High School - Gateway 1

The instructional materials reviewed for the High School CPM Traditional series meet the expectation that materials attend to the full intent of the mathematical content contained in the high school standards for all students. For this indicator, all lessons were examined for evidence of each standards presence and the extent to which the full depth was met. Overall, all of the standards are addressed at some point during the course of this high school series.

• Many of the standards were addressed to their full depth in the instructional material. For example, the Statistics and Probability standards were represented throughout the series. This series places the S-CP cluster in Geometry. The S-ID cluster is in the Algebra 1 text, and the S-IC cluster is in Algebra 2.
  • For example, chapters 6, 10 and 11 of Algebra 1 teach all the standards in the S-ID cluster. Students collect data and model with mathematics as they are learning to quantify variability and describe associations, using common sense, residuals and statistics to interpret categorical and quantitative data.
• N-Q.1 is partially addressed in the materials. There are questions and activities where units were used which included interpreting units associated with graphs. However, students were not required to use units as a guide to solving all problems, nor were they required to interpret the origin in all data displays.
• F-IF.2 is partially addressed in that students use function notation and evaluate functions for the inputs in their domains. However, there were no questions or activities found such that students interpret statements that use function notation in terms of a context.

The instructional materials reviewed for the High School CPM Traditional series meet the expectation that the materials attend to the full intent of the modeling process when applied to the modeling standards. Modeling standards are well-integrated throughout the entire series. Overall the modeling process is used to reach the full depth of the modeling standards. Furthermore, the materials provide students guided support as they develop their understanding of the modeling process.

• The first problem in many lessons is a real-world modeling question. The lesson then develops by investigating several aspects related to the modeling content standard. The lesson scaffolds the modeling problem to provide additional support for students to work through the modeling process. For example, in Algebra 2, section 2.1.1 begins with a modeling problem involving a disk and its radius and mass. Problem 2-1 provides scaffolding to help students breakdown the steps necessary to solve the problem. Building on this, problems 2-2 and 2-3 provide extensions to the original modeling problem.
• The modeling process and every listed modeling standard was evident in the materials. Some examples include:
  • The Burning Candle problem, 11-74, in Algebra 1, section 11.3.1, asks students to gather data and make a prediction using best-fit lines. This is an example of standards S-ID.1 and S-ID.6
  • The Line Factory Logo problem, 2-88, in Algebra 1 asks students to model a logo design and then to have other students use the model to recreate the design. This is an example of modeling standard F-LE.2.
  • The Down on the Farm problem, 2-75, in Algebra 1 asks students to use multiple representations to model the weight of chickens since they were hatched. This is an example representing many standards: N-Q.2, A-CED.2, F-IF.4, F-IF.6, F-IF.7.A, F-BF.1.A, F-LE.1.B, F-LE.2, and F-LE.5.
  • The Sandy Dandy Dune Buggies problem found in Algebra 2 section 4.2.3 models a linear programming situation, including thinking about constraints associated with the situation. This is an example of standards A-CED.3 and F-IF.5.
  • The Blood Splatter problem found in Algebra 2 section 7.1.1 models a swinging pendulum, resulting in a sine curve. This is an example of standard F-TF.5.
  • The Cookie Cutter problem, 8- 115, in Geometry is an authentic, modeling problem in that no specific directions as to what tools, processes, or mathematics should be used are given, yet students must use their knowledge of ratios in Geometry to solve the problem. The standards covered in this problem are G-MG.1 and G-MG.3.
  • The Interior Design problem in Geometry section 7.1.3 models an optimization problem, scaffolding for additional support in problems beyond the original problem. This is an example of standard G-MG.3.

The Instructional materials reviewed for the High School CPM Traditional series 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 in the teacher's resources suggest a timeline and shows a strong focus on widely applicable prerequisites.
  • In Algebra 1, the majority of the 121- 128 days focus on the widely applicable prerequisites.
  • In Geometry, the majority of the 140 days focus on the widely applicable prerequisites, with 37 of those days spent on optional lessons (lessons that can be omitted depending on students prior geometry knowledge).
  • In Algebra 2, the majority of the 129-134 days focus on the widely applicable prerequisites.
• The prerequisites from Grades 6 - 8 were not seen as distracting, but as helpful. For example, in Algebra 1, section 1.2.3 includes the Grade 8 standards on functions. This is helpful in building the high school function objectives of the WAPs.
• Box plots, a middle school learning prerequisite, are found in the Algebra 1 "math notes" in section 11.2.1. The S.ID cluster of standards in Chapter 11 builds on this middle school standard in supporting the statistics and probability standards in the WAPs.
• In the Review & Preview sections of each lesson in Geometry, there are problems that focus on Algebra standards, reinforcing and continuing to build on these important skills from the WAPs.

The instructional materials reviewed for the High School CPM Traditional series meet the expectation that the materials require students to engage in mathematics at a level of sophistication appropriate to high school. Overall, the materials meet the full depth of the non-plus standards and give all students an opportunity to have extensive work with the non-plus standards.

