 3
    • Unit 8, Module 3, Sessions 1 and 6
    • Unit 8, Module 4, Session 5
  • Bridges sessions require 60 minutes. A total of 137 sessions are focused on major work grade work of the grade. Bridges sessions devote 8,220 minutes of 9,600 minutes to major work of the grade. A total of 150 days of Number Corner activities address major work of the grade. Number Corner activities are 20 minutes each adding another 3,000 minutes to this total. In all 11,220 of 13,000 minutes, approximately 86 percent, is devoted to major work of the grade.
) [5] => stdClass Object ( [code] => coherence [type] => component [report] =>

The instructional materials reviewed for Grade 1 meet the expectations for coherence. The materials use supporting content as a way to continue working with the major work of the grade. For example, students count shapes in categories and then compare the quantities. The materials include a full program of study that is viable content for a school year, including 160 days of lessons and assessment. All students are given extensive work on grade-level problems, even students who are struggling, and this work progresses mathematically. However, future grade-level content is not consistently identified. These instructional materials are visibly shaped by the cluster headings in the standards; for example, one session is called "Sorting & Graphing Shapes." Connections are made between domains and clusters within the grade-level. For instance, materials make connections between operations and algebraic thinking and measurement and data. Overall, the Grade 1 materials support coherence and are consistent with the progressions in the standards.

) [6] => stdClass Object ( [code] => 1c1f [type] => criterion ) [7] => stdClass Object ( [code] => 1c [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Grade 1 meet expectations that supporting content enhances focus and coherence by engaging students in the major work of the grade.

Supporting standard 1.G.2 is connected to 1.NBT.1 throughout the instructional materials. For example, in Unit 5, Module 1, Session 4 when filling outlines of shapes with pattern blocks, students are asked to count and record in a table the number of each type of shape used. Students practice writing numerals as they record information in the table.

Supporting standard 1.MD.4 is connected to major work of the grade throughout the instructional materials. For example, in Unit 1, Module 1, Session 2 students are creating popsicle graphs and then counting to determine the differences between the flavors; sums and differences are primarily within 10. Within Unit 2, Module 3, Session 3, Workplace 2E, 1.MD.4 supports the major work of 1.OA.6 with the game "Spin & Add;" students are spinning two spinners, adding, and recording totals on chart. Another example is Unit 5, Module 4, Session 2; students make a graph after sorting shapes into two categories and then answer questions such as how many more than, fewer than, and in all. This work with 1.MD.4 supports standard 1.OA.1 and practice with standard 1.OA.6. Also, in the September-May Calendar Collector 1.MD.4 is connected to 1.OA.4, 1.NBT.1, and 1.NBT.3. The first few weeks are spent collecting the data. In the fourth week, students compare and order, estimate and count collections from the previous three weeks. Most months also have students applying concepts to story problem contexts.

) [8] => stdClass Object ( [code] => 1d [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Grade 1 meet the expectations for this indicator by providing a viable level of content for one school year. Overall, the materials have expectations for teachers and students that are reasonable.

  • Materials provide for 160 days of instruction. Each Unit has 20 sessions = 20 days. There are eight Units. (20x8=160)
  • The prescribed daily instruction includes both Unit session instruction and a Number Corners session. (170 days). There are no additional days built in for re-teaching.
  • Assessments are incorporated into sessions and do not require an additional amount of time. Instead, they are embedded into module sessions one on one as a formative assessment.
  • The Number Corner Assessments/Checkups (a total of 10 assessments, one interview and one written, in each of the following months: September, October, January, March and May) would require additional time to conduct a 7-10 minute interview with each student.
  • A Comprehensive Growth Assessment is completed at the end of the year and will require additional number of days to administer.
  • There are no additional time/days built in for additional support, intervention or enrichment in the pacing guide. The publisher recommends re-teaching of strategies, facts, and skills take place in small groups while the rest of the class is at work places (math stations) or doing some other independent task. There is a concern that if a particular session’s activities take up most of the 60 minutes allotted, there will be no time for the remediation and enrichment to take place.
  • Based on the Bridges Publisher Orientation Video and Guide provided to the reviewers, Unit sessions are approximately 60 minutes of each instructional day.
    • Each Unit session contains: Problems & Investigations (whole group), Work Places (math stations), Assessments (*not found in each session), and Home Connections (homework assignments *not found in each session).
  • Based on the Introduction section in the Number Corners Teacher Guide, as well as the Bridges Publisher Orientation Video, Number Corners sessions are approximately 20 to 25 minutes of each instructional day.
  • Approximately 80-85 minutes is spent on the Bridges and Number Corner activities daily.
) [9] => stdClass Object ( [code] => 1e [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Grade 1 are partially consistent with the progressions in the standards. Although students are given extensive grade-level problems and connections to future work are made, off-grade level content is not always clearly identified to the teacher or student.

At times, the session materials do not concentrate on the mathematics of the grade. Some of the sessions within each module concentrate on below or above-grade level concepts. Examples of this include counting objects one-by-one, growing patterns, rotational symmetry, building shapes from nets, fraction notation, and fractions of a set. The inclusion of off-grade level concepts takes away from the number of sessions that could be spent more fully developing the work on the mathematics of the grade.

In some cases, the below or above-grade level content is identified as such by the publishers, and in other cases it is not. On the first page of every session, the skills and concepts are listed along with the standard to which it has been aligned by the publisher. In some cases, this alerts the user to the inclusion of off-grade level concepts. Examples include:

  • Unit 1, Module 2, Session 1: One of the skills listed is "Count objects one by one, saying the numbers in standard order and pairing each object with only one name." The publisher lists this standard as K.CC.4.A, alerting teachers to the fact that this session involves below-grade level standards.
  • Unit 1, Module 2, Session 1: One of the skills listed is "Count up to 20 objects arranged in a line or array to answer "how many?" questions." This skill is listed as K.CC.5, alerting teachers to the fact that this session involves below-grade level standards.
  • Unit 1, Module 2, Session 3: One of the skills listed is "recognize the number of objects in a collection of 6 or fewer, arranged in any configuration." This skill is listed as "supports K.CC" which alerts the teacher that this is a below-grade level concept.
  • Unit 2, Module 1, Session 1: One of the skills listed is "count up to 20 objects arranged in a line, rectangular array, or circle to answer "how many." This skill is listed as K.CC.5 which alerts the teacher that this is a below-grade level concept.
  • Unit 2, Module 1, Session 1: One of the skills listed is "identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group for groups of up to 10 objects." This standard is listed at K.CC.6 which alerts the teacher that this is a below-grade level standard.
  • Unit 5, Module 3, Session 2: The About This Session note acknowledges that the activity dives into symmetry and that symmetry isn’t formally studied until fourth grade, so this is simply an “exploration.”

