Alignment

Focus & Coherence

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Rigor & Mathematical Practices

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Meets Expectations
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Usability

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Alignment

The instructional materials reviewed for the CPM Integrated series meet expectations for alignment to the CCSSM for high school. The materials meet the expectations for focus and coherence. The materials attend to the full intent of the mathematical content standards and also attend fully to the modeling process when applied to the modeling standards. The materials also meet the expectations for rigor and the Mathematical Practices as they reflect the balances in the Standards and help students meet the Standards’ rigorous expectations and meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice.

GATEWAY ONE

Focus & Coherence

MEETS EXPECTATIONS

Criterion 1a-1f

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Focus and Coherence: The instructional materials are coherent and consistent with "the high school standards that specify the mathematics which all students should study in order to be college and career ready" (p. 57 of CCSSM).

The instructional materials reviewed for the High School CPM Integrated series meet the expectation for focusing on the non-plus standards of the CCSSM and exhibiting coherence within and across courses. Overall, the instructional series attends to the full intent of the non-plus standards and the modeling process, spends a majority of time on the widely applicable prerequisites from the CCSSM, and requires students to engage at a level of sophistication appropriate to high school.

Indicator 1a

The materials focus on the high school standards.*

4/4
Indicator 1a.i

The materials attend to the full intent of the mathematical content contained in the high school standards for all students.

The instructional materials reviewed for the High School CPM Integrated series meet the expectations for attending to the full intent of the mathematical content contained in the high school standards for all students. The series is designed to spiral. Overall, all of the standards are addressed within the Integrated I, Integrated II, and Integrated III courses.

Most standards are addressed to the full intent throughout the entire series. The following are examples of standards addressed in the series:

  • A-APR.1 is addressed in Integrated II and Integrated III. In Integrated II, Lesson 4.1.4 provides evidence explaining how integers are closed under the operations of addition, subtraction, and multiplication; Lesson 4.1.5 has students determine whether polynomials are also closed under these operations by creating several examples to support this notion and then generalize beyond the examples. In Integrated III, Lesson 8.3.1 has students divide polynomials.
  • A-REI.3 is addressed in Integrated I. Lesson 3.3.1 utilizes several methods for solving linear equations: rewriting, undoing, and looking inside. Also in Integrated I, Lessons 9.1.1 and 9.1.2 have students solve linear inequalities.

The Number and Quantity domain occurred throughout Integrated I and Integrated II; however, N-Q.1 was only partially addressed. Students were not explicitly expected to use units as a guide to solve problems. Also, students were not required to interpret the origin in every data display.

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Indicator 1a.ii

The materials attend to the full intent of the modeling process when applied to the modeling standards.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials attend to the full intent of the modeling process when applied to the modeling standards. The series includes modeling tasks throughout the materials. Frequently, tasks include significant scaffolding or support to focus students on specific mathematics, but scaffolding of modeling tasks decreases within a course and over the series helping to develop students' abilities to work with modeling tasks. In the series, students have opportunities to develop their own solution strategies, select the best tools for solving a problem or set of problems, create their own charts, graphs, and/or equations, evaluate and revise answers, and report on their work.

  • In Lesson 2.1.3 of Integrated I, the problem “How Steep Is It?” has students use a model, stairs, to represent the slope of a function. This problem asks students to make, use, and describe a model, but it does not engage them in the full modeling process as defined by the Modeling, High School Progressions Document. This problem addresses standards F-IF.4, F-IF.6, F-IF.7a, F-LE.1a, F-LE.2 and F-LE.5.
  • In Integrated I, “The Big Race - Finals” problem in Lesson 2.2.3 has students engage in parts of the modeling process, such as defining variables, interpreting data, validating their conclusions, and reporting out to other teams. However, the students do not come to develop the question themselves, and they do not collect the data for the investigation themselves. All of this is explicitly given to them by the materials. This problem addresses standards N-Q.2, A-CED.2, F-IF.4, F-IF.7a, F-BF.1a, F-LE.1b, F-LE.2 and F-LE.5.
  • In “The Burning Candle” problem in Lesson 11.2.3 of Integrated I, students are presented a real-life situation that should be engaging for Integrated I students. The students are asked to design an experiment, collect data and analyze data in order to predict how long a birthday candle will stay lit. The problem appropriately engages students in all aspects of the modeling process. This problem effectively engages students in applying the modeling process to standards N-Q.2, N-Q.3, A-CED.1, F-IF.7a, F-BF.1a, F-LE.2, S-ID.6a, S-ID.6c, S-ID.7 and S-ID.8
  • In Integrated II, Lesson 1.2.1 uses a bracelet task to have students perform an experiment and then collect, record and analyze data. Next, they are prompted to modify the experiment, collect, record and analyze new data, and compare the new data set to their first set of results. Finally, the students are prompted to design their own experiment “spin off” of the original and then collect, record and analyze their data. If the lesson is followed through to the end, every aspect of the modeling process would be completed by the students. Right below this task is a flowchart example of modeling with mathematics.
  • In Integrated II, “Standards to Maintain” the “Shrinking Targets Lab” in Lesson 9.1.1 has students define variables, collect and analyze data and then use their data to extrapolate. They are provided a significant level of support, but they are still actively engaged in the modeling process. This lab addresses standards A-CED.2, F-IF.4, F-IF.5, F-IF.7a and F-BF.1a.
  • In Integrated III, Lesson 6.1.1 has students use coins to model whether a child is born a girl or a boy. The students design the experiment and then record and analyze their results. They are given many parameters that prevent students from determining their own variables. The students share and compare their data with other teams, and they also compile their data and analyze if/how the data changes when the sample size is larger. This lesson addresses standards S-IC.2 and S-MD.6(+).
  • In Integrated III, Lesson 6.2.1 has students design a computer simulation to model a real-life situation and then collect and analyze data. This lesson addresses standards S-IC.1, S-IC.4 and S-IC.5.
  • In Integrated III, Lesson 9.1.1 (F-TF.5) introduces a task entitled "Emergency!" The experiment procedure is outlined in the materials. The specific directions provided allow for students to focus on the appropriate mathematics and do not detract from the modeling process. Students are asked guiding questions that require them to develop their own strategies for solving the problem and reflect on the difference between their process and the process of others.
Indicator 1b

The materials provide students with opportunities to work with all high school standards and do not distract students with prerequisite or additional topics.

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Indicator 1b.i

The materials, when used as designed, allow 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.

The materials, when used as designed, meet the expectation 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).

The materials show a strong focus on widely applicable prerequisites.

