# 2.1a Homework Proportional Relationships In Tables

### Alignment

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for alignment. The materials spend the majority of the time on the major work of the grade, and the assessments are focused on grade-level standards. Content is aligned to the standards and progresses coherently across the grades and within each grade. The lessons include conceptual understanding, fluency and procedures, and application. There is a balance of these aspects for rigor. The Standards for Mathematical Practice (MPs) are used to enrich the learning.

GATEWAY ONE

### Focus & Coherence

MEETS EXPECTATIONS

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation for being focused on and coherent with the Common Core State Standards in Mathematics. The Unit Assessments do not assess above grade-level topics, and the instructional materials devote over 65 percent of class time to major work. Supporting work is connected to the major work of the grade, and the amount of content for one grade level is viable for one school year and will foster coherence between the grades. The materials explicitly relate grade-level concepts to prior knowledge from earlier grades, and the materials foster coherence through connections at a single grade, where appropriate and required by the standards.

#### Focus

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for focus within the grade. The Unit Assessments address grade-level material, and the instructional materials meet the expectations for focus within major clusters. Approximately 82 percent of the instructional days are on major work of the grade, including days in which work addressing supporting clusters directly reinforces major work of the grade. Overall, students and teachers using the materials as designed devote the majority of class time to the major work of the grade.

**Criterion 1a**

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Materials do not **assess** topics **before **the grade level in which the topic should be introduced.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation for not assessing topics before the grade-level in which the topic should be introduced. The materials did not include any assessment questions that were above grade-level.

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##### Indicator 1a

The instructional material ** assesses** the grade-level content

**, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.**

__and__,__if applicable__The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for focus within assessment. Overall, the instructional material does not assess content from future grades within the assessment sections of each unit.

There are multiple Self-Assessments within each unit. Each assessment includes a scoring rubric that helps students articulate their understanding of key concepts being assessed. All assessments have answer keys provided in the Teacher Workbook.

On grade-level examples include:

- Chapter 2 Section 2.3- Students demonstrate their knowledge of 7.NS.2 by applying and extending previous understandings of multiplication and division of fractions to multiply and divide rational numbers. Question 2c on the Self-Assessment states: “Estimate each product or quotient. Then find the actual product or quotient of -89(0.5).”
- Chapter 6 Section 6.3- Students solve word problems leading to linear inequalities demonstrating their knowledge of 7.EE.4b. Question 3b on the Self-Assessment states: “Write an inequality to represent each of the following word problems. Solve each problem. Explain your solution in context. 'Jeremy is two years older than Rachel. The sum of the ages of Jeremy and Rachel is less than 46. How old could Jeremy be?'”

**Criterion 1b**

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for having students and teachers using the materials as designed, devoting the large majority of class time to the major work of the grade. Overall, the materials devote approximately 82 percent of class time to major work.

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##### Indicator 1b

Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for Grade 7 meet expectations for focus by spending a majority of class time on the major clusters of the grade including all clusters in 7.RP, 7.NS, and 7.EE. To determine this, three perspectives were evaluated: 1) the number of chapters devoted to major work, 2) the number of lessons devoted to major work, and 3) the number of weeks devoted to major work. Of the three perspectives, the number of lessons is most representative and was used to determine the score for this indicator.

Overall, the materials spend approximately 82 percent of instructional time on the major clusters of the grade. The Grade 7 materials have 8 chapters that contain 139 lessons, which accounts for a total of 31 weeks of class time including Anchor Problems and Self-Assessments.

- Grade 7 instruction is divided into eight chapters. More than half of Chapter 1 addresses 7.NS. Chapter 2 addresses 7.NS. Chapter 3 addresses 7.EE. Chapter 4 addresses 7.RP. More than half of Chapter 6 addresses 7.EE. Therefore, approximately 4.5 out of 8 chapters (56 percent) focus exclusively on the major work of the grade.
- Grade 7 instruction consists of 139 lessons. Approximately 114 lessons out of 139 (82 percent) focus on the major work of the grade level, which includes supporting work that connects to the major work of the grade.
- Grade 7 instruction is divided into 31 weeks. Approximately nineteen out of 31 weeks (61 percent) focus exclusively on the major work of the grade.

#### Coherence

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for being coherent and consistent with the standards. Supporting work is connected to the major work of the grade, and the amount of content for one grade level is viable for one school year and fosters coherence between the grades. Content from prior or future grades is clearly identified, and the materials explicitly relate grade-level concepts to prior knowledge from earlier grades. The objectives for the materials are shaped by the CCSSM cluster headings, and they also incorporate natural connections that will prepare a student for upcoming grades.

**Criterion 1c-1f**

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

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##### Indicator 1c

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The Instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation for the supporting content-enhancing focus and coherence simultaneously by engaging students in the major work of the grade. Overall, the lessons that focus on supporting content also engage students in major work where natural and appropriate.

The following examples demonstrate where the supporting work enhances understanding of the major work of Grade 7.

- Chapter 5: Sections 5.2b, 5.2c, and 5.4b work with 7.G.1, and 7.G.6 supports 7.RP.2 by having students create and solve ratios and proportions to find similar figures. For example, in the overview teachers are told that the central idea of Section 5.2 is scale and its relationship to ratio and proportion. The standard for ratio and proportion are not listed. Ratio language is used in the Activities and Homework problems.
- Chapter 6: Section 6.1 supports 7.EE.4 and 7.NS.1 by having students find angle pairs which involves working with rational numbers and creating/solving equations.
- Chapter 7: Activities 7.1a and 7.1b support 7.NS.1 by having students compare populations which involves working with rational numbers.
- Chapter 7: Sections 7.1 and 7.2 include problems that are related to ratio and proportions (7.RP.2) while working with statistics. For example, Chapter 7, Class Activity 7.2b, Teacher Workbook, page 7WB7 – 43, students use the ratio of colors of jellybeans for a statistical experiment.
- Chapter 8: Activity 8.1c supports 7.EE.4 by having students create and solve equations in real-life mathematical problems based on composite area.

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##### Indicator 1d

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials reviewed for Grade 7 meet the expectations for the amount of content designated for one grade level being viable for one school year in order to foster coherence between grades. The instructional materials are designed to take approximately 155 days. According to the publisher, completing the work would take a total of 31 weeks. That includes days for Anchor Problems, Class Activities, Homework, and Spiral Review. According to the Preface, “Each lesson covers classroom activity and homework for a 50-minute class. Sometimes the demands of the material exceed this limitation; when we recognize this, we say so; but some teachers may see different time constraints, and we defer to the teacher to decide how much time to devote to a lesson, how much of it is essential to the demands of the relevant standard. What is important are the proportions dedicated to the various divisions, so that it all fits into a year’s work. Within a lesson, the activities for the students are graduated, so that, in working the problems, students can arrive at an understanding of a concept or procedure. In most cases there is an abundance of problems, providing the teacher with an opportunity to adapt to specific needs.” The number of weeks was converted to days for this review. Each chapter has built-in days for Self Assessments. Overall, the amount of content that is designated for this grade level is viable for one school year.

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##### Indicator 1e

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The materials reviewed for The Utah Middle School Math Project Grade 7 meet expectations for consistency with the progressions in the Standards. In general, materials develop according to the grade-by-grade progressions in the Standards and provide extensive work with grade-level problems. Materials consistently relate grade-level concepts explicitly to prior knowledge from earlier grades.

Content from prior and future grade levels is identified in Connections to Content at the beginning of each student and teacher workbook chapter. Chapter overviews/summaries, as well as section overviews, include a written explanations of what students will be doing throughout the chapter. Summaries explain what students will learn and how they will use this knowledge in future learning.

- Chapter 1 explains that “throughout the chapter, students are provided with opportunities to review fractions, decimals and percents.” (page 7WB1 - 2)
- Chapter 2: “The development of rational numbers in 7th grade is a progression in the development of the real number system that continues through 8th grade. In high school students will move to extending their understanding of number into the complex number system.” (pages 7WB2 – 3)
- Chapter 4: “The chapter begins by reviewing ideas from 6th grade as well as 7th grade chapters 1-3 and transitioning students to algebraic representations. Student will rely on knowledge developed in previous chapters and grades in finding unit rates, proportional constants, comparing rates and situations in multiple forms, writing expressions and equations, and analyzing tables and graphs.“ (page 7WB4 – 2)
- Chapter 6: “Work on inequalities in this chapter builds on Grade 6 understandings where students were introduced to inequalities represented on a number line. The goal in Grade 7 is to move to solving simple one-step inequalities, representing ideas symbolically rather than with models.” (page 7WB6 - 2)
- Chapter 8: “In 8th grade, students will continue working with volume, formalizing algorithms for volume of cylinders and adding methods for finding the volume of cones and spheres.” (page 7WB8 – 3)

Materials consistently relate grade-level concepts explicitly to prior knowledge from earlier grades. Connections between concepts are addressed in the Connections to Content, chapter overviews/summaries, and Math Textbook. Examples of these explicit connections include:

- Chapter 4, Class Activity 4.1a: “Equivalent Ratios, Fractions, and Percents (Review from 6th grade): They should know a ratio expresses a numerical relation between two quantities. Students studied ratios extensively in 6th grade.” (p. 7WB4 – 14)
- Chapter 8, Connections to Content: “Towards the end of this section students review the use of nets (a concept from 6th grade) to find surface area of prisms and cylinders and then to differentiate this measure from volume, which they will also find.” (page 7WB8 – 2)

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##### Indicator 1f

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.

In the teacher's workbook, the CCSSM are identified on the introduction page of each chapter. Each chapter correlates to a Grade 7 domain, with sections within the chapter focusing on standards within the domain. There is a section titled, “Concepts and Skills to Master," which identifies specific learning objectives for each section in the teacher, parent, and student workbooks.

- “Investigate chance processes, develop/use probability models, as well as the work within the section,” a learning objective from Chapter 1 Section 1.1, reflects Cluster 7.SP.C (Investigate chance processes and develop, use, and evaluate probability models).
- In Chapter 3, students are engaged in activities aligned to Cluster 7.EE.A (Use properties of operations to generate equivalent expressions). In Section 3.1a (page 7WB3-7 - 12), the first Activity, “Naming Properties of Arithmetic,” has an objective to “recognize properties of arithmetic and use them in justifying work when manipulating expressions.” Students are engaged in using the identified properties and identifying pairs of equivalent expressions. In Anchor Problem 3.0 (page 7WB3 - 6), a teacher’s note reflects the cluster heading: “A big idea you’re after right now is that one can write equivalent expressions in a number of ways and that different ways shed light on different thinking.”

The materials include problems and activities that serve to connect two or more clusters in a domain where connections are natural and important.

- The Chapter 3 Overview connects 6.EE.1 and 6.EE.2 through this statement: “By the end of this section (3.1) students should be proficient at simplifying expressions and justifying their work with properties of arithmetic. Section 3.2 uses the skills developed in the previous section to solve equations…Section 3.3 ends the chapter with application contexts.” (pages 7WB3 - 2 and 7WB3 - 3)

The materials include problems and activities that serve to connect two more domains in a grade where connections are natural and important.

- Chapter 3 Section 3.1 and 3.2 connects 7.NS.A and 7.EE.A as students transfer integer properties to algebraic expressions. Students use the Distributive Property of Multiplication and Division over Addition and Subtraction to write equivalent algebraic expressions and to develop an understanding of combining coefficients of like terms and calculating the product of two numbers. (pages 7WB3 – 6 and 7WB3 - 93 through 7WB3 - 109)
- In Chapter 4, Class Activities 4.3d and 4.3e 7.RP.A, 7.NS.A and 7.EE.B are connected as students write equations and compute to solve percent problems. (pages 7WB4 - 173 through 7WB4 - 181)

GATEWAY TWO

### Rigor And Mathematical Practices

MEETS EXPECTATIONS

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for rigor and mathematical practices. The materials meet the expectations for rigor as they balance and help students develop conceptual understanding and procedural skill and fluency. The materials meet the expectations for mathematical practices as they identify and use each of the MPs and support the Standards' emphasis on mathematical reasoning.

#### Rigor and Balance

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance in all three aspects of rigor.

**Criterion 2a-2d**

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance in all three aspects of rigor.

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##### Indicator 2a

Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Each chapter starts with an Anchor problem which poses a mathematical situation that students will learn to solve. Many of these are conceptual in nature. For example, Chapter 3 Anchor Problem 3.0 engages students in the understanding that there are different ways to write equivalent expressions and that the different ways shed light on ways of thinking about the problem. The Teacher’s Notes for that problem emphasize developing understanding.

Many Class Activity problems involve hands-on activities or models. In Chapter 3, students use the properties of operations to generate equivalent expressions. This chapter gives students practice with algebra tiles to build a conceptual understanding of equivalent expressions. In Chapter 3 Class Activity 3.1d, students learn how to use algebra tiles to build a representation of factoring. Later in Class Activity 3.1h, students are shown two ways to factor. Method 1 encourages the use of a model, and in Method 2 students use the greatest common factor.

The teacher notes for each lesson describe the purpose of the lesson and how to guide students to develop their conceptual understanding. The notes include prompts and questions during instruction that lead to conceptual understanding.

Chapters 1 and 2 address 7.NS.A.

- The Chapter 1 Section 1.2 Overview states: “The concept of equivalent fractions naturally leads students to the issues of ordering and estimation. Students will represent order of fractions on the real number line.” Students understand where rational numbers are placed on a number line and use models to solve multi-step problems.
- The Chapter 2 Section 2.1 Overview summarizes the use of hands-on manipulatives and number lines so that students can eventually “reason through addition and subtraction of integers without a model.” Students develop a conceptual understanding of negative numbers and additive inverse by adding integers on a number line, using chips to model addition problems, and using the number line to model subtraction problems.

Chapters 3 and 6 address 7.EE.A.

- In Chapter 3 Class Activity 3.1a and Homework, students determine if two expressions are equivalent and justify their conclusions, consolidating their understanding of the properties of operations.
- In Chapter 3 Class Activity 3.1c and Homework, students use algebra tiles to rewrite algebraic expressions.
- Students demonstrate conceptual understanding to solve Chapter 6 Class Activity 6.2c Problem 1: “Matt, Rosa, and Kathy are cousins. If you combine their ages, they would be 40 yrs. old. Matt is one-third of Rosa's age. Kathy is five years older than Rosa. How old are they? Show several ways to solve the problem. Be able to explain how you came to your answer.”

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##### Indicator 2b

Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, when the intention is that procedural skill and fluency be developed, the materials offer opportunities for their development.

There are examples and repetition in practice in each lesson and homework. Spiral Reviews are found in each chapter that set an expectation of procedural skill and fluency. For example, the Chapter 2 Spiral Review (page 7WB3 - 24) addresses a number of computational standards from previous grades as well as 7.NS.A. Question 5 asks students to solve 5 × (−9).

The following standards are addressed within the course:

- 7.NS.A: Section 2.1 and 2.3 give students practice adding and subtracting rational numbers. Students begin to describe situations in which opposite quantities combine to make 0, and as teachers introduce the properties of arithmetic, students use these properties to add and subtract fluently. Students practice multiplying and dividing rational numbers. Extra Practice sections are also provided.
- 7.EE.1: In Chapter 3, students begin the concept of generating equivalent expressions through the use of concrete models. Students use the commutative property as well as the distributive property to generate an equivalent expression. Students continue procedural practice with solving equations. By the end of Chapter 3, students are expected to be fluent with the properties of operations.
- 7.EE.4: Students move from translating contexts to numeric expressions in Chapter 3 Class Activity 3.1b and Homework to translating contexts to algebraic expressions in Class Activity 3.1c Homework and Additional Practice. In Section 3.2 students build procedural skill and fluency by modeling two-step equations and by using their knowledge of properties. Class Activities 3.2a-c give students practice using models (algebra tiles) to solve two-step equations with and without rational numbers. Class Activities 3.2d and 3.2.e provide opportunities to continue working on this skill to gain fluency.

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##### Indicator 2c

Attention to Applications: Materials are 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

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation that teachers and students spend sufficient time working with engaging applications of mathematics, without losing focus on the major work of the grade. Overall, the materials have opportunities for students to apply mathematical knowledge and/or skills in a real world context.

Throughout the materials students engage in application problems in Class Activity and Anchor Problems. These problems are contextual, and some include multiple representations and steps. Students are asked to present their solutions in ways that demonstrate their understanding of the mathematics in the context.

