## enVision Mathematics Common Core

##### v1
###### Usability
Our Review Process

#### Additional Publication Details

Title ISBN Edition Publisher Year
enVision Mathematics Common Core Grade 5 9780134959054 Digital Pearson Education 2020
enVision Mathematics Common Core Kindergarten 9780134958996 Digital Pearson Education 2020
enVision Mathematics Common Core Grade 3 9780134959023 Digital Pearson Education 2020
enVision Mathematics Common Core Grade 4 9780134959030 Digital Pearson Education 2020
enVision Mathematics Common Core Grade 2 9780134959016 Digital Pearson Education 2020
enVision Mathematics Common Core Grade 1 9780134959009 Digital Pearson Education 2020
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## Report for 7th Grade

### Overall Summary

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for alignment to the CCSSM. ​The instructional materials meet expectations for Gateway 1, focus and coherence, by focusing on the major work of the grade and being coherent and consistent with the Standards. The instructional materials meet expectations for Gateway 2, rigor and balance and practice-content connections, by reflecting the balances in the Standards and helping students meet the Standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor and meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

###### Alignment
Meets Expectations
###### Usability
Meets Expectations

### Focus & Coherence

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for Gateway 1, focus and coherence. The instructional materials meet the expectations for focusing on the major work of the grade, and they also meet expectations for being coherent and consistent with the standards.

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

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

​The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for not assessing topics before the grade level in which the topic should be introduced. The materials assess grade-level content and, if applicable, content from earlier grades.

##### Indicator {{'1a' | indicatorName}}
The instructional material assesses the grade-level content and, if applicable, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that they assess grade-level content.

Each Topic contains diagnostic, formative, and summative assessments. Summative assessments provided by the program include: Topic Assessments Forms A and B, Topic Performance Tasks Forms A and B, and Cumulative/Benchmark Assessments. Assessments can be administered online or printed for paper/pencil format. No above grade-level assessment questions are present. Examples of grade-level assessment aligned to standards include:

• Topic 1, Assessment Form A, Question 2, “Four out of nine dogs weigh less than 20 pounds. What is the decimal equivalent for the number of dogs weighing under 20 pounds? A) 0.$$\overline{2}$$ B) 0.24 C) 0.$$\overline{4}$$  D) 0.49. (7.NS.2)
• Topic 2, Assessment Form A, Question 3, “The graph shows how many bottles a machine fills in a certain number of seconds. Part A: What is the constant of proportionality, and what does it mean in this situation? Part B: Choose one ordered pair on the graph. What does it represent in this situation?” (7.RP.2)
• Topic 3, Performance Task Form A, Question 6, “The sanctuary takes out a $12,500 loan to renovate its gift shop. At 5% simple interest, how much would the sanctuary need to pay back in total after 10 years?” (7.RP.3) • Topics 1-4, Cumulative/Benchmark Assessment, Question 20, “The temperature of chicken soup is 192.7° F. As it cools, the temperature of the soup decreases 23° F per minute. Part A: What is the temperature of the soup after 25 minutes? Part B: How many minutes will it take for the soup to cool to 100.7° F?” (7.EE.3) • Topic 8, Performance Task Form A, Question 3, “Dave decides to add a strip of wood diagonally in each frame for extra support. The wood is sold in lengths of 10 feet, 15 feet, or 20 feet. If Dave wants to use one strip for each diagonal with the least amount of waste, which length of wood should he buy? Explain.” (7.G.2) #### Criterion 1.2: Coherence 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 enVision Mathematics Common Core Grade 7 meet expectations for students and teachers using the materials as designed devoting the large majority of class time to the major work of the grade. The instructional materials devote approximately 75% of instructional time to the major clusters of the grade. ##### Indicator {{'1b' | indicatorName}} Instructional material spends the majority of class time on the major cluster of each grade. The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for spending a majority of instructional time on major work of the grade. • The approximate number of Topics devoted to major work of the grade (including assessments and supporting work connected to the major work) is 5.5 out of 8, which is approximately 69%. • The number of lessons (content-focused lessons, 3-Act Mathematical Modeling tasks, projects, Topic Reviews, and assessments) devoted to major work of the grade (including supporting work connected to the major work) is 67 out of 89, which is approximately 75%. • The number of days devoted to major work (including assessments and supporting work connected to the major work) is 144 out of 186, which is approximately 77%. A lesson- level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each Topic. As a result, approximately 75% of the instructional materials focus on major work of the grade. #### Criterion 1.3: Coherence Coherence: Each grade's instructional materials are coherent and consistent with the Standards. The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for being coherent and consistent with the standards. The instructional materials have supporting content that engages students in the major work of the grade and content designated for one grade level that is viable for one school year. The instructional materials are also consistent with the progressions in the standards and foster coherence through connections at a single grade. ##### Indicator {{'1c' | indicatorName}} Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Materials are designed so supporting standards/clusters are connected to the major standards/clusters of the grade. Examples from the Teacher Resource include: • Lesson 6-2, Draw Inferences from Data, Visual Learning, Do You Know How?, Item 6, students use proportional relationships and equations to make predictions based on given data, “In the dot plot above, 3 of 20 players made all 5 baskets. Based on this data, how many players out of 300 players will make all 5 baskets.” This question connects the supporting work of 7.SP.2, use data from a random sample to draw inferences about a population with an unknown characteristic of interest to the major work of 7.RP.2, recognize and represent proportional relationships between quantities and 7.EE.3, solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form. • Lesson 6-3, Make Comparative Inferences about Populations, Visual Learning, Example 3, students calculate measures of center and measures of variability to compare two data representations, “Mr. Bunsen had students grow the same type of plant in two different rooms to test the growing conditions. The box plots show the heights of all the plants after 3 weeks. How do the two populations compare? What inferences can be drawn?” This example connects the supporting work of 7.SP.3, informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability to the major work of 7.EE.3, solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form. • Lesson 7-3, Understand Experimental Probability, Visual Learning, Example 2, students use ratios and percents to make predictions using experimental probability while analyzing proportional relationships, “Joaquin also kept track of players and winners for his game during the fair. Based on the results shown in the table, how many winners should he expect if 300 people play his game?” This example connects the supporting work of 7.SP.6 supports approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency and predict the approximate relative frequency given the probability to the major work of 7.RP.3, use proportional relationships to solve multistep ratio and percent problems. • Lesson 8-1, Solve Problems Involving Scale Drawings, Visual Learning, Example 1, students use proportional relationships to solve problems involving scale drawings, “The island in the blueprint is 6 inches long. What is the actual length of the island in the diagram?” This example connects the supporting work of 7.G.1, solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale to the major work of 7.RP.2, recognize and represent proportional relationships between quantities. • Lesson 8-4, Solve Problems Using Angle Relationships, Visual Learning, Example 1, students find the measure of angles using angle relationships and recognize the relationship between different angles formed by intersecting lines and rays, “Why might a civil engineer be concerned if the intersection of roads is skewed?” This example connects the supporting work of the 7.G.5, use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure, to the major work of 7.EE.3, solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form. ##### Indicator {{'1d' | indicatorName}} 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 enVision Mathematics Common Core Grade 7 meet expectations that the amount of content designated for one grade-level is viable for one year. As designed, the instructional materials can be completed in 162-186 days. According to the Pacing Guide in the Teacher Resource, Program Overview, “enVision Mathematics 6-8 was designed to provide students rich opportunities to build understanding of important new mathematical concepts, develop fluency with key skills necessary for success in algebra, and to gain proficiency with the habits of mind and thinking dispositions of proficient mathematical students. To achieve these goals, the program includes content-focused lessons, 3-Act Mathematical Modeling lessons, STEM projects, and Pick a Project. All of these instructional activities are integral to helping students achieve success, and the pacing of the program reflects this. Teachers are encouraged to spend 2 days on each content-focused lesson, giving students time to build deep understanding of the concepts presented, 1 to 2 days for the 3-Act Mathematical Modeling lesson, and 1 day for the enVisions STEM project and/or Pick a Project. This pacing allows for 2 days for each Topic Review and Topic Assessment, plus an additional 2 to 4 days per topic to be spent on remediation, fluency practice, differentiation, and other assessment.” For example: • There are 8 Topics with 57 content-focused lessons for a total of 114 instructional days. • Each of the 8 Topics contains a 3-Act Mathematical Modeling Lesson for a total of 8-16 instructional days. • Each of the 8 Topics contains a STEM Project/Pick a Project for a total of 8 instruction days. • Each of the 8 Topics contains a Topic Review and Topic Assessment for a total of 16 instructional days. • Materials allow 16-32 additional instructional days for remediation, fluency practice, differentiation, and other assessments. ##### Indicator {{'1e' | indicatorName}} 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 instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for the materials being consistent with the progressions in the Standards. The instructional materials clearly identify content from prior and future grade-levels and use it to support the progressions of the grade-level standards. According to the Teacher Resource, Program Overview, “Connections to content in previous grades and in future grades are highlighted in the Coherence page of the Topic Overview in the Teacher’s Edition.” These sections are labeled Look Back and Look Ahead. Examples from the Teacher Resource include: • Topic 1 Overview, Rational Number Operations, Math Background, Coherence, “In Grade 6 students graphed integers and their opposites on a number line to understand their equal distances from zero. Students examined positive and negative numbers in real world contexts and related the meaning of the distance from the number to zero. Students began exploration into the concept of the opposite of the opposite of a number as being the number itself. In Grade 6, students performed operations with decimals and fractions by following an algorithm or utilizing a number line. Students recognized that opposite signs in rational numbers indicated locations on opposite sides of zero on the number line. In Grade 8, students will continue to use operations with positive and negative integers and rational numbers to solve equations by using inverse relationships and operation rules. In Grade 8, students will use operations with positive and negative integers and rational numbers to find solutions for a system of linear equations using algebraic methods. In Grade 8, students will begin to examine the use of square and cube roots and will discover that square roots of negative numbers cannot be found in the real number system.” • Topic 2 Overview, Analyze and Use Proportional Relationships, Math Background, Coherence, “In Grade 6 students learned to reason about ratios by using equivalent ratios, tables of equivalent ratios, bar diagrams, and double-number-line diagrams. In Topic 6, they used what they learned about ratios to work with a special type of ratio called a percent. In Topic 5, students learned about a special type of ratio called a rate. In Grade 8, students will understand the connections among proportional relationships, lines, and linear equations. Students will also graph proportional relationships and compare proportional relationships represented in different ways. In Grade 8, students will use similar triangles to investigate slope, and they will derive the equations y = mx and y = mx + b for lines.” • Topic 5 Overview, Solve Problems Using Equations, Math Background, Coherence, “In Grade 6, students learned to evaluate expressions, and then applied this knowledge to write and solve one-step equations and inequalities. Students represented and analyzed the quantitative relationship between dependent and independent variables and established an understanding that a change to one quantity directly affects the other. In Grade 6, students learned to use graphs, tables, and number lines. They analyzed solutions and related mathematical models to expressions, equations, and inequalities. In Grade 8, students will extend their understanding of expressions, equations, and inequalities to more complex versions. They begin to explore equations that contain two variables. In Grade 8, students continue to make connections between models and equations including proportional relationships, lines, and linear equations.” The instructional materials attend to the full intent of the grade-level standards by giving all students extensive work with grade-level problems. The Solve & Discuss It! section presents students with high-interest problems that embed new math ideas, connect prior knowledge to new learning and provide multiple entry points. Example problems are highly visual, provide guided instruction and formalize the mathematics of the lesson. Try It! provides problems that can be used as formative assessment following Example problems and Convince Me! provides problems that connect back to the Essential Understanding of the lesson. Do You Understand?/Do You Know How? problems have students answer the Essential Question and determine students’ understanding of the concept and skill application. Examples from the Teacher Resource include: • Lesson 1-5, Add and Subtract Rational Numbers, Solve & Discuss It!, students extend their knowledge of positive and negative rational numbers to adding and subtracting with rational numbers and apply their knowledge to solve real-world problems, “Malik hikes Castle Trail from point A to point B. The elevation at Point A is below sea level. What are the possible beginning and ending elevations of Malik’s hike?” (7.NS.1) • Lesson 3-6, Solve Simple Interest Problems, Visual Learning, Example 1, students understand what simple interest is and how it is calculated, “Victoria opens a savings account with a deposit of$300. She will earn 1.6% simple interest each year on her money. How much interest will she earn over 5 years (assuming she does not add or take out any money?).” (7.RP.3)
• Lesson 4-5, Factor Expressions, Do You Know How?, Item 3, students find common factors of linear expressions using the distributive property and recognize factoring is the opposite of expanding expressions, “Sahil is putting together supply kits and has 36 packs of x pencils, 12 packs of y crayons, and 24 erasers. a. Write an expression to show the total number of items. b. Use factoring to show how many kits Sahil can make while putting every type of item in each kit. c. Use the factored expression to find the number of each item in each kit.” (7.EE.1, 7.EE.2)
• Lesson 6-4, Make More Comparative Inferences About Populations, Visual Learning, Example 1, students use center and variability to compare populations, “Quinn collects data from a random sample of 20 seventh grade students who participate in a youth fitness program. She compares the number of curl-ups each student completed in thirty seconds last year and this year. What can Quinn infer from her comparison of the data sets?” (7.SP.3, 7.SP.4)
• Lesson 5-1, Write Two-Step Equations, Visual Learning, Example 2, Try It!, students use a variable to represent an unknown in a real-world context and construct an equation, “Marcia and Tamara are running a race. Marcia has run 4 kilometers. Tamara has completed $$\frac{3}{4}$$ of the race and is 2.5 kilometers ahead of Marcia. Write an equation that represents the relationship between the distances each girl has run. Let k represent the total length of the race in kilometers.” (7.EE.4)
• Lesson 8-5, Solve Problems Involving Circumference of a Circle, Visual Learning, Example 1, Convince Me!, students use given information about diameter to make deductions about circumference, “If the diameter is doubled, what happens to the circumference? Explain.” (7.G.4)

The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Each Lesson Overview contains a Coherence section that connects learning to prior grades. Examples from the Teacher Resource include:

• Lesson 2-1, Connect Ratios, Rates, and Unit Rates, Lesson Overview, Coherence, “Students will be able to use ratios and rates to describe the relationship between two quantities. Students find equivalent ratios and use unit rates to solve multi-step problems.” (7.RP.1, 7.RP.3) “In Grade 6, students learned the concepts of ratios and rates and used math models and ratio reasoning to solve problems.”
• Lesson 6-1, Population and Samples, Lesson Overview, Coherence, “Students will be able to understand the difference between a population and a sample. Students establish whether a sample is representative of a population. Students will be able to generate random samples that represent the entire population.” (7.SP.1). “In Grade 6, students learned that statistical questions include, and account for variability in the data as part of the answers.”
• Lesson 8-2, Draw Geometric Figures, Lesson Overview, Coherence, “Students will be able to draw geometric shapes with given conditions. Students name and classify quadrilaterals according to their properties.” (7.G.2) “In Grade 6, students drew polygons in the coordinate plane using given conditions. Students found the area of special quadrilaterals and applied this understanding to solve real-world problems.”
##### Indicator {{'1f' | indicatorName}}
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 enVision Mathematics Common Core Grade 7 meet expectations that materials foster coherence through connections at a single grade, where appropriate and required by the Standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings. Topics are divided into Lessons focused on domains. Grade 7 standards are clearly identified in each Topic Planner found in the Topic Overview. Additionally, each lesson identifies the Content Standards in the Mathematics Overview. Examples from the Teacher Resource include:

• Lesson 1-6, Multiply Integers, Lesson Overview, Mathematics Objective, “Multiply positive and negative integers. Use models and mathematical properties to develop a deep understanding of and fluency with multiplying integers.” (7.NS.2a, 7.NS.2c)
• Lesson 2-2, Determine Unit Rates with Ratios of Fractions, Lesson Overview, Mathematics Objective, “Find unit rates with ratios of fractions. Use unit rates to solve multi-step problems.” (7.RP.1, 7.RP.3)
• Lesson 4-7, Subtract Expressions, Lesson Overview, Mathematics Objective, “Identify the similarity of the procedure between subtracting integers and subtracting linear expressions. Simplify linear expressions involving subtraction.” (7.EE.1, 7.EE.2)
• Lesson 6-1, Populations and Samples, Lesson Overview, Mathematics Objective, “Understand the difference between a population and a sample. Establish whether a sample is representative of a population. Generate random samples that represent the entire population.” (7.SP.1)
• Lesson 8-7, Describe Cross Sections, Lesson Overview, Mathematics Objective, “Describe and sketch cross sections of right rectangular prisms and right rectangular pyramids. Solve problems involving cross sections.” (6.G.A.3)

Materials include problems and activities that 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. Examples include:

• Lesson 2-5, Graph Proportional Relationships, Visual Learning, Example 2, students solve real-world problems involving the four operations with rational numbers, “The graph shows a proportional relationship between the distance and the amount of time Mr. Brown drives. a. What does each of these points represent in this situation: (0, 0), (1, 55), and (5, 275)? b. What is the constant of proportionality? c. What equation relates the distance, y, and the time, x?” This example connects the work of 7.RP to the work of 7.NS.
• Lesson 3-2, Connect Percent and Proportions, Visual Learning, Example 1, students use their understanding of proportional relationships to solve equations with percentages from a diagram (shown are 9 blue dots representing shots made and 3 purple dots representing shots missed), “The basketball team statistician tracked the shots Emily made and the shots she missed during the last game. What percent of attempted shots did she make? Draw a bar diagram and write a proportion to represent the number of shots made and the total number of shots. Solve the proportion to find the percent of shots made during the last game.” This example connects the work of 7.RP to the work of 7.EE.
• Lesson 4-4, Expand Expressions, Practice & Problem Solving, Item 15, students generate equivalent expressions as they use properties of operations to add, subtract, multiply, or divide rational numbers, “A grocery store has 13%-off sale on all bread. You decided to purchase 6 loaves of bread. Let b be the original price of a loaf of bread. Expand the expression 6(b - 0.13b). Once the expression is expanded, what do the terms represent?” This example connects the work of 7.EE.A, use properties of operations to generate equivalent expressions to the work of 7.NS.A, apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
• Lesson 5-1, Write Two-Step Equations, Practice and Problem Solving, Item 7, students use the full range of rational numbers as they work with algebraic expressions and equations, “A farmer ships oranges in wooden crates. Suppose each orange weighs the same amount. The total weight of the crate with g oranges is 24.5 pounds. Write an equation that represents the relationship between the weight of the crate and the number of oranges it contains.” Students are provided a diagram that shows the weight of the crate itself as 15 pounds. This question connects the work of 7.NS to the work of 7.EE.
• Lesson 8-3, Draw Triangles with Given Conditions, Visual Learning, Example 3, students draw triangles with a combination of given side lengths and angle measures, “Can more than one triangle be drawn with the following conditions? a. side lengths of 5 inches and 6 inches with an angle of 45°. b. a side length of 6 inches with angles at each end measuring 40° and 60°.” This example connects the work of 7.G.A, draw, construct, and describe geometrical figures and describe the relationships between them to the work of 7.G.B, solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

### Rigor & Mathematical Practices

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for Gateway 2, rigor and balance and practice-content connections. The instructional materials meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations by giving appropriate attention to the three aspects of rigor, and they meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

##### Gateway 2
Meets Expectations

#### Criterion 2.1: Rigor

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 enVision Mathematics Common Core Grade 7 meet expectations for reflecting the balances in the standards and helping students meet the standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications. The instructional materials also do not always treat the aspects of rigor separately or together.

##### Indicator {{'2a' | indicatorName}}
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 enVision Mathematics Common Core Grade 7 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “The Solve & Discuss It in Step 1 of the lesson helps students connect what they know to new ideas embedded in the problem. When students make these connections, conceptual understanding takes seed. In Step 2 of the instructional model, teachers use the Visual Learning Bridge, either in print or online, to make important lesson concepts explicit by connecting them to students’ thinking and solutions from Step 1.” Examples from the Teacher Resource include:

• Lesson 1-4, Subtract Integers, Visual Learning, Example 2, students use number lines to build their understanding of integers and writing matching equations. “Ian’s football team lost 2 yards on a running play. Then they received a 5-yard penalty. What is the team’s total change in yards? Write a subtraction expression to represent the change in yards. Write an equivalent addition expression.” (7.NS.1)
• Lesson 2-2, Determine Unit Rates with Ratios of Fractions, Solve & Discuss It!, students extend their understanding of rates and ratios as they explore real-world problems, “Allison and her classmates planted bean seeds at the same time as Yuki and her classmates in Tokyo did. Allison is video-chatting with Yuki about their class seedlings. Assume both plants will continue to grow at the same rate. Who should expect to have the taller plant at the end of the school year?” (7.RP.1, 7.RP.3)
• Lesson 4-5, Factor Expressions, Visual Learning, Example 1, students develop conceptual understanding of factor expressions by using area models, “Kiana painted a rectangular wall blue to start an ocean mural. She used 3 cans of paint, each of which covered x square meters, and a different-sized can that covered 12 square meters. What are possible length and height dimensions of Kiana’s mural?” Teachers ask, “In the area model diagram, what does the green area labeled 12 represent? What does the blue area labeled 3x represent? How does the area model relate to the original and factored expressions?” (7.EE.1 and 7.EE.2)
• Lesson 6-1 Populations and Samples, Visual Learning, Example 2, students develop an understanding of the importance of random sampling when generalizing a population, “Morgan decides to survey a sample of the town’s voting population. How can she know that the survey results from the sample of voters represent the population of the entire town’s population? How much of the population should be sampled in a ‘representative’ sample? Explain. Why do you think a random sample is usually also a representative sample?” (7.SP.1)
• Lesson 8-6, Solve Problems Involving Area of a Circle, Visual Learning, Example 3, students develop conceptual understanding as they use the formula for the area of the circle, "Elle needs new grass in the circular pen for her chickens. What is the area of the pen?" (7.G.4)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice and Problem Solving exercises found in the student materials provide opportunities for students to demonstrate conceptual understanding. Try It! provides problems that can be used as formative assessment of conceptual understanding following Example problems. Do You Understand?/Do You Know How? problems have students answer the Essential Question and determine students’ understanding of the concept. Examples include:

• Lesson 1-1, Relate Integers and Their Opposites, Do You Know How?, Item 6, students independently demonstrate understanding of how positive and negative numbers relate to zero by using models and by combining opposite quantities, “The scores of players on a golf team are shown in the table. The team’s combined score was 0. What was Travis’s score?” (7.NS.1)
• Lesson 2-4, Describe Proportional Relationships: Constant Proportionality, Do You Know How?, Item 4, students independently demonstrate understanding of proportional relationships and their constant of proportionality, “Determine whether each equation represents a proportional relationship. If it does, identify the constant of proportionality. a. y = 0.5x - 2; b. y = 1,000x; c. yx + 1.”(7.RP.A.2)
• Lesson 4-6, Add Expressions, Practice & Problem Solving, Item 13, students independently demonstrate understanding of writing and simplifying expressions, “An art class is making a mural for the school that has a triangle drawn in the middle. The length of the bottom of the triangle is x. Another side is 1 more than three times the length of the bottom of the triangle. The last side is two more than the bottom of the triangle. Write and simplify an expression for the perimeter of the triangle.” (7.EE.2)
• Lesson 7-2, Understand Theoretical Probability, Practice & Problem Solving, Item 7, students independently demonstrate understanding of theoretical propropbality, “A spinner has 8 equal-sized sections.To win the game, the pointer must land on a yellow section.” (7.SP.6, 7.RP.2)
• Lesson 8-5, Solve Problems Involving Circumference of a Circle, Visual Learning, Example 1, Try It!, students independently demonstrate understanding of circumference of a circle as they use the formula, "What is the circumference of the rim of a basketball hoop with a radius of 9 inches? First, multiply the radius by ____ to get the diameter, ___ inches. Then multiply the diameter by 3.14 (an approximation for π) to get a circumference of about ____ inches." (7.G.4)
##### Indicator {{'2b' | indicatorName}}
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 enVision Mathematics Common Core Grade 7 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. According to the Teacher Resource Program Overview, “Students develop skill fluency when the procedures make sense to them. Students develop these skills in conjunction with understanding through careful learning progressions.” Try It! And Do You Know How? Provide opportunities for students to build procedural fluency from conceptual understanding. Examples Include:

• Lesson 1-2, Understand Rational Numbers, Visual Learning, Example 2, Try It!, students write rational numbers in decimal form to develop and maintain fluency in dividing whole numbers and decimals, “What is the decimal form of $$\frac{100}{3}$$, $$\frac{100}{5}$$, and $$\frac{100}{6}$$? Determine whether each decimal repeats or terminates.” (7.NS.2)
• Lesson 2-3, Understand Proportional Relationships: Equivalent Ratios, Do You Know How?, Item 6, students determine whether quantities are proportional by testing for equivalent ratios, “Is the relationship between the number of tickets sold and the number of hours proportional? If so, how many tickets were sold in 8 hours?” Students are provided a table with hours (h) and tickets sold (t). (7.RP.2)
• Lesson 4-3, Simplify Expressions, Visual Learning, Example 2, Try It!, students develop procedural skill when they simplify expressions, “Simplify each expression. a. 59.95m - 30 + 7.95m + 45 + 9.49m; b. -0.5p + $$\frac{1}{2}$$p- 2.75 + $$\frac{2}{3}$$p.” (7.EE.1)
• Lesson 6-3, Make Comparative Inferences About Populations, Visual Learning, Example 1, Try It, students use box plots to compare and make inferences about populations, “Kono gathers the heights of a random sample of sixth graders and seventh graders and displays the data in box plots. What can he say about the two data sets? The median of the ___ grade is greater than the median of the ___ grade sample. The ____ grade sample has a greater variability.” Box Plots of 6th and 7th grade students’ heights are shown. (7.SP.3 and 7.SP.4)
• Lesson 8-3, Draw Triangles with Given Conditions, Do You Know How?, Item 4, students draw triangles when given information about their side lengths and angle measures, “How many triangles can be drawn with side lengths 4 centimenters, 4.5 centimenters, and 9 centemeters? Explain.” (7.G.2)

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice and Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate procedural skill and fluency. Additionally, at the end of each Topic is a Fluency Practice page which engages students in fluency activities. Examples include:

• Lesson 1-5, Add and Subtract Rational Numbers, Practice & Problem Solving, Item 11, students use the same procedure for adding and subtracting signed rational numbers as they do when adding and subtracting integers, “Simplify each expression. a) 50$$\frac{1}{2}$$ + (-12.3) b) -50$$\frac{1}{2}$$ + (-12.3) c) -50$$\frac{1}{2}$$ + 12.3.” (7.NS.1)
• Lesson 2-2, Determine Unit Rates with Ratios of Fractions, Practice & Problem Solving, Item 12, students calculate unit rate with fractional measurements, “A box of cereal states that there are 90 Calories in a $$\frac{3}{4}$$ - cup serving. How many calories are there in 4 cups of the cereal?” (7.RP.1)
• Lesson 4-5, Factor Expressions, Practice & Problem Solving, Item 8, students factor the GCF from expressions, “Factor the expression. 14x + 49.” (7.EE.1 and 7.EE.2)
• Lesson 7-6, Find Probabilities of Compound Events, Practice & Problem Solving, Item 9, students find the sample space and probability of compound events, “The organized list shows all the possible outcomes when three fair coins are flipped. The possible outcomes of each flip are heads (H) and tails (T). What is the probability that at least 2 fair coins land heads up when 3 are flipped?” (7.SP.8)
• Topic 8 Review, Fluency Practice, students use the percent equation to solve problems, “Pathfinder: Shade a path from START to FINISH. Follow the answers to the problems so that each answer is greater than the one before. You can only move up, down, right, or left.” The starting box states, “Amy deposits $360 in an account that pays 1.2% simple annual interest. How much interest will she earn in 6 years?” (7.RP.3) • Lesson 8-5, Solve Problems Involving Circumference of a Circle, Practice & Problem Solving, Item 15, students calculate the circumference, radius, or diameter of a circle, “What is the radius of a circle with a circumference of 26.69 centimeters?” (7.G.4) ##### Indicator {{'2c' | indicatorName}} 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 enVision Mathematics Common Core Grade 7 meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which mathematics is applied. The instructional materials include multiple opportunities for students to independently engage in routine and non-routine application of mathematical skills and knowledge of the grade level. According to the Teacher Resource Program Overview, “In each topic, students encounter a 3-Act Mathematical Modeling lesson, a rich, real-world situation for which students look to apply not just math content, but math practices to solve the problem presented.” Additionally, each Topic provides a STEM project that presents a situation that addresses real social, economic, and environmental issues. For example: • Topic 1, 3-Act Mathematical Modeling: Win Some, Lose Some, Question 14, students predict the winner of a trivia game and the final score, “If there were one final round where each contestant chooses how much to wager, how much should each person wager? Explain your reasoning." (7.NS.1 and 7.NS.3) • Topic 1, STEM Project, How Cold is Too Cold?, students collect and organize their data related to temperatures and plot their data on graphs, find the range in temperatures, and form conclusions, "There are many regions of the world with cold temperatures and extreme conditions. How do the inhabitants of these regions adapt and thrive? Do conditions exist that make regions too cold for human living? You and your classmates will explore and describe the habitability of regions with low temperatures.” (7.SP.1, 7.SP.2, and 7.SP.3) • Topic 2, 3-Act Mathematical Modeling: Mixin' It Up, Question 15, students attempt to make the liquid in a water glass have the same flavor like that of a large water cooler, “A classmate usually adds 6 drops to 16 ounces of water. Use your updated model to predict the number she would use for the large container." (7.RP.1 and 7.RP.2) • Topic 5, STEM Project, Water is Life, students research filtration systems, decide which one they would purchase, and plan a fundraiser. Part of planning is writing an equation to represent the amount of money they will earn from a fundraiser to purchase the filtration system, "You have water to drink, to use to brush your teeth, and to bathe. You and your classmates will research the need for safe, clean water in developing countries. Based on your research, you will determine the type, size, and cost of a water filtration system needed to provide clean, safe water to a community.” (7.EE.3 and 7.EE.4) • Topic 5, 3-Act Mathematical Modeling: Digital Downloads, Question 14, students determine how many songs a person can purchase using the balance of a gift card, “If all single tracks were on sale for 10% off, how would your model change? How would the answer to the Main Question change?" (7.EE.3 and 7.EE.4) • Topic 8, STEM Project, Upscale Design, students make scale drawings of existing paths or create plans for new walking paths or bikeways, "Choose an existing path or bikeway and make a scale drawing of the route. Add improvements or extensions to your drawing that enhance the trails and better meet the needs of users. If your area lacks a trail, choose a possible route and make a scale drawing that proposes a new path.” (7.G.1 and 7.G.2) The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Pick a Project is found in each Topic and students select from a group of projects that provide open-ended rich tasks that enhance mathematical thinking and provide choice. Additionally, Practice and Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example: • Lesson 1-10, Solve Problems with Rational Numbers, Practice & Problem Solving, Item 15, students solve problems using rational numbers operations, “The table shows the relationship between a hedgehog's change in weight and the number of days of hibernation. a. What number represents the change in weight for each day of hibernation? b. What number represents the change in weight in ounces for the hedgehog in 115 days of hibernation?” (7.NS.3 and 7.EE.3) • Lesson 2-1, Connect Ratios, Rates, Unit Rates, Practice & Practice Solving, Item 9, students apply knowledge of solving multi-step problems with rational numbers to solving problems with ratios, rates, and unit rates. Given 3 bags of rice, “Which package has the lowest cost per ounce of rice?” (7.RP.1 and 7.RP.3) • Topic 3, Pick a Project 3A, students use coupons to calculate the best price for several items, “Select at least three items that you want to purchase from one or more stores. One should be something on sale or available at a discount. One should be something from the sporting goods section. One should be a toy or game. Research the selling price of each item. Use the coupons below to calculate the best price for each item. Use each coupon only once. Make a collage with pictures of the items and copies of the coupons. Write up your calculations, explain how you found the best price, and include this information with your collage.” (7.RP.3) • Lesson 4-8, Analyze Equivalent Expressions, Practice & Problem Solving, Item 15, students apply the meaning of equivalent expressions to simplify a problem in a new way, “A customer at a clothing store is buying a pair of pants and a shirt. The customer can choose between a sale that offers a discount on pants, or a coupon for a discount on the entire purchase. Let n represent the original price of the pants and s represent the price of the shirt. a. Write two expressions that represent the ‘15% off sale on all pants’ option. b. Write two expressions that represent the ‘10% off her entire purchase’ option. c. If the original cost of the pants is$25 and the shirt is $10, which option should the customer choose? Explain.” (7.EE.2) • Topic 6, Pick a Project 6A, students conduct a survey and analyze their results, “Conduct a survey on an improvement or change you want to see in your community. Write a letter to your representative or local council about the changes you would like to see in your community. In your letter, include data and conclusions from your survey to support your position. What might a better sample be?” (7.SP.1, 7.SP.2) • Lesson 8-9, Solve Problems Involving Volume, Practice & Problem Solving, Item 15, students solve real-world problems involving the volume of three-dimensional objects, “A cake has two layers. Each layer is a regular hexagonal prism. A slice removes one face of each prism, as shown. a. What is the volume of the slice? b. What is the volume of the remaining cake?” (7.G.6, 7.G.3, 7.EE.3, and 7.EE.4) ##### Indicator {{'2d' | indicatorName}} 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 enVision Mathematics Common Core Grade 7 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present independently throughout the program materials. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, and application include: • Lesson 1-3, Add Integers, Explore It!, students extend their conceptual understanding of positive and negative numbers as they use number lines and absolute value to solve problems, “Rain increases the height of water in a kiddie pool, while evaporation decreases the height. The pool water level is currently 2 inches above the fill line. A. Look for patterns in the equations in the table so you can fill in the missing numbers. Describe any relationships you notice. B. Will the sum of 2 and (-6) be a positive or negative number? Explain.” (7.NS.1) • Topic 3 Review, Fluency Practice, students find unit rates with ratios of fractions, “Riddle Rearranging: Find the value of x in each unit rate. Then arrange the answers in order from least to greatest. The letters will spell out the answer to the riddle below.” Box K states, “$$\frac{\frac{3c}{4}}{\frac{1h}{3}} = \frac{xc}{1h}$$”. (7.RP.1) • Lesson 8-8, Solve Problems Involving Surface Area, Practice & Problem Solving, Item 12, students use application of surface area knowledge to solve real-world problems, “A box has the shape of a rectangular prism. How much wrapping paper do you need to cover the box?” Illustration dimensions provided are h = 3 inches, w = 15 inches, and l = 16 inches. (7.G.6) Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. Examples include: • Lesson 1-2, Understand Rational Numbers, Practice & Problem Solving, Item 18, students solve real-world problems with rational numbers, “Aiden has one box that is 3$$\frac{3}{11}$$ feet tall and a second box that is 3.27 feet tall. If he stacks the boxes, about how tall will the stack be?” This question develops conceptual understanding and application of 7.NS.2, apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. • Lesson 4-1, Write and Evaluate Algebraic Expressions, Do You Know How?, Item 5, students use algebraic expressions to represent and solve problems with an unknown value, “Write an algebraic expression that Marshall can use to determine the total cost of buying a watermelon that weighs w pounds and some tomatoes that weigh t pounds. How much will it cost to buy a watermelon that weighs 18$$\frac{1}{2}$$pounds and 5 pounds of tomatoes?” This question develops procedural skill and fluency of 7.EE.3, solve real-life and mathematical problems using numerical and algebraic expressions and equations. • Lesson 5-5, Solve Inequalities Using Multiplication or Division, Practice & Problem Solving, Item 12, students write inequalities to solve real-world problems, “Brittney can spend no more than$15 for new fish in her aquarium. a. Let f be the number of fish she can buy. What inequality represents this problem? b. How many fish can Brittney buy?” This question develops conceptual understanding and application of 7.EE.4, use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
• Lesson 7-3, Understand Theoretical Probability, Do You Know How?, Item 4, students find the theoretical probability for an event, “Kelly flips a coin 20 times. The results are shown in the table where “H” represents the coin landing heads up and “T” represents the coin landing tails up.” This question develops conceptual understanding and procedural skill of 7.SP.6, approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
• Lesson 8-4, Solve Problems Using Angle Relationships, Do You Know How?, Item 6, students use understanding of angle relationships to find the value of a given angle, “Use diagram 4-6. If ∠1 and ∠3 are the same measure, what is the value of x? ” This question develops conceptual understanding and procedural skill of 7.G.5, use facts about supplementary, complementary, vertical, and adjacent angles to write and solve simple equations for an unknown angle in a figure.

