High School - Gateway 2

The instructional materials reviewed for the High School CPM Traditional series meet the expectation that materials support the intentional development of students' conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters. Overall, the clusters and standards that specifically relate to conceptual understandings are thoroughly addressed. The materials develop conceptual understandings across the series by building on a tactile form.

• 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.
• A-REI.10 addresses conceptual understanding, and the materials offer opportunities for students to develop a deep understanding and ability to communicate or demonstrate that understanding of A-REI.10. Sections 10.3.1 and 10.3.2 in Algebra 1 and section 4.1.2 in Algebra 2 explore this standard at length, guiding students to make predictions, solve in multiple ways, and explain intersections and intercepts of graphs. The Review & Preview sections of each course spiral this concept throughout the series for all types of functions (i.e., linear, polynomial, rational, absolute value, exponential and logarithmic as stated in the standards).
• Multiple representations are embedded throughout the series, reinforcing students' ability to verbalize and recognize connections graphically, analytically, and numerically. Algebra 1 begins building multiple representations in section 1.1.4, and Algebra 2 frames this again starting in chapter 2. Multiple representations continue throughout the remainder of the materials to draw connections among parent graphs from all types of functions. Additionally, the materials use a common resource called a representation web to reinforce four ways to look at functions.
• Conceptual understanding is a strength of this series. Concepts grow over many lessons within and between each course in the series. Specific clusters and domains that are represented include N-RN.1, A-APR.B, A-REI.A, A-REI.10, A-REI.11, F-IF.A, F-LE.1, S-ID.7, G-SRT.2 and G-SRT.6. Some specific examples are:
  • A-APR.A: Students use algebra tiles to build conceptual understanding of adding, subtracting, and multiplying polynomials beginning in Chapter 3 of Algebra 1. Students make connections between the tactile and the algebraic methods of performing arithmetic operations on polynomials by moving from algebra tiles to generic rectangles to algebraic computation. In lesson 3.2.3, students are specifically told what is meant by "a closed set" and asked to explore if integers are closed under addition given that whole numbers are closed under addition. Then, they explore and explain whether polynomials are closed under addition and subtraction, extrapolating from what they know about whole numbers and integers.
  • F-TF.A: In Chapter 7 of Algebra 2 a conceptual model is used to develop the concept of a unit circle, radian measures, and trigonometric functions. First, students are introduced to models for cyclic relationships through an experiment in 7.1.1. Then, in 7.1.2 students create a sine graph using experimental data based on a Ferris wheel. In the following lesson, students use the same Ferris wheel to develop a unit circle and discover reference angles at various points on the circle and their relationships to each other. In lesson 7.1.4, students create a cosine function and calculate horizontal distances in a unit circle to draw conclusions about relationships between sine and cosine and their functions. The next lesson introduces students to radian measures. All of this leads to the conceptual understanding in 7.1.6 in which they use radian measures to determine the exact values of the coordinates on a unit circle.

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

• Checkpoints are scattered throughout the lessons in the series. The checkpoints are designed as fluency problems. If students work through a checkpoint and see that they are not fluent with that problem, then there are more problems at the back of each of the student editions that provide more practice for students to work through until they reach fluency.
• The spaced nature of the problems helps build the fluency since students are expected to know how to solve them '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: There are many opportunities to develop procedural fluency with operations of polynomials in Algebra 1 Chapter 3 and again in Algebra 2 in section 3.1. Additionally, there are checkpoints in the reference section of each series that address this standard. These checkpoints include 6B in Algebra 1, 5A in Geometry and 5A in Algebra 2.
  • A-APR.6: There are many opportunities to develop procedural fluency in rewriting simple rational expressions in Chapter 8 of Algebra 2. Additionally, there is Polydoku (similar to Sudoku) and checkpoints 6A and 6B add to the already present opportunities throughout Chapter 8, and the following chapters.
  • F-BF.3: Effects of parameters of functions is found throughout the Algebra 1 course, and again in chapter 2 of Algebra 2. There is lots of practice throughout both books and as spaced practice in Geometry.

The instructional materials reviewed for the High School CPM Traditional 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. The cluster(s) or standard(s) that specifically relate to applications are thoroughly addressed multiple times. The materials include numerous applications across the series.

• In every lesson, students are solving non-routine problems, from simple to complex. Students are provided opportunities to make their own assumptions, question, investigate, critically analyze and communicate their thinking in groups, independently and in learning logs as they model mathematical situations.
• The lessons are built upon application and modeling problems. The lessons do not show several, worked examples of various problems followed by a set of problems where subsets align to each type of the worked examples.
• Modeling builds across high school courses, with applications that are relatively simple when students are first encountering new content. For example, Algebra 1 problem 1-38 has students recall patterns, a tool that students have been learning since elementary school, and begins teaching families of functions. Families of functions are then developed throughout the series. This is an example of a problem that provides opportunities for students to make their own assumptions in order to model a situation mathematically. Students must reason about what happens as the pattern continues, thus beginning early in the series to challenge themselves in making assumptions.
• Examples of select cluster(s) or standard(s) that specifically relate to applications include, but are not limited to:
  • F-IF.7a: The Saint Louis Gateway Arch in 9.1.3 in Algebra 1.
  • G-SRT.8: Statue of Liberty problem in 4-45 and 4-46; Wheelchair Ramp problem in 5-24.
  • S-IC.1: Charity Race in 9.3.2 of Algebra 2; ACT Scores in 9.3.3 of Algebra 2.