- Chapter 3 Class Activity 3.3a (7WB3 - 119) includes contextual problems. For example, “Today is Rosa’s 12th birthday. She has a savings account with $515 in it, but her goal is to save $10,000 by the time she turns 18. How much money should she add to her savings account each month to reach her goal of $10,000 between now and her 18th birthday?” Students draw a model, write an equation that represents the model, solve the equation, and answer the question in a full sentence.
- Chapter 4’s Anchor Problem, “Tasting Lemonade,” is a multi-step, real-world, contextual problem that develops analysis of proportional relationships (7.RP.A). It emphasizes solving the problem using a variety of strategies. Students are presented with the context that “you want to sell lemonade in a park” and have five different recipes to choose from, consisting of different concentrate and water ratios. The following problems develop students’ understanding on how the different ratios would affect the flavor of the lemonade, and the Teacher Notes that follow provide a variety of solution strategies to share with students to help them develop flexibility in their application of mathematics.
- “Which one would be the most 'lemony'?"
- “Which would use 10 cups of water?”
- “How much would you need to make 50 cups of each recipe?”

- Chapter 4 Class Activity 4.3c has a variety of multi-step and contextual problems. For example, Question 1 reads as follows: “Ginger and her brother Cal have red and green planting buckets in the ratio of 3:1. a. If there are 5 green buckets, how many red buckets are there? b. Ginger and Cal bought more buckets because they have more to plant. They purchased the buckets in the same red:green ratio of 3:1. If they now have 28 buckets total, how many red and green buckets do they have? c. How are the problems different?”

In Grade 7, some specific standards that include application are 7.NS.3 and 7.EE.3. Examples of problems that address these standards include:

- On page 7WB1 – 55, students are asked to solve problems involving investment rates, target heart rates, and the cost of dinner with tax and tip. (7.NS.3) “Rico's resting heart rate is 50 beats per minute. His target exercise rate is 350% of his resting rate. What is his target rate?” Students use a model and write a number sentence to solve the multi-step problem.
- In Chapter 2 Lesson 2.3a Problem 25 students use a number line to model situations, answer questions using their knowledge of the number line, write an addition equation, and explain their thinking. (7WB2 - 20) (7.EE.3)
- In Chapter 6 Section 6.2 students work in two “different directions.” In some sections, students are given a context and asked to find relationships and solutions while in other sections students are given relationships and asked to write contexts.
- “Write a context that models the following equation. 2L + 2(3L) = 990.” (7WB6 - 66) (7.EE.3)
- “Martha divides $94 amongst her four friends. Leon gets twice as much money as Kokyangwuti. Jill gets five more dollars than Leon. Isaac gets ten less dollars than Kokyangwuti. How much money does each friend get?” (7WB6 - 67) (7.EE.3)

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##### Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation that the materials balance all three aspects of rigor with the three aspects not always combined together nor are they always separated. Every chapter includes all three aspects of rigor. In some lessons the aspects of rigor are addressed separately, and in some lessons multiple aspects of rigor are addressed. Overall, the three aspects of rigor are balanced in this program.

There are lessons where the aspects of rigor are not combined.

- In Homework 4.3b students practice their procedural skill in solving proportions.
- Spiral Reviews throughout the materials provide opportunities for students to reinforce their procedural skills and fluencies from previous standards and lessons.

There are multiple lessons where two or all three of the aspects are interwoven.

- Class Activity 2.1a (page 7WB2 - 7) begins with exploring additive inverses in contexts. For example, “A hydrogen atom has one proton and one electron.” Students demonstrate their understanding by creating a model/picture, writing the net result in words, and answering how many zero pairs exist in the context.
- In Class Activity 2.2c students make connections between multiplication and division of integers. Students solve division problems and also solve contextual problems. At the end of this lesson/homework there is an extra practice section for students to gain fluency with integer operations as well as more contextual problems with integers.

#### Mathematical Practice-Content Connections

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectations for practice–content connections. The materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialized language of mathematics. The instructional materials also support the Standards' emphasis on mathematical reasoning.

**Criterion 2e-2g**

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

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##### Indicator 2e

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade.

The Standards for Mathematical Practices are identified in both the Teacher and Student Workbooks in most lessons. The MPs are explained in the beginning of the chapter and are identified using an icon within the lessons where they occur.

Overall, the materials clearly identify the MPs and incorporate them into the lessons. All of the MPs are represented and attended to multiple times throughout the year, and MPs are used to enrich the content and are not taught as a separate lesson.

- Chapter 1 Class Activity 1.1c Question 5 asks students to "look for and express regularity in repeated reasoning" as students determine patterns emerging in the previous examples of probability (MP8).
- The Chapter 2 Anchor Problem presents a number line with 0, 1, and variables (a) and (b). Students are asked: “Which of the following numbers is negative? Choose all that apply. Explain your reasoning.” Students reason abstractly and quantitatively (MP2) as well as construct viable arguments (MP3).
- Chapter 4 Class Activity 4.1f asks students to "attend to precision" as they find the unit rate in word problems and compare two quantities (MP6). For example, “Frosted Flakes has 11 grams sugar per ounce and Raisin Bran 13 grams per 1.4 ounces. Which cereal has more sugar per ounce?

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##### Indicator 2f

Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 partially meet the expectations for attending to the full meaning of each Mathematical Practice Standard. The MPs are most frequently identified in Teacher Notes where they are aligned to a particular practice activity or question. Many times the note is guidance on what the teacher does or says rather than engaging students in the practice.

The intent of the MPs is often not met since teachers engage in the MPs as they demonstrate to students how to solve the problems.

- Many problems marked MP1 do not ensure that students have to make sense of problems and persevere in solving them. For example, Chapter 2 Class Activity 2.1 directs students to “use the idea of 'zero pairs' to complete the worksheet.” Students are not making sense of problems but answering problems based on how the teacher models the problem.
- MP4 is identified throughout the program; however, it is rarely identified in situations where students are modeling a mathematical problem and making choices about that process. In many situations, it is labeled when directions are provided for how the teacher models. For example, in Chapter 3 Class Activity 3.1i students are given a number line as the model.
- Where MP5 is labeled, the materials suggest a specific tool for students to use which does not lead students to develop the full intent of MP.5. For example, in Chapter 4 Class Activity 4.2b students are told to use the graph and table to model the context.

##### Indicator 2g

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:

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##### Indicator 2g.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation for prompting students to construct viable arguments concerning grade-level mathematics detailed in the content standards.

In many cases, students are asked to construct arguments and justify their thinking.

- Throughout the materials students are asked to justify their thinking. For example, Chapter 4 Homework 4.2b Question 3b asks, “Which solution is saltier, Solution A or Solution B? Justify your answer with at least two pieces of evidence.”
- There are instances where students are asked to make conjectures. For example, in Chapter 1 Class Activity 1.1a Question 7 students are asked to “make a conjecture about how many GREEN tiles are in your bag if the bag contains 12 total tiles.”
- Students are asked to engage in Error Analysis in some of the lessons. For example, in Chapter 4 Class Activity 4.1b Question 6 students must identify the error in the table of values that is represented. In the given solution the error was in adding 2 to each value in column A to get Column B rather than multiplying by 3/2 in Column A.

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##### Indicator 2g.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for The Utah Middle School Math Project Grade 7 meet the expectation of 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. Many of the directions for MP3 are the same as those found written in the Student Workbook. Guidance is given on how to assist students in expressing arguments.

A few examples of guidance provided for teachers include:

- In Chapter 6 Homework 6.1a Questions 11-15 the Teacher Notes state: “Note: constructing an argument to
*disprove*a statement only requires one counterexample, while constructing an argument to 'prove' something is more involved. In other words, one affirmative example does not prove a statement. In 7th grade attention to precision in making statements is an important first step towards building arguments. So, for #14, press students to explain why the statement is true; look for statements that build on understanding of supplementary angles and transitivity.” - In Chapter 4 Class Activity 4.1b the students are given: “The values in the table below represent the lengths of corresponding sides of two similar figures. The side lengths are proportional to one another. Darcy filled in the remaining values in the table and has made a mistake. Find her mistake and fix it by filling in the correct values in the table on the right. Then provide an explanation as to what she did wrong.” The Teacher Note says: ”This problem allows students to critique Darcy’s reasoning and then make their own conjectures about the proportional constant.”
- There are some prompts for the teachers in the form of questions to ask or problems to present. For example, in Chapter 1 Class Activity 1.1b the students roll dice to simulate a horse race. Students determine a specific answer about which horse won most often and why. The Teacher Notes clarify the question and prompt the teacher to ask follow up questions: “Have students justify their arguments. Ask them for evidence to support their claims. Do you think that this game is fair? Why or why not?”