#### Criterion 2.2: Math Practices

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

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for meaningfully connecting the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs). The MPs are identified and used to enrich mathematics content, and the instructional materials support the standards’ emphasis on mathematical reasoning.

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

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade level.

All 8 MPs are clearly identified throughout the materials, with few or no exceptions. Math Practices identification in this program according to the Teacher Resource Program Overview include:

• Materials provide a Math Practices and Problem Solving Handbook for students, “A great resource to help students build on and enhance their mathematical thinking and habits of mind.” This handbook explains math practices in student-friendly language and digital animation videos for each math practice are also available.
• Opportunities to apply math practices are found in the Explore It, Explain It, and Solve & Discuss It portions of the lesson. “The Solve & Discuss It calls on students to draw on nearly all of the math practices, but especially sense-making and solution formulation as well as abstract and quantitative reasoning. The Explore It focuses students on mathematical modeling, generalizations, and structure of mathematical models. The Explain It emphasizes mathematical reasoning and argumentation. Students construct arguments to defend a claim or critique an argument defending a claim.”
• The Math Practices and Problem Solving Handbook Teacher Pages, “provide overviews of the math practices, offer instructional strategies to help students refine and enhance their thinking habits, and include student behaviors to listen and look for for each standard.”
• Each Topic Overview contains Math Practices Teacher Pages which include, “Two highlighted math practices with student behaviors to look for, and questions to help students become more proficient with these thinking habits.” For example in Topic 7, mathematical reasoning and explanation questions state, “How would you describe the problem in your own words? What are some other strategies you might try in order to determine the different outcomes?”
• Math Practices boxes found in the student text provide, “Reminders to be thinking about the application of the math practices as they solve problems.”
• Math Practices Run-in Heads found in the Practice & Problem Solving questions, “Remind students to apply the math practices as they solve problems.”

The majority of the time the MPs are used to enrich the mathematical content and are not treated separately. Examples include:

• MP1: Make sense of problems and persevere in solving them. Lesson 3-2, Connect Percent and Proportion, Practice & Problem Solving, Item 11, students examine the relationships between the quantities and solve for the whole, “A restaurant customer left 3.50 as a tip. The tax on the meal was 7% and the tip was 20% of the cost including tax. What was the total bill?” • MP2: Reason abstractly and quantitatively. Lesson 6-3, Make Comparative Inferences About Populations, Practice & Problem Solving, Item 9, students interpret and compare statistical measures and reason about data sets in both qualitative and quantitative forms, “A family is comparing home prices in towns where they would like to live. The family learns that the median home price in Hometown is equal to the median home price in Plainfield and concludes that the homes in Hometown and Plainfield are similarly priced. What is another statistical measure that the family might consider when deciding where to purchase a home?” • MP4: Model with mathematics. Lesson 7-1, Understand Likelihood and Probability, Practice & Problem Solving, Item 15, students use tools to determine the likelihood of an event occurring, “Henry is going to color a spinner with 10 equal-sized sections. Three of the sections will be orange and 7 of the sections will be purple. Is this spinner fair? If so, explain why. If not, explain how to make it a fair spinner.” • MP5: Use appropriate tools strategically. Lesson 7-7, Simulate Compound Events, Practice & Problem Solving, Item 10, students choose an appropriate tool (e.g., spinner, coin, number cube) to stimulate the outcome of a compound event, “Julie used a number cube to simulate a flower seed sprouting, for which the success rate is 50%. She used even numbers to represent success and odd numbers to represent failure. The results of 8 trials that simulate the sprouting of five seeds are shown below. Based on the simulated results, what is the probability that none of the next five flower seeds will sprout successfully?” • MP6: Attend to precision. Lesson 2-1, Connect Ratios, Rates, and Unit Rates, Practice & Problem Solving, Item 11, students use formal mathematical vocabulary to communicate ratio concepts and reasonings to solve problems, “An arts academy requires there to be 3 teachers for every 75 students and 6 tutors for every 72 students. How many tutors does the academy need if it has 120 students?” • MP7: Look for and make use of structure. Lesson 3-2, Connect Percent and Proportion, Practice & Problem Solving, Item 15, students use structure to identify and align the part, whole, and percent to set up a proportion to solve real-world problems, “A school year has 4 quarters. What percent of a school year is 7 quarters?” • MP8: Look for and express regularity in repeated reasoning. Lesson 8-3, Draw Triangles with Given Conditions, Practice & Problem Solving, Item 12, students analyze triangles and generalize that its side and angle conditions determine if it results in one triangle, more than one triangle, or no triangle, “Given two side lengths of 15 units and 9.5 units, with a non included angle of 75°, can you draw no triangles, only one triangle, or more than one triangle?” ##### Indicator {{'2f' | indicatorName}} Materials carefully attend to the full meaning of each practice standard The instructional materials reviewed for enVision Mathematics Common Core Grade 7 partially meet expectations that the instructional materials carefully attend to the full meaning of each practice standard. The materials do not attend to the full meaning of MP5: Use appropriate tools strategically. Examples include: • Topic 1: Rational Number Operations, Topic Performance Task, Problem 3, “Rhiannon digs a 6$$\frac{4}{8}$$ inches deep hole in the ground. She places a tomato plant into the hole. This leaves 3.25 inches of the plant above the ground. She says the height of the plant can be found using the expression $$|-6\frac{7}{8}+3.25|$$. Part A. Is Rhiannon correct? Why or why not? Use the number line, and then explain.” Students are given a vertical number line to solve the problem. • Lesson 1-3, Add Integers, Practice & Problem Solving, Item 10, “An airplane flying at an altitude of 30,000 feet flies up to avoid a storm. Immediately after passing the storm, the plane returns to its original altitude.” Part C states, “Draw a number line to represent the airplane’s change in altitude.” Use Appropriate Tools states, “Draw a number line to represent the airplane’s change in altitude.” Students are instructed to use a number line to represent changes in altitude. • Lesson 7-6, Find Surface Areas of Prisms, Visual Learning, Visual Learning, Example 1, “Kelly wants to cover a shoebox with decorative paper without overlapping the paper. How much paper will she need to cover the box?” Use Appropriate Tools states, “Would a net help you find the surface area (SA) of the rectangular prism?” Students are instructed to use nets to find the surface area. • Lesson 8-2, Draw Geometric Figures, Do You Understand?, Item 2, “How can you decide whether to draw a shape freehand, with a ruler and protractor, or using technology?” Pose Purposeful Questions states, “Give an example of when you want to use technology to draw figures and when a sketch is adequate.” The materials do attend to the full meaning of the following MPs. For example: • MP1: Make sense of problems and persevere in solving them. Lesson 2-1, Connect Ratios, Rates, and Unit Rates, Visual Learning, Example 1, “Nathan and Dan were both hired as lifeguards for the summer. They receive their paychecks for the first week. Who earns more per hour?” An image of Dan Jones shows 9 hours and total earnings of78.75 and Nathan Smith shows 5 hours and total earnings of $46.25. Students use proportional reasoning to make sense of the problem and preserve as they find ratios, rates, and unit rates to solve multi-step problems. • MP2: Reason abstractly and quantitatively. Lesson 5-1, Write Two-Step Equations, Practice & Problem Solving, Item 15, “In a certain country, the life expectancy of a woman born in 1995 was 80.2 years. Between 1995 and 2005, the life expectancy increased 0.4 year every 5 years. If L represents the life expectancy of a woman born in 2005, what equation could you use to represent the situation? Could two different equations be used to find the value of L? Explain?” Students analyze word problems to write two-step equations. • MP4: Model with mathematics. Lesson 3-3, Represent and Use the Percent Equation, Practice & Problem Solving, Item 15, “There are 4,000 books in the town’s library. Of these, 2,600 are fiction. Write a percent equation that you can use to find the percent of books that are fiction. Then solve your equation.” Students identify important quantities, use equations to represent their relationships, and interpret the results using mathematical models in a real-world situation. • MP6: Attend to precision. Lesson 5-4, Solve Inequalities Using Addition or Subtraction, Do You Understand?, Item 2, “How do the solutions of the two inequalities differ? Are any of the solutions the same? Explain. A. x + 5 < 8 and x + 5 > 8 B. x + 5 ≤ 8 and x + 5 ≥ 8.” Students solve inequalities using the Addition and Subtraction Properties of Inequality. • MP7: Look for and make use of structure. Lesson 4-8, Analyze Equivalent Expressions, Practice & Problem Solving, Item 13, “The area of a rectangular playground has been extended on one side. The total area of the playground, in square meters, can be written as 352 + 22x. Rewrite the expression to give a possible set of dimensions for the playground.” Students analyze relationships between quantities in real-world situations for equivalency. • MP8: Look for and express regularity in repeated reasoning. Lesson 6-4, Make More Comparative Inferences about Populations, Practice & Problem Solving, Item 9, “Brianna asks 8 classmates how many pencils and erasers they carry in their bags. The mean number of pencils is 11. The mean number of erasers is 4. The MAD of both data sets is 2. What inference could Brianna make using this data?” Students apply patterns in how measures of center and variability are calculated to make assumptions about samples and populations. ##### Indicator {{'2g' | indicatorName}} Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by: ##### Indicator {{'2g.i' | indicatorName}} 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 enVision Mathematics Common Core Grade 7 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. Student materials consistently prompt students to construct viable arguments. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include: • Lesson 3-1, Analyze Percents of Numbers, Practice & Problem Solving, Item 18, students use their understanding of percents of numbers and construct arguments to support their response, “Brad says that if a second number is 125% of the first number, then the first number must be 75% of the second number. Is he correct? Justify your answer.” • Lesson 4-2, Generate Equivalent Expressions, Explore It!, students construct arguments as they compare and contrast representations of real world situations, “How can you represent the total number of eggs in the shipment using diagrams or images? Explain your diagram. How can you represent the total number of eggs in the shipment using expressions? What variables do you use? What do they represent? How do the two representations compare? How are they different?” • Lesson 5-3, Solve Equations Using the Distributive Property, Explain It!, students use their understanding of the Distributive Property to construct arguments, “Six friends go jet skiing. The total cost for the adventure is$683.88, including a $12 fee per person to rent flotation vests. Marcella says they can use the equation 6r + 12 = 683.88 to find the jet ski rental cost, r, per person. Julia says they need to use equation 6(r + 12) = 683.33. A. Whose equation accurately represents the situation? Construct an argument to support your response. B. What error in thinking might explain the inaccurate equation?” • Lesson 6-4, Make More Comparative Inferences About Populations, Do You Understand?, Item 3, students use their understanding of inferences to construct arguments, “Two data sets have the same mean but one set has a much larger MAD than the other. Explain why you may want to use the median to compare the data sets rather than the mean.” • Lesson 8-4, Solve Problems Using Angle Relationships, Explore It!, students analyze problems and use angle relationships to construct and justify arguments, “C. How does the sum of the measures of 1 and 2 change when one ski moves? Explain. Why does the sum of all four angle measures stay the same when one of the skis moves? Explain.” Student materials consistently prompt students to analyze the arguments of others. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include: • Lesson 1-9, Divide Rational Numbers, Practice & Problem Solving, Item 17, students analyze the arguments of others as they explain the errors of dividing and multiplying rational numbers, “Kayla wants to find 2$$\frac{2}{3}$$ $$\div$$ (-1$$\frac{7}{3}$$). She first rewrites the division as (2$$\frac{2}{3}$$)(-1$$\frac{7}{3}$$). What is wrong with Kayla’s reasoning?” • Lesson 2-5, Graph Proportional Relationships, Do You Understand?, Item 3, students analyze the arguments of others by interpreting if points contain a proportional relationship. “Makayla plotted two points (0,0) and (3,33), on a coordinate grid. Noah says that she is graphing a proportional relationship. Is Noah correct? Explain.” • Lesson 4-7, Subtract Expressions, Practice & Problem Solving, Item 16, students analyze the arguments of others using properties of operations to subtract expressions, “Tim incorrectly rewrote the expression 1/2p - (1/4p + 4) as 1/2p + 1/4p - 4. Rewrite the expression without the parenthesis. What was Tim’s error?” • Lesson 5-7, Solve Multi-Step Inequalities, Practice & Problem Solving, Item 10, students analyze the arguments of others using inequalities. “Sierra says that she can simplify the left side of the inequality 2(-3 + 5) + 2 -4(x - 2) - 3 by combining the terms within the parentheses, but that she can’t do the same on the right side. Is Sierra correct? Explain.” • Lesson 6-2, Draw Inferences from Data, Do You Understand?, Item 3, students analyze the arguments of others as they make inferences about a population from sample data, “Darrin surveyed a random sample of 10 students from his science class about their favorite types of TV shows. Five students like detective shows, 4 like comedy shows, and 1 like game shows. Darrin concluded that the most popular type of TV shows among students in his school is likely detective shows. Explain why Darrin’s inference is not valid.” ##### Indicator {{'2g.ii' | indicatorName}} 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 enVision Mathematics Common Core Grade 7 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. Teacher materials assist teachers in engaging students in constructing viable arguments frequently throughout the program. Examples include: • Lesson 3-6, Solve Simple Interest Problems, Visual Learning, Example 1, “Victoria opens a savings account with a deposit of$300. She will earn 1.6% simple interest each year on her money. How much interest will she earn over 5 years (assuming she does not add or take out any money)?” ETP (Effective Teaching Practices) Use and Connect Mathematical Representations teacher prompt states, “Does Victoria earn the same amount of interest every year? Explain.”
• Lesson 5-5, Solve Inequalities Using Multiplication or Division, Example 1, Convince Me!, “Frances solved the inequality 5g $$\ge$$ 35. She says that 7 is a solution to the inequality. Is Frances correct? Explain.” ETP Elicit and Use Evidence of Student Thinking teacher prompt states, “What other values make the inequality true?”
• Lesson 7-3, Understand Experimental Probability, Visual Learning, Example 1, Try It!, “During the second day of the school fair, Talia and Yoshi recorded 43 winners out of a total of 324 players. How does the actual number of winners compare to the expected number of winners? ETP Convince Me! teacher prompt states, “Is it possible for the theoretical probability to be $$\frac{1}{2}$$ while the experimental probability is 1? Give an example.”