The instructional materials reviewed for the High School CPM Traditional 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 of all three aspects of rigor 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.

• Fluency is specifically targeted in the Checkpoint problems and also in the spaced practice.
• Concepts are systemically developed in the lessons and are intentionally reinforced in the closure activities. The closure at the end of each section and the spaced nature of the homework intentionally connects conceptual learning with problem solving and procedural fluency.
• Most lessons start out with an application or investigation. For example, Algebra 2, section 7.2.1 begins with students investigating functions and Algebra 2, section 7.1.2 begins with students engaged in an application, the roller coaster problem.
• The Review & Preview sections of each lesson also intentionally interweave problems of all three types of rigor together.
• As an example of this balance, Chapter 2 in Algebra 1 focuses on linear relationships. The first two lessons use investigations of tile patterns with various representations to introduce and develop conceptual understanding of linear relationships. The third lesson has students investigating "steepness" using a variety of representations. Finally, in the fourth lesson, these explorations are formalized with mathematical definitions for slope and slope-intercept form of a line. In the remaining five lessons of the chapter, students apply this understanding to a variety of real-world situations and contextual problems that further develop connections and lead to procedural fluency.
• As another example, Chapter 4 in Geometry maintains this balance in a different way. The first lesson begins with a real-world investigation of the relationship between angles in right triangles and the ratios of side lengths. The next two lessons use conceptual understanding to solve basic problems and develop procedural fluency, even though the students do not have the formal understanding of a tangent function. In lesson 4, the investigation culminates in the formal tangent function with additional procedural practice. Then lesson 5 provides a variety of real-world problems to apply the conceptual understanding and procedural fluency. Chapter 5 follows a similar pattern to introduce students to sine and cosine functions, Pythagorean Theorem, and a wide variety of problems involving triangles.

The instructional materials reviewed for the High School CPM Traditional 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. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

• There was no evidence of instances where MP1 and MP6 are treated as separate from the mathematics content throughout the series. Many lessons require students to "make sense" of problems and "persevere" when solving them. Students are constantly asked to explain or compare representations or solutions paths, hence a natural opportunity to practice precision.
• Students are encouraged to persevere and attend to precision as they reflect on and correct their assessments. They may partner together to discuss mistakes and revise their work. Students are asked questions like: Why did you miss the problem? What did you learn from revising the problem? Who (if anyone) helped you and what did they say to help you better understand the problem? Make up a new, similar problem to show and explain to a new student (in writing) how to solve the problem. Analyzing errors in thinking or computations incorporates many of the MPs.
• It is important to note that MP1 and MP6 are represented in many lessons and assessments throughout the series. Listed below are a few examples of where MP1 and MP6 are used to enrich the mathematical content:
  • In Algebra 1, section 2.2.3, a challenging team puzzle is incorporated into the lesson that encourages making sense of problems and persevering in problem solving and attending to precision as part of the solving of this puzzle.
  • Geometry, section 8.3.3, includes 4 core problems, of which 3 are recommended for students to make sense of the problems and persevere in solving them as well as attending to precision while solving the problems. This particular lesson is an extended lesson, designed so that students have extra time to read, interpret, strategize, engage in the problem solving, and solve the problems, attending to these specific mathematics practices.
  • Algebra 2, section 8.1.3, includes opportunities for students to use these math practices as they are making connections and discoveries about higher level polynomial functions. They are given opportunities to check their answers with multiple methods and monitor and evaluate their progress in solving problems as they persevere in solving problems.

The instructional materials reviewed for the High School CPM Traditional 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 and are not treated as individual practices. Throughout the materials, students are expected to reason abstractly and quantitatively as well as construct viable arguments and critique the reasoning of others. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