  • The majority of the lessons in Integrated I focus on the WAPs. Chapters 1, 2, 3, 5, 6, 7, 8, 9 and 11 had lessons in which the majority of the time was spent on the WAPs. There are some lessons that review middle school mathematics standards, but this does not occur in a way that is distracting or in a way that takes time away from the WAPs. For example Section 1.3 is projected to last approximately two days and reviews rewriting expression with integer exponents (8.EE.1). Also, Appendix A provides review activities for rewriting expressions (7.EE.1). These sections provide opportunities to support struggling learners and clear up misconceptions but could easily be omitted if not needed by the students.
  • The majority of the lessons in Integrated II are spent on the WAPs. Chapters 1, 2, 3, 4, 5, 6, 7 and 9 included lessons which were primarily focused on the WAPs. Chapters 8, 10, 11 and 12 are not focused on the WAPs, but their content does fit the flow of content through the materials. Section 1 of Chapter 1 spends time reviewing attributes of polygons which aligns to middle school standards.
  • In Integrated III, approximately half of the lessons focus on WAP standards. The progression and flow of the materials are logical and support a deep understanding of the mathematics. Chapters 1, 2, 3, 5 and 7 included lessons in which the majority of the work was related to the WAPs.

Overall, the majority of student time is spent on the widely applicable prerequisites.

2/4
Indicator 1b.ii

The materials, when used as designed, allow students to fully learn each standard.

The materials, when used as designed, partially meet the expectation that students are provided with opportunities to fully learn each non-plus standard. Overall, the lessons are structured in a way that students will fully learn all aspects of most standards and do not distract students with prerequisite or additional topics. However, there are a few missed opportunities for students to make every connection and fully learn all aspects of every standard.

  • A-SSE.3b: There are several instances where the student is asked to complete the square and to find the vertex. For example, in Integrated II, Lesson 9.1.3 has students complete the square to find the vertex, but there is no mention of maximum or minimum.
  • A-REI.4a: Problem 6-79 in Integrated II outlines the derivation of the Quadratic Formula by completing the square. The materials suggest students should follow the algebraic steps and explain how each step is justified.
  • A-REI.7: In Integrated II, Lesson 10.1.1 has one example where students find the intersection between a line and a circle. The intersection is found graphically, not algebraically. In Integrated III, Lesson 3.1.3 has two examples in which students graph a system consisting of a parabola and a circle.
  • G-CO.2: Although Lessons 3.1.1-3.1.6 from Integrated I are aligned to this standard, there is no explicit instruction about functions that take points in the plane as inputs and give points as outputs, but there are problems for students to solve in the homework (8-49 and 9-63 from Integrated I).
  • G-CO.13: Lessons and problems provide limited opportunities for students to construct a square inscribed in a circle.
  • G-GPE.6: There were a limited number of problems that partitioned segments in a ratio other than 1:1.
  • F-IF.2: Students use function notation and evaluate functions for the inputs in their domains. Problems requiring students to interpret statements that use function notation in terms of a context such as problem 2-53 in Integrated I are infrequent.
  • F-IF.6: There was limited evidence found of finding the average rate of change in non-linear situations. In Integrated II, Lessons 9.3.1 and 9.3.2 and problems 9-73 and 9-74 have students work with parabolas. In problem 9-94, students work with linear, quadratic and exponential situations. These questions and activities have students calculate and interpret the average rate of change of a function over a specified interval. These problems do not require estimation or interpretation. In Integrated III, lesson 2.2.5 and problem 2-111 have students work with non-linear situations with graphs and tables and require them to do some interpretation.
  • F-IF.4 and F-IF.7c: Key features are interpreted throughout the series; however, there was little evidence of the actual verbiage of "end behavior" used.
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Indicator 1c

The materials require students to engage in mathematics at a level of sophistication appropriate to high school.

The instructional materials reviewed fully meet the expectations for students to engage in mathematics at a level of sophistication appropriate to high school. The materials give extensive opportunities to work with course-level problems and exercises appropriate to high school and relate new concepts to students' prior skills and knowledge.

The Universal Access section allows all students an opportunity for entry points to the content. There is appropriate guidance for the teacher to help scaffold for different students, and the level of scaffolding and support is appropriate and does not impede students from engaging in the full intent of the mathematics.

Contextual problems are appropriate for high school students. Several contextual problems complement content that students learn in other core classes, such as farming and sustainability, exercise, and genetics.

Scenarios presented in application problems are authentic, as well as adjustable to different interests. Examples of authentic application and/or real world problems include the following:

  • In Integrated I, Lesson 1.1.2 uses three contextual scenarios, placement of tiles in a yard, modeling the spread of a flu epidemic, and time it takes to sign your name on multiple documents, in order to teach growth of patterns.
  • In Integrated II, Lesson 3.1.3 uses a rock-paper-scissors game with scoring rules which may or may not be fair. Students have to decide the fairness based on their knowledge of probability models.
  • In Integrated III, Lesson 6.2.1 focuses on statistical testing using sampling variability. The lesson poses a question of whether students support keeping or canceling a winter formal dance.

Students work with appropriate numbers for high school and see a wide variety in equation/expression formats.

2/2
Indicator 1d

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.

The instructional materials reviewed meet the expectations that the materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and required by the standards.

All conceptual categories are addressed over each of the courses. Each course contains work in number and quantity, algebra, functions, geometry, and statistics and probability. Topics are addressed when they are developmentally appropriate. More concrete ideas are examined in Integrated I while more abstract ideas are examined in Integrated III. The progression of difficulty is logical.

The following are examples of connections made between the books in the series:

  • Connections are made throughout every course in the Review & Preview portion of every Section. These problems connect with prior work in both the current course and past courses (if any), the current topic, and future topics (usually using the preview problems to review skills and concepts for work that is immediately upcoming).
  • In Integrated II, Chapter 2 provides students the opportunity to make connections from their work on congruence in Integrated I with a brief review of congruence theorems to the work that they will do with similarity and dilations.

The following are examples of connections made within the books in the series:

  • In Integrated I, Chapter 7 starts by engaging students in what it means for two figures to be congruent, and then it engages in determining the least amount of information needed for proving two figures to be congruent. It proves triangle congruence criteria using rigid motions. Then it has the opportunity to connect that work to the coordinate plane. The students study polygons on the coordinate grid by proving statements about the figures using coordinate geometry and relationships for distance, slope, and midpoint.
  • In Integrated II Chapter 9, Modeling with Functions, the study of quadratic associations in statistics and probability builds on students' understanding of quadratic relationships, from Chapters 2, 5 and 6.

The materials are designed to spiral concepts throughout the chapters and courses. Some topics included within the same chapter are disconnected. These were placed this way intentionally to allow students more time with the first concepts in Review & Preview before the concept is developed further in a future chapter.

