Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings, meeting the expectations for this indicator.

• Evidence for this indicator is found throughout all of the chapters, including the chapter assessments.
• Generally, lessons develop understanding through the group work that students complete in the lessons.
• There are extensive suggestions in the Teacher's Guide for every lesson describing the purpose of the lesson and how to guide study teams to develop their understanding of a concept.
• Teacher questioning during instruction is designed to lead to conceptual understanding, with questions like "How do you see it?" "How can you tell if it’s correct?" "What is the pattern?" "Is there a different way?"
• Students are consistently being asked to communicate with their group and explain their understanding.
• Chapters 2, 3, 4 and 7 all include work directly related to the clusters that address conceptual understanding (7.NS.A, 7.EE.A). Examples of this are:
  • Percent bars;
  • Ratio tables;
  • Multiple representations of the constant of proportionality;
  • Cecil, the tightrope walker, for adding integers;
  • Algebra Tiles for combining like terms, variables, comparing expressions; and
  • Positive/Negative chips for integers.
• The materials provide evidence of high-quality conceptual problems using concrete representation, algebra tiles, experimenting, verbalization, online activities/tools, multiple representations, and interpretation.
• Students are required to use previous learning to construct new learning.

Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency, thereby meeting the expectations for this indicator.

• There is evidence of the opportunity to develop fluency and procedural skills in every chapter.
• Fluency is especially evident in the constantly spiraling homework. According to the publisher, about one-fifth of the homework is on new material and the rest of the homework reviews previous concepts.
• The skills are frequently embedded in an engaging activity such as the silent board game or the human graph.
• Procedural skill and fluency, from the clusters that emphasize it (7.NS.A, 7.EE.A.1, 7.EE.B.4), is evident in Chapters 2, 3, 4, 6 and 7. Procedural skill and fluency is developed through strategies such as:
  • Examples and repetition in practice;
  • Chapter closures have problems with solutions - if students miss them, they are directed back to the relevant lesson ("Need Help?") and to additional practice problems that align with what they missed ("More Practice");
  • Math Note boxes reinforce vocabulary and concrete examples (Integers, Proportional Relationships in Graphs and Tables);
  • Learning logs (Evaluating Algebraic Expressions, Unit Rate);
  • Spiral homework; and
  • Checkpoint problems - with extra practice if not mastered (Unit Rates & Proportions, Simplifying Expressions).
• Beyond the lessons, the suggested chapter tests also require students to demonstrate fluency and procedural skill.

Materials meet the expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade.

• There is evidence of the opportunity to work with engaging applications of the mathematics in every chapter.
• There are multiple non-routine problems throughout the chapters such as Robert’s New Hybrid Car, What Are You Eating, Fencing the Basketball Court, Community Service, Renting the Hall, and Birthday Sweets.
• Chapters 2, 4, 5 and 7 all explicitly provide opportunity for students to engage in application in the standards of Grade 7 (7.RP.A, 7.NS.A.3, 7.EE.B.3) that specify application. Application is developed through non-routine problem solving such as:
  • Shopping Deals - (finding and using percentages);
  • The Yogurt Shop - charged by weight (proportional relationships);
  • Maverick Movie Theater - sizes of popcorn (volume and scaling); and
  • Alvin, the Deep Submergence Vehicle -- ocean depths (integers).
• Students are frequently presented with problems in real world situations that are relevant to them.
• Students must also apply their understanding through teaching others.
• Beyond the lessons, the suggested chapter tests also require students to apply their knowledge.

The three aspects of rigor are not always treated together and are not always treated separately, thereby meeting the expectations for this indicator. There is a balance of the three aspects of rigor within the grade.

• Conceptual and procedural knowledge as well as application of knowledge and skills are balanced throughout the course.
• There are multiple lessons where two, or all three, of the aspects are interwoven.
  • A mixed lesson, application and conceptual, is 7.1.5. The community service club is painting senior citizen apartments and need to figure out how many can be done in a day. Previously only 4/7 of the club was able to do 9 apartments, how many can the whole club complete? Students are guided through a possible process with the president of the club visualizing with algebra tiles and being able to create an equation to solve the problem.
• There are also multiple lessons where one aspect is the clear focus, which is almost equally split among all three aspects, with perhaps a slight emphasis on conceptual development.
  • An example of a fluency lesson is 2.2.1. Cecil, the tightrope walker, has to combine lengths to get across. In various ways of presenting the problem, students solve about 35 addition of integer problems in the lesson.
  • An example of a conceptual lesson is 3.2.2. It begins with an exploration with integer tiles to find multiple ways to represent a value and identifying efficient steps, followed by creating an argument to answer the question, "Do you think that every subtraction problem can be rewritten as an addition problem that gives the same result?" In the next problem, they have to justify, "When would rewriting subtraction problems into addition problems be useful?" The answers to these are shared/discussed in class. The lesson ends with a Learning Log entry called "Connecting Addition and Subtraction" which needs to include examples and diagrams.
• In addition, there is a balance of the three aspects of rigor in included assessments, all pre-made individual assessments contain questions on conceptual understanding, procedural skill and fluency, and application.