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##### Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation for attending to the specialized language of mathematics. Overall, the materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols. When students are introduced to new mathematical vocabulary, it is explained, and teachers are encouraged to tell students to use the new terms.

- Each chapter in the workbook begins with a vocabulary list of words used in the chapter that includes words from previous learning as well as new terms.
- Throughout the chapter, these terms are used in context during Class Activities, Homework, and Self-Assessments.
- Vocabulary is bold in the context of the lesson.
- Vocabulary is presented throughout the Textbook: Mathematical Foundations along with accurate definitions. For example on 7MF2 - 17, “A
*golden rectangle*is a rectangle that is not a square, but has this property: if we remove the square of whose side is the length of the smaller side of the rectangle, the remaining rectangle is a smaller version of the original.” - Students are encouraged to use vocabulary appropriately. For example, Class Activity 1.1c Question 2f asks: “Have you been computing
*theoretical*or*experimental probability*? Explain.” Class Activity 3.2a, between questions #10 and #11 asks: “What do the terms*evaluate*and*solve*mean? What is the difference between an*equation*and an*expression*?” - At times the Teacher Notes give suggestions for using vocabulary in a lesson. For example, in Chapter 1 Class Activity 1.1a, students are learning about
*experimental probability*, and the Teacher notes recommend, “Discuss again as a group. Compare their thinking now with their thinking before the experiment. Formalize the definition.”. - The terminology that is used in the course is consistent with the terms in the standards.

Although it is not included in the CCSSM, the word *simplify* is used throughout the instructional materials. For example, in Chapter 3 Class Activity 3.1e, between Questions 8 and 9: “Your friend is struggling to understand what it means when the directions say, '*simplify* the expression.' What can you tell your friend to help him? Teacher Note: Answers will vary. Discuss *'simplify'* vs. *'evaluate'* vs. *'solve'* and *'expression'* vs. *'equation.'* Also discuss why we simplify—when does it help and when is it easier to not simplify? You might refer back to Activity 2 above."

GATEWAY THREE

### Usability

PARTIALLY MEETS EXPECTATIONS

The instructional materials for The Utah Middle School Math Project Grade 7 partially meet the expectations for usability. In reviews for use and design, the problems and exercises are developed sequentially, and each activity has a mathematical purpose. Manipulatives and models are used to enhance learning, and the purpose of each is explained well. All materials include support for teachers in using questions to guide mathematical development, and the teacher editions have many annotations and examples on how to present the content and an explanation of the math of each unit and the program as a whole. Although materials include opportunities for ongoing review and practice, there are no assessments that purposely identify prior knowledge within and across grade levels.The teacher materials identify common misconceptions and errors, but there are no specific strategies to address these when they arise. Tasks provide students with multiple entry points that can be solved with a variety of solutions and representations, and there are suggestions for students to monitor their own progress. Activities provide few ELL strategies, support strategies for special populations, or strategies for advanced students to investigate mathematics content at greater depth.

**Criterion 3a-3e**

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The instructional materials for The Utah Middle School Math Project Grade 7 meet the expectations for use and design. Materials are well-designed, and lessons are intentionally sequenced. Students are presented with an Anchor Problem at the beginning of each chapter to introduce new concepts. Anchor Problems are sometimes referenced throughout the chapter. Students produce a variety of types of answers including both verbal and written answers. Manipulatives are used in the instructional materials as mathematical representations and to build conceptual understanding.

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##### Indicator 3a

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

The instructional materials for Grade 7 meet the expectation that the underlying design of the materials distinguishes between problems and exercises.

The chapters begin with a non-routine problem that introduces new concepts and is labeled as an Anchor Problem. The chapters are subsequently sectioned into Class Activities, Homework, Spiral Reviews, and Assessments.

Generally, each Class Activity has problems to solve together as a class with instructor guidance. Occasionally, they are intended to review previous grades' concepts in order to connect them to seventh grade concepts. Most often, the Class Activities are for the students to apply what they have already learned.

The mathematics taught in each Class Activity is reinforced by an accompanying Homework component.

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##### Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.

The instructional materials for Grade 7 meet the expectation that the design of assignments is not haphazard; exercises are given in intentional sequences.

Students are presented with an Anchor Problem at the beginning of each chapter to introduce new concepts. Anchor Problems are sometimes referenced throughout the chapter.

Within each chapter, concept development is sequential. During Class Activities, the teacher introduces new concepts or builds upon prior knowledge. Students work individually or as a whole class when engaged in the Class Activities. The Homework component reinforces the mathematical concepts taught during the previous Class Activity. Spiral Reviews are used to provide continued practice of newly learned mathematical concepts throughout the year.

The progression of lessons taught is intentional and assists students in building their mathematical understanding and skill. Students begin with activities to build conceptual understanding and procedural skill, and progress to applying the mathematics with more complex problems and procedures.

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##### Indicator 3c

There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

The instructional materials for Grade 7 meet the expectation for the variety in what students are asked to produce.

Throughout the Class Activities, students are asked to produce answers and solutions, discuss ideas, make conjectures, explain solutions and justify reasoning, make sketches and diagrams, and use appropriate models. These aspects are found individually within problems as well as in combination with others, such as provide an explanation of a solution and include a diagram.

- Chapter 3 Class Activity 3.1b: Models are used to represent the quantities and relationships stated in contextual problems. Students examine the models and write numeric expressions that represent the quantities and relationships represented by the models as well as explain why the various correct representations are equivalent. In subsequent tasks students determine and explain which expressions are equivalent, write contexts for expressions, and explain how they determined if various expressions adequately represent given contexts. The final problems provide opportunity for application.

- Chapter 3 Class Activity 3.1c: Students transition from writing numeric expressions to algebraic expressions using the same types of tasks and problem formats as those presented in 3.1b. The materials provide additional practice for students to draw models, define variables, and write expressions that model given situations.

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##### Indicator 3d

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The instructional materials for Grade 7 meet the expectation that manipulatives are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models.

Colored tiles are used when students work with probability. In Class Activity 1.1a, students are learning the difference between experimental and theoretical probability with the activity, “How Many Green Tiles Are In Your Bag?” Students draw several tiles out of a bag and record the color each time. By using the tiles, students are able to make conjectures, as well as compare theoretical and experimental probability.

The Anchor Problem in chapter 7, “The Teacher Always Wins,” uses teacher-created colored number cubes to create data through a game between the students and the teacher. Students use the data collected from the game to analyze the probability of winning when using different colored cubes.

##### Indicator 3e

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The instructional materials for Grade 7 meet the expectation that the visual design is not distracting or chaotic and supports students in engaging thoughtfully with the subject.

- The student materials are clear and consistent between activities within a grade level as well as across grade levels.

- Each Class Activity and Homework is clearly labeled and provides consistent numbering for each investigation and problem set with both a lesson number and page number.

- The examples shown in the Textbook: Mathematical Foundation are consistently labeled and numbered within each section.

**Criterion 3f-3l**

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

The instructional materials for The Utah Middle School Math Project Grade 7 meet the expectations for supporting teachers’ learning and understanding of the standards. The instructional materials provide questions that support teachers in delivering quality instruction. The teacher’s edition is easy to use and consistently organized and annotated. The teacher’s edition explains the mathematics in each unit as well as the role of the grade-level mathematics within the program as a whole. The instructional materials are all aligned to the standards, and the instructional approaches and philosophy of the program are clearly explained.

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##### Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

The instructional materials for Grade 7 meet the expectation for supporting teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

### Alignment

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for alignment. The materials spend the majority of the time on the major work of the grade, and the assessments are focused on grade-level standards. Content is aligned to the standards and progresses coherently across the grades and within each grade. The lessons include conceptual understanding, fluency and procedures, and application. There is a balance of these aspects for rigor. The Standards for Mathematical Practice (MPs) are used to enrich the learning.

GATEWAY ONE

### Focus & Coherence

MEETS EXPECTATIONS

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation for being focused on and coherent with the Common Core State Standards in Mathematics. The Unit Assessments do not assess above grade-level topics, and the instructional materials devote over 65 percent of class time to major work. Supporting work is connected to the major work of the grade, and the amount of content for one grade level is viable for one school year and will foster coherence between the grades. The materials explicitly relate grade-level concepts to prior knowledge from earlier grades, and the materials foster coherence through connections at a single grade, where appropriate and required by the standards.