• Each lesson throughout the series is designed to provide students with "learning and practicing a math skill at spaced intervals" (page 31, Teacher Resource, Team Support & Universal Access). The method of spaced intervals allows students to demonstrate mastery over time and to make connections between concepts.
• Each course is also designed to allow for students at a variety of learning levels to access and engage with grade-level, non-plus standards. Using the Teacher Resource binders, teachers are provided with a variety of instructional strategies to assist students who may struggle with aspects of the course work (page 32). There are additional suggestions and supports for students who need additional help (page 33), students who are unprepared for the course (page 34), special needs students (page 34 - 35), English Language Learners (page 35 - 36), and advanced learners (page 37).
• The instructional materials contain a tutorial website for homework help through the eBook. Support is provided for both parents and students who need additional assistance at home. The homework help includes free access via the Internet to all of the Review & Preview problems from the student text. Some of the problems include hints and complete solutions.
• The materials contain a Literacy Support Guidebook within the Teacher Resource Binders (pages 39-49).
• Any guidance for differentiation stresses that pacing is the key to success, rather than reducing the concepts to be learned. The pacing in the Teacher Edition is designed for instructing students at grade level. Below-grade-level students should be provided more time with the concepts by "concentrating on the core problems" that "teach sub-skills and the conceptual understanding needed to progress towards mastery of the course objectives" (page 37). Advanced students should complete the challenges (enrichments) and extensions within the latter parts of the lessons, in addition to core problems and homework.
• Each lesson includes teacher instructions for facilitating discussion around the lesson's core concepts and the connections students have made with other mathematical concepts in earlier coursework.
• The context of the problems are relevant to high school students. Some examples of high school sophistication include, but are not limited to:
  • Algebra 1: Problem 6-109 requires students to work with a piecewise function, find regression lines using a calculator and discuss residual plots. Students are using both exponential best fit and linear best fit models, and are expected to make predictions and identify domains.
  • Geometry: Sections 2.1.2 and 2.1.4 continue to build on problem 2-14 so that a level of sophistication is developed in a real-life and relevant situation that is appropriate for high school students. Students have ample opportunities throughout Chapter 2 to engage deeply with the CCSSM of G-CO.9 and G-CO.10.
  • Algebra 2: In Chapter 6, problem 6-137 involves a case of the cooling corpse. It is a high school level forensic science problem that is sophisticated in both context and content, involving log modeling, and appropriate for Algebra 2.

The instructional materials reviewed for the High School CPM Traditional series meet the expectation that the materials are mathematically coherent and make meaningful connections in a single course and throughout the series, where appropriate and where required by the standards. Overall, the materials include connections that are intentional and thoughtful, and they consistently point out places where students are expected to connect their learning to previous lessons. The sequence of the materials is designed to spiral concepts throughout the chapters and courses.

There are several examples of connections made within the books in the series:

• The homework section within each lesson includes distributed practice of previously learned skills. Each lesson has a Review & Preview section that includes several problems from previous courses and previous lessons.
• The homework problems allow students to apply previously-learned concepts and skills in new contexts. For example, the Team Challenge problem in section 5.1.4 of Geometry extends the trigonometric application of the Climbing in Yosemite problem introduced as an early problem in 5.1.4, and builds on the lessons taught in sections 5.1.1 and 5.1.2.
• Throughout the materials, there are checkpoint problems to determine if students have understanding of previous skills at the expected level. For example, problem 2-53 in Algebra 2 asks students to determine the distance between two points and to write an equation for the line between the points. Checkpoint 2A at the end of the Algebra 2 materials provides more problems of this type for students who need more practice of this skill learned in a previous course.
• Algebra 1 demonstrates strong connections between the conceptual categories of the standards. The materials connect linear functions, exponential functions, arithmetic and geometric sequences, and recursive and explicit representations. In Chapters 1 and 2, students develop a foundational understanding for both the general idea of a function and linear functions. Chapters 3 and 4 focus on solving, simplifying and solving systems of equations but continue to spiral back with problems involving functions and linear functions to deepen and reinforce these foundational skills. In Chapter 5, students begin to work with both arithmetic and geometric sequences which is deliberately connected to students understanding of functions and linear functions. Chapter 5 also builds a foundation for exponential relationships, which is explored more formally in Chapter 7 when exploring and examining geometric sequences. Additionally, the problems use both explicit and recursive representations to further push and connect these concepts.

There are several examples of connections made between the books in the series.

• One example of this connection exists within Geometry and in the connections between Algebra 1 and Geometry. The Geometry book introduces students to similarity early (Chapter 3) and uses the concept of similarity throughout many of the remaining chapters. Chapter 4 carefully develops the major concepts of trigonometry through similarity and slopes of lines. In fact, students are not presented with the formal tangent function until they have had extensive work exploring and solving trigonometry related problems using only their conceptual understanding of slope, similarity, and proportional reasoning, thus making the connections to key concepts from the Algebra 1 materials and previewing concepts in the Algebra 2 materials.
• Another specific instance of connections among standards is in Chapter 2 of Algebra 1, when linear relationships are built using slope triangles from Grade 8. This continues in Geometry in lesson 4.1.3 when students connect slope triangles to trigonometric ratios. Then, Algebra 2 continues with slope triangles in 1.1.2 in the "Math Note" section as a review of linear functions.