Teacher materials assist teachers in engaging students in analyzing the arguments of others frequently throughout the program. Examples include:

• Lesson 1-5, Add and Subtract Rational Numbers, Example 1, Try It!, and Convince Me!, “A dolphin is at the surface of the water and then descends to a depth of 4$$\frac{1}{2}$$ feet. Then the dolphin swims down another 2$$\frac{3}{4}$$ feet. What is the location of the dolphin relative to the surface of the water?” ETP Elicit and Use Evidence of Student Thinking teacher prompt states, “Why is the first number of the addition statement -4$$\frac{1}{2}$$? How is the subtraction of a positive number from a negative number changed to solve the problem?”
• Lesson 4-5, Factor Expressions, Explain It!, “Tasha is packing gift bags that include the same items. She has 72 glow sticks, 36 markers, and 24 bottles of bubbles. Tasha believes that she can pack no more than 6 bags using all of her supplies. Do you agree with Tasha? Explain. ETP Observe Students at Work teacher prompt states, “How do students decide whether or not they agree with Tasha? Students might find the GCF of the three numbers (12) and say that it is the greatest number of gift bags Tasha can pack.”
• Lesson 6-2, Draw Inferences From Data, Visual Learning, Example 3, “Margo and Ravi are also trying to get their parents to let them stay up later. They collect data about the number of hours of sleep a random sample of seventh graders get each night. The two box plots show their data. Do Margo’s and Ravi’s data support Sasha’s inference about the number of hours of sleep that seventh graders get?” ETP Pose Purposeful Questions teacher prompt states, “Why is it important that Ravi and Margo’s data corroborate, or support, Sasha’s data?”

Teacher materials assist teachers in engaging students in both the construction of viable arguments and analyzing the arguments or reasoning of others frequently throughout the program. Each Topic Overview highlights specific Math Practices and suggests look fors in student behavior and provides questioning strategies. Examples include:

• Topic 2, Analyze and Use Proportional Relationships, look fors, “Mathematically proficient students: Use what they have previously learned about ratios and rates in constructing arguments and explanations. Construct arguments by using accurate definitions of proportional quantities. Justify and support mathematical reasoning by using diagrams, tables, graphs, and equations. Ask questions to clarify others’ reasoning in order to decide whether arguments make sense or to improve the arguments.”
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Materials explicitly attend to the specialized language of mathematics.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols. Each Topic Overview provides a chart in the Topic Planner that lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow. Lesson practice includes questions to reinforce vocabulary comprehension and students write using math language to explain their thinking. Each Topic Review contains a Vocabulary Review section for students to review vocabulary taught in the Topic. Students have access to an Animated Glossary online in both English and Spanish. Examples include:

• Lesson 2-3, Understand Proportional Relationships: Equivalent Ratios, Visual Learning, Example 3, “A proportion is an equation that represents equal ratios.”
• Lesson 3-3, Represent and Use the Percent Equation, Visual Learning, Example 2, “Jane earns a 5.5% commission on the selling price of each home she sells. She earned $9,020 in commission on the sale of a home. What was the selling price of the home? Instead of a salary some workers earn a percent of the value of a transaction, called a commission.” • Topic 4, Solve Problems involving Geometry, Mid-Topic Checkpoint, Question 1, “How are adjacent angles and vertical angles alike? How are they different?” • Lesson 6-1, Populations and Samples, Visual Learning, Example 1, “Morgan and her friends could ask every registered voter, or the entire population of voters in town, how they play to vote. Morgan and her friends could ask a subset, or a sample of the registered voters in town how they plan to vote.” • Topic 7, Probability, Use Vocabulary in Writing, “A restaurant serves either skim milk or whole milk in glasses that are small, medium, or large. Use vocabulary words to explain how you could determine all the possible outcomes of milk choices at the restaurant. Use vocabulary words in your explanation.” Students are provided a word bank containing, “event, relative frequency, outcome, sample space, probability, and simulation.” The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them. A Vocabulary Glossary is provided in the back of Volume 1 and lists all the vocabulary terms and examples. Teacher side notes, Elicit and Use Evidence of Student Thinking and Pose Purposeful Questions, provide specific information about the use of vocabulary and math language. Examples include: • Lesson 1-9, Divide Rational Numbers, Visual Learning, Example 2, Pose Purposeful Questions, “Why is 3$$\frac{2}{3}$$ $$\div$$ (-$$\frac{2}{3}$$) equivalent to the given complex fraction? What do you notice about the signs of a fraction and its reciprocal? Explain why this is true.” • Lesson 3-2, Connect Percent and Proportion, Visual Learning, Example 3, Pose Purposeful Questions, “Multiply the numerator and denominator of $$\frac{20}{100}$$ by an integer to get an equivalent fraction with a numerator of 260. What is the multiplier? What is the equivalent fraction?” • Lesson 4-2, Generate Equivalent Expressions, Visual Learning, Example 2, Pose Purposeful Questions, “What does the Commutative Property of Addition mean? What does the Associative Property of Addition mean?” • Lesson 5-4, Solve Inequalities Using Addition and Subtraction, Visual Learning, Example 1, Elicit and use Evidence of Student Thinking, ‘What does h represent in this scenario? Why is the $$\ge$$ inequality symbol used to represent this scenario? Explain.” • Lesson 8-4, Solve Problems Using Angle Relationships, Visual Learning, Example 1, Convince me!, Elicit and Use Evidence of Student Thinking, “How can you use the definition of vertical angles to write an equation?” • Student Edition, Glossary, “constant of proportionality: In a proportional relationship, one quantity y is a constant multiple of the other quantity x. The constant multiple is called the constant of proportionality. The constant of proportionality is equal to the ratio $$\frac{y}{x}$$. Example, In the equation y =4x, the constant of proportionality is 4.” ###### Overview of Gateway 3 ### Usability ##### Gateway 3 Meets Expectations #### Criterion 3.1: Use & Design Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing. The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for being well-designed and taking into account effective lesson structure and pacing. The instructional materials include an underlying design that distinguishes between problems and exercises, assignments that are not haphazard with exercises given in intentional sequences, variety in what students are asked to produce, and manipulatives that are faithful representations of the mathematical objects they represent. ##### Indicator {{'3a' | indicatorName}} 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 reviewed for enVision Mathematics Common Core Grade 7 meet expectations that 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. Materials engage students in both problems and exercises through the grade level. Problems where students learn new mathematics are typically found in the Lesson’s Visual Learning Bridge. This portion of the lessons consists of visual examples that formalize the mathematics of the lesson by providing guided instruction of the math concepts with one example stepped-out. Examples from the Teacher Resource include: • Lesson 2-5, Graph Proportional Relationships, Visual Learning, Example 1, students learn how to recognize a proportional relationship using graphs, “Tanya exercise for 30 minutes. She noted the Calories burned at three times during her workout. How can Tanya use this information to find how many Calories she burned after 15 minutes of exercise?” • Lesson 4-4, Expand Expressions, Visual Learning, Example 3, students learn two methods for expanding complex expressions, “Simplify the expression -$$\frac{1}{3}$$ (2 - 3x + 3). One way: Use the Distributive Property first to distribute the coefficient -$$\frac{1}{3}$$. Another way: Simplify within the parentheses first. Then distribute the coefficient -$$\frac{1}{3}$$. • Lesson 8-4, Solve Problems Using Angle Relationships, Visual Learning, Example 1, students learn to solve problems involving adjacent and vertical angles, “A skewed intersection has two roads that intersect at more than 20 degrees away from 90°. Determine whether the road intersection shown is skewed by finding the measures of ∠ABC and ∠DBE.” Exercises, where students apply learning to build mastery, are typically found in the Practice and Problem Solving section. These exercises build independent proficiency, challenge higher-order thinking, and simulate high-stakes testing questions. Examples from the Teacher Resource include: • Lesson 1-3, Add Integers, Practice & Problem Solving, Item 18, students add two negative integers to find a solution, “A deep-sea diver dives 81 feet from the surface. He then dives 14 more feet. The diver’s depth can be represented by -81 + (-14). What is the diver’s present location?” • Lesson 5-2, Solve Two-Step Equations, Practice & Problem Solving, Item 9, students write and solve two-step equations, “While shopping for clothes, Tracy spent$38 less than 3 times what Daniel spent. Write and solve an equation to find how much Daniel spent. Let x represent how much Daniel spent.”
• Lesson 7-6, Find Probabilities of Compound Events, Practice & Problem Solving, Item 10, students find the probability of a compound event using a table, tree diagram, or organized list, “Gary spins two game wheels at the carnival. He will win a prize if both of the wheels land on any red section. How does the chance of winning change if different game wheels are used with more sections that aren’t red?”
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Design of assignments is not haphazard: exercises are given in intentional sequences.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that the design of assignments is not haphazard: exercises are given in intentional sequences.

