8th Grade - Gateway 3

The materials meet the expectation for the underlying design of the materials distinguishing between problems and exercises.

Each student lesson begins with “core mathematical content (page 4)” - usually a non-routine, problem-solving team challenge or game, then includes further learning (i.e., math notes box, learning log, more practice), then homework, called “review/preview” that allows them to practice mathematics concepts and to prepare for future mathematics concepts because this is a spiral text.

New mathematics is usually taught through completion of core problems, sometimes referred to as tasks, by collaborative groups with the teacher acting as the facilitator, leading the students through guiding questions and activities. Each lesson is designed so that students solve only a few complex problems.

Following the core problems is the “Review & Preview” section that includes 4-7 problems, which may also be scaffolded. Even within this spiral review, there are non-routine problems as well as routine exercises, designed to build mastery over the course of the year. Students have multiple opportunities to be exposed to previously learned concepts throughout the year.

The design of assignments is not haphazard; exercises are given in intentional sequences meeting the expectations for this indicator.

The sequence of lessons develops in a coherent flow that naturally builds students’ mathematical foundations. The chapters are divided by similar content. For example, in Chapter 9, Angles and Pythagorean Theorem, section 9.1 is angles, section 9.2 is Pythagorean theorem, clearly indicating to students that, while connected in the big picture, this is different than what they just worked on.

• Within each section, topic development is sequential.
• Lessons are numbered by chapter.topic.lesson: for example, 4.1.3.
• Problems are numbered sequentially throughout the entire chapter, classwork, and homework: for example, 6-83 is the 83rd problem in Chapter 6.

Homework spirals and includes practice of what was learned in that lesson, review of previous concepts, and introduction of what’s coming. This is intentionally spaced practice to constantly review key concepts and lay foundation for what’s coming. The review and preview section problems build on each other, starting with simpler problems and building to more complex problems, most problems are scaffolded with parts a and b allowing students to have an entry point into the problem and then moving to more complex math. The program’s philosophy is to embed the mixed, spaced practice throughout the course of the year, teaching students to rely on both recall of how to solve a problem and to identify what type of problem it is.

Each chapter ends with Closure for students to consolidate understanding or correct misunderstandings. Full mastery of a concept is not always intended after the daily lesson. Partial mastery of material is often reviewed and revisited in further lessons, building to full mastery.

There is a variety in what students are asked to produce, which meets the expectations for this indicator.

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 all work, including: checking of solutions, drawing visual representations in the form of figures, tables, graphs, etc., writing equations, explaining steps and reasoning, and justifying responses.

Students are asked to represent their thinking using multiple representations. Within the collaborative class structure, students are constructing viable arguments and critiquing each other's reasoning, using equation mats, drawing tables and creating graphs, writing equations, looking for patterns, explaining their reasoning both verbally and in writing, and making explicit connections between the variety of representations. There is a Learning Log in each lesson that allows students to make connections and reflect on their learning.

The manipulatives are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models, which meets the expectations for this indicator.

Manipulatives are appropriately used and explained. Algebra tiles are the primary manipulative used at this level. This program makes heavy use (especially during Grades 6 and 7) of algebra tiles to make concrete, accurate representations of very abstract concepts. The tiles are used so often that students would really learn these concepts well.

Most of the manipulatives used in CPM are commonly available, and many schools may already have them such as rulers, protractors, angle rulers, cubes, square tiles, counters, spinners, and dice. There is a materials list provided on page 7 that includes general supplies like graph paper, sticky dots, masking tape as well as grade-level specific materials for activities like ribbon, straws, tracing paper, models of solids. Manipulatives and/or models accurately and consistently represent the related mathematics.

Materials support teachers in planning and providing effective learning experiences through teacher questioning, which meets expectations for this indicator.

This is a strength of the program with multiple questions for each lesson/problem throughout all of the Grade 8 units. Prompts and questions are provided in the teacher edition for both the lesson activity and daily closure. These questions are not generic, but are designed to focus students on the core concept of the lesson.

Some lessons contain a narrative titled “Team Strategies” that offers suggestions for management of the collaborative teams during activities. Teachers are often given suggested teaching strategies that incorporate higher level thinking.

Sample questions are also provided to assist ELL students and students who have more difficulty with the mathematics concepts in the Universal Access section of each lesson. Additionally, teachers are prompted to reflect on student learning after the lesson in order to prepare for the next daily lesson or unit.

However, this guidance is provided in long, narrative form and requires teachers to spend significant time “digging in” in order to prepare the lesson. If teachers do not use narrative information fully in preparation of each lesson, they may struggle to guide the instruction and use questioning techniques effectively.

The materials meet the expectation for the teacher’s edition having many suggestions on presenting the content to students.

The preparatory materials presented for each lesson are extensive - generally two to five narrative pages of suggestions, scripts, prompts, things to look for, tips for strategies, universal access, homework, pacing, and guiding questions. The multi-page lesson overview also gives narrative information about the mathematics content, including connections to MPs, information to assist teachers in understanding connections being made within and across chapters, and the core problems to focus on for each lesson. The teacher’s edition, text, assessments and additional resources are also online and some include videos that may generate interest in the lesson.

The Universal Access guide gives teachers support on helping struggling students, but it is simplistic and doesn’t help teachers in identifying the root cause of why students maybe be struggling or how to address the needs of the struggling students. Extensive information is offered on management of collaborative teams, including specific methods/techniques that can be used for various activities and helping students to work together and share their thought process and mathematics learning.