In other cases, the below or above-grade level concepts are not identified as such within the sessions in the "Skills and Concepts" listing or at the beginning of the units in the "Skills Across the Grade Levels" sections. Examples of unidentified below or above-grade level content include:

  • Unit 1, Module 1, Session 2: Counting by 5's is a Grade 2 standard (2.NBT.2).
  • Unit 1, Module 1, Session 3: The Work Place focus is on Polydrons, 1.G.2; however, work involves Grade 6 standards constructing triangular prisms and pyramids.
  • Unit 1, Module 1, Session 5: Counting by 2's is the focus of this lesson and is a Grade 2 standard (2.OA.3).
  • Unit 1, Module 3 and Unit 1, Module 4, Session 3: Counting by 5's is a Grade 2 standard.
  • Unit 2, Module 4, Session 1: Students are making triangular quilt pieces to represent the five arms of a sea star. Counting by 5's is a Grade 2 standard.
  • Unit 2, Module 4, Session 1: Students are assembling a quilt making rows of five which is a Grade 2 standard (2.OA.4).
  • Unit 2, Module 4, Session 3: The focus is counting by 5s, a Grade 2 standard.
  • Unit 3, Module 1, Session 2 and Session 5: Counting by 2’s is a skip counting strategy/skill that is not introduced until Grade 2 (2.NBT.2).
  • Unit 4, Module 4, Session 2 and Session 5: Both sessions involve counting by 5’s, which is a skip counting strategy/skill that is not introduced until Grade 2 (2.NBT.2).
  • Unit 6, Module 4, Session 5: The publisher identifies count by twos and number patterns as “supporting 1.NBT” as an indicator that this is not Grade 1 expectation, but it doesn’t specifically call out for which grade level it would be appropriate.

Materials provide students opportunities to work with grade-level problems. The majority of differentiation/support provided is on grade-level. Extension activities are embedded within Sessions and allow students to engage more deeply with grade-level work. Additional Extension activities are also provided online.

) [10] => stdClass Object ( [code] => 1f [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Grade 1 meet the expectations for fostering coherence through connections at a single grade, where appropriate and when the standards require. The standards are referred to throughout the materials. Overall, materials include learning objectives that are visibly shaped by CCSSM cluster headings and include problems and activities that connect two or more clusters in a domain or two or more domains when these connections are natural and important.

Instructional materials shaped by cluster headings include the following examples:

  • The Unit 5 Module titles are loosely informed by the K.G cluster headings.
  • Unit 5, Module 3, “Putting Shapes Together & Taking Them Apart,” is informed by 1.G.A.
  • Unit 5, Module 4, “Sorting & Graphing Shapes,” is similar to 1.G.A.
  • Unit 6 Module titles are informed by the 1.OA cluster heading.
  • Unit 6, Module 1, “Story Problems for Basic Addition & Subtraction,” is shaped by 1.OA.A.
  • Unit 6, Module 3, “Solving for the Unknown in Penguin Stories,” is informed by 1.OA.A.
  • Unit 6, Module 4, “Measuring & Comparing Emperor and Little Blue Penguins,” is informed by 1.MD.A.

Units, Modules, and Sessions that connect two or more clusters in a domain or two or more domains include the following examples:

  • Unit1, Module1, Session 2: "Popsicle Graph" connects 1.MD.C cluster to 1.OA.A as students are adding up popsicles and determining how many more/many fewer/in all.
  • Unit1, Module 2, Session 2: "Making 5 & 10" connects 1.OA.B with 1.OA.C as students are counting on their number racks to make combinations of 5 and 10, then writing the combinations in the form of equations.
  • Unit 1, Module 3, Session 3: "Which Coin Will Win" connects cluster 1.NBT.A with 1.MD.C as students are counting and writing numerals to represent the number of coins on the graph.
  • Unit 1, Module 3, Session 4: "Quick Look!" connects cluster heading 1.OA.C with 1.NBT.A as students are counting and adding beads on the number rack then writing a numeral that represents the number of beads.
  • Unit 1, Module 3, Session 5: "Measuring with Popsicle Sticks" connects cluster 1.MD.A with 1.NBT.A as students measure the length of their hands, use tallies to record the amount on a frequency chart, then write down the numeral that represents the number of tally marks.
  • Unit 1, Module 4, Session 1: "Number Rack Detectives" connects clusters1.OA.B, 1.OA.C, and 1.OA.D as students play a game of building addends on the number rack and then only share one row of the number rack with a partner, who then determines how many beads must be on the bottom rack.
  • Unit 1, Module 4, Session 3- "How Long is the Jump Rope" -connects cluster 1.NBT.A with 1.MD.A as students are measuring a jump rope using steps and then representing the number of steps with a numeral.
  • Unit 1, Module 4, Session 4: "Quick! Look! Plus One, Minus One" connects cluster 1.OA.C with 1.NBT.A as students are shown various quantities of beads on the number rack, record the number of objects with a numeral, and then write down the number that comes before and after it.
  • Unit 2, Module 1, Session 3: "Domino Add & Compare" connects cluster 1.OA.C to 1.NBT.B as students count dominos, compare amounts, and represent with a greater than, less than, or equal to symbol.
  • Unit 2, Module 2 connects cluster-level headings 1.OA.A and 1.OA.C.
  • Unit 2, Module 2, Session 1: "Introducing Double-Flap Dot Cards" connects cluster 1.OA.B with 1.OA.C as students identify number of dots on the dot cards, count on to find the total, and then write an equation to represent the dots shown as fact families.
  • Unit 2, Module 2, Session 2: "Double-Flap Picture Cards" connects cluster 1.OA.A to 1.OA.B as students add objects on picture cards to solve word problems and write equations that represent fact families.
  • Unit 2, Module 2, Session 3, Introducing Work Place 2C: "Sort the Sum" connects cluster 1.OA.C to 1.NBT.C as students find the sum of dots and then compare the totals. *Note - totals are under 10 which is not the full intent of 1.NBT.3 which calls for comparing two two-digit numbers.
  • Unit 2, Module 2, Session 4: "Double Flap Number Cards" connects 1.OA.B to 1.OA.D as students add numbers on the number cards applying strategies to add and subtract as well as determining the unknown whole number in equations.
  • Unit 2, Module 3, Session 3: connects cluster 1.OA.C with 1.MD.4 as students are counting numbers and dots on spinner, counting on to get the sum, and then recording amount on a graph
  • Activities in Unit 5, Module 1, Session 4: "Pattern Block Puzzles: How Many Ways?" connects 1.G.2 with 1.NBT.1 and 1.MD.4. When composing shapes within a shape outline, students count and record the number of each shape in a table and answer questions regarding which shape is fewer or more.
  • Activities in Unit 6, Module 1 connect addition and subtraction story problems 1.OA.A to 1.OA.C.
  • Unit 6, Module 2: “Combinations and Story Problems” connects 1.OA.A and 1.OA.C.
  • Activities in Unit 6, Module 3 connect work in story problems (1.OA.A) to solving for an unknown in an addition equation involving 3 whole numbers (1.OA.D)
  • Activities in Unit 6, Module 4 connect 1.NBT with 1.MD.
  • Unit 6, Module 2: “Combinations and Story Problems” connects 1.OA.A and 1.OA.C.
) [11] => stdClass Object ( [code] => rigor-and-balance [type] => component [report] =>

The materials reviewed for Grade 1 meet the expectations for this criterion by providing a balance of all three aspects of rigor throughout the lessons. To build conceptual understanding, the instructional materials include concrete materials, visual models, and open-ended questions. In the instructional materials students have many opportunities to build fluency with adding and subtracting within 20. Application problems occur throughout the materials. The three aspects are balanced within the instructional materials.

) [12] => stdClass Object ( [code] => 2a2d [type] => criterion ) [13] => stdClass Object ( [code] => 2a [type] => indicator [points] => 2 [rating] => meets [report] =>

The materials reviewed in Grade 1 for this indicator meet the expectations by attending to conceptual understanding within the instructional materials.

The instructional materials often develop a deeper understanding of clusters and standards by requiring students to use concrete materials and multiple visual models that correspond to the connections made between mathematical representations. The materials encourage students to communicate and support understanding through open-ended questions that require evidence to show their thinking and reasoning.

The following are examples of attention to conceptual understanding of 1.OA.4:

  • In Unit 3, Module 1, Session 5, students use the number rack as a tool to model and solve subtraction word problems. Problem types include Take From-Change Unknown, Take From-Start Unknown, and Compare-Difference Unknown. These varying problem structures provide opportunities for students to develop a deep understanding of the relationship between addition and subtraction.

The following are examples of attention to conceptual understanding of 1.OA.7:

  • In Unit 6, Module 1, Session 2, students are provided with equations with a box for the missing addend. They solve various equations and also determine if equations are true.

The following are examples of attention to conceptual understanding of 1.NBT.B:

  • In Unit 4, Module 4, Session 2, students build conceptual understanding of bundles of ten within 100 using concrete materials and the structure of a ten-frame.
  • In Unit 7, Module 1, Session 2, students count popsicle sticks, grouping 10 ones into groups of 10. Then, two index cards are labeled "10's" and "1's," and students reorganize their sticks so that they can be counted more easily. Students collaborate to model groupings in our base ten number system helping them develop a deep understanding of place value.
  • In the February Number Corner Calendar Collector, a ten-strip model is used to build conceptual understanding of place value with tens and ones.

The following are examples of attention to conceptual understanding of 1.NBT.C:

  • In Unit 2, Module 2, Session 4, the 100s Number Grid is observed, and students share some things they notice (you can count by 10s in the last column, there are all 0's in that column, there are 1's under 1's and 2's under 2's, etc.). Students use the Number Grid as a scaffold/tool to help solve the "Change Cards" game problems. As students play the game, "Change Cards," they are adding and subtracting multiples of 10 (Cards: 25 and 35 = rule of +10). They then discuss the "rule" and pair-share to make predictions for the next group of cards.
  • In Unit 3, Module 3, Sessions 1-4, students are using cube trains of ten and single cubes to represent addition two-digit equations equations.
  • In Unit 8, Module 3, Sessions 3-6, students use unifix cube trains
) [14] => stdClass Object ( [code] => 2b [type] => indicator [points] => 2 [rating] => meets [report] =>

The Grade 1 materials meet the expectations for procedural skill and fluency by giving attention throughout the year to individual standards which set an expectation of procedural skill and fluency.

  • Students spend a significant amount of time and have a variety of opportunities to fluently add and subtract throughout Number Corner activities.
    • In the Number Corner September Computational Fluency, in Activity 1, students are using the double 10-frame to solve addition problems by matching the equation with the cards.
    • In the Number Corner October Computational Fluency, in Activity 2, students play Ten-Frame Flash and show, with their fingers, how many dots are on the 10-frame cards.
    • In the December Number Corner Computational Fluency Routine, the routine for this month involves investigating double facts greater than 10 using a double ten-frame.
    • In the March Number Corner Days in School Routine, the hundreds grid is used as a visual model to assist students in explaining their mental reasoning for ten more or ten less.
  • Fluency is developed throughout the sessions of the Grade 1 instructional materials.
    • In Unit 1, Module 2, Session 3, students respond to representations of the 10-frame as the teacher flashes the 10-frame cards with totals within 10.
    • In Unit 4, Module 1, Session 5, students solve addition and subtraction combinations such as 3+2 and 5-2. Students are using their fluency to help deepen their understanding about the relationship between addition and subtraction.
    • In Unit 2, Module 2, Session 3, students identify strategies they used in adding numbers represented on a domino. Teacher led discourse elicits student thinking using counting on, doubling, and decomposing strategies.
    • In Unit 1, Module 3, Session 1, students solve missing-addend problems on their number racks. They use 5 as a landmark and find doubles and then count on.
    • In Unit 3, Module 4, Sessions 1 and 2, students use unifix cubes to represent equations up to 10 with various combinations of two or three addends.
    • In Unit 6, Module 2, Session 2, students use ten and double ten frames to represent addition up to twenty in the game "I have, who has" to create the addition equations.
) [15] => stdClass Object ( [code] => 2c [type] => indicator [points] => 2 [rating] => meets [report] =>

Materials meet the expectations for having engaging applications of mathematics as they are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade.

Materials include multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. The materials provide single step contextual problems that revolve around real world applications. Major work of the grade level is addressed within most of these contextual problems. The majority of the application problems are done with guiding questions elicited from the teacher through whole group discussions that build conceptual understanding and show multiple representations of strategies. Materials could be supplemented to allow students more independent practice for application and real-world contextual problems that are not teacher guided within discussions.

The instructional materials include problems and activities aligned to 1.OA.1 and 1.OA.2 that provide multiple opportunities for students to engage in application of mathematical skills and knowledge in new contexts. Examples of these applications include the following:

  • In Unit 2, Module 2, Session 2, the materials provide story problems to investigate the relationship between addition and subtraction equations. Students write their own story problems and equations as an extension of the learning.
  • In Unit 3, Module 1, Session 5, students use the number rack as a tool to model and solve subtraction word problems (problem types - Take From-Change Unknown, Take From-Start Unknown, and Compare-Difference Unknown). These varying problem structures provide opportunities for students to develop a deep understanding of the relationship between addition and subtraction and apply mathematical knowledge and skills in a real-world context.
  • In Unit 4, Module 1, Session 3, the teacher provides frog word problem to students and they solve using the number line.
  • In Unit 4, Module 2, Session 3, the lesson utilizes a floor number line to investigate the use of addition and subtraction on the number line through contextual story problems.
  • In Unit 7, Module 3, Session 2, students solve addition story problems with sums to 20 involving adding to, put together with unknowns in all positions.
  • In the Number Corner October Calendar Grid, the teacher provides a word problem (add to, result unknown problem type), and students solve.
  • In the Number Corner February Computational Fluency, students began to add to ten in the context of themed story problems and application within the Number Corner computational fluency routine. Thinking within these contextual situations is extended toward building conceptual understanding of subtraction as a missing addend problem.
) [16] => stdClass Object ( [code] => 2d [type] => indicator [points] => 2 [rating] => meets [report] =>

The materials reviewed in Grade 1 meet the expectations for providing a balance of rigor. The three aspects are not always combined nor are they always separate.

In the Grade 1 materials all three aspects of rigor are present in the instructional materials. All three aspects of rigor are used both in combination and individually throughout the unit sessions and in Number Corner activities. For example, in Unit 3 Module 3 Session 1 and Unit 6 Module 1 Session 4 all aspects of rigor are present. Application problems are seen to utilize procedural skills and require fluency of numbers. Conceptual understanding is enhanced through application of previously explored clusters. Procedural skills and fluency learned in early units are applied in later concepts to improve conceptual understanding.

) [17] => stdClass Object ( [code] => mathematical-practice-content-connections [type] => component [report] =>

The materials reviewed for Grade 1 meet this criterion. The MPs are often identified and often used to enrich mathematics content. There are, however, several sessions that are aligned to MPs with no alignment to Standards of Mathematical Content. The materials usually attend to the full meaning of each practice. The materials reviewed for Grade 1 attend to the standards' emphasis on mathematical reasoning. Students are prompted to explain their thinking, listen to and verify the thinking of others, and justify their own reasoning. Although the materials often assist teachers in engaging students in constructing viable arguments, more guidance about how to guide students in analyzing the arguments of others is needed.

) [18] => stdClass Object ( [code] => 2e2g [type] => criterion ) [19] => stdClass Object ( [code] => 2e [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Grade 1 meet the expectations for identifying the MPs and using them to enrich the mathematical content. Although a few entire sessions are aligned to MPs without alignment to grade-level Standards for Mathematical Content, the instructional materials do not over-identify or under-identify the MPs and the MPs are used within and throughout the grade.

The Grade 1 Assessment Guide provides teachers with a Math Practices Observation Chart to record notes about students' use of MPs during Sessions. The chart is broken down into four categories: habits of mind, reasoning and explaining, modeling and using tools, and seeing structure and generalizing. The publishers also provide a detailed, "What Do the Math Practices Look Like in Grade 1?" guide for teachers (AG, page17).

Each session clearly identifies the MPs used in the Skills & Concept section. Some sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation of the connection between the indicated MP and the Standards for Mathematical Content for the teacher. Examples of the MPs in the instructional materials include the following:

  • Unit 1, Module 4, Session 1 references MP2 and MP5. Unit 1, Module 4, Session 2 references MP5 and MP6. Unit 1, Module 4, Session 3 addresses MP4 and MP6. In Unit 1, Module 4, Sessions 4 and 5 both address MP4 and MP7. There is a "Math Practices In Action" reference in Session 1 and Session 3.
  • In Unit 2, Module 1 in the Skills & Concepts section, two sessions (3 and 4) list MP2, Session 4 lists MP3, Session 5 lists MP6, two sessions (1 and 2) list MP7, and Session 2 lists MP8.
  • In Unit 6, Module 1, Sessions 1 and 3 reference the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
  • In Unit 7, Module 2, four of the five sessions address MP7.
  • In Unit 7, Module 3, Session 1 references the MPs within the Problems and Investigations portion of the session as, "Math Practices in Action."
  • In the September Number Corner, MP4 is addressed in Calendar Grid, Days in School, and Computational Fluency; MP6 is addressed in Calendar Collector, MP7 is addressed in Calendar Grid, Days in School, Computational Fluency, and Number Line, and MP8 is addressed in Number Line.

Lessons are aligned to MPs with no alignment to Standards of Mathematical Content. These sessions that focus entirely on MPs include the following:

  • Unit 1, Module 2, Session 1
  • Unit 6, Module 4, Session 4
  • Unit 8, Module 4, Session 5
) [20] => stdClass Object ( [code] => 2f [type] => indicator [points] => 2 [rating] => meets [report] =>

The materials meet the expectations for attending to the full meaning of each practice standard. Each Session clearly identifies the MPs used in the Skills & Concept section of the session. Typically there are two MPs listed for each session, so there is not an overabundance of identification.

Some sessions contain a "Math Practice In Action" sidebar that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. Although the MPs are listed at the session level, the MPs are not discussed or listed in unit overviews or introductions (major skills/concepts addressed); however, they are listed in the section 3 of the Assessment Overview. With limited reference in these sections, overarching connections were not explicitly addressed.

In Number Corners, the MPs are listed in the Introduction in the Target Skills section with specific reference to which area of Number Corner in which the MP is addressed (Calendar Grid, Calendar Collector, Days in School, Computational Fluency, Number Line). The MP are also listed in the assessment section of the introduction as well. Although the MPs are listed in these sections, there is no further reference to or discussion of the MPs within Number Corner.

The following are examples of times that the instructional materials attend to the full meaning of the MPs:

  • MP7 is addressed in Unit 1 Module 3 Session 2. This session focuses on the pattern and structure of the unit of ten using number lines, 10-frames and the place value relationship between the ones unit and tens unit.
  • MP1 is addressed in Unit 3, Module 1, Session 5. Students are presented with very challenging number rack story problems in the Problems & Investigations section of this lesson. The wide range of problem types makes this session cognitively challenging for students in Grade 1. They are supported in their efforts to solve the problems and grow accustomed to devoting significant time and effort to persevere in solving them.
  • MP3 is addressed in Unit 6, Module 2, Session 5. Students are working together to play the game "Pick Two to Make Twenty." Students have to pick two cards that total a number closest to 20. Students share their ideas and learn to construct viable arguments and critique the reasoning of others. Playing together as a team against the teacher motivates students to listen carefully to one another so they have the best chance at winning (Math Practices In Action, page 31). Students are invited to make a case for the combination they thinks works best by explaining their thinking to the class, and they can demonstrate on the number rack to help justify their thinking.
  • MP5 is addressed in Unit 8, Module 4, Session 2. This session is titled, "How We Have Grown." Students work to compare the lengths of two strings that represent the length of an average baby with the length of an average student in Grade 1. Students are asked what strategies they might used to figure out the difference between the two lengths, and possible strategies are discussed, such as using Unifix cubes to represent lengths and then laying them next to each and count the extras on the big one, a number line, and models. Students are using appropriate tools strategically when they compare the lengths using strings, Unifix cubes, and the number line. They think carefully about how to use Unifix cubes and how to use efficient jumps on the number line. They are making choices about which tools to use and how to use them based on the problem at hand. (Math Practices in Action).

However, at times the materials only partially attend to the meaning of MP4. The intent of this practice standard is to apply mathematics to contextual situations in which the math arises in everyday life. Often when MP4 is labeled students are simply selecting a model to represent a situation. For example, in Unit 3, Module 1, Session 1, MP4 is indicated, but students are simply representing a number on a 10-frame. The Math Practices in Action note states that "Students will use drawings, numbers, expressions, and equations to model with mathematics."

) [21] => stdClass Object ( [code] => 2g [type] => indicator [report] => ) [22] => stdClass Object ( [code] => 2g.i [type] => indicator [points] => 2 [rating] => meets [report] =>

The materials reviewed for Grade 1 meet the expectations of this indicator by attending to the standards' emphasis on mathematical reasoning.

Students are asked to explain their thinking, listen to and verify other's thinking, and justify their reasoning. This is done in interviews, whole group teacher lead conversations, and in student pairs. For the most part, MP3 is addressed in classroom activities and not in Home Connection activities.

  • In Unit 2, Module 1, Session 4, students work with the teacher to create a class "Addition Strategies Chart." As they review the Domino Add & Compare game, students discuss some of their strategies they can use to find the total number of dots on a domino. After the chart is created, students are asked to share the advantages and disadvantage of each strategy. Since all of the strategies are written on the chart, students are able to critique and compare the strategies of others in a safe manner.
  • In Unit 3, Module 4, Session 2, students are asked to show their thinking about how to decompose the number seven into two addends in multiple ways using addition and subtraction.
  • In Unit 4 Module 3 Session 5, students are given a contextual word problem to compare two number sets involving the unit of one with the tool of pennies. They have to show their thinking with a visual representation.
  • In Unit 6, Module 2, Session 5, students are introduced to a new game, "Pick Two to Make Twenty." When students share their ideas about how to get as close to 20 as possible, they are learning to construct viable arguments and critique the reasoning of others.
  • In Unit 6, Module 4, Session 2, students are asked to show how they solved a contextual problem using pictures, numbers and words to explain their thinking.
  • In Unit 7, Module 2, Session 1, students are making trains of Unifix cubes that total 120. Before beginning, students are asked to discuss with their partner about how many groups of 20 cubes will be needed to make the train. Volunteers share and explain their thinking. The next part of the lesson students are asked again to explain their thinking regarding locating a specific cube within the train.
) [23] => stdClass Object ( [code] => 2g.ii [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

The instructional materials reviewed for Grade 1 partially meet the expectations for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards. Although the instructional materials often assist teachers in engaging students in constructing viable arguments, there is minimal assistance to teachers in how to guide their students in analyzing the arguments of others.

There are Sessions containing the "Math Practice In Action" sidebars that explicitly states where the MP is embedded within the lesson and provides an in-depth explanation for the teacher. A few of the sessions contain direction to the teacher for prompts and sample questions and problems to pose to students.

  • In Unit 1, Module 2, Session 1, the teacher asks students to share strategies of how they found different combinations of numbers on the number rack to equal the number the teacher gave students to create.
  • In Unit 7, Module 2, Session 1, students construct and share their thinking about units of 20 within a total of 120. A number path with discrete units is used to analyze and explain their thinking.
  • In Unit 7, Module 4, Session 4, the Problems & Investigations lesson involves critiquing the reasoning of others in using place value understanding with units of 5 and 10 with number bonds, equations and coins. Students represent their thinking in multiple ways and make connections between the reasoning of others in comparison to their own chain of reasoning.

Although teachers are provided guidance to help students construct arguments and students are provided many opportunities to share their arguments, more guidance is need to support teachers in guiding their students through the analysis of arguments once they are shared. For example, in Unit 5, Module 4, Session 2, students are introduced to the Work Place "Shape Sorting & Graphing." The teacher ask students to talk with their neighbors about how the two shapes are alike and how they are different. Within the sample dialogue, students are asked, "Can you explain more about that?" and "Can someone tell me more about that?" The teacher is provided with sample dialogue that encourages students to construct viable arguments; however, little direction is provided for students to continue the discourse and analyze the arguments of others. Although this activity allows students to analyze the arguments of classmates, the teacher is not provided enough support to help students with this analysis.

) [24] => stdClass Object ( [code] => 2g.iii [type] => indicator [points] => 2 [rating] => meets [report] =>

The instructional materials reviewed for Grade 1 meet the expectations for explicitly attending to the specialized language of mathematics. Overall, the materials for both students and teachers have multiple ways for students to engage with the vocabulary of mathematics that is present throughout the materials.

The instructional materials provide explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols. Students have opportunities to explain their thinking while using mathematical terminology, graphics, and symbols to justify their answers and arguments in small group, whole group teacher directed, and teacher one-to-one settings.

The materials use precise and accurate terminology and definitions when describing mathematics and support students in using them. Examples of this include using geometry terminology such as rhombus, hexagon and trapezoid and using operations and algebraic thinking terminology such as equation, difference, and 10-frame.

  • Many sessions include a list of mathematical vocabulary that will be utilized by students in the session.
  • The online teacher materials component of Bridges provides teachers with "Word Resources Cards" which are also included in the Number Corner Kit. The Word Resources Cards document includes directions to teachers regarding the use of the mathematics word cards. This includes research and suggestions on how to place the cards in the room. There is also a "Developing Understanding of Mathematics Terminology" included within this document which provides guidance on the following: providing time for students to solve problems and ask students to communicate verbally about how they solved them, modeling how students can express their ideas using mathematically precise language, providing adequate explanation of words and symbols in context, and using graphic organizers to illustrate relationships among vocabulary words
  • At the beginning of each section of Number Corner, teachers are provided with "Vocabulary Lists" which lists the vocabulary words for each section.
  • Unit 3 Module 4, Session 1 investigates the relationship between numbers with the commutative property of operations. The language within the lesson reinforces and contextually uses the terms equations, equal and strategies to explain the reasoning between visual representations.
  • In Unit 5, Module 2, Session 4, during a Problems & Investigations activity with 3-dimensional shapes, the teacher asks students how they identified a cube just by touch. The teacher guide gives specific direction to the teacher: "Model the vocabulary of geometry as you field their responses." The student's response includes the word "corner," and the teacher restates the response by saying, "What do you mean the corners - in math we call them vertices..."
  • The Number Corner, October Calendar Collection investigates patterns of geometrical shapes using the terms hexagon, rhombus and trapezoid.
) [25] => stdClass Object ( [code] => alignment-to-common-core [type] => component [report] =>

The instructional materials reviewed for Grade 1 are aligned to the CCSSM. Most of the assessments are focused on grade-level standards, and the materials spend the majority of the time on the major work of the grade. The materials are also coherent. The materials follow the progression of the standards and connect the mathematics within the grade level although at times off-grade level content is not identified. There is also coherence within units of each grade. The Grade 1 materials include all three aspects of rigor, and there is a balance of the aspects of rigor. The MPs are used to enrich the learning, but additional teacher assistance in engaging students in constructing viable arguments and analyzing the arguments of others is needed. Overall, the materials are aligned to the CCSSM.

[rating] => meets ) [26] => stdClass Object ( [code] => usability [type] => component [report] =>

The materials reviewed meet the expectations for usability. In reviews for use and design, the problems and exercises are developed sequentially and each activity has a mathematical purpose. Students are asked to produce a variety of assignments. Manipulatives and models are used to enhance learning and the purpose of each is explained well. The visual design is not distracting or chaotic and supports learning. The materials support teachers in learning and understanding the standards. All materials include support for teachers in using questions to guide mathematical development. Teacher editions have many annotations and examples on how to present the content and an explanation of the mathematics of each unit and the program as a whole.

A baseline assessment allows teachers to gather information on student's prior knowledge, and teachers are offered support in identifying and addressing common student errors and misconceptions. Materials include opportunities for ongoing review and practice. All assessments include information on standards alignment and scoring rubrics. There are no systems or suggestions for students to monitor their own progress. Activities provide ELL strategies, support strategies, challenge strategies, and grouping strategies to assist with differentiating instruction. A chart at the beginning of each unit indicates places in the instructional materials where suggestions for differentiating instruction can be found. Most activities allow opportunities for differentiation. The materials provide many grouping strategies and opportunities. Support and intervention materials are also available online.

All of the instructional materials available in print are also available online. Additionally, the Bridges website offers additional resources such as Whiteboard files, interactive tools, virtual manipulatives, and teacher blogs. Digital resources, however, do not provide additional technology-based, assessment opportunities, and the digital resources are not easily customized for individual learners. Overall, the materials meet the expectations for usability. ) [27] => stdClass Object ( [code] => 3a3e [type] => criterion [report] =>

Materials are well-designed, and lessons are intentionally sequenced. Typically students learn new mathematics in the Problems & Investigations portion of Sessions while they apply the mathematics and work towards mastery during the Work Station portion of Sessions and Number Corner. Students produce a variety of types of answers including both verbal and written answers. Manipulatives such as 10-frames, craft sticks, and Unifix cubes are used throughout the instructional materials as mathematical representations and to build conceptual understanding.

) [28] => stdClass Object ( [code] => 3a [type] => indicator [points] => 2 [rating] => meets [report] =>

The sessions within the units distinguish the problems and exercises clearly. In general, students are learning new mathematics in the Problems & Investigations portion of each Session. Students are provided the opportunity to apply the mathematics and work toward mastery during the Work Station portion of the session as well as in daily Number Corners.

For example, in Unit 5, Module 1 in Session 3, students initially learn about the trapezoid. They work with the teacher to describe a trapezoid, with a focus on using accurate and precise geometrical language. Students then work with the teacher using pattern blocks to fill in a shape three different ways and discuss the differences, the number of blocks used, and the composition of new shapes. Students are introduced to the Pattern Block Puzzle. They observe a sheet with the puzzle and work together to fill in the shape with various pattern blocks. The teacher uses student input from the class to record the solution to solving the puzzle. Students continue in this manner until they have filled in the pattern three different ways and solutions are recorded. In the Work Places activity, students work independently to complete additional Pattern Block Puzzles.

In the February Calendar Grid, students observe various shapes and report out the number of sides and vertices on each of the shapes. Then, students confirm that these shapes are, in fact, triangles. They fill out the Calendar Grid Observation Chart, adding the triangle and it's attributes to the chart.

) [29] => stdClass Object ( [code] => 3b [type] => indicator [points] => 2 [rating] => meets [report] =>

The assignments are intentionally sequenced, moving from introducing a skill to developing that skill and finally mastering the skill. After mastery, the skill is continued to be reviewed, practiced and extended throughout the year.

The "Skills Across Grade Level" table is present at the beginning of each Unit. This table shows the major skills and concepts addressed in the Unit. The table also provides information about how these skills are addressed elsewhere in the grade, including Number Corner, and in the grade that follows. Finally, the table indicates if the skill is introduced (I), developed (D), expected to be mastered (M), or reviewed, practiced or extended to higher levels (R/E).

Concepts are developed and investigated in daily lessons and are reinforced through independent and guided activities in Work Places. Number Corner, which incorporates the same daily routines each month (not all on the same day) has a spiraling component that reinforces and builds on previous learning. Assignments, both in class and for homework, directly correlate to the lesson being investigated within the unit.

The sequence of the assignments is placed in an intentional manner. First, students complete tasks as a whole group in a teacher-directed setting. Then students are given opportunities to share their strategies used in the tasks completed in the Problems & Investigations. The Work Places activities are done in small groups or partners to complete tasks that are based on the problems done as a whole group in the Problems & Investigations. The students then are given tasks that build on the Session skills learned for the Home Connections.

For example, 1.OA.1 and 1.OA.2 are introduced in Unit 2 and continue to be developed in Units 3 and 4. Mastery is expected in Units 6 and 7. The standard is developed further in Number Corners in October, January, and February. Another example is 1.NBT.3. This standard is Introduced in Unit 2 and continues to be developed in Units 4, 6, 7 and 8 and Number Corners in October, November, January and February.

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There is variety in what students are asked to produce. Throughout the grade, students are asked to respond and produce in various manners, often by working with concrete and moving to more abstract models as well as verbally explaining their strategies. Students are asked to produce written evidence using drawings, representations of tools or equations along with a verbal explanation to defend and make their thinking visible.

For example, in Unit 5, Module 2, Session 4, students are given many opportunities to work with 3-dimensional shapes concretely, abstractly, verbally, and then back to concrete by actually creating cubes using Polydrons. First, students play the game "Guess My Shape." They initially observe 3-dimensional shape cards and geoblocks, and then they match up the geoblocks with the 3-dimensional shape cards, naming the shapes as they go. Next, the teacher gives a series of clues to students as they listen carefully and act like detectives to guess the teacher's secret shape. The clues are characteristics of the shape. As clues are given, students remove the shapes that don't match the clues, ending up with the last shape which should be the secret shape. The next activity requires students to identify a cube just by touch as they place their hand in a bag and feel for the cube. Students are asked how they identified the cube, and the teacher models the vocabulary of geometry as students share. For example, as student share "There are a lot of corners on the cube," the teacher replies, "In math, we call the corners, vertices." Then students construct cubes using Polydrons. Finally, students work with a worksheet to identify pictures of Polydrons and determine if they can make a cube. They first make a prediction and then use the actual Polydrons to test their predictions.

Also, in Unit 3, Module 3, Session 1, "Ten & Some More," students are working with activities that focus on the teen numbers. They move from concrete to abstract and explain their strategies verbally. Students look at double 10-frame cards with matching equations, build the sum with Unifix cubes, and write their own equations. Next, the teacher presents students with a teen number, and students imagine what it will look like with Unifix cubes and then write an equation to match.

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Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods. Manipulatives are used and provided to represent mathematical representations and provide opportunities to build conceptual understanding. Some examples are the 10-frames, number lines, Unifix cubes, number racks, coins, craft sticks and tiles. When appropriate, they are connected to written representations.

For example, in Unit 2, Module 3, Session 1, in the Problems & Investigation section, students are using dominoes and number racks to write and solve addition problems. Students begin with having the dominoes flashed at them then use their number racks to represent the number of dots shown. Finally, they write an equation to represent the amounts.

Although manipulatives are faithful representations, there are some that are unusual and difficult to understand representations.

For example, In Unit 7, Module 2 is called "Hansel & Gretel's Path on the Number Line." The symbols/images used are difficult representations of the number line. A picture is used to display pebbles, breadcrumbs, and pinecones on a path. Students are asked to fill in the empty boxes on the path or "Number Line." There is a key that shows the symbols that represent breadcrumbs (every 1 step), pinecones (every 5 steps), and pebbles (every 10 steps). The images may be confusing for students, and this theme continues for all of Module 2.

Also, in Unit 7, Module 3 uses fences, benches, trash cans, and flowerpots to have students write and solve addition equations that represent how long each path section is. As in Module 2, a key is used to show how many steps each image represents. The choice and use of images are difficult to understand and make the connection to a number line difficult to make.

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The material is not distracting and does support the students in engaging thoughtfully with the mathematical concepts presented. The visual design of the materials is organized and enables students to make sense of the task at hand. The font, size of print, amount of written directions and language used on student pages is appropriate for Grade 1. The visual design is used to enhance the mathematics problems and skills demonstrated on each page. The pictures match the concepts addressed such as having the characters that are in the story problems placed in picture format on the page as well. Some problems may even require students to use the pictures to solve the story problems.

For example, in the Number Corner April, "Days In School", students are working with two separate hundreds grids to mark and represent the number of days they have been in school. The Day 2, Activity 1 task has students marking another day on the second hundreds grid, and then the teacher represents the total number of days in digit form, word form and expanded form. The charts are easy to see and read using a the familiar hundreds grid and a large, clear chart for expanded notation.

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The instructional materials support teacher learning and understanding of the standards. The instructional materials provide questions and discourse that support teachers in providing quality instruction. The teacher's edition is easy to use and consistently organized and annotated. The teacher's edition explains the mathematics in each unit as well as the role of the grade-level mathematics within the program as a whole. The instructional materials are all aligned to the standards, and the instructional approaches and philosophy of the program are clearly explained.

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Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development. Lessons provide teachers with guiding questions to elicit student understanding and conduct discourse that allows student thinking to be visible. Discussion questions provide a context for students to communicate generalizations, find patterns, and draw conclusions.

Each unit has a Sessions page, which is the Daily Lesson Plan. The materials have quality questions throughout most lessons. Most questions are open-ended and prompt students to higher-level thinking.

In Unit 1, Module 2, Session 1, the teacher is prompted to ask the following questions:
  • "How many red beads do you see? How many white?"
  • "I heard you counting. Can you come show us on the big rack how you were counting those beads? And can you show us how you knew there were the same number of white as red?""
In Unit 3, Module 1, Session 2, students are working on number lines. The teacher is prompted to ask the following questions:
  • "What if you could choose - you could jump by either a 5 or 10. Would that make the problem any shorter?
  • Could we use fewer jumps to get from 0 to our target of 15?"
  • "How do you know that?"
  • "Does anyone have a different solution?"

In Unit 4, Module 1, Session 4, students are working on measuring using their "Inchworm Rulers." The teacher asks the following questions:

  • "How long do you think each strip will be when it's cut out? Why?"
  • "Will they stretch out the length of the poster board strip? How do you know?"
  • "Who is likely to use this tool, and when?"
  • "How does a ruler help us measure the length of something?"
In Unit 5, Module 2, Session 1 teachers are prompted to ask the following questions:
  • "Who can tell us what they saw?"
  • "Did anyone else see something different, or have a different way to describe what they saw?"

In Number Corner December Calendar Grid, the following questions are provided to help students as they work with shapes:

  • "What shape do you think you'll see on the next marker - why?"
  • "If there were 32 days in December, what kind of shape would be on marker 32? How do you know?"
  • "What observations can you make about today's shape?"
  • "How many sides does it have?"
  • "How many vertices?"
) [35] => stdClass Object ( [code] => 3g [type] => indicator [points] => 1 [rating] => partially-meets [report] =>

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials; however, additional teacher guidance for the use of embedded technology to support and enhance student learning is needed.

There is ample support within the Bridges material to assist teachers in presenting the materials. Teacher editions provide directions and sample scripts to guide conversations. Annotations in the margins offer connections to the MPs and additional information to build teacher understanding of the mathematical relevance of the lesson.