• Many lessons require students to "reason abstractly and quantitatively" and "construct viable arguments and critique the reasoning of others." Students are constantly asked to represent situations symbolically, create their own conjectures, determine if their answers make sense, and communicate this in various ways, hence a natural opportunity to engage in these two mathematical practices.
• It is important to note that MP2 and MP3 are represented in many of the lessons and assessments throughout the series. Listed below are a few examples of where MP2 and MP3 are used to enrich the mathematical content:
  • The Chapter 7 sample chapter test in Algebra 1 has five problems. These five problems are representative of problems taught in that course. This particular sample test requires students to explain, justify, show or provide evidence for all five problems on the test. Most of the problems also require students to represent the situations symbolically and to understand the relationships between problem scenarios and mathematical representations. One specific problem asks students to determine if their answer makes sense.
  • The teacher's notes of section 6.1.2 in Geometry lists at least five different probing and clarifying questions as a guide for the teacher to help students who may be struggling at different parts of the lesson. Teachers are encouraged to use this questioning strategy in connection to MP2 and MP3 to enrich the mathematics content. For example, one of the questions, "Is SSA a valid similarity conjecture? Why or Why not?" helps students to construct viable arguments as they are learning about and understanding similarity and congruence of triangles.
  • Chapter 2 of Algebra 1 focuses on helping students move between real world situations, graphs, tables, and equations. In doing this, students begin to reason abstractly and represent situations symbolically. The chapter also asks students to move in the other direction, giving them an equation or graph and asking them to contextualize the abstract representations. This is the essence of MP2 and the materials provide ample opportunity to develop this type of thinking. This chapter also has students creating and critiquing arguments. Some examples of where students are asked to justify or create arguments to support their answers are problems 2-1, 2-9, 2-12, 2-13, 2-14, 2-23, 2-26, 2-28, 2-31, 2-32, 2-35, 2-53, 2-54, 2-68, 2-69, 2-74, 2-77, 2-78, 2-80, 2-87 and 2-96. Some examples of where students are asked to critique the reasoning of others are problems 2-18, 2-25, 2-38, 2-43 and 2-97.

The instructional materials reviewed for the High School CPM Traditional 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 MP. Overall, the majority of the time MP4 and MP5 are used to enrich the mathematical content and are not treated as individual practices. Throughout the materials, students are expected to model with mathematics and use tools strategically. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

• Many lessons require students to "model with mathematics" and "use appropriate tools strategically." The modeling is done with multiple representations and using various tools. Digital resources, calculators, algebra tiles, blocks, graph paper, and various other tools are used throughout the series. Students are not told which tools to use in solving problems.
  • For example, in sections 7.1.1 - 7.1.2 of Algebra 2, students use tools to build their own models/tools to investigate cyclical behavior. In section 5.1.3 of Algebra 1, students use measuring tools, coordinate grids, and technology to model exponential decay. There are many times when students use tools in smaller, non-modeling situations such as homework problems or investigations and ample opportunities for students to use tools strategically in full, modeling situations.
• It is important to note that MP4 and MP5 are represented in many of the lessons and assessments throughout the series. Listed below are a few examples of where MP4 and MP5 are used to enrich the mathematical content:
  • In section 9.1.3 of Geometry, students are encouraged to make use of appropriate tools as they engage with investigating solids. The teacher's resources recommend a few tools and manipulatives to have available for use. This same lesson has students use graph paper to create and use a model involving a net and a solid that has multiple solutions. Additionally, students solve problems applying prior knowledge to new problems. Students make assumptions and reason if those assumptions work with additional solid figures, leading to drawing conclusions pertaining to surface area and volume of solids.
  • Technology is extensively used across the series. Many lessons and problems are linked to Desmos.com to investigate patterns or connections between graphs and equations or to work with data in modeling situations.

The instructional materials reviewed for the High School CPM Traditional 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 MP. Overall, the majority of the time MP7 and MP8 are used to enrich the mathematical content and are not treated as individual practices. There is an increasing expectation that these practices will lead students to experience the full intent of the standards.

• Since students are developing contextual understanding in most lessons, the units often bring closure in asking students to generalize and make use of structure in their closure sections.
• Students are constantly extending the structures used when solving problems that build on one another and, as a result, are able to solve increasingly complex problems. Repeated reasoning allows for increasingly complex mathematical concepts to be developed from simpler ones, and this series has the expectation that this will occur.
• It is important to note that MP7 and MP8 are represented in many of the lessons and assessments throughout the series. Listed below are a few examples of where MP7 and MP8 are used to enrich the mathematical content.
  • In lesson 5.3.1 and 5.3.2 of Algebra 1, students are asked to compare growth rates in tables to explore the structure of linear and exponential equations and to differentiate between them. They then use their observations to make generalizations about linear and exponential equations to create equations from data tables.
  • In Chapter 5 of Geometry, students use what they know about similarity to develop definitions for sine and cosine functions. Then they further use the structure of similarity and trigonometry to generalize the relationship between angles and sides of triangles by creating shortcuts for special right triangles.
  • In section 2.1.3 of Algebra 2, the lesson calls for students to use their prior knowledge of quadratics and parabolas to go to greater depths by prompting students to look at patterns and structures of quadratic equations to make more generalizations than have occurred prior to this lesson. Though students are not directly asked to decompose "complicated" things to "simpler" things, this is something that students may discover through the investigation in the lesson. One of the main goals of the lesson is to be able to quickly graph and identify key parts of a graph based on an algebraic representation, without a graphing calculator. In essence, students are looking for shortcuts and general methods to do this as these processes are repeated.