#### Focus

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for focus within the grade. The Unit Assessments address grade-level material, and the instructional materials meet the expectations for focus within major clusters. Approximately 80 percent of the instructional days are on major work of the grade, including days in which work addressing supporting clusters directly reinforces major work of the grade. Overall, students and teachers using the materials as designed devote the majority of class time to the major work of the grade.

**Criterion 1a**

- 0
- 22

Materials do not **assess** topics **before **the grade level in which the topic should be introduced.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation for not assessing topics before the grade-level in which the topic should be introduced. The materials did not include any assessment questions that were above grade-level.

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##### Indicator 1a

The instructional material ** assesses** the grade-level content

**, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.**

__and__,__if applicable__The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for focus within assessment. Overall, the instructional material does not assess content from future grades within the assessment sections of each unit.

There are multiple Self-Assessments within each unit. Each assessment includes a scoring rubric that helps students articulate their understanding of key concepts being assessed. All assessments have answer keys provided in the Teacher Workbook.

On grade-level examples include:

- Chapter 4 Section 4.1- Students demonstrate their knowledge of 8.EE.8 by graphing or solving simultaneous, linear equations by substitution or elimination. Question 3 on the Self-Assessment states: “One equation in a system of linear equations is ???? = −2???? + 4. a. Write a second equation for the system so that the system has only one solution.”
- Chapter 5 Section 5.3- Students demonstrate their knowledge of 8.F.5 by analyzing and then describing a graph. Question 1 Concept 3 on the Self-Assessment states: “Below are two graphs that look the same. Note that the first graph shows the distance of a car from home as a function of time and the second graph shows the speed of a different car as a function of time. Describe what someone who observes the car’s movement would see in each case.”
- Chapter 8 Section 8.2- Students demonstrate their knowledge of 8.EE.4 by subtracting, adding, multiplying, and dividing numbers in scientific notation and then converting the answer to standard form. Question 2a on the Self-Assessment states: “Change the numbers below into scientific notation. 3,450,000,000.” Question 2c states: “Change the number given below into standard form. 6.03 x 108.”

**Criterion 1b**

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for having students and teachers using the materials as designed, devoting the large majority of class time to the major work of the grade. Overall, the materials devote approximately 80 percent of class time to major work.

4/4

##### Indicator 1b

Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for Grade 8 meet the expectation for focus by spending a majority of class time on the major clusters of the grade including all clusters in 8.EE, 8.F, 8.GA, and 8.G.b. To determine this, three perspectives were evaluated: 1) the number of chapters devoted to major work, 2) the number of lessons devoted to major work, and 3) the number of weeks devoted to major work. Of the three perspectives, the number of lessons is most representative and was used to determine the score for this indicator.

Overall, the materials spend approximately 80 percent of instructional time on the major clusters of the grade. The Grade 8 materials have 10 chapters that contain 164 lessons, which accounts for a total 33 weeks of class time including Anchor Problems and Self-Assessments.

- Grade 8 instruction is divided into 10 chapters. Approximately 8 out of 10 chapters (80 percent) focus exclusively on the major clusters of Grade 8, while the other 2 chapters focus primarily on supporting work that does not often support major work.
- Grade 8 instruction consists of 139 lessons. Approximately 131 out of 164 lessons (80 percent) focus on the major clusters of the grade.
- Grade 8 instruction is divided into 33 weeks. Approximately 24.5 out of 33 weeks (74 percent) focus exclusively on the major work of the grade.

#### Coherence

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for being coherent and consistent with the standards. Supporting work is connected to the major work of the grade, and the amount of content for one grade level is viable for one school year and fosters coherence between the grades. Content from prior or future grades is clearly identified, and the materials explicitly relate grade-level concepts to prior knowledge from earlier grades. The objectives for the materials are shaped by the CCSSM cluster headings, and they also incorporate natural connections that will prepare a student for upcoming grades.

**Criterion 1c-1f**

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

2/2

##### Indicator 1c

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The Instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation for the supporting content enhancing focus and coherence simultaneously by engaging students in the major work of the grade. Overall, the lessons that focus on supporting content also engage students in major work where natural and appropriate.

The following examples demonstrate where the supporting work enhances understanding of the major work of Grade 8.

- Chapter 6: Section 6.1 focuses on fitting a straight line to the scatter plot. Students then write a prediction function for the line of best fit and explain the meaning of the slope and y-intercept of the function in context. This work uses 8.SP.A to support 8.F.A and 8.F.B.
- Chapter 6: Section 6.2a and 6.2b uses 8.SP.A to support 8.EE through the use of trend lines and finding the line of best fit.
- Chapter 7: In Activity 7.1a 8.NS.A supports 8.G.B. Through creating squares of different areas, the idea of the Pythagorean Theorem emerges as the sides of a right triangle form three squares and the two sides of a right triangle place together equals the longest side square.
- Chapter 7: Activity 7.2a uses 8.NS.A to support 8.EE through students creating and solving expressions and equations based on powers and roots.
- Chapter 7: Section 7.3 8.NS.A is used to support major work as it focuses on rational and irrational numbers. Under Concepts and Skills to be Mastered, it lists, “Know that the square root of a non-perfect square is an irrational number,” which is a part of 8.EE.2.
- Chapter 10: This chapter links geometry with both number system and expressions and equations, 7.EE.4, as students write and solve Pythagorean Theorem equations where solutions are approximated.

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##### Indicator 1d

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials reviewed for Grade 8 meet the expectations for the amount of content designated for one grade level being viable for one school year in order to foster coherence between grades. The instructional materials are designed to take approximately 165 days. According to the publisher, completing the work would take a total of 33 weeks. (There is some discrepancy in the material regarding Chapter 3. The Chapter overview states that the Chapter is designed for 4 weeks, but the title of the Chapter folder indicates 3 weeks.) Completing the work includes days for Anchor Problems, Class Activities, and Homework. According to the Preface, “Each lesson covers classroom activity and homework for a 50-minute class. Sometimes the demands of the material exceed this limitation; when we recognize this, we say so; but some teachers may see different time constraints, and we defer to the teacher to decide how much time to devote to a lesson, how much of it is essential to the demands of the relevant standard. What is important are the proportions dedicated to the various divisions, so that it all fits into a year’s work. Within a lesson, the activities for the students are graduated, so that, in working the problems, students can arrive at an understanding of a concept or procedure. In most cases there is an abundance of problems, providing the teacher with an opportunity to adapt to specific needs.” The number of weeks was converted to days for this review. Each chapter has built-in days for Self Assessments. Overall, the amount of content that is designated for this grade level is viable for one school year.

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##### Indicator 1e

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The materials reviewed for The Utah Middle School Math Project Grade 8 meet the criteria for consistency with the progressions in the Standards. In general, materials develop according to the grade-by-grade progressions in the Standards and provide extensive work with grade-level problems. Materials consistently relate grade-level concepts explicitly to prior knowledge from earlier grades.

Content from prior and future grade levels is identified in Connections to Content at the beginning of each student and teacher workbook chapter. Chapter overviews/summaries, as well as section overviews, include written explanations of what students will be doing throughout the chapter. Summaries explain what students will learn and how they will use this knowledge in future learning.

- Chapter 1: The Section 1.1 Overview explains that Section 1.1 “involves a review of algebraic expressions.” The teacher notes emphasize this further, telling teachers that the first section is 7th grade material and to work through it faster with an honors class or assign it as homework. (page 8WB1 - 8)
- Chapter 2: “Students begin this chapter by reviewing proportional relationships from 6th and 7th grade, recognizing, representing, and comparing proportional relationships. In 8th grade, a shift takes place as students move from proportional linear relationships, a special case of linear relationships, to the study of linear relationships in general.” (page 8WB2 – 2)
- Chapter 3: “In Chapter 5 students will solidify the concept of function, construct functions to model linear relationships between two quantities, and interpret key features of a linear function. This work will provide students with the foundational understanding and skills needed to work with other types of functions in future courses.” (page 8WB3 - 2)
- Chapter 5: “This chapter builds an understanding of what a function is and gives students the opportunity to interpret functions represented in different ways, identify the key features of functions, and construct functions for quantities that are linearly related. This work is fundamental to future coursework where students will apply these concepts, skills, and understandings to additional families of functions.” (page 8WB5 – 2)

Materials consistently relate grade-level concepts explicitly to prior knowledge from earlier grades. Connections between concepts are addressed in the Connections to Content, chapter overviews/summaries, and Math Textbook. Examples of these explicit connections include:

- Chapter 1, Math Textbook: “The first three chapters of grade 8 form a unit that completes the discussion of linear equations started in 6th grade and their solution by graphical and algebraic techniques.” (page 8MF1 - 1)
- Chapter 6, Class Activity 6.1a: “Problems 1 and 2 provide students with an opportunity to connect what they have learned in 6th/7th grade with what they will learn in 8th grade. Problem 1 is a review of 6th and 7th grade content where students learned to display and analyze univariate data. Students have learned…In 8th grade, students connect this learning with bivariate data.” (page 8WB6 - 10)

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##### Indicator 1f

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. Materials include problems and activities that serve to connect two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.