In-print and online, teacher guidance is available for use of various technologies, including examples such as graphing calculators and an online transformation tool available to the teacher and students. Individual problems include notes on when to use a form of technology.

Materials do provide strategies for gathering information about students' prior knowledge within and across grade levels.

• There is a pre-course diagnostic assessment to gather information about skills from prior grade levels.  
  • Questions are aligned to standards from the previous grade to assess background content knowledge
  • There is a good balance of content with 4-5 questions per cluster

• Within the grade level, there are multiple opportunities to check for student understanding before they move to the next chapter.  

• There is an assessment bank resources with a test generator online providing teachers with flexibility to create assessments to meet their needs.

Materials meet the expectation for providing strategies for teachers to identify and address common student errors and misconceptions. Within each lesson of the Teacher’s Guide, trouble spots for students are embedded within the “Suggested Lesson Activity” and ideas about how to address those possible misunderstandings are provided.

Often, possible questions to ask students are offered that allow for students to discuss misconceptions and to lead them toward deeper, more accurate understanding of the math concepts.

In the student tasks, students are presented with situations which are based on common misconceptions and asked to identify mistakes or provide rationales, which presents opportunities for “teachable moments.”

Materials meet the expectation for providing opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

This program is very strong in on-going review; the program’s philosophy of how to build mastery is research-informed, where mastery is best achieved over time. The materials provide several opportunities for ongoing review and practice, particularly in the daily spiral homework and mid/end of year course reflections.

Each chapter ends with a “Chapter - Closure” lesson. This lesson contains several activities which provide students with multiple ways to review and prove what they have learned. Each chapter has a checkpoint problem that is material that students are supposed to have mastered at this point, generally a review from content learned in prior years. Students are provided with additional practice in the back of the book if the checkpoint concept is not mastered.

Practice of both basic skills and major concepts is spaced throughout the curriculum. Teacher Planning offers prompts to orchestrate discussion which gives students instant feedback on their thinking and processes. The teacher materials do not offer provisions on how to provide feedback to students regarding homework assignments. A concern is that because of the structure of study teams for learning students may not routinely receive specific individual feedback.

The materials meet the expectation for providing strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners:

• Scaffolding and sequencing is built in to lesson development and specific problems to help students master concepts.
• Each suggested lesson activity gives support to teachers to help engage all students through use of different strategies.

• Throughout the Teacher's Guide, many lessons include specific scaffolding suggestions, including probing questions, for concepts that students may struggle with. These can often be found in the Suggested Lesson Activity and/or the Universal Access section that is included with many lessons. Ex:  Lesson 4.1.2 has a suggestion for students who are having a hard time seeing pattern growth to use one color to shade each level of growth.

• Every lesson also has a section of “Team Strategies” with tips on how to facilitate collaboration.
• Each lesson identifies the “core problems” in case some of the work needs to be reduced.
• Specific problems that may need scaffolding are identified. Example: “Academic Literacy & Language Support: A useful sentence starter for the Learning Log entry in problem 4-57 could be ‘In order to tell if a shape has been enlarged correctly…’”
• Specific problems that may provide additional challenge are identified. Example: “Additional Challenge: Problem 4-56 is provided for students or teams who have time or the inclination to reduce by a more challenging multiplier.”

The materials meet the expectation for providing teachers with strategies for meeting the needs of a range of learners:

• The program is designed with varying levels of problems. Students who need more time can focus on the core problems of the lesson (which are listed in each lesson plan). On-level students can solve more problems, and advanced learners can additionally incorporate the enrichment and extension problems built into the latter part of each lesson.
• There is a “Universal Access” tab in the teacher’s edition. It provides suggestions for Special Education students, ELL, and Advanced students. They are non-specific suggestions, though there is repeated emphasis on the benefits of the study team structure.
• Some lessons contain a narrative titled “Universal Access” with suggestions to support ELL, low-level, and high-level students on specific concepts. These tips are specific to the lesson's content.
• Students have access to an online tutorial website for homework help. It provides hints and step-by-step solutions intended to help students understand homework concepts.
• Homework also includes varying levels of problems that could challenge more advanced students.

The materials meet the expectation for embedding tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

A variety of solution strategies is always encouraged in the problem tasks, and students are given opportunities to create solution paths on their own.

Study teams are encouraged to use collaboration to solve tasks that lead to understanding and mastery of concepts in most lessons. Students are encouraged to make observations and brainstorm to solve the tasks within their study team.

Lessons include richer problems with multiple entry points and teacher suggestions for scaffolding lower performing students consistently through teacher’s edition.

The materials meet the expectation for providing opportunities for advanced students to investigate mathematics content at greater depth.

Problems that are not identified as “core problems” in each lesson are considered enrichment or extension problems, allowing students to investigate deeper application. In the back of the book, there is a section of Puzzle Investigator Problems that offers challenging situations.

There is a pacing option provided to accelerate students through Courses 1-3 to be able to enroll in Algebra I in Grade 8.

The teacher materials contain a “Universal Access Guidebook” offering general information about how the CPM series supports students at all levels. Some lessons contain a narrative titled “Universal Access” with suggestions to support high-level students on specific concepts. The study team structure provides opportunity for advanced students to deepen their understanding by helping their group. Students are frequently encouraged to reflect on their learning, providing opportunities for advanced students to make connections.

The materials meet the expectation for providing a balanced portrayal of various demographic and personal characteristics.

No examples of bias were found.

Pictures, names, and situations present a variety of ethnicities and interests.