  • In Integrated III, Chapter 7 is on logarithms and triangles. The connection between logarithms and triangles is not evident.
  • In Integrated II, Chapter 7 is about Proof and Conditional Probability, and Chapter 3 is about Probability and Trigonometry. The connection between these topics is not evident.
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Indicator 1e

The materials explicitly identify and build on knowledge from Grades 6--8 to the High School Standards.

The instructional materials reviewed partially meet the expectations that the series explicitly identify and build on knowledge from Grades 6-8 to the High School Standards. The content that is identified as being from previous grades is appropriate and develops as a natural progression into high school, but it is not always clearly connected to a specific middle school standard.

The following are examples of connections made between content from Grades 6-8 and the high school content:

Integrated I:

  • Chapter 1 begins with a comment that "in previous courses you may have learned about relationships between two quantities that could be graphed using a straight line." It goes on to discuss what will be learned in this chapter. Also, the Overview of the Chapter states that Section 1.3 should be a review from Grade 8. Students are reminded that "in previous courses, you might have looked at patterns, in tables, graphs, equations, and situations that were linear."
  • Before Chapter 2, it is highly recommended to look through Appendix A to determine if the students need the additional lessons before continuing. In the overview of the chapter for Chapter 2, the Teacher Edition states that the chapter will build on the study of linear functions from previous courses. Chapter 2 is mostly middle grades work that examines patterns of growth, slope, and writing and using linear equations that aligns to 8.F.4 and 8.F.5, with a few exceptions and without being identified. Lesson 2.2.4 teaches unit conversion with ratios (6.RP.3d). As students progress through Chapter 2, connections are made to F-BF.1a, F-LE.1a,b and N-Q.1,2.
  • Lesson 3.1.1 uses the notions of reflections, rotations, translations (8.G.1), and nets (6.G.4) to introduce ideas of visualization in three dimensions (G-GMD.4) but does not identify the middle grade standard.
  • In Chapter 3, Lesson 3.2.1 mentions that in previous courses, students worked with areas and perimeters of shapes made up of different rectangles.
  • The Overview of the Chapter for Chapter 7 states that students start the chapter by reviewing what they know from previous courses about similarity. There is also a Math Notes box that reviews similar triangles and scale factor.
  • The teacher's notes for Lesson 10.1.2 state that, in previous courses, students will have computed mean and five-number summaries and will have described the shape of a distribution. They will have also chosen between mean and median based on the shape of the distribution and will have calculated the mean absolute deviation and interquartile range.
  • Appendix A is appropriately identified as middle school content. Specific standards are not presented.

Integrated II:

  • In Chapter 1, Lesson 1.1.1 states that in previous courses students have studied polygons like triangles and quadrilaterals. Lessons 1.1.1-1.1.2 have students composing polygons and examining attributes of polygons. The material is reminiscent of various elementary standards, such as 3.G.1, 4.G.2 and 5.G.3. No standards of any level are identified for these lessons. Lesson 1.3.1 reviews angle pairs (complementary, supplementary, vertical, linear pairs - 7.G.5). The work supports the high school work but is not identified as middle school. Lessons 1.3.2 and 1.3.3 do the same with angles formed by parallel lines intersected by a transversal (8.G.5).
  • The Chapter 3 Overview states that students will be expanding their understanding of simple probability studied in middle school. Lesson 3.1.1 states that in previous courses students studied probability.
  • In Chapter 5, Lesson 5.1.1 states that in previous courses students have investigated linear and exponential functions.
  • In Chapter 6, Lesson 6.1.4 states that students learned about the laws of exponents in a previous course.
Indicator 1f

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.

The instructional materials reviewed for the High School CPM Integrated series explicitly identify the plus standards, when included, and the plus standards coherently support the mathematics which all students should study in order to be college- and career-ready.

In most cases, it would be difficult to separate out the plus standards material from the non-plus standards. However, work with the plus standards does not deter from the work with the non-plus standards. An example of where it could easily be separated is in Integrated II, Chapter 12, which is comprised of primarily plus standards; this section could be omitted. The materials, however, offer no guidance or pacing suggestions should teachers not wish to use the plus standards.

The plus standards that are identified in the teacher resource binder and addressed to reach the full intent of the standard are listed below:

  • Integrated I does not address any plus standards.
  • Integrated II: N-CN.8 (Lessons 5.2.6 and 6.2.4); N-CN.9 (Lesson 6.2.4); G-C.4 (Lesson 10.2.5); S-CP.8 (Lesson 7.2.3); S-CP.9 (Lessons 12.1.1-12.1.4 and 12.2.4); S-MD.6 (Lessons 3.1.5, 12.1.1-12.1.3 and 12.2.4); S-MD.7 (Lessons 6.2.6, 7.2.3, 12.1.1, 12.2.1 and 12.2.4)
  • Integrated III: N-CN.8 (Lessons 8.2.2 and 8.3.4); N-CN.9 (Lessons 8.1.1, 8.1.2, 8.2.1, 8.2.2, and 8.3.2); A-APR.5 (Lessons 10.3.1-10.3.3); A-APR.7 (Lessons 11.1-11.1.4 ); F-TF.9 (Lessons 12.2.1, 12.2.2 and 12.2.3); G-SRT.9 (Lessons 7.2.1 and 7.2.2); G-SRT.10 (Lessons 7.2.1-7.2.4); G-SRT.11 (Lesson 7.2.5); S-MD.6 (Lessons 6.1.1, 6.1.2 and 6.3.1); S-MD.7 (Lesson 6.2.3)

One plus standard is assessed on checkpoints. Standard N-CN.8 is assessed on Integrated III Checkpoint 11.

GATEWAY TWO

Rigor And Mathematical Practices

MEETS EXPECTATIONS

Rigor and Balance
MEETS EXPECTATIONS

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The instructional materials reviewed for the High School CPM Integrated series meet the expectations for rigor and balance. Overall, all three elements of rigor are thoroughly attended to and interwoven in a way that focuses on addressing specific standards as well as balancing procedural skill and fluency, application, and conceptual understanding within individual courses and across the series as a whole.

Criterion 2a-2d

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Rigor and Balance: The instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by giving appropriate attention to: developing students' conceptual understanding; procedural skill and fluency; and engaging applications.

2/2
Indicator 2a

Attention to Conceptual Understanding: The materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

The instructional materials reviewed for the High School CPM Integrated 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. Overall, the clusters and standards that specifically relate to conceptual understandings are thoroughly addressed.

Most of the lessons across the series are exploratory in nature and encourage students to develop understanding through questioning and activities. Each chapter has a closure section that recaps the concepts of the chapter. It includes reflections on and synthesis of the connections to what the learning targets were for the chapter.