In the teacher's workbook, the CCSSM are identified on the introduction page of each chapter. Each chapter correlates to a Grade 8 domain, with sections within the chapter focused on standards within the domain. There is a section titled “Concepts and Skills to Master" which identifies specific learning objectives for each section in the teacher, parent, and student workbooks.

- The Chapter 4 Overview reflects 8.EE.C (Analyze and solve linear equations and pairs of simultaneous linear equations) as students work with and “discuss intuitive, graphical, and algebraic methods of solving simultaneous linear equations; that is, finding all pairs (if any) of numbers (
*x, y*) that are solutions of both equations.” (page 8WB4 - 2) - In Chapter 5 Cluster 8.F.A students work with functions (define, evaluate, and compare functions). In Section 5.2 Explore Linear and Nonlinear Functions, students distinguish between linear and nonlinear functions given a context, table, graph, or equation. In Section 5.3 Model and Analyze Functional Relationships, the objective is to analyze functional relationships between two quantities given different representations.

The materials include problems and activities that serve to connect two or more clusters in a domain where connections are natural and important.

- Chapter 3 Section 3.1 Classroom Activity 3.1f problems 1-6 include content from both 8.F.A and 8.F.B as students review the slope-intercept form of a linear equation and then use their understanding to model relationships between quantities. (page 8WB3 – 49)
- Chapter 5 Section 5.3 Classroom Activity 5.3a connects 8.F.A and 8.F.B. Students construct functions to model linear relationships while they are comparing properties of functions that are represented in different ways. (page 8WB5 - 76)

The materials include problems and activities that serve to connect two more domains in a grade where connections are natural and important.

- Chapter 2 Section 2.3d-g connects 8.EE.5 and 8.F.A as students determine the rate of change from graphs. Students compare the rates of graphs, compare the steepness of several lines on the same graph, and relate the steepness of the lines to their rates of change. (pages 8WB2 - 84, 8WB2 - 85 & 8WB2 - 126)
- Chapter 8 Section 8.3 connects 8.EE.A and 8.G.C as students use square root and cube root symbols and evaluate square roots and cube roots while solving problems involving the volume of cylinders, cones, and spheres. (page 8WB8 - 72)

GATEWAY TWO

### Rigor And Mathematical Practices

MEETS EXPECTATIONS

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for rigor and mathematical practices. The materials meet the expectations for rigor as they balance and help students develop conceptual understanding and procedural skill and fluency. The materials meet the expectations for mathematical practices as they identify and use each of the MPs and support the Standards' emphasis on mathematical reasoning.

#### Rigor and Balance

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance in all three aspects of rigor.

**Criterion 2a-2d**

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for rigor and balance. The materials meet the expectations for rigor as they help students develop conceptual understanding, procedural skill and fluency, and application with a balance in all three aspects of rigor.

2/2

##### Indicator 2a

Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Each chapter/section starts with an Anchor Problem which poses a mathematical situation that students will learn to solve. Many of these are conceptual in nature and also provide explicit connections to prior knowledge. For example, Chapter 3 Anchor Problem 3.0: Solutions to a Linear Equation specifically refer to prior learning from Chapter 1 involving writing and solving linear equations with one variable. Students then move on to more complex linear equations as the activity guides them toward problems with infinite solutions and plotting ordered pairs on the coordinate plane. The Teacher Notes say, “Project the grid on the board and ask students to come up and plot an ordered pair that is a solution to the equation. They will soon see that the ordered pairs follow a pattern. Some students may even come up with solutions that include fractions. If not, ask them if there are solutions that fall between integer ordered pairs. Begin filling in all of these solutions as well. Soon a line will start to appear because all of the fractional solutions will start to 'merge' together.” The Teacher Notes emphasize developing understanding.

Many Class Activity problems involve hands-on activities or models. In Chapter 10 Homework 10.2a students use grid paper to draw squares adjacent to the given triangle sides showing a proof of the Pythagorean theorem.

The Teacher Notes for each lesson describe the purpose of the lesson and how to guide students to develop their understanding of a concept. The notes include prompts and questions during instruction that lead to conceptual understanding.

Chapter 2 addresses 8.EE.B as students make the connection between proportional relationships, lines, and linear equations. Students explain, generalize, and connect ideas using supporting evidence; make and justify conjectures; compare information within or across data sets; and generalize patterns. Chapter 2 builds conceptual development as students make the connection between proportional relationships, lines and linear equations.

- The Section 2.1 Concepts and Skills to Master lists conceptual objectives for the students.
- "Graph and write equations for a proportional relationship and identify the proportional constant or unit rate given a table, graph, equation, or context."
- "Compare proportional relationships represented in different ways."

- Throughout Chapter 2 Class Activities and Homework, students are asked to identify correspondences between contexts, tables, graphs, and equations. Students explain, generalize, and connect ideas using supporting evidence (Question 10a, page 8WB2 - 13 and Question 2e-f, page 8WB2 - 46); compare information within or across data sets (Question 7, page 8WB2 - 19), and generalize patterns (2.2a Class Activity & Homework).
- A Teacher Note provides the following directive and explanation to assist teachers in engaging students in the conceptual understanding of the work: “Please refer to the Mathematical Textbook for Chapter 2, as this will help the teacher understand why it is important to approach Standard 8.EE.6 from the perspective that the slope is the same between any two distinct points on a line because of dilations. Transformational geometry is integrated with slope by understanding that a dilation produces figures with proportional parts. Right triangles that are formed from any two distinct points on a line are dilations of one another. Since they are dilations of one another they have corresponding parts that are proportional and parallel. This is why the rise/run ratio is the same from any of these triangles and thus the slope is the same between any two distinct points.” (page 8WB2 - 85)

Chapter 9 addresses 8.G.A.: Lessons develop conceptual understanding of translations, reflections, rotations, dilations, their properties, as well as their roles in determining if/when two figures are congruent, and if/when they are similar. Students examine preimages and images, use patty paper to perform reflections and rotations, use tables to record coordinates of images and preimages, describe transformations, draw figures congruent and/or similar to ones shown, and write coordinate rules for transformations shown. Properties of translations, reflections, rotations, and dilations are experimentally verified and compared/contrasted.

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##### Indicator 2b

Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Overall, when the intention is that procedural skill and fluency be developed, the materials offer opportunities for their development.

There are examples and repetition in practice in each lesson and homework.

- 8.EE.7: In Lessons 1.1a, 1.2a, 1.2b, and 1.2d students have opportunities to develop procedural skill and fluency in solving linear equations. Students work with rational numbers throughout. In the Homework students are tasked with problems that promote fluency, including being able to identify common mistakes (pages 8WB1 - 35 and 36). Throughout Chapter 1 and in other chapters in the grade (Chapters 3, 4, and 6), students are solving linear equations in one variable that includes rational coefficients. This leads them to fluency by the end of the year.
- 8.G.9: In the student content in Chapter 8, students are introduced to volume in a conceptual way, as they are asked to describe what volume is and its importance. Attention is given to 8.G.9 in Section 8.3. In Lessons 8.3b, 8.3c, and 8.3d, students find the volume of spheres, cones, and cylinders using the correct formulas. Students derive the equations for the volume of cones, cylinders, and spheres and practice problems related to them in Section 8.3. In Class Activity 8.3b Questions 7-12 and Homework Questions 1-6, students find the volume and/or missing measurement of each cylinder. In Class Activity 8.3c Questions 7-12 and Homework Questions 1-6, students find the volume and/or missing measurement of each cone. In Class Activity 8.3d8 - 13 and Homework 1-6 students are given the directions to find the volume and/or missing measurement of each sphere.

