The materials reviewed for Eureka Math² Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The Learn portion of the lesson presents new learning through instructional segments to develop conceptual understanding of key mathematical concepts. Students independently demonstrate conceptual understanding in Exit Tickets, formative assessments that close the learning, and Practice, additional practice problems aligned to the lesson’s learning objectives.

Materials develop conceptual understanding throughout the grade level. Examples include:

• Module 2, Topic A, Lesson 5: Decomposing Rational Numbers to Make Addition More Efficient, Learn, Problem 3, students identify the additive inverse and write number sentences. “Use the provided number line to answer the following questions (a number line from -4 to 4 is provided). a. What is the additive inverse of 3.4? Plot a point at the additive inverse on the number line. b.Using a different color, plot a point at −1.2 on the number line. c. What is the additive inverse of −1.2? Plot a point at the additive inverse on the number line in the same color as part (b). d.Write a number sentence showing that 3.4 and −3.4 are additive inverses. e. Write a number sentence showing that −1.2 and 1.2 are additive inverses. f. What do you notice about the additive inverses on the number line?” This activity supports the conceptual understanding of 7.NS.1 (Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram).

• Module 4, Topic E, Lesson 22: Understanding Planes and Cross Sections, Learn, students identify cross sections formed by cutting a right rectangular prism with a plane that is parallel or perpendicular to its bases. “Hold up a stick of butter for the class. What are the bases of this right rectangular prism? What shapes are the lateral faces? Cut the butter parallel to the bases as shown (show is a cut parallel to the base). Did I cut the stick of butter parallel or perpendicular to the bases? If I lift the top piece of the butter up and show you where I cut it, what shape do you expect to see? The rectangular regions that resulted from cutting the butter are identical to the bases. Now consider that the square faces of the prism are the bases. If we made the same horizontal cut, would that cut be parallel or perpendicular to the square bases? Did I cut this parallel or perpendicular to the bases? How do you know? If I separate the butter along the new cut, what shape do you expect to see? The square regions that result from cutting the butter are identical to the square lateral faces. Now consider that the square faces of the prism are the bases. If we made the same vertical cut, would that cut be parallel or perpendicular to the square bases? Is there another way that I could cut the butter that would be perpendicular to the original bases? If so, what shape do you expect to see when the cut is made? What shape do you expect to see after this cut? The rectangular regions that result from cutting the butter are not identical to the square lateral faces. There are many ways to cut a right rectangular prism perpendicular to its bases; however, this does not mean that the figures formed by the cuts will be the same. The figure formed by cutting the butter is known as a cross section. A cross section is the figure formed by the intersection of a plane and a three-dimensional solid. In this demonstration, what represents a three-dimensional figure? What represents the plane? What represents the cross section?” This activity supports the conceptual understanding of 7.G.3 (Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids).

• Module 6, Topic C, Lesson 11: Populations and Samples, Learn, Problem 14, students answer statistical questions by sampling. “In 2018, the ACS asked a sample of US workers, age 16 and older, about their commuting times to work. a. In this sample, 9.5% reported commuting more than an hour to work each day. Is 9.5% a sample statistic or a population characteristic? Explain. b. Do you think people use the ACS data to learn about the workers in the sample or the population of the country? c. How do you think people could use the ACS data? d. If there were 146,357,588 commuting workers in the entire population, about how many traveled more than an hour to work each day?” This activity supports conceptual understanding of 7.SP.1 (Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. understand that random sampling tends to produce representative samples and support valid inferences).

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Examples include:

• Module 1, Topic B, Lesson 8: Relating Representations of Proportional Relationships, Exit Ticket, students identify the constant of proportionality. “The number of minutes Logan spends showering is proportional to the number of gallons of water used. a. Complete the table to represent this relationship (a table with time spent showering and water used is provided). b. Graph this relationship. c. Write an equation to represent the number of gallons of water w used when Logan showers for t minutes.” Students independently demonstrate conceptual understanding of 7.RP.2 (Recognize and represent proportional relationships between quantities).

• Module 3, Topic A, Lesson 6: Comparing Expressions, Practice, Problem 7, students explain if two expressions are equivalent. “Yu Yan uses properties of operations to claim that $-6-2(5x-76+3)$ and $-40x+56y-24$ are equivalent. Explain why you agree or disagree.” Students independently demonstrate conceptual understanding of 7.EE.1 (Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients).

• Module 5, Topic A, Lesson 5: Common Denominators or Common Numerators, Exit Ticket, students solve percent problems by using strategies that involve finding common denominators or common numerators to solve proportions. “Jonas waters 10 plants, which is 25% of the plants in a greenhouse. How many plants are in the greenhouse? a. What does the problem ask you to find-the part, the whole or the percent? Explain your thinking. b. Estimate the number of plants that are in the greenhouse. Is the unknown number less than, equal to, or greater than 10? Why? c. Determine the number of plants in the greenhouse by solving an equation.” Students independently demonstrate understanding of 7.RP.3 (Use proportional relationships to solve multistep ratio and percent problems).