Lesson activities within each Topic are intentionally sequenced developing student understanding and leading towards mastery of the content. Students are introduced to concepts and procedures with a problem-solving experience, Solve & Discuss it. The Visual Learning Bridge provides direct instruction that makes the important mathematics explicit through class discussion of student thinking and solutions. Examples from the Teacher Resource include:

• Lesson 2-2, Determine Unit Rates and Ratios of Fractions, Solve & Discuss It!, students are presented with the problem and work independently to solve using various strategies, after which discussion ensues to develop key concepts important in finding fractions and decimals in unit rates. “Allison and her classmates planted bean seeds at the same time as Yuki and her classmates in Tokyo did. Allison is video-chatting with Yuki about their class seedlings. Assume that both plants will continue to grow at the same rate. Who should expect to have the taller plant at the end of the school year?” The picture shown states that Allison’s class plant grew 2.5 inches in 5 days and Yuki’s class plant grew 5.5 centimeters in 4 days.
• Lesson 4-8, Analyze Equivalent Expressions, Solve & Discuss It!, students are presented with the problem and work independently to solve using various strategies, after which discussion ensues to develop key concepts important in using equivalent expressions. “How many toothpicks make a triangle? Two triangles? Write an expression that represents the number of toothpicks needed to make x triangles that appear side-by-side in a single row, as shown. Explain your reasoning.”
• Lesson 8-8, Solve Problems Involving Surface Area, Solve & Discuss It!, students are presented with a situation allowing them to compare the surface area of a box and connect it to finding the area of a rectangular prism. “Alaya will paint the outside of a box with three different colors. Decide how she could paint the box. What is the total area that each color will cover?”
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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 reviewed for enVision Mathematics Common Core Grade 7 meet expectations that 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. Examples from the Teacher Resource include:

• Lesson 1-9, Divide Rational Numbers, Practice & Problem Solving, Item 17, students critique the work of another and provide an justify their thinking, “Kayla wants to find 2$$\frac{2}{3}$$ $$\div$$ (-1$$\frac{3}{7}$$). She first rewrites the division as (2$$\frac{2}{3}$$)(-1$$\frac{3}{7}$$). What is wrong with Kayla’s reasoning?”
• Lesson 3-4, Solve Percent Change and Percent Error Problems, Practice & Problem Solving, Item 17, students visually represent an expression, “You have 20 quarters. You find 40% more in your room. Then you go shopping and spend 50% of the total number of quarters. Write an expression that represents the total number of quarters you take with you when you go shopping.”
• Lesson 7-7, Simulate Compound Events, Practice & Problem Solving, Item 11, students justify their understanding of simulated probability and theoretical probability events, “How is the difference between simulated probability and the theoretical probability of an actual event related to the number of simulated trials conducted?”
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Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

Students have access to Anytime Math Tools powered by Desmos to build understanding and are accessible from the Tools panel online. Desmos tools include a graphing calculator, a scientific calculator, and a geometry construction tool. In addition, students have access to digital math tools such as algebra tiles, integer chips, area models, and bar diagrams. Students see an icon with a wrench when tools are suggested for use during examples and questions. Examples from the Teacher Resource include:

• Lesson 1-6, Multiply Integers, Visual Learning, Example 1, students use a number line to represent integers, “While playing a board game, unlucky Lawrence had to move back 2 spaces for 4 turns in a row. What integer represents his change in position? Use a number line to represent the change in position on the gameboard.”
• Lesson 4-6, Add Expressions, Solve & Discuss It!, students use blank number lines as they add expressions, “The Smith family took a 2-day road trip. On the second day, they drove ¾ the distance they traveled on the first day. What is a possible distance they could have traveled over the 2 days? Is there more than one possible distance? Justify your response.”
• Lesson 7-1, Understand the Likelihood and Probability, Solve & Discuss It, students use grid paper or squares of paper to add the possibilities of winning. “For a game show, Jared has to choose 1 of 8 boxes to win a prize. One of the boxes has a big prize, 3 boxes have a medium prize, 3 boxes have smaller prizes, and 1 box is empty. How confident should Jared be that whatever box he chooses, he will win a prize? Support your response with a mathematical argument.”
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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 reviewed for enVision Mathematics Common Core Grade 7 have a visual design (whether in print or online) that is not distracting or chaotic, and supports students in engaging thoughtfully with the subject.

The font size, graphics, amount of directions, and language used on student pages and in Digital Lessons is appropriate for students. Graphics promote understanding of the mathematics being learned. The digital format is easy to navigate and is engaging. There is ample “white space” for students to calculate and write answers in the student materials.

#### Criterion 3.2: Teacher Planning

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

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for supporting teacher learning and understanding of the CCSSM. The instructional materials include: quality questions to support teachers in planning and providing effective learning experiences, a teacher edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials; full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons; and explanations of the role of the specific grade-level mathematics in the context of the overall mathematics curriculum.

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

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students’ mathematical development.

Effective Mathematics Teaching Practices (ETP) side notes provide quality questions that are designed to promote reasoning and problem solving, support productive struggle, and engage students in mathematical discourse. Establish the Mathematical Goal provides questions related to the Essential Question. Use and Connect Mathematical Representations and Pose Purposeful Questions provide probing questions to enrich the mathematics. Elicit Student Thinking is an opportunity to formatively assess students to determine their understanding of concepts learned. Examples from the Teacher Resource include:

• Lesson 3-6, Solve Simple Interest Problems, Visual Learning, Example 2, Try It, Elicit and Use Evidence of Student Thinking, “How can you find the amount of interest paid in one year? What ratio gives the interest rate?”
• Lesson 5-2, Solve Two-Step Equations, Visual Leaning, Example 2, Pose Purposeful Questions, “What does Jon want to do with his gift card? What does he want to find out? What steps did he take to isolate the variable? Why is the variable positive when the expression and number on each side of the equal sign are negative?”
• Lesson 8-4, Solve Problems Using Angle Relationships, Visual Learning, Example 1, Use and Connect Mathematical Representations, “What is another name for ∠ABE? What is the difference between adjacent and vertical angles formed by two intersecting lines? What are the possible measures of the acute angle and obtuse angle formed by a skewed intersection? Why might a civil engineer be concerned if the intersection of roads is skewed?”
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Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student materials and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

Effective Mathematics Teaching Practices (ETP) side notes provide Before, During, and After suggestions regarding lesson implementation. Examples from the Teacher Resource include:

• Lesson 1-2, Understand Rational Numbers, Solve & Discuss It!, ETP: Before, “1. Introduce the Problem: Provide blank number lines, as needed. 2 Check for Understanding of the Problem: Ensure students understand the problem by asking: ‘Would you rather surf on a shorter narrower surfboard or longer wider one? Why?’”
• Lesson 4-2, Generate Equivalent Expressions Explore It!, ETP: During, “3. Observe Student Work: How do students represent the total number of eggs with a diagram? Students might draw a diagram that indicates an unknown number of cartons with 6 eggs each and unknown number of cartons with 12 eggs each. How do students represent the total number of eggs using an expression? Students might let x represent the number of cartons with a half-dozen eggs and y represent the number of cartons with a dozen eggs and write the expression 6x + 12y. If needed, ask, What is known about the situation? What is unknown?”
• Lesson 7-4, Use Probability Models, Explain It!, ETP: After, “4. Discuss Solution Strategies and Key Ideas: Have students present their analyses of the models. Encourage them to use mathematical terms, such as outcome, chance, and probability. Have students discuss if all the team members have the same probability of being chosen caption (they do). Then have them discuss if the probability would be the same or different if their caption was not drawn randomly; if the name is not drawn randomly, each team member would not have the same chance of being picked. 5. Consider Instructional Implications: When presenting Example 1, make connections between the random drawing of a name in the Explain It, and the random drawing of marbles. In the Explain It, each team member had only one instance of their name in the draw, so each probability was equal. In Example 1, however, there are multiple marbles of each color. Have students discuss how the chess team draw could be altered to have uneven outcomes for team members; for example, some team members may be able to enter a name more than once.”
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Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials contain a teacher’s edition (in print or clearly distinguished/ accessible as a teacher’s edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

Each Topic contains a Topic Opener, Math Background: Focus section that provides a discussion of the math content in the topic along with sample work and strategies that illustrate the underlying concepts to help teachers anticipate the works students will do. The Topic Opener also contains Advanced Concepts for the Teacher that provides examples and adult-level explanations of more advanced mathematical concepts related to the topic with explanations and examples to support teacher understanding of the underlying mathematical progressions. Examples from the Teacher Resource include:

• Topic 2, Analyze and Use Proportional Relationships, Math Background, “Solving Proportions by Inspection: Equivalent ratios are ratios that express the same relationship between numbers. When two equivalent ratios have the same first or second term, then the other terms are equal. For example, if $$\frac{z}{x}$$ is equivalent to $$\frac{z}{y}$$, then x must equal y. For some equivalent ratios, one term is a multiple of a corresponding term of the other ratio. For example, the ratio $$\frac{2}{5}$$ shows a relationship of 2 to 5, shown in the diagram as 2 green for every 5 blue. The ratio $$\frac{4}{10}$$ shows a relationship that is equivalent to 2 groups of 2 and 5 groups of 2. Therefore the ratio $$\frac{4}{10}$$ is equivalent to $$\frac{2}{5}$$. Any ratio in the form of $$\frac{2n}{5n}$$, where n $$\not=$$ 0 is equivalent to $$\frac{2}{5}$$. This is generalized as a proportion, $$\frac{a}{b}$$ = $$\frac{an}{bn}$$.” Visual representations are provided.
• Topic 4, Generate Equivalent Expressions, Math Background, “Equivalence of Algebraic Expressions: Two expressions are equivalent if, for any value in the domain of the expression, the expressions represent the same value. Expressions that are not equivalent for all rational numbers may be equivalent on more restricted domains. For example, consider a single-element domain of (1). The following expressions are equivalent on the domain. Restricted domains can also be infinite. For all positive rational numbers, the expressions x and |x| have equal values. Therefore on the domain of $$\ge$$ 0 the expressions x and |x| are equivalent. These same expressions are not equivalent on a domain of all rational numbers. Disproving equivalence over a domain requires only a single counterexample. Proving equivalence over infinite domains cannot be done by example. The properties of real numbers and the properties for simplifying algebraic expressions are given as a set of rules that always produce or verify equivalent expression for the set of all rational numbers.” Visual representations also provided.
• Topic 7, Probability, Math Background, “Likelihood: In Lesson 7-1, students analyze how likely an event is to occur. They use probability to examine fairness. In Lesson 7-2, students determine theoretical probability and then make predictions. Experimental Probability: In Lesson 7-3 and 7-4, students compare and contrast theoretical and experimental probability and construct probability models. Students use experimental probability to make predictions, estimate and evaluate a situation.” An example of likelihood outcomes is provided.
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Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials contain a teacher’s edition (in print or clearly distinguished/ accessible as a teacher’s edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

Each Topic Opener contains a section Math Background: Coherence that summarizes the content connections through the materials to prior and future grades. Look Back illustrates connections to previously taught concepts and skills include those within the grade, across content, or across grades. Look Ahead illustrates connections within or across grades. Examples from the Teacher Resource include:

• Topic 3, Analyze and Solve Percent Problems, Math Background, Look Back, “Grade 6: Ratio Reasoning - In Grade 6, students learned to reason about ratios by using equivalent ratios and tables of equivalent ratios, and used their understanding of ratios to work with a special type of ratio called a percent. Rates- In Grade 6, students learned about a special type of ratio called a rate.”
• Topic 5, Solve Problems Using Equations and Inequalities, Math Background, Look Back, “Earlier in Grade 7: Define and Evaluate Expressions - Students identified parts of an expression and learned to view the expression as one entity. They performed arithmetic operations and substituted letters for unknown quantities to create algebraic operations.”
• Topic 7, Probability, Math Background, Look Ahead, “Grade 8: Two-Way Frequency Tables - In Grade 8, students will continue to find probabilities of simple and compound events. They will extend this knowledge to finding probabilities and making inferences and predictions using a two-way frequency table.”
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Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 provides a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher’s edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

Each Topic Opener contains a Topic Planner that provides an overview of the Learning Objective, Essential Understanding, and Standards. The Content Overview Introduction also contains a breakdown of each Topic into lessons, objectives, and standards. Finally, the Teacher Resource Program Overview contains a Pacing Guide with Topic titles and number of instruction days required, “Teachers are encouraged to spend 2 days on each content-focused lesson, giving students time to build deep understanding of the concepts presented, 1 to 2 days for the 3-Act Mathematical Modeling lesson, and 1 to 2 days for the enVisionSTEM project and Pick a Project. This pacing allows for 2 days for each Topic Review and Topic Assessment, plus an additional 2 to 4 days per topic to be spent on remediation, fluency practice, differentiation, and other assessment.”