Some examples of the intentional development of conceptual understanding are as follows.

  • In Integrated II, Lesson 3.1.1 introduces students to probability area models by analyzing the frequency of inherited traits. This lesson builds on concepts that would be introduced and developed in a middle school science course and relates that knowledge to the new mathematics concept. The materials begin with simple scenarios and guide students to more complex situations to apply area models.
  • Lesson 8.3.1 in Integrated II, which contains the Mighty Mascot problem, provides students with a simple real world example of scale factor to which students can easily visualize and relate. Then, the students are asked to make observations and look for patterns in the relationships between scale factor, area, and perimeter. Finally, the students are asked to apply these observations and patterns to other situations.
  • In Integrated III, Chapter 6, Simulating Sampling Variability, begins with simple probability examples that students should be familiar with (tossing coins, playing cards) to introduce the new concept. Then, the materials expose students to more complicated situations in which to apply the concept.
  • In Integrated III, Lesson 10.1.1 helps students develop the concept of arithmetic sequences and prepares them to determine the sums of arithmetic sequences. To introduce this concept, the materials begin with the real world situation of saving money for college, a topic very appropriate for this age group. As they build on this basic concept, they provide students with multiple visual interpretations to deepen conceptual understanding and prevent misconceptions.

Examples of select cluster(s) or standard(s) that specifically relate to conceptual understanding include, but are not limited to:

  • A.REI.1: Students have the opportunity to conceptualize this standard in Integrated I, Lessons 3.3.1, 3.3.2 and 3.3.3. In these lessons, students examine the connections between different methods of solving the same equation and construct arguments to show that the two methods are equivalent.
  • F-IF.A: In the first chapter of Integrated I, students develop the idea of what is a function and what is not a function through questions, exercises and diagrams.
  • F-LE.1: In Integrated I, Lesson 2.1.1 provides students with a picture of a tile pattern to help them develop a deep understanding of the relationship between arithmetic sequences and linear equations. The students are prompted to analyze the pattern, extend the pattern, make an extrapolation and summarize these observations by developing a linear equation. The way the activity provides students with a simple visual and guides them through the discovery process will help them develop a deep understanding of the concept. The next problem guides students through the same process but with a more complicated tile pattern.
  • G-SRT.6: In Integrated II, Lesson 3.2.1 introduces students to constant ratios in right triangles by starting with a problem regarding the Leaning Tower of Pisa. Then, the students are guided to use graph paper to model the real-life situation.
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Indicator 2b

Attention to Procedural Skill and Fluency: The materials provide intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that materials provide intentional opportunities for students to develop procedural skills and fluency, especially where called for in specific content standards or clusters. The clusters and standards that specifically relate to procedural skills and fluency are thoroughly addressed multiple times. The materials develop procedural skills and fluency across the series.

The exercises have been thoughtfully placed in a progression of learning that provides students the opportunity to make connections between topics and to "build procedural fluency from conceptual development." The instructional materials are set up so that students review and preview throughout each chapter, connect skills learned in that chapter with skills learned in previous and future chapters, and have the practice needed to become fluent with those skills. Checkpoint problems incorporate skills that students develop in previous courses and continue to use in the mathematics they are learning, providing students the opportunity to build procedural fluency for those skills. The spiral nature of the materials helps build fluency since students are expected to know how to solve problems "on demand" and not just after the section on that standard.

Examples of select cluster(s) or standard(s) that specifically relate to procedural skill and fluency include, but are not limited to:

  • A-APR.1: Students practice operations on polynomials in many lessons including Integrated II, Lessons 1.2.3, 4.1.3-4.1.4 and 5.2.6, and Integrated III, Lessons 1.1.4 and 8.3.1.
  • F-BF.3: Students use transformations on functions in Integrated I, Lesson 10.2.1, by examining the effect of adding a constant to a function. In Integrated II, Lessons 9.1.2-9.1.4 expand transformations to include dilations and shifts in any direction on parabolas and absolute value functions. Integrated III expands to include additional types of functions. Lessons 2.2.1-2.2.4 expand to include cubic, rational and root functions and non-functions such as circles. Lesson 5.2.4 includes logarithmic functions and Lessons 9.2.1-9.2.4 include periodic functions.
  • G-SRT.5: Ample practice is provided in Integrated II, Lessons 2.1.1, 2.1.2 and 2.3.1-2.3.4, in determining similarity/congruence and using these characteristics to find missing sides and angles.
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Indicator 2c

Attention to Applications: The materials support 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.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that materials support 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 frequently solve non-routine application problems that develop the mathematics of the standards. Students are provided opportunities to make their own assumptions, question, investigate, critically analyze and communicate their thinking in teams, independently and in Learning Logs as they model mathematical situations.

  • In Integrated I, Lesson 2.2.2 students graph and write equations to describe the real-life situation of runners in a race. The materials use the context to provide opportunities for students to apply contextual meaning to interpret parts of an expression in terms of its context. (A-SSE.1b)
  • In Integrated II, Lesson 5.1.2 students apply what they have learned about quadratic functions to the context of a water balloon contest. The students relate the intercepts and vertex of a parabola to the launch, landing, and maximum height of a launched water balloon. (F-IF.4)
  • In Integrated III, Lesson 4.1.2 students compare the representative nature of samples selected using intentional choice with those selected randomly by applying this concept in a real-life scenario involving astronomers determining the average diameter of asteroids captured by a satellite image. (S-IC.1)
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Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the three aspects of rigor are not always treated together and are not always treated separately. The three aspects are balanced with respect to the standards being addressed. Overall, there is clear evidence that all three aspects of rigor are present in the materials. Additionally, the materials engage in multiple aspects of rigor in order to develop students’ mathematical understanding of a single topic/unit of study.

The materials contain a balance of opportunities for students to develop fluency in new mathematics concepts and to apply these skills in engaging tasks. The materials consistently present students with problems that include real world application and significance whenever appropriate. As needed, students are provided opportunities to practice skills for procedural fluency. This balance is maintained throughout the course. The balance of procedural skill development and application is not rigid throughout the materials and changes based on the targeted concept.

  • In Integrated I, Chapter 1, Lesson 1.1.1 builds conceptual understanding and fluency by having students work in teams to evaluate expressions, build “function machines” to connect inputs and outputs, and make observations and generalizations about functions. Lesson 1.1.2 has students work in collaborative teams to complete labs where they collect and analyze data.
  • In Integrated I, Chapter 6, Lessons 6.1.1 and 6.1.2 focus on procedural skills such as rearranging linear equations to y = mx+b form, solving equations, and finding the missing terms in a sequence. Then, in 6.1.3, students engage in tasks that apply these skills in real world context.
  • In Section 3.1 of Integrated II, every lesson provides context and application for the skills the students are exploring. The materials did not decontextualize to teach the probability models. The materials address the concepts within real world examples and scaffold students to a level of deep conceptual understanding. In Lesson 3.2.1, students are introduced to a new concept (constant ratios in right triangles) within a rich context, and then in Lesson 3.2.2, the materials have students practice the procedural skill and build fluency for connecting slope ratios to specific angles.
  • In Chapter 1 of Integrated III, Investigations and Functions, the lessons in Section 1 focus mainly on procedural fluency with functions, and Section 2 focuses on application in a rich context.
  • In Chapter 4 of Integrated III, Natural Distributions and Geometric Modeling, the lessons include many real world application problems and fewer procedural fluency problems as is appropriate for the concept.

Mathematical Practice-Content Connections
MEETS EXPECTATIONS

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The instructional materials reviewed for the High School CPM Integrated series meet the expectation that materials support the intentional development of all eight MPs, in connection to the high school content standards. The teacher resources for each course includes a "Correlation of CPM Core Connections" document that is designed to illustrate "how the practices are integrated into a few typical lessons," but the program is designed to use the MPs "as the foundation of each of the daily lessons." The teacher's notes also list the MPs that are a focus of each lesson. Overall, many of the lessons in the series incorporate the MPs as an integral part of the materials.

Criterion 2e-2h

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Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

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Indicator 2e

The materials support the intentional development of overarching, mathematical practices (MPs 1 and 6), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that materials support the intentional development of making sense of problems and persevering in solving them as well as attending to precision (MP1 and MP6), in connection to the high school content standards. Overall, the majority of the time MP1 and MP6 are used to enrich the mathematical content and are not treated as individual mathematical practices. Throughout the materials, students are expected to make sense of problems and persevere in solving them while attending to precision.

Some examples of MP1 and MP6 are as follows:

  • In Lesson 1.3.3 of Integrated II, students attend to precision as they continue to identify and justify angle pair relationships.
  • In Lesson 11.1.3 of Integrated I, students explore constructions of parallel lines and squares. Students devise a strategy for completing the constructions.
  • In Integrated I, Lesson 2.1.1 has students make sense of tile pattern investigation problems to see linear growth.
  • In Integrated I, Lesson 2.2.3 uses six card clues to make sense of a series of linear equations which helps them solve who wins "The Big Race," a puzzle.
  • In Lesson 3.1.4 of Integrated II, "Students make sense of problems involving the probabilities of independent events and attend to precision as they differentiate between unions and intersections."
  • In Integrated II, Lesson 9.3.1 has students make a number of connections between the tables and graphs of parabolas and relate both to the average rate of change (average velocity) calculation. Finding the velocity at the vertex prompts questions about whether the answer makes sense.
  • In Integrated III, Lesson 3.1.4 gives situations and asks students to create systems of equations, determine solutions, and think about the meaning of the solution.
  • In Lesson 7.1.4 of Integrated III, students solve a murder mystery using logarithms.
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Indicator 2f

The materials support the intentional development of reasoning and explaining (MPs 2 and 3), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that materials support the intentional development of reasoning and explaining (MP2 and MP3), in connection to the high school content standards, as required by the MPs. Overall, the majority of the time MP2 and MP3 are used to enrich the mathematical content inherently found in the text, and these practices are not treated as isolated experiences for the students. Throughout the materials, students are expected to reason abstractly and quantitatively as well as construct viable arguments and critique the reasoning of others.

Some examples of MP2 and MP3 are as follows:

  • In Integrated I, Lesson 2.3.1 has students decontextualize and examine the situation numerically, and they recontextualize by examining the numbers in terms of the original problem situation.
  • In Lesson 10.1.2 of Integrated I, students must make choices about data displays and defend their choices.
  • In Integrated II, Lesson 3.2.4 connects the tangent ratio to the slope of a line.
  • In Integrated II, Lesson 7.2.3 has students construct arguments as they explain their thinking about independent situations.
  • In Lesson 3.2.4 of Integrated III, students construct viable arguments and critique the reasoning of others about a hypothetical mathematics contest.
  • In Integrated III, Lesson 4.4.3 introduces a 3-D printing design problem, and students must reason about the quantities and what they represent. They must also formulate a rationale for their choices.
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Indicator 2g

The materials support the intentional development of modeling and using tools (MPs 4 and 5), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that materials support the intentional development of addressing mathematical modeling and using tools (MP4 and MP5), in connection to the high school content standards, as required by the MPs. Overall, the majority of the time MP4 and MP5 are used to enrich the mathematical content inherently found in the text, and these practices are not treated as isolated experiences for the students. Throughout the materials, students are expected to model with mathematics and use tools strategically.

Some examples of MP4 and MP5 are as follows:

  • In Integrated I, Lesson 1.1.2 has students determine an organized way to record their data.
  • In Lesson 6.4.1 of Integrated I, students decide which strategy is most efficient when solving systems of equations.
  • In Integrated I, Lesson 10.1.2 models a golf game by tossing pennies and measuring the distance from a hole. Students collect data using this model and must make decisions about the tools involved in measurement, data collection, and data display.
  • In Integrated II, Lesson 6.2.1 has students model a tennis serve.
  • In Lesson 10.2.5 of Integrated II, students model orbiting satellites and perform constructions.
  • In Integrated III, Lesson 7.2.1 presents problems with missing parts of triangles. Students determine the information necessary to find the missing measurements. Students may need to try multiple solution paths before finding one which will be successful, and they also need to consider multiple cases and combinations of known and unknown parts in both right and non-right triangles.
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Indicator 2h

The materials support the intentional development of seeing structure and generalizing (MPs 7 and 8), in connection to the high school content standards, as required by the mathematical practice standards.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that materials support the intentional development of seeing structure and generalizing (MP7 and MP8), in connection to the high school content standards, as required by the MPs. Overall, the majority of the time MP7 and MP8 are used to enrich the mathematical content inherently found in the text, and these practices are not treated as isolated experiences for the students. Throughout the materials, students are expected to see structure and generalize.

Some examples of MP7 and MP8 are as follows:

  • In Integrated I, Lesson 3.1.6 has students summarize and generalize the symmetry of figures.
  • In Integrated I, Lesson 5.3.1 examines growth rates.
  • In Lesson 4.1.3 of Integrated II, students attend to aspects of MP7 and MP8 while factoring general and special quadratics.
  • In Integrated II, Lesson 8.2.1 has students "use repeated reasoning to generalize a process for determining the sum of the interior angles of a polygon and then make use of structure to calculate individual interior and exterior angle measures in regular polygons."
  • In Integrated III, Lessons 2.2.1 and 2.2.4 examine the structure of equations in conjunction with repeated reasoning to make sense of transformations of both functions and non-function equations. In Lesson 2.2.1, students "look for and make use of structure and look for and express regularity in repeated reasoning as they make connections between the transformations of parabolas and the transformations of other parent graphs."
  • In Lesson 8.3.4 of Integrated III, students consider factoring patterns for special polynomials and compare differences of powers as special cases of differences of squares.

GATEWAY THREE

Usability

MEETS EXPECTATIONS

Use and design facilitate student learning
MEETS EXPECTATIONS

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The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials are designed well and take into account effective lesson structure and pacing. The design and layout of the materials, in print and in the eBook, are simple, easy to use and not distracting.

Criterion 3a-3e

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Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

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Indicator 3a

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the underlying design of the materials distinguish between lesson problems and student exercises for each lesson.

Students are learning new mathematics in each lesson and then applying what they have learned in order to build knowledge. Within lessons, both problems and exercises are found. Core problems and suggested lesson activities are problems that develop concepts targeted in the lesson. Review & Preview are exercises that allow students to apply their learning. Overall, the spaced and spiraling nature of the series helps build mastery. Each chapter, section and lesson has a variety of problems and exercises and has intentional purpose in developing learning and thinking.

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Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the design of assignments is not haphazard and that exercises are given in intentional sequences.

The basic structure of each lesson includes core mathematical content followed by a Review & Preview homework section. Lessons may include additional components such as Discussion Points, Further Guidance, Stoplight Problems, Calculator or No Calculator Problems, Learning Log Entries, and Math Notes. The exercises are given in intentional sequences.

2/2
Indicator 3c

There is variety in how students are asked to present the mathematics. For example, students are asked to produce answers and solutions, but also, arguments and explanations, diagrams, mathematical models, etc.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that students are asked to produce a variety of products during the lessons in each chapter to demonstrate their learning.

Throughout various lessons and within the problem sets, students are asked to produce answers and solutions as well as to describe their answers, discuss ideas, make conjectures, explain their work and reasoning, make sketches and diagrams, justify their reasoning, and use appropriate models. Students are asked to show work including checking of solutions, drawing visual representations, explaining steps and reasoning and justifying responses.

For example, in Integrated II Lesson 5.1.3, students investigate the minimum number of points to sketch a parabola. Students examine x- and y-intercepts of parabolas and relate these to the equations, make sense of the zero product property, and complete a Learning Log entry. Students also sketch a graph of a quadratic equation. Also, in Integrated II Lesson 6.2.3, students investigate shortest distance problems using physical models and diagrams. Students decide whether answers make sense and explain their rationale.

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Indicator 3d

Manipulatives, both virtual and physical, are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The series makes use of a wide range of virtual manipulatives. The materials have their own collection of virtual manipulatives including algebra tiles, probability tools, data representation tools, transformation tools, similarity toolkit, number lines and graphing tools. The materials also make regular use of pre-made Desmos.com graphs and other applets. There are general manipulatives and tools that the materials recommend always having available. A few examples of these include, but are not limited to: colored pencils, graph paper, markers, masking tape, meter sticks, rulers, scissors and tape. Then, there are specific manipulatives and tools that will be needed for specific lessons. A few examples of these include, but are not limited to: linguini, coffee straws, and pennies. Overall, students are exposed to a number of manipulatives and virtual tools and expected to use these as necessary.

Indicator 3e

The visual design (whether in print or digital) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The instructional materials reviewed for the High School CPM Integrated series have a visual design that is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The layout of both the online text and the print materials is not distracting. The layout is the same for every lesson and every course. The design of the teacher materials is consistent across lessons and courses. Materials are presented in a manner which aligns to the student textbook.

The online materials are contained in the largest box on the screen. A small frame on the left side of the screen allows the user to navigate quickly to any section within the textbook and contains links to the index, glossary and other reference materials. Across the top of the window is a tab that allows student to toggle between the Spanish and English translations of the lesson. Below the textbook are links to an online dictionary and translation tools and the online mathematics tools (algebra tiles, probability tools, Desmos, data analysis tools, and transformation tools). These open in a new browser tab. Illustrations in the online textbook are minimal and appear to the right of the problems. They do not distract from the textbook itself.

Teacher Planning and Learning for Success with CCSS
MEETS EXPECTATIONS

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The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials support teacher learning and understanding of the standards. Overall, the materials provide the teacher necessary supports using adult-level expectations, the student with guiding questions for appropriate mathematical development and the parents with resources.

Criterion 3f-3l

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Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

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Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

Guiding questions are provided in the teacher materials. These are usually found in the "Suggested Lesson Activity" section, although sometimes can be found in the "Universal Access" or "Team Strategies" sections. For example, in Integrated III Lesson 8.1.1, teacher materials focus on questions connecting the number of roots of a polynomial with the degree of the polynomial. Example questions include the following: “What if you multiply by another factor?” “How will multiplying by another factor affect the graph?” “How will it affect the equation?” “What might the graph look like if the highest power of x were five or six?”.

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Indicator 3g

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. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials provide ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials.

The teacher edition contains a lesson guide for every lesson, with the main part being "Suggested Lesson Activity." The Suggested Lesson Activity section contains a detailed description of the lesson activities and how teachers may choose to teach the material. This is always followed with a "Closure" section which describes how to close out the lesson. Nearly every lesson contains sections for "Universal Access" and suggestions for student teams.

The teacher edition has a section for TI-83/84 support. Lessons which require the use of a graphing calculator (specifically, TI-83/84) include a section entitled "Technology Notes," which describes how to use the calculator for the activities. The online textbook includes links to additional information and instructions, as well as videos which show how the calculator steps are performed. For example, in Integrated I, Lesson 2.2.1 uses a graphing calculator and a TI CBL or CBR motion detector. Clear directions are given for the use of the motion detector and calculator.

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Indicator 3h

Materials contain a teacher's edition that contains full, adult--level explanations and examples of the more advanced mathematics concepts and the mathematical practices so that teachers can improve their own knowledge of the subject, as necessary.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials contain 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.

Lesson videos and lesson teacher notes (both printed and in the eBook) provide teachers with full preparation for each lesson, including historical notes, video models, mathematical background and adult-level explanations to guide the teacher. The series provides a newsletter with lesson ideas and up-to-date strategies and best practices to guide teachers in planning and in advanced learning of the mathematics. Past editions of the newsletter are archived in the teacher support of the teacher materials.