The materials reviewed for Eureka Math² Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The Learn portion of the lesson presents new learning through instructional segments to develop procedural skill of key mathematical concepts. Students independently demonstrate procedural skill in Exit Tickets, formative assessments that close the learning, and Practice, additional practice problems aligned to the lesson’s learning objectives.

Materials develop procedural skills and fluency throughout the grade level. Examples include:

• Module 1, Topic A, Lesson 2: Exploring Tables of Proportional Relationships, Learn, Problem 2, students determine whether relationships represented in a table are proportional. “Have students work in pairs to complete the Try These Tables problem. Circulate as students work, and listen for pairs who identify equivalent ratios or a constant unit rate between all pairs of values in a table. Given the following tables, determine whether each relationship is proportional (Table 1: Number of Days to Number of Hours, Table 2: Number of At Bats to Number of Hits, and Table 3: Area in Square Feet to Total Cost of Tile in Dollars). When most students are finished, call the class back together to discuss the following questions. Highlight student reasoning that identifies a constant unit rate for every pair of values in a table. Which tables did you identify as representing proportional relationships? Why? Which table was the most difficult to classify as either a proportional relationship or not a proportional relationship? Why was that? Why were other tables less difficult to classify?” Students develop procedural fluency of 7.RP.2 (Recognize and represent proportional relationships between quantities).

• Module 3, Topic B, Lesson 7: Angle Relationships and Unknown Angle Measures, Learn Problem 3, students find unknown angle measures. “In the diagram, $\angle DAB$ is a straight angle. a. Describe the relationship between $\angle DAC$ and $\angle CAB$. Write an equation for the angle relationship and solve for x.” The straight line shown has angle measures x and 23° Students develop procedural fluency of 7.EE.4a (Solve word problems leading to equations of the form $px+q=r$ and $p(x+q)=r$ where p, q, and r are specific rational numbers).

• Module 5, Topic D, Lesson 15: Tips and Taxes, Learn, Problems 2-3, students calculate subtotal, tax, tip, and the total amount of the bill. “Divide students into groups of three. Have groups complete problems 2 and 3. Circulate and verify that students are finding the tax and tip from the subtotal. Pretend your group members are ordering lunch from Vic’s Diner. Complete the chart by stating what each group member orders and the price for each item. Refer to the information about guest 1’s order from problem 2. a. What is the subtotal of guest 1’s order? b. If the tax rate is 8%, how much tax needs to be added? c. If guest 1 leaves a 17% tip, how much is the tip? What is the total amount of guest 1’s bill? Use the following prompts to discuss efficiency. To determine the amount of tax, you found 8% of what amount? To determine the amount of tip, you found 17% of what amount? Can we find the total of the tip and tax more efficiently? Explain your thinking. I can add these percents to calculate tip and tax, but I could not add the percents when applying two percent-based discounts. Why does adding the percents work in one situation but not the other? We could not add the percents when applying two percent-based discounts because we calculated each discount with a different subtotal. We calculate tip and tax with the same subtotal. For any subtotal x, we can add the tip and tax percents together because of the distributive property. For this problem, the sum of the tax and tip is $0.08x+0.17x$, or $(0.08+0.17)x$, which is equivalent to $0.25x$.” Students develop procedural skill of 7.RP.3 (use proportional relationships to solve multistep ratio and percent problems).

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

• Module 2, Topic D, Lesson 19: Rational Numbers as Decimals, Part 1, Exit Ticket, Problem 1, students calculate quotients of integers and express them as terminating decimals. “For problems 1-5, write the expression as a decimal, ‘$\frac{3}{10}$.’” Students independently demonstrate procedural skill of 7.NS.2d (Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats).

• Module 3, Topic B, Lesson 10: Problem Solving with Unknown Angle Measures, Practice, Problem 3, students write and solve equations that use angle relationships to find unknown angle measures. “Four angles that form a right angle have measures of $55\degree$, $x\degree$, $(6x)\degree$, and $21\degree$. Determine the unknown angle measures.” Students independently demonstrate procedural skill of 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).

• Module 6, Topic C, Lesson 16: Sampling Variability When Estimating a Population Proportion, Practice, Problem 1, students find population proportions. “Find the following population proportions. Write your answer as a fraction and as a decimal. a. In a population of 50 parrots, 30 of the parrots have green feathers. What is the population proportion of parrots that have green feathers? b. In a population of 85 seventh graders, 17 of the seventh graders play golf. What is the population proportion of seventh graders who play golf? c. In a population of 125 vehicles, 100 of the vehicles have at least four doors. What is the population proportion of vehicles that have at least four doors?” Students independently demonstrate procedural skill of 7.SP.2 (Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions).

Each lesson begins with Fluency problems that provide practice of previously learned material. The Implementation Guide states, “Fluency provides distributed practice with previously learned material. It is designed to prepare students for new learning by activating prior knowledge and bridging small learning gaps. Fluency activities are included with each lesson, but they are not accounted for in the overall lesson time. Use them as bell ringers, or, in a class period longer than 45 minutes, consider using the facilitation suggestions in the Resources to teach the activities as part of the lesson.” For example, Module 2, Topic B, Lesson 9: Subtracting Integers, Part 2, Fluency, Problem 5, