High School - Gateway 2

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.