The following are examples of the type of information included in the Mathematical Background:

  • In Integrated I Lesson 6.2.1, Mathematical Background shows how an equal values method of solving a system of two linear equations where the x-values are 8 apart at the same y-value (rather than finding the x-value where the y-values are equal) may be used instead of the method suggested in the lesson guide and answer key, and it also shows why that method is mathematically valid. The Mathematical Background in Lesson 6.2.2 contains a similar explanation, except looks for the x-value where the y-values are 78 apart.
  • In Integrated III Lesson 7.2.2, Mathematical Background describes the proof of the law of sines in the case where the altitude falls outside of the triangle (a case which is not proved in the course).
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Indicator 3i

Materials contain a teacher's edition that explains the role of the specific mathematics standards in the context of the overall series.

The instructional materials reviewed for the High School CPM Integrated series partially meet the expectation that the materials explain the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through Grade 12.

The introduction of each chapter contains a section in the teacher materials entitled "Where is this going?" which describes how the work in the chapter connects to future chapters within the same book. Infrequent connections are made to "future courses," but specifics are not provided. This section does not contain reference to specific standards.

Within the lessons, few specific connections are made to future work. For example, F-IF.5 is addressed over all three courses, beginning in Integrated I, Lesson 1.2.3. The materials do not explain how the work will continue through the next two courses.

References are made to past work in both the student and teacher editions. For instance, in Integrated II, Lesson 5.1.2 refers specifically to a multiple representations web completed in Integrated 1. However, these connections are also infrequent.

Indicator 3j

Materials provide a list of lessons in the teacher's edition, cross-- referencing the standards addressed and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The instructional materials reviewed for the High School CPM Integrated series provide a list of lessons, cross-referencing the standards covered. The materials in the series provide several different pacing guides, lesson sequences, and standards (both practice and content) correlations. Charts are provided at the beginning of all chapters across all courses. These charts include the number of days for each lesson, the lesson objectives, and the materials and tools required for the lesson.

Indicator 3k

Materials contain strategies for informing students, parents, or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The instruction materials reviewed for the High School CPM Integrated series have an ancillary resource book for each course designed to provide parents with additional practice problems for students, including explanations for parents. The Parent Guide is available both in print and online. It contains mathematical explanations, definitions, examples, and extra practice problems with annotated solutions. The Parent Guide is written in a more traditional way than the text, which is written to guide students through problems rather than explain and provide examples.

Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research--based strategies.

The instructional materials reviewed for the High School CPM Integrated series provide many different instructional strategies, support on how to effectively implement these strategies, and extensive research literature supporting the use of these strategies and the design of the materials. The teacher materials include research summaries for the use of cooperative learning, problem-based learning, and mixed, spaced practice. All of these are foundational practices for CPM Integrated and are evident throughout the courses.

Assessment
PARTIALLY MEETS EXPECTATIONS

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The instructional materials reviewed for the High School CPM Integrated series partially meet the expectation that the materials offer teachers resources and tools to collect ongoing data about student progress on the standards. The materials provide assessments that offer evidence of students’ knowledge of the CCSSM. However, there are limited strategies for gathering information about students’ prior knowledge, and guidance to teachers for interpreting student performance and suggestions for follow-up are limited.

Criterion 3m-3q

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Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

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Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels/ courses.

The instructional materials reviewed for the High School CPM Integrated series partially meet the expectation that the materials provide strategies for gathering information about students' prior knowledge within and across grade levels. The materials have pre-assessments for Integrated I. The pre-assessments do not list specific standards that are being addressed, and there is no indication of what to do with the information that is collected. The materials do provide the opportunity within lessons to see prior knowledge being addressed.

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Indicator 3n

Materials provide support for teachers to identify and address common student errors and misconceptions.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials in the series provide strategies for teachers to identify and address common student errors and misconceptions.

The teacher notes provide tips for teachers to address common errors. Teacher notes often include areas in which common student errors or misconceptions are highlighted. The instructional materials provide questions a teacher can pose to students in an effort to address errors or misconceptions. For example, in Integrated II Lesson 4.1.1, the materials suggest asking students, “How did you decide how to split up the x’s? How did you decide whether to use 2x times 3x, or 6x times 1x in part (b)? Do you see any patterns in the product and the sum?” as they are introduced to factoring. Emphasis is heavily placed on discussing errors/misconceptions between teacher and students or even among students through Think-Pair-Share, Teammates Consult, or other team strategies. The Closure sections provide opportunities for discussion of common errors and misconceptions, along with the cooperative learning tasks.

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Indicator 3o

Materials provide support for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The instructional materials reviewed for the High School CPM Integrated series meets the expectation that the materials in the series provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The Review & Preview questions incorporated at the end of every lesson provides the opportunity for students to continue practicing and reviewing mathematical concepts as they work toward mastery of the content. The Learning Log entry incorporated into many lessons also allows a student to convey their understanding (entry topics are designed for students to demonstrate procedural and/or conceptual knowledge). These two used in tandem can provide multiple strategies for providing feedback that address a student’s knowledge of both skills and concepts.

The teacher materials provide rubrics along with information on giving feedback on portfolios, team tests, presentations and participation quizzes. Students also have feedback available through the eBook homework help and the animated explanations.

Indicator 3p

Materials offer ongoing assessments:

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Indicator 3p.i

Assessments clearly denote which standards are being emphasized.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that assessments clearly denote which standards are being emphasized. Each assessment lists the standards emphasized for each problem on the Answers pages.

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Indicator 3p.ii

Assessments provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The instructional materials reviewed for the High School CPM Integrated series partially meet the expectation that assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. The materials in the series offer ongoing formative and summative assessments. The assessments include some generic rubrics. However, the rubrics are typically very general in nature and may not provide enough guidance to teachers to interpret current student performance. Assessments have answer keys but lack any guidance to the teacher on how to score or how to interpret the results.

Indicator 3q

Materials encourage students to monitor their own progress.

The materials in the series encourage students to monitor their own progress through Learning Logs, concept webs, closure, team activities and portfolios. Each of the Review & Preview problems have a Homework Help link in which students can click a button and receive a hint. This can provide guidance to students as they work and self-assess their understanding by asking questions: How confident am I in my approach/method? How confident and I in my solution? Furthermore, each unit ends with a “Closure” lesson in which students can reflect on their learning and make connections with prior learning as they complete tasks and more practice problems. At the conclusion of the “Closure” lesson, solutions are provided for each practice problem as well as guidance as to where to refer to earlier in the chapter if a student has questions or needs additional practice. This feature is a monitoring tool as students prepare for the unit summative assessment.