Note that there are no spiral reviews in Grade 8 to provide additional procedural skill and fluency practice.

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##### Indicator 2c

Attention to Applications: Materials are 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

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation that teachers and students spend sufficient time working with engaging applications of mathematics, without losing focus on the major work of the grade. Overall, the materials have multiple opportunities for application.

The materials incorporate Anchor Problems at the beginning of each chapter which provide students multi-step questions where they solve problems by using a variety of paths.

- In Anchor Problem 1.0 students use their own assumptions to model several situations mathematically. Students are directed to “(c)onsider the following situations. Then answer the questions below. Include any pictures, models, or equations you used to solve the problem and clearly explain the strategy you used.” For example, students explore the following situation: “Two students, Theo and Lance, each have some chocolates. They know that they have the same number of chocolates. Theo has four full bags of chocolates and five loose chocolates. Lance has two full bags of chocolates and twenty-nine loose chocolates. Determine the number of chocolates in a bag. Determine the number of chocolates each child has.” (page 8WB1 - 7)
- Chapter 4’s Anchor Problem, “Chickens and Pigs,” has multiple ways to solve (trial and error, a table, pictures or symbolic representations, and equations and graphs). There is flexibility for students in this contextual problem to apply their mathematical understanding. “A farmer saw some chickens and pigs in a field. He counted 30 heads and 84 legs. Determine exactly how many chickens and pigs he saw. There are many different ways to solve this problem, and several strategies have been listed below. Solve the problem in as many different ways as you can and show your strategies below.“ (page 8WB4 – 6)

In Grade 8, a specific standard and cluster that include application are 8.EE.8c and 8.F.B. Examples of problems that address these standards include:

- Chapter 4 "Who will win the race?" Class Activity and Homework is a multi-step problem leading to two linear equations in two variables that encourage students to use their own methods of problem solving so that there are multiple paths of entry. (8.EE.8c)
- In Chapter 4 Section 4.2 students solve simultaneous linear equations that have one, no, or infinitely many solutions using algebraic methods. For example, Homework 4.2e Question 4 asks the following: “Sarah has $400 in her savings account, and she has to pay $15 each month to her parents for her cellphone. Darius has $50, and he saves $20 each month from his job walking dogs for his neighbor. At this rate, when will Sarah and Darius have the same amount of money? How much money will they each have?” (8.EE.8c)
- In Chapter 2 students calculate the cost for attending the state fair and riding various rides given the costs for the fair and each ride. Students determine the total number of rides that can be be purchased, given a specific amount, and create a table and graph to represent the situation. (8.F.B)
- Chapter 5 Section 5.3 includes real-world contexts such as a height vs. time function, pennies earned per day, and the half-life of Carbon-14. (8.F.B)

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##### Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation that the materials balance all three aspects of rigor with the three aspects not always combined together nor are they always separate. Every chapter includes all three aspects of rigor. In some lessons the aspects of rigor are addressed separately, and in some lessons multiple aspects of rigor are addressed. Overall, the three aspects of rigor are balanced in this program.

There are lessons where the aspects of rigor are not combined.

- Homework 3.1a provides a variety of problems for students to practice their procedural skill of writing equations in slope-intercept form.
- In Class Activity 9.1a students develop their conceptual understanding of translations by answering a variety of questions designed to illicit similarities and differences between the properties of translations.

There are multiple lessons where two or all three of the aspects are interwoven.

- In Class Activity 4.2c (page 8WB4 - 44) students are first asked to solve, in any way they choose, a couple of contextual problems that can be modeled and solved using systems of linear equations. For example, “Carter and Sani each have the same number of marbles. Sani’s little sister comes in and takes some of Carter’s marbles and gives them to Sani. After she has done this, Sani has 18 marbles and Carter has 10 marbles. How many marbles did each of the boys start with? How many marbles did Sani’s sister take from Carter and give to Sani?” Students then determine the values of "shape-addends" that form two shape equation systems designed to match the contextual problems. This work is followed by more shape equation systems for students to represent algebraically and to solve using the elimination method. Students apply their understanding of the elimination method and develop procedural skills as they solve several systems of linear equations. In the final task students create a context for a given system, solve the system, and write the solution in a complete sentence.
- In Class Activity 1.1c students use models to solve linear equations by combining like terms. Students develop an understanding of like terms through the models leading to development of procedural skill. By the end of the lesson, students are asked to solve equations and verify their solutions.

#### Mathematical Practice-Content Connections

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectations for practice–content connections. The materials show strengths in identifying and using the MPs to enrich the content along with attending to the specialized language of mathematics. The instructional materials also support the Standards' emphasis on mathematical reasoning.

**Criterion 2e-2g**

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

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##### Indicator 2e

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade.

The Standards for Mathematical Practices are identified in both the Teacher and Student Workbooks in most lessons. The MPs are listed in the beginning of the chapter and are also identified using an icon within the lessons where they occur.

Overall, the materials clearly identify the MPs and incorporate them into the lessons. All of the MPs are represented and attended to multiple times throughout the year, and MPs are used to enrich the content and are not taught as a separate lesson.

- Chapter 3 Homework 3.1g Questions 1-4 ask students to "reason quantitatively" as they write equations from graphs of linear representations and then tell the story that relates to the equation (MP2). For example, “The graph below shows a trip taken by a car where
*x*is time (in hours) the car has driven and*y*is the distance (in miles) from Salt Lake City. Label the axes of the graph. Use your graph and equation to tell the story of this trip taken by the car.” - Chapter 4 Class Activity 4.2c highlights MP1 as students make sense of problems. The problem states: “Ariana and Emily are both standing in line at Papa Joe’s Pizza. Ariana orders 4 large cheese pizzas and 1 order of breadsticks. Her total before tax is $34.46. Emily orders 2 large cheese pizzas and 1 order of breadsticks. Her total before tax is $18.48. Determine the cost of 1 large cheese pizza and 1 order of breadsticks. Explain the method you used for solving this problem..” The Teacher Note also emphasizes that “there are a variety of ways to solve this problem” and gives examples of methods which reinforce the goal of students making sense and persevering in solving the problem as students can solve the problem many different ways.
- Chapter 5 Class Activity 5.1b, “The Function Machine,” asks students to “attend to precision” by figuring out the rule from a table of values. Students rely upon accurately calculating the values to figure out the rules (MP6).

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##### Indicator 2f

Materials carefully attend to the full meaning of each practice standard

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 partially meet the expectations for attending to the full meaning of each Mathematical Practice Standard. The MPs are most frequently identified in Teacher Notes where they are aligned to a particular practice activity or question. Many times the note is guidance on what the teacher does or says rather than engaging students in the practice.

The intent of the MPs is often not met since teachers engage in the MPs as they demonstrate to students how to solve the problems.

- Many problems marked MP1 do not ensure that students have to make sense of problems and persevere in solving them. For example, in Chapter 4 Class Activity 4.2b and Homework, problems are heavily scaffolded and centered around using systems of equations. This does not give students an opportunity to engage in making sense of the problem or persevering in solving them.
- MP4 is identified throughout the program; however, it is rarely identified in situations where students are modeling a mathematical problem and making choices about that process. In many situations, it is labeled when directions are provided for how the teacher models. For example, in Chapter 6 Class Activity 6.1a: “In 10-13 above the adjacent angle pairs are also examples of supplementary angles. Are adjacent angles always supplementary? Why or why not?” The Teacher Notes add: “No, have students draw a counterexample.” “Also, begin to talk about simple equations. For example, #12 can be written as: B + 123 = 180 or 180 – 123 = B. In other words, you’re beginning to discuss modeling with mathematics.”
- Where MP5 is labeled, the materials suggest a specific tool for students to use which does not lead students to develop the full intent of MP5. For example, in Chapter 3 Class Activity 3.1c students are told to graph each equation by hand and then use a graphing calculator to check their line.

##### Indicator 2g

Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:

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##### Indicator 2g.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation for prompting students to construct viable arguments concerning grade-level mathematics detailed in the content standards.