The materials meet the expectation for ensuring that the MPs are identified and used to enrich mathematics content within and throughout each applicable grade.

• There is a clear articulation of connection between MPs and content. Materials regularly and meaningfully connect MPs throughout the lessons.
• There is a chart in the teacher’s guide that aligns the MPs with the course, including an in-depth explanation of how they are “deeply woven into daily lessons.”
• Every unit identifies the MPs used on the teacher chapter overview page.
• In the Teacher's Guide, each unit specifically relates how the listed standards are used in the unit and for each lesson. These are logical connections and integrated with the content.
• Teachers are reminded to encourage the use of mathematical practices in team discussions even if they aren’t identified.
• Most lessons incorporate multiple practice standards as students have the opportunity to deeply engage with authentic mathematics of the grade.
• All eight MPs are represented throughout the course.
• Connections are NOT made in the student materials until the end of the book in the end-of-course reflection when students are asked to discuss/reflect on the entire course about them. The questions and problems in this section clearly facilitate students understanding and making connections to the MPs, though there is concern that the reflection could easily be skipped by teachers if instructional time for the regular lessons runs short.

The materials meet the expectation for attending to the full meaning of each practice standard.

• Each practice is addressed multiple times throughout the year. Over the course of the year, students have ample opportunity to engage with the full meaning of every MP. Examples include:
  • MP1; 9.3.1 Maverick Movie Theater Popcorn, 9.1.1 Bubble Madness
  • MP2; 2.2.1 Composing Integers, 8.3.1 Intro to Angles
  • MP3; 7.2.1 Proportional Relationships, 2.3.1 Choosing a Scale
  • MP4; 9.3.3 Estimating Fish Populations, 7.1.2 Scaling Quantities
  • MP5; 4.3.1 Combining Like Terms, 6.2.1 Solving Equations
  • MP6; 8.1.1 Measurement Precision, 3.1.2 Scale Drawings
  • MP7; 3.1.1 Grouping Expressions, 5.2.4 Probability Tables
  • MP8; 9.3.2 Delightful Design, 3.2.2 Addition and Subtraction, 9.1.1 Circumference, Diameter and Pi
• MPs are embedded in lessons, assessments, mid-year and end-of-year reflection, and puzzle investigator problems.
• There are clear definitions for all the practices as well as where they are addressed in the curriculum.
• The core structure and components section of the Teacher's Guide defines each MP and provides a rationale of how the program addresses each math practice. On page 55 a chart identifies problem tasks that integrate multiple MPs.
• In the Teacher Guide the prep section for every lesson identifies the MP(s) and connects the MP(s) to the content of the lesson. For example, in 4.2.3 "Students will use tables and graphs to make sense of problems...Attending to precision is important today as students must be careful when specifying units of measure."

The materials meet the expectation for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

• The materials have questions throughout every lesson to encourage students to construct viable arguments and critique each other’s reasoning with heavy emphasis on group work.
• Students are consistently being asked to verify their work, find mistakes and look for patterns or similarities.
• Students construct viable arguments through activities such as explaining their thinking or justifying steps. For example in a single lesson (5.3.2), students are asked to justify their answers, present work so that someone could make sense of it, explain their reasoning, and show their solution.

The materials meet the expectation for assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

• Teachers are encouraged throughout the Teacher’s Guide to ask students questions.
  • Teachers ask questions such as “Who agrees? Who disagrees? Why is there a disagreement?”
  • For example in 3.3.3, the Teacher's Guide states "As you circulate, listen for the abstract and quantitative reasoning, the construction of viable arguments and critiquing of others’ reasoning and attention to precision in team discussions."
  • In 3.3.3, the Teacher's Guide prompts teachers throughout the lesson description to ask questions such as "How can you tell? What does your answer represent? and Do you have enough information to keep working?" There are also prompts about having teams do Think-Pair-Share and Hot Potato with different problems so they have to explain their thinking and asking students to justify solutions.
  • This is true for every lesson. Any page you flip to includes questions prompts like, “What information do you need? How can you check? What does this mean? Help me understand how? Why did you? Did anyone else?”
• Teachers also are encouraged to assign tasks that require students to consistently engage in debate.
• The course is designed for students to work in teams and have them collaborate and explain their thinking to each other.

The materials meet the expectation for explicitly attending to the specialized language of mathematics.

• Each chapter ends with a vocabulary list of words used in the unit that includes words from previous learning as well as new terms. Students are referred to the glossary, and it is suggested that they record unfamiliar words in the Learning Log.
• Each chapter includes a resource page of concept map cards with the vocabulary of the chapter.
• Throughout the unit, these terms are used in context during instruction, practice, and assessment.
• Vocabulary is bolded in the context of the lesson, and then, it is pulled out specifically in “Math Notes” sections in each chapter.
• There are suggestions like “Encourage students to use appropriate vocabulary, referencing the word wall when necessary.” Sometimes they even list specific words that should be included.
• There is vocabulary that seems unique to CPM such as the “Giant 1” or “fraction busting” or “the 5-D process” as strategies.
• The terminology that is used in the course is consistent with the terms in the standards.