Differentiated Instruction
MEETS EXPECTATIONS

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The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials support teachers in differentiating instruction for diverse learners within and across grades. The materials offer lessons that are scaffolded to allow a range of learners to gain entry into the problems. The materials provide multiple language translators so that students of various backgrounds can engage in the mathematics in their native language. Additionally, teachers are offered ideas and suggestions for grouping all learners in a variety of ways.

Criterion 3r-3y

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Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

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Indicator 3r

Materials provide teachers with strategies to help sequence or scaffold lessons so that the content is accessible to all learners.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials provide strategies to help teachers sequence or scaffold lessons so the content is accessible to all learners. Each course in the series comes with a Teacher Resource Binder. In this binder, there is a tabbed section called Team Support and Universal Access that provides strategies to assist English Language Learners and other special populations. It also includes information for pacing and complexity for advanced learners. One strategy included is to focus on core or essential problems for struggling learners so that they have access to the same level of rigor but with fewer problems.

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Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials provide teachers with strategies meeting the needs of a range of learners. Each volume in the series comes with a Teacher Resource Binder. In this binder, there is a tabbed section called Team Support and Universal Access that provides strategies to assist English Language Learners, and other special populations. These strategies include the suggestion of study teams, outside the class tutoring, and the Parent Guide with Extra Practice resource provided with each course. The Literacy Support Guidebook highlights the features of the textbook that support student reading of the text and has specific reading strategies and suggestions for working with individual students as well as teams.

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Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations.

Throughout the chapters in each course, there are tasks that are based on real-world situations. Students are encouraged to reflect on their work and to use different strategies to arrive at solutions. Appropriate scaffolding is evident as there are multiple entry points for students with a range of learning abilities to have access so they are able to solve the problems. Many problems encourage multiple representations (graph, verbal, analytical, numerical).

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Indicator 3u

Materials provide support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

Each volume in the series comes with a Teacher Resource Binder. In this binder, there is a tabbed section called Team Support and Universal Access that provides strategies to assist English Language Learners, and other special populations. There are explanations on planning for Special Needs Students, English Language Learners, as well as differentiation in pacing and complexity for Advanced Learners.

The series clearly identifies checkpoint problems and core problems with guidance on how to modify the pacing for special populations so all students have access to course-level resources. "Math Note" sections throughout the lesson provide definitions and examples with regard to vocabulary. Additionally, every lesson is provided in Spanish and, through the eBook translator, can be translated into most languages.

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Indicator 3v

Materials provide support for advanced students to investigate mathematics content at greater depth.

The instructional materials reviewed for the High School CPM Integrated series meet the expectation that the materials provide opportunities for advanced students to investigate mathematics content at greater depth. Pacing charts are provided, and each of these charts suggest time intervals for not only the struggling student but also the more advanced student. There are also problems that are provided as enrichment opportunities for advanced students. The teacher materials explicitly state that advanced students also benefit from the richness of the problems in the text and will often be able to develop considerable depth in their work.

Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.

The instructional materials reviewed for the High School CPM Integrated series provide a neutral portrayal of various demographic and personal characteristics. The names and situations portrayed in the book are diverse and representative of many different cultures and heritages.

Indicator 3x

Materials provide opportunities for teachers to use a variety of grouping strategies.

The materials provide opportunities and directions for teachers to use a variety of grouping strategies. The materials focus on team cooperative learning. The materials provide activities, discussions, and tasks tailored for whole team, and individual work. For example, in Integrated I Lesson 1.1.3, the teacher materials refer to three different grouping strategies to use throughout the lesson. When the user clicks on the strategy, a pop-up box appears that defines the purpose of the strategy and gives a full explanation of how to use the strategy. This full explanation includes materials needed, classroom management tips, the full protocol of how to use the strategy, and sometimes links to other resources for more information about the strategy. The materials also include a Team Support section. This section "describes the value of student interactions in study teams and offers suggestions for creating and maintaining a learning environment that supports effective study teams."

Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.

The materials encourage teachers to draw upon home language and culture to facilitate learning. The eBook materials include a translation link for every lesson. All lessons in the eBook have tabs for both English and Spanish.

Effective Technology Use

The instructional materials reviewed for the High School CPM Integrated series support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms. The materials provide access to many e-tools through the eBook. Additionally, the materials allow teachers to create their own assessments as well as collaborate with other teachers who are also using the same materials.

Criterion 3z-3ad

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

Indicator 3z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

The materials integrate technology, manipulatives, and dynamic software in ways that engage students in the MPs.

The materials have their own collection of virtual manipulatives which includes algebra tiles, base ten blocks, probability tools, data representation tools, transformation tools, similarity toolkit, number lines and graphing tools. Additionally, the materials make regular use of pre-made Desmos.com graphs and other pre-made applets. Examples include, but are not limited to:

  • Tile eTools/square tiles: Introduced in Integrated I Lesson 1.1.1 and used throughout Integrated I, the tile pattern tool allows the student to investigate tile patterns without the use of physical tiles. The tools include algebra tiles, base 10 blocks, number lines, and units and unit squares for area and perimeter measurement.
  • The probability eTools are introduced in Integrated II Lesson 3.1.2 and used throughout Chapter 3. The probability tools allow for different probability simulations: spinners, bags, dice, coins, cards, and a random number generator.
  • The data representations eTool is introduced in Integrated III Lesson 1.2.3. The data representations tool allows students to investigate univariate data with histograms, box plots, and stem and leaf plots.
Indicator 3aa

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Mac and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

The digital materials are web-based and compatible with multiple Internet browsers. CPM eTools and Desmos are web-based and work on Mac browsers (Safari and Chrome), PC browsers (Chrome and Firefox), and iPad browsers (Safari and Chrome). Additionally, Desmos has apps for iPad, iPhone, Android phones and tablets, and Chrome (for ease of use with Chromebooks).

The series uses Quicktime, YouTube, and Internet resources that are accessible on multiple devices and browsers. Quicktime player needs to be installed on computers. HTML5 works on most mobile devices and on many computers. YouTube may not be allowed at all schools.

Indicator 3ab

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. Teachers have an assessment bank available through the eBook. Assessments are not available for students through the eBook and are not adaptive, but teachers can customize assessments using the test bank.

Indicator 3ac

Materials can be easily customized for individual learners.

Indicator 3ac.i

Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations.

The materials in the series do not provide adaptive technology.

Indicator 3ac.ii

Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

The technology components offered in the materials are not customizable for students based on their needs or interests.

Indicator 3ad

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

The materials provide opportunities for teachers to collaborate with each other through the eBook resources via the Sharing Tab. There are not opportunities for student-to-student or student-to-teacher collaboration via the eBook.