In many cases, students are asked to construct arguments and justify their thinking,

- Throughout the materials students are asked to justify their thinking. For example, in Class Activity 1.2c Questions 9-13 students are asked to “plot each fraction on the number line. Fill in the blank with < , > or = . How do you know your answer is correct? Justify your answer.”
- There are instances where students are asked to make conjectures. For example, in Chapter 5 Class Activity 5.1a Questions 1-3 the directions state: “Make a conjecture (an educated guess) about what kind of relationship makes a function and what disqualifies a relation from being a function.”
- Students are asked to engage in Error Analysis in some of the lessons. For example, in Chapter 10 Homework 10.2c Question 15 students identify the error in using the Pythagorean Theorem Formula to calculate the leg between hypotenuse and other leg. In the given solution, the error was in subtracting, instead of adding the area of the squares before finding the square root.

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##### Indicator 2g.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation of 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. Many of the directions for MP3 are the same as those found written in the Student Workbook. Guidance is given on how to assist students in expressing arguments.

A few examples of guidance provided for teachers include:

- There is assistance for the teacher in engaging students in constructing viable arguments. Chapter 6 Class Activity 6.1a Question 1a asks students to “make some observations about the data shown in the dot plot.” The Teacher Notes state the following: “Listen to what students say. They may say things like, the average amount she makes is around $100. The data does not appear to be very spread out. The point 55 appears to be an outlier and may pull the average down. What could have caused this outlier? She can usually expect to make between $75 and $120 a day.”
- There is assistance for the teacher in engaging students in analyzing the arguments of others. For example, in Chapter 8 Class Activity 8.1d the Teacher Notes say,
*“*When they are finished have them discuss their answers with a neighbor before moving on to a group discussion. Ask for people to come to the board to show and justify how they fixed the mistake in each statement.” - There are some prompts for the teacher in the form of questions to ask or problems to present. For example, in Chapter 5 Class Activity 5.2b students are prompted to determine if given situations can be modeled with linear functions and to provide evidence to back their claim. Teachers are given this guidance in the notes: “For the answers below, students can provide various pieces of evidence (constant/changing rate of change; first difference is the table is constant/not constant; graph is/is not a line; form of the equation). Accept all valid explanations.”

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##### Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for The Utah Middle School Math Project Grade 8 meet the expectation for attending to the specialized language of mathematics. Overall, the materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols. When students are introduced to new mathematical vocabulary, it is explained, and teachers are encouraged to tell students to use the new terms.

- Each chapter in the workbook begins with a vocabulary list of words used in the chapter that includes words from previous learning as well as new terms.
- Throughout the chapter, new terms are used in context during Class Activities, Homework, and Self-Assessments.
- Vocabulary is bold in the context of the lesson.
- Vocabulary is presented throughout the textbook, Mathematical Foundations, along with accurate definitions. For example, on page 8MF2 - 3: “Given two quantities x and y, they are said to be proportional if, whenever we multiply one by a factor r, the other is multiplied by the same factor, r. For example, if we double the variable x, then y also doubles.”
- Students are encouraged to use vocabulary appropriately. For example, Chapter 9 Class Activity 9.1 introduces the terms:
*translation,**pre-image, image,*and*corresponding vertice**s*. These terms are introduced, defined, and taught to the students. They are used throughout the chapter. - At times the Teacher Notes give suggestions for using vocabulary in a lesson. For example, Chapter 5 Class Activity 5.1a says, “Talk to the students about the term
*unique*and how it is used in mathematics, as they will see it in many definitions in the future.” - The terminology that is used in the course is consistent with the terms in the standards.

GATEWAY THREE

### Usability

PARTIALLY MEETS EXPECTATIONS

The instructional materials for The Utah Middle School Math Project Grade 8 partially meet the expectations for usability. In reviews for use and design, the problems and exercises are developed sequentially, and each activity has a mathematical purpose. Manipulatives and models are used to enhance learning, and the purpose of each is explained well. All materials include support for teachers in using questions to guide mathematical development, and the teacher editions have many annotations and examples on how to present the content and an explanation of the math of each unit and the program as a whole. Although materials include opportunities for ongoing review and practice, there are no assessments that purposely identify prior knowledge within and across grade levels.The teacher materials identify common misconceptions and errors, but there are no specific strategies to address these when they arise. Tasks provide students with multiple entry points that can be solved with a variety of solutions and representations, and there are suggestions for students to monitor their own progress. Activities provide few ELL strategies, support strategies for special populations, or strategies for advanced students to investigate mathematics content at greater depth.

**Criterion 3a-3e**

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The instructional materials for The Utah Middle School Math Project Grade 8 meet the expectations for use and design. Materials are well-designed, and lessons are intentionally sequenced. Students are presented with an Anchor Problem at the beginning of each chapter to introduce new concepts. Anchor Problems are sometimes referenced throughout the chapter. Students produce a variety of types of answers including both verbal and written answers. Manipulatives are used in the instructional materials as mathematical representations and to build conceptual understanding.

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##### Indicator 3a

The underlying design of the materials distinguishes between problems and exercises. In essence, the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Each problem or exercise has a purpose.

The instructional materials for Grade 8 meet the expectation that the underlying design of the materials distinguishes between problems and exercises.

The chapters begin with a non-routine problem that introduces new concepts and is labeled as an Anchor Problem. The chapters are subsequently sectioned into Class Activities, Homework, and Assessments.

Generally, each Class Activity has problems to solve together as a class with instructor guidance. Occasionally, they are intended to review previous grades concepts in order to connect them to eighth grade concepts. Most often, the Class Activities are for the students to apply what they have already learned.

The mathematics taught in each Class Activity is reinforced by an accompanying Homework component.

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##### Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.

The instructional materials for Grade 8 meet the expectation that the design of assignments is not haphazard; exercises are given in intentional sequences.

Students are presented with an Anchor Problem at the beginning of each chapter to introduce new concepts. Anchor Problems are sometimes referenced throughout the chapter.

Within each chapter, concept development is sequential. During Class Activities, the teacher introduces new concepts or builds upon prior knowledge. Students work individually or as a whole class when engaged in the Class Activities. The Homework component reinforces the mathematical concepts taught during the previous Class Activity. Spiral Reviews are used to provide continued practice of newly learned mathematical concepts throughout the year.

The progression of lessons taught is intentional and assists students in building their mathematical understanding and skill. Students begin with activities to build conceptual understanding and procedural skill, and progress to applying the mathematics with more complex problems and procedures.

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##### Indicator 3c

There is variety in what students are asked to produce. For example, students are asked to produce answers and solutions, but also, in a grade-appropriate way, arguments and explanations, diagrams, mathematical models, etc.

The instructional materials for Grade 8 meet the expectation for the variety in what students are asked to produce.

Throughout the Class Activities, students are asked to produce answers and solutions, discuss ideas, make conjectures, explain solutions and justify reasoning, make sketches and diagrams, and use appropriate models. These aspects are found individually within problems as well as in combination with others, such as provide an explanation of a solution and include a diagram.

- Chapter 1 Class Activity 1.1a: Students examine the models and write the simplified form of the expressions. Subsequent tasks require students to evaluate expressions and produce a model for a given expression.

- Chapter 1 Homework 1.1a: Students are asked to “identify the mistake, explain it, and simplify the expression correctly.”

- Chapter 1 Class Activity 1.1c: Students are asked to “model and solve the following equations. Show the solving action, and verify your solution.”

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##### Indicator 3d

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The instructional materials for Grade 8 meet the expectations that manipulatives are faithful representations of the mathematical objects they represent and, when appropriate, are connected to written models.

Algebra tiles are used to model and simplify linear expressions. In Class Activity 1.1a, Question 5 reads, “Using your tiles, model the expression 2(2???? − 1). a. Write the simplified form of this expression. b. Evaluate this expression when ???? = 0.”

Number lines are used to model the approximate location of irrational numbers. In Class Activity 7.3d, Question 1 reads, “Between which two integers does the square root of 5 lie? a. Which integer is it closest to? b. Show its approximate location on the number line below. c. Now find the square root of 5 accurate to one decimal place. Show its approximate location on the number line below.”

##### Indicator 3e

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The instructional materials for Grade 8 meet the expectation that the visual design is not distracting or chaotic and supports students in engaging thoughtfully with the subject.

- The student materials are clear and consistent between investigations within a grade-level as well as across grade-levels.

- Each Class Activity and Homework is clearly labeled and provides consistent numbering for each investigation and problem set with both a lesson number and page number.

- The examples shown in the Textbook: Mathematical Foundation are consistently labeled and numbered within each section.