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.

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.

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)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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?”.

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.

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).

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.

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.

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).

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.

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.