Each of the eight units also have an Introductory section that describes the mathematical content of the unit and includes charts for teacher planning. Teachers are given an overview of mathematical background, instructional sequence, and the ways that the materials relate to what the students have already learned and what they will learn in the future units and grade levels. There is a Unit Planner, Skills Across the Grade Levels Chart, Assessment Chart, Differentiation Chart, Module Planner, Materials Preparation Chart. Each unit has a sessions page, which is the Daily Lesson Plan.

The Sessions contain:

  • sample teacher/student dialogue;
  • Math Practices In Action icons as a sidebar within the sessions - These sidebars provide information on what MP is connected to the activity;
  • a Literature Connection sidebar - These sidebars list suggested read-alouds that go with each session;
  • ELL/Challenge/Support notations where applicable throughout the sessions;
  • A Vocabulary section within each session - This section contains vocabulary that is pertinent to the lesson and indicators showing which words have available vocabulary cards online.

Technology is referenced in the margin notes within lessons and the notes suggests teachers go to the online resource. Although there are no embedded technology links within the lessons, there are technology resources available on the Bridges Online Resource page such as videos, whiteboard files, apps, blogs, and online resource links (virtual manipulatives, images, teacher tip articles, games, references). However, teacher guidance on how to incorporate these resources is lacking within the materials. It would be very beneficial if the technology links were embedded within each session, where applicable, instead of only in the online teacher resource. For instance, the teacher materials would be enhanced if a teacher could click on the embedded link, (if using the online teacher manual) and get to the Whiteboard flipchart and/or the virtual manipulatives.

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Materials contain adult-level explanations of the mathematics concepts contained in each unit. The introduction to each unit provides the mathematical background for the unit concepts, the relevance of the models and representations within the unit, and teaching tips. When applicable to the unit content, the introduction will describe the algebra connection within the unit.

At the beginning of each unit, the teacher's edition contains a "Mathematical Background" section. This includes the mathematics concepts addressed in the unit. For example, in Unit 2 the following is provided: "Throughout Unit 2, students explore base ten concepts and models within 1,000. The unit is designed to promote measuring concepts even as students are learning about our base ten number system. As they count, total, and compare units, they are encouraged to think about and apply base ten concepts."

The mathematical background also includes sample models with diagrams and explanations, strategies, and algebra connections. There is also a Teaching Tips section following the Mathematical Background that give explanations of routines within the sessions such as: "Like any mathematical tool, the more teachers are aware of both the benefits and constraints of the number line, the more likely they are to use it effectively with students..."

In the Implementation section of the Online Resources, there is a "Math Coach" tab that provides the Implementation Guide, Scope & Sequence, Unpacked Content, and CCSS Focus for Grade 1 Mathematics.

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Materials contain a teacher’s edition (in print or clearly distinguished/accessible as a teacher’s edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

In the Unit 1 binder, there is a section called "Introducing Bridges in Mathematics." In this section, there is an overview of the components in a day (Problems & Investigations, Work Places, Assessments, Number Corner). Then there is an explanation of the Mathematical Emphasis in the program. Content, Practices, and Models are explained with pictures, examples and explanations. There is a chart that breaks down the mathematical practices and the characteristics of children in that grade level for each of the MPs. There is an explanation of the Skills Across the Grade Levels chart, the assessments chart, and the differentiation chart to assist teachers with the use of these resources. The same explanations are available on the website. There are explanations in the Assessment Guide that go into the Types of Assessments in Bridges Sessions and Number Corner.

The CCSSM Where to Focus Grade 1 Mathematics document is provided in the Implementation section of the Online Resources. This document lists the progression of the major work in grades K-8.

Each unit introduction outlines the standards within the unit. A “Skills Across the Grade Level” table provides information about the coherence of the mathematics standards that are addressed in the previous grade as well as in the following grade. The "Skills Across the Grade Level" document at the beginning of each Unit is a table that shows the major skills and concepts addressed in the Unit and where that skill and concept is addressed in the curriculum in the previous grade as well as in the following grade.

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The materials provide a list of lessons in the teacher's edition cross-referencing the standards covered and providing an estimated instructional time for each lesson and unit. The "Scope and Sequence" chart lists all modules and units, the CCSS standards covered in each unit, and a time frame for each unit. There is a separate "Scope and Sequence" chart for Number Corners.

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Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

Home connection materials and games sometimes include a “Note to Families” to inform them of the mathematics being learned within the unit of study.

Additional Family Resources are found at the Bridges Educator's Site.

  • Grade 1 Family Welcome letter in English and Spanish - This letter introduces families to Bridges in Mathematics, welcomes them back to school, and contains a broad overview of the year's mathematical study.
  • Grade 1 Unit Overviews for Units 1-8, in English and Spanish.
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Materials contain explanations of the instructional approaches of the program. In the beginning of the Unit 1 binder, there is an overview of the philosophy of this curriculum and the components included in the curriculum. There is a correlation of the CCSSM and MPs as the core of the curriculum in the mathematical emphasis section. The assessment philosophy is given in the beginning of the Assessment binder. The types of assessments and their purpose is laid out for teachers. In the Introductory section of Unit 1, there is a reference to the number rack and research as follows: "Research has consistently identified the importance of helping students visualize number quantities. The number rack is instrumental in this endeavor, allowing them to 'see inside' numbers."

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The instructional materials offer teachers resources and tools to collect ongoing data about student progress. The September Number Corner Baseline Assessment allows teachers to gather information on student's prior knowledge, and the Comprehensive Growth Assessment can be used as a baseline, quarterly, and summative assessment. Checkpoint interviews and informal observation are included throughout the instructional materials. Throughout the materials, Support sections provide common misconceptions and strategies for addressing common errors and misconceptions. Opportunities to review and practice are provided in both the Sessions and Number Corner routines. Checkpoints, Check-ups, Comprehensive Growth Assessment, and Baseline Assessments clearly indicate the standards being assessed and include rubrics and scoring guidelines. There are, however, limited opportunities for students to monitor their own progress.

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Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

The September Number Corner Baseline Assessment is designed to ascertain students' current levels of skills. The Comprehensive Growth Assessment contains interview items and written items and addresses every Common Core standard for Grade 1. The Comprehensive Growth Assessment can be administered as a baseline assessment, an end-of-the-year summative assessment or quarterly assessment to monitor students' progress.

Informal observation is used to gather information. Many of the Sessions and Number Corner workouts open with a question prompt: a chart, visual display, a problem, or even a new game board. Students are asked to share comments and observations, first in pairs and then as a whole class. This gives the teacher an opportunity to check for prior knowledge, address misconceptions, as well as review and practice with teacher feedback. There are daily opportunities for observation of students during whole group and small group work as well as independent work when they work in Work Places.

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Materials provide strategies for teachers to identify and address common student errors and misconceptions.

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