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Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 contain some strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The online Teacher’s Resource Masters have Home School Connection Letters, in English and Spanish, for each Topic. The letters include information on the mathematical content and activities parents can do with their child to support the mathematical content. For example, Grade 7, Topic 7, Probability, “Dear Family, Your child is studying probabilities of simple and compound events. He or she is learning to use precise terms to describe actions and their possible outcomes, and to distinguish between and determine theoretical and experimental probabilities. You can help your child understand probability concepts by playing the following game.”

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Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 contain explanations of the instructional approaches of the program and identification of the research-based strategies.

EnVision is based on research-based strategies. According to the Teacher Resource Program Overview, “enVision Mathematics embraces time-proven research principles for teaching mathematics with understanding. One understands an idea in mathematics when one can connect that idea to previously learned ideas (Hiebert et al., 1997). So, understanding is based on making connections, and enVision Mathematics was developed on this principle.” Additionally, the core instructional model is based in research, “Over the past twenty years, there have been numerous research studies measuring the effectiveness of problem-based learning, a key part of the core instructional approach used in enVision Mathematics. These studies have found that students taught partly or fully through problem-based learning showed greater gains in learning. However, the interaction of problem-based learning, which fosters informal mathematical learning, and more explicit visual instruction that formalizes mathematical concepts with visual representations leads to the greatest gains for students. The enVision Mathematics instructional model is built on the interaction between these two instructional approaches.”

#### Criterion 3.3: Assessment

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for offering teachers resources and tools to collect ongoing data about student progress on the CCSSM. The instructional materials provide strategies for gathering information about students’ prior knowledge, strategies for teachers to identify and address common student errors and misconceptions, opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills, and assessments that clearly denote which standards are being emphasized.

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

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials provide strategies for gathering information about students’ prior knowledge within and across grade levels.

Materials provide strategies for gathering students’ prior knowledge. Examples include:

• Grade Level Readiness Test diagnoses students’ readiness for learning by assessing prerequisite content. This assessment is also available online and is autoscored. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource.
• Topic Readiness Assessment diagnoses students’ proficiency with Topic prerequisite concepts and skills. This assessment is available online and is autoscored. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource.
• Review What You Know, found at the beginning of each Topic, checks for understanding of key math concepts previously learned. An Item Analysis is provided for diagnosis and remediation in the Teacher Resource.
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Materials provide strategies for teachers to identify and address common student errors and misconceptions.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials provide strategies for teachers to identify common student errors and misconceptions.

Materials provide strategies to identity student errors. Prevent Misconceptions are found in the Teacher Resource sidenotes for the Visual Learning portion of the lesson and Error Interventions are found in the Practice & Problem Solving Section. Examples from the Teacher Resource include:

• Lesson 2-1, Connect Ratios, Rates, and Unit Rates, Do You Understand/Do You Know How?, Prevent Misconceptions, Item 4, “If students start by comparing $134.97 for Plan 2 with$34.99 for Plan 1, remind them that when comparing Internet service plans, they must compare the cost for the same number of months for each plan. Q: What is the unit rate for Plan 1? For Plan 2?”
• Lesson 5-3, Solve Equations Using the Distributive Property, Practice & Problem Solving, Error Intervention, Item 14, “Students often make mistakes with the signs of terms. Q: State in your own words the rules for multiplying terms with like or opposite signs. [Sample answer: Multiplying numbers with like signs results in a positive number. Multiplying numbers with opposite signs results in a negative number.] Q: Why is it useful to first change the subtraction expression to addition? [So it is obvious to multiply -3 and -r. It is easy to overlook the sign of the term when this first step is omitted.]”
• Lesson 8-5, Solve Problems Involving Circumference of a Circle, Do You Understand/Do You Know How?, Prevent Misconceptions, Item 3, “Make sure students understand that $$\frac{22}{7}$$ is a rational approximation but is not equal to . Q: Is the rational number $$\frac{22}{7}$$ equal $$\pi$$? Explain. [No; Sample answer: is not a rational number. It never terminates or repeats. The rational number $$\frac{22}{7}$$ is a useful estimate but is only an approximation of $$\pi$$.”
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Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

Materials provide opportunities for ongoing review of concepts and skills. Examples Include:

• Each Topic includes Review What You Know to activate prior knowledge and and review prerequisite skills needed for the Topic. Both vocabulary and practice problems are provided.
• The Cumulative/Benchmark Assessments are found at the end of Topics 2, 4, 6 and 8 assess students’ understanding and proficiency with concepts and skills taught throughout the year. An item analysis is provided for diagnosis and intervention. Students can take the assessment online, with differentiated intervention automatically assigned to students based on their scores.
• The Math Diagnosis and Intervention System has practice pages which are specific to a skill or strategy (i.e. Markups and Markdowns and Mental Math).
• There are multiple pages of extra practice available at Pearson Realize online that give students extra opportunities to review skills assigned by the teacher. Each of these pages is able to be customized by the teacher or used as is.
• Different games online at Pearson Realize support students in practice and review of skills, as well procedural fluency.
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Materials offer ongoing formative and summative assessments:
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Assessments clearly denote which standards are being emphasized.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials offer ongoing formative and summative assessments, clearly denoting which standards are being emphasized.

Formative and summative assessments clearly denote standards being assessed. Examples include:

• Try It! and Convince Me! are found following the Visual Learning Examples and assess students’s understanding of concepts and skills presented in each Example and results can be used to modify instruction. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• Do You Understand? And Do You Know How? are found after the Visual Learning instruction and assess students’ conceptual understanding and procedural fluency and results can be used to review content. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• Following each lesson is a Lesson Quiz that assesses students’ conceptual understanding and procedural fluency with the lesson content. Results can be used to determine differentiated instruction. Standards assessed are listed in the Lesson Overview, Mathematics Overview, Common Core Standards, Content Standards.
• At the end of each Topic there is a Topic Assessment with 2 forms, Form A and Form B, that assesses students’ conceptual understanding and procedural fluency with the topic content. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.
• At the end of each Topic there is a Performance Task with 2 forms, Form A and Form B, that assess students’ ability to apply concepts learned and proficiency with math practices. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.
• Cumulative/Benchmark Assessments found at the end of Topics 2, 4, 5, and 8 assess students’ understanding and proficiency with concepts and skills taught throughout the school year; results can be used to determine intervention. Standards for these assessments are found in the teacher side matter under Item Analysis for Diagnosis and Remediation.
##### Indicator {{'3p.ii' | indicatorName}}
Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials offer ongoing formative and summative assessments, which include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Following Lesson Quizzes, Topic Assessments, Topic Performance Task and Cumulative/Benchmark Assessments Scoring Guides are provided. Teachers can also assign these assessments online where they are auto-scored and differentiated intervention is automatically assigned to students based on their scores. Examples from the Teacher Resource include:

• Lesson 1-9, Divide Rational Numbers, Lesson Quiz, “Use the student scores on the Lesson Quiz to prescribe differentiated assignments. Intervention 0-3 Points. On-Level 4 Points. Advanced 5 Points. You may opt to have students take the Lesson Quiz online. The Lesson Quiz will be automatically scored and appropriate remediation, practice, or enrichment will be assigned based on student performance.”
• Topic 3, Analyze and Solve Percent Problems, Topic Assessment, Form A, “Greater Than 85%: Assign the corresponding MDIS for items answered incorrectly. Use Enrichment activities with the student. 70% - 85%: Assign the corresponding MDIS for items answered incorrectly. You may also assign Reteach to Build Understanding and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly. Less Than 70%: Assign the corresponding MDIS items answered incorrectly. Assign appropriate intervention lessons available online. You may also assign Reteach to Build Understanding, Additional Vocabulary Support, Build Mathematical Literacy, and Virtual Nerd Video assets for the lessons correlated to the items the student answered incorrectly.”
• Topic 7, Probability, Performance Task, Form A, Item 1, “Luca decides to play Wheel of Letters. To play the game, contestants spin a wheel with 26 letters equal sections, lettered A through Z. If the pointer lands on any letter in the phrase “COUNTRY FAIR” the contestant wins a prize. Part A: What is the probability that Luca will win Wheel of Letters? Is your answer a theoretical probability or an experimental probability? Explain.” Two charts are provided for the teacher, Item Analysis for Diagnosis and Intervention and Scoring Rubric for forms A and B. The Item Analysis for Diagnosis and Intervention Chart contains information to help the teacher with RTI such as DOK, MDIS, and standard. The scoring rubric provides the teacher with solutions and scoring explanations. “Item 1, Form A 2 Points:Correct probability and explanation. 1 Point: Correct probability or explanation.”
##### Indicator {{'3q' | indicatorName}}
Materials encourage students to monitor their own progress.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 encourage students to monitor their own progress.

Each Topic contains a Mid-Topic Checkpoint for students to monitor their understanding of concepts and skills taught in the first lessons of the Topic. Following the assessment students are asked, “How well did you do on the mid-topic checkpoint? Fill in the stars.” Three stars are provided.

#### Criterion 3.4: Differentiation

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

​The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations for supporting teachers in differentiating instruction for diverse learners within and across grades. The instructional materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners and strategies for meeting the needs of a range of learners. The materials embed tasks with multiple entry points that can be solved using a variety of solution strategies or representations, and they provide opportunities for advanced students to investigate mathematics content at greater depth. The instructional materials also suggest support, accommodations, and modifications for English Language Learners and other special populations and provide a balanced portrayal of various demographic and personal characteristics.

##### Indicator {{'3r' | indicatorName}}
Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The Topic Overview in the Teacher Resource provides a coherence section which enhances the opportunity to scaffold instruction by identifying prerequisite skills needed. All lessons include instructional notes and classroom strategies in the side matter labeled ETP, Effective Teaching Practices. ETP notes provide teachers with sample questions, differentiation strategies, discussion questions, possible misconceptions, and student “look fors” to assist in making content accessible to all learners. Additionally, the Solve and Discuss It! Section provides teachers with Before, During, and After instruction notes to help scaffold learning for students. Examples from the Teacher Resource include:

• Lesson 3-1, Analyze Percents of Numbers, Solve & Discuss It!, ETP: Before, “How would you find the total cost of a bill plus a 15% tip?” During: “How do you find an equal share among four people?” After: “What did Caleb assume about the amount each person would pay? Did Caleb and Jackie use the same steps to find one share with a 15% tip? Explain.”
• Lesson 5-2, Solve Two-Step Equations, Example 1, ETP: Use and Connect Mathematical Representations, “Q: What does each block labeled m in the bar diagram represent? [The cost of one movie ticket]. Q: Why was 6 subtracted from both sides of the equation? [Sample answer: In order to write an equivalent equation, an operation performed on one side of an equation must also be performed on the other side of the equation.] Q: Why was each side divided by 3? [To find the cost of 1 ticket.]”
• Lesson 8-1, Solve Problems Involving Scale Drawings, Example 3, ETP: Pose Purposeful Questions, “Q: Why is 4 the constant of proportionality? [The scale factor is 1 in. = 4 ft. The value, 4, that relates the scale drawing to actual measures is constant.] Q: Why are the length and height of the landscape drawing each multiplied by 4? [To find the length and width of the actual landscape.] Q: What is another way to solve this problem? [Sample answer: Relate the measures of the original landscape drawing to the measures of the drawing on the mural. 10 in. = 80 in., so the scale factor is 8. The height of the drawing on the mural is 8 x 8 in. = 64 in.] Have students mark the dimensions for the mural on the wall.”
##### Indicator {{'3s' | indicatorName}}
Materials provide teachers with strategies for meeting the needs of a range of learners.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials provide teachers with strategies for meeting the needs of a range of learners.

Each lesson contains Response to Intervention and Enrichment strategies in each lesson. Additional Examples and Additional Practice are provided if students need more support. At the end of each lesson Differentiated Intervention is provided for Intervention, On-Level, and Advanced learners. Examples from the Teacher Resource include:

• Lesson 3-1 Analyze Percents of Numbers, Response to Intervention, “Use with Example 2: Some students may need reinforcement on the concept of percents greater than 100%. Remind students that a fraction with a numerator greater than the denominator is greater than 1. Q: How is $$\frac{3}{2}$$ written as a decimal? Is this greater than or less than 1? Connect this idea to a percent greater than 100%. Q: How do you write a percent as a decimal? Q: How is 150% written as a decimal? Is this greater or less than 1?”
• Lesson 5-2 Solve Two-Step Equations, Enrichment, “Use with Example 3, Challenge students to write an algebraic and arithmetic solution to the following scenarios involving the band in Example 3. Q: The number of percussion players is 2 more than twice the number of students who play trumpet. How many percussion players are there? Q: The number of flute players is 3 fewer than $$\frac{1}{8}$$ of the total number of students in the band. How many flute players are there?”
• Lesson 7-6, Find the Probabilities of Compound Events, Differentiate Intervention, Reteach to Build Understanding, Problem 2, “List all the possible outcomes to describe the side of the coin that may be facing up when the coin lands in the container. Are the outcomes equally likely?”
##### Indicator {{'3t' | indicatorName}}
Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

Each lesson begins with a Problem-Based Learning activity, Solve & Discuss It, Explore It or Explain it! that offer multiple entry-points. 3-Act Mathematical Modeling tasks and Performance Tasks also include questions with multiple entry points that can be solved using a variety of representations. Examples from the Teacher Resource include:

• Topic 1, Rational Number Operations, 3-Act Mathematical Modeling, Win Some, Lose Some. Students are shown a video and then encouraged to consider the situation and ask any questions that come to mind. Teachers pose the Main Question, “Who will win the game? What is the final score?” Teachers are given questions and tips to facilitate discussion about the 3-Act Mathematical Modeling activities, “Why do you think your prediction is the answer to the Main Question? Who had a similar prediction? How many agree with that prediction? Who has a different prediction?”
• Lesson 4-2, Generate Equivalent Expressions, Explore It!, “A shipment of eggs contains some cartons with a dozen eggs and some cartons with half-dozen eggs. A. How can you represent the total number of eggs in the shipment using diagrams or images? Explain your diagram. B. How can you represent the total number of eggs in the shipment using expressions? What variables do you use? What do they represent? How do the two representations compare? How are they different?”
• Topic 7, Probability, Performance Task Form A, Item 4, “In Toss-N-Turn, contestants flip a fair coin and then spin the pointer of a spinner with seven equal sections, numbered 1-7. Contestants win if their coin lands heads up and the pointer lands on an even number. If Paulina wants to win a prize, should she play Toss-N-Turn or Rubber Duckie? Explain.”
##### Indicator {{'3u' | indicatorName}}
Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics.

Each lesson contains instructional strategies for Emerging, Developing, and Expanding English Language Learners. Additionally, the Language Support Handbook provides Topic and Lesson instructional support and online academic vocabulary activities. Examples from the Teacher Resource include:

• Lesson 2-4, Describe Proportional Relationships: Constant of Proportionality, English Language Learners, “Emerging: Complete Example 1. Write the equation y = 0.8x on the board. Q: Discuss with a partner what x, y, and 0.8 mean in this equation. Listen for students who use content vocabulary and build academic language proficiency. Q: What could you use the equation y = 0.8(5) to do? [Find the amount of water the sponge filters in 5 hours.] Q: Discuss with a partner how you could find the time it takes the sponge to filter 6 liters, using either equivalent ratios or an equation.”
• Lesson 5-6, Solve Two-Step Inequalities, English Language Learners, “Developing: Ask students questions as they work through Example 2, in order to make sure they understand the terms in the problem. Q: What does exceed mean? What symbol will you use to represent an amount that exceeds the amount raised last year? [Sample answer: Exceeds means greater than. Use the greater than symbol in the inequality.] Q: What does raised mean in this problem? [Sample answer: To collect.]”
• Lesson 7-3, Understand Experimental Probability, English Language Learners, “Expanding: Read Example 1. Have students work with a partner and discuss the following concepts. Then have a volunteer share the results with the class. Q: Why does the experimental probability value change each time the experiment is carried out? Q: Does the experimental probability equal the theoretical probability in every probability model? Explain.”
##### Indicator {{'3v' | indicatorName}}
Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials provide opportunities for advanced students to investigate mathematics content at greater depth.

Each lesson provides an Enrichment side note with instructional strategies for advanced learners. The Problem-Based Learning activity provides instructional strategies During the lesson for Early Finishers. A Challenge question is presented in the teacher side notes for Practice & Problem Solving. Examples from the Teacher Resource include:

• Lesson 2-2, Determine Unit Rates with Ratios of Fractions, Enrichment, “Use with Example 3, Challenge advanced students to recognize that a unit rate on a map is the scale of the map. Have them reason about the unit rate for the map in the Try It. Q: What is the unit rate? Q: What does the unit rate tell you about the map? Q: How could you use the unit rate, or scale of the map, to find any actual distance from the map?”
• Lesson 4-5, Factor Expressions, Explain It!, ETP: During, “Early Finishers, How would the problem change if Tasha has 54 bottles of bubbles? Explain.”
• Lesson 8-4, Solve Problems Using Angle Relationships, Practice & Problem Solving, Item 13, “Some students can be challenged to find angle measures of complementary angles. Q: What is the measure of ∠A, and ∠B if the two angles were complementary, not supplementary? Explain.”
##### Indicator {{'3w' | indicatorName}}
Materials provide a balanced portrayal of various demographic and personal characteristics.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 meet expectations that materials provide a balanced portrayal of various demographic and personal characteristics.

Different cultural names and situations are represented. Role names are used instead of pronouns referencing gender. Objects, animals, and cartoon drawings are used in place of actual people. Examples from the Teacher Resource include:

• Lesson 2-6, Apply Proportional Reasoning to Solve Problems, Visual Learning, Example 4, four kids are shown: two caucasion (female and male) and two African Americans (female and male). Students use the picture with the data to answer the question.
• Lesson 4-5, Factor Expressions, Visual Learning, Example 2, students factor expressions with negative coefficients, “Rodrigo and Jordan each factor the expression -2x - 6. Who factored the expression correctly?”
• Lesson 7-3, Understand Experimental Probability, Practice & Problem Solving, Item 15, students identify similar and different frequencies, “A basketball player makes 65% of all free throws in her first 5 seasons. In her 6th season she makes 105 out of 150 free throws. How does the observed frequency of her 6th season compare to the expected frequency? Provide a possible explanation for any similarities or differences in the frequencies.”
##### Indicator {{'3x' | indicatorName}}
Materials provide opportunities for teachers to use a variety of grouping strategies.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 provide opportunities for teachers to use a variety of grouping strategies.

Each lesson begins with a Problem-Based Learning activity which is introduced to the whole class. Then students break into small groups to work on the activity and come back together to discuss solutions and strategies as a whole class. Independent practice is found in the Problem & Practice Solving portion of the lesson. Icons in the Teacher’s Edition indicate whether the activity should be completed with Whole Class or Small Group.

##### Indicator {{'3y' | indicatorName}}
Materials encourage teachers to draw upon home language and culture to facilitate learning.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 encourage teachers to draw upon home language and culture to facilitate learning.

The Language Support Handbook provides research-based support strategies for English Language Learners, Academic Vocabulary Activities, a list of key vocabulary in 6 languages, and specific language support for each Topic Lesson. Digital and Student Edition Glossaries are in both English and Spanish. Assessments in Spanish can be accessed online. Each Topic’s Home-School Connection Letter explains the content of the Topic in English or Spanish.

#### Criterion 3.5: Technology

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7: integrate technology in ways that engage students in the Mathematical Practices; are web-­based and compatible with multiple internet browsers; include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology; can be easily customized for individual learners; and include or reference technology that provides opportunities for teachers and/or students to collaborate with each other.

##### Indicator {{'3aa' | indicatorName}}
Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

The digital instructional materials reviewed for enVision Mathematics Common Core Grade 7 are web-­based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are “platform neutral” (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

##### Indicator {{'3ab' | indicatorName}}
Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology. Examples include:

• Digital games that enhance fluency and provide opportunities for students to use procedural skills to solve problems are available online.
• Virtual Nerd offers tutorials on a variety of math concepts with procedural skill emphasised.
• The online Readiness Assessment tab for each topic includes a Remediation link that has tutorials and opportunities for students to practice procedural skills using technology.
• Fluency Practice Pages for each Topic are available online.
##### Indicator {{'3ac' | indicatorName}}
Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

The digital materials reviewed for enVision Mathematics Common Core Grade 7 include opportunities for teachers to personalize learning for all students. Adaptive technology is not provided by digital materials.

Digital materials include opportunities for teachers to personalize learning for all students. Examples include:

• Teachers can select and assign individual practice items for student remediation based on the Topic Readiness assessment. If students take the test online it is automatically scored and students are automatically assigned enrichment or remediation activities.
• Teachers can create online classes and assignments for students.
• Interactive Student Edition is accessible online and can be assigned to students.

The digital materials reviewed for enVision Mathematics Common Core Grade 7 can easily be customized for local use. Digital materials provide online materials for teachers to assign to students. Examples include:

• Interactive media lessons are accessible that cover all learning standards
• Lesson plans can be customized by day, week, or month or resequenced to match the district curriculum map.
• Outside content can be uploaded and Teacher Resource Masters can be customized.
##### Indicator {{'3ad' | indicatorName}}
Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

The materials reviewed for enVision Mathematics Common Core Grade 7 include technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).

Teachers can create Online Discussion Boards and monitor student participation.

##### Indicator {{'3z' | indicatorName}}
Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

The instructional materials reviewed for enVision Mathematics Common Core Grade 7 integrate technology including interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices. Examples include:

• The Math Practices and Problem Solving Handbook is an online reference available for students.
• Digital Desmos Activities provide embedded technology with engaging instruction of real-world content.
• Visual Learning Animation Plus provides scaffold animations of learning with real aloud options to support English learners.
• Animated Glossary in digital resources provides math terms with support in English and Spanish.
• Math Practice Animations are online videos explaining the Practices and sample problems supporting the Practices.
• A variety of Interactive Math Tools are available online for students and teachers.
• Topic Readiness Tests and Lesson Quizzes taken online are automatically graded and remediation and enrichment activities are automatically assigned to students.

## Report Overview

### Summary of Alignment & Usability for enVision Mathematics Common Core | Math

#### Math K-2

​The instructional materials reviewed for enVision Mathematics Common Core Kindergarten-2 meet expectations for alignment to the Standards and usability. The instructional materials meet expectations for Gateway 1, focus and coherence, Gateway 2, rigor and balance and practice-content connections, and Gateway 3, instructional supports and usability indicators.

##### Kindergarten
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

#### Math 3-5

​The instructional materials reviewed for enVision Mathematics Common Core Grade 3-5 meet expectations for alignment to the Standards and usability. The instructional materials meet expectations for Gateway 1, focus and coherence, Gateway 2, rigor and balance and practice-content connections, and Gateway 3, instructional supports and usability indicators.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

#### Math 6-8

​The instructional materials reviewed for enVision Mathematics Common Core Grade 6-8 meet expectations for alignment to the Standards and usability. The instructional materials meet expectations for Gateway 1, focus and coherence, Gateway 2, rigor and balance and practice-content connections, and Gateway 3, instructional supports and usability indicators.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

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### Overall Summary

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###### Usability
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