High School - Gateway 3

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 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.

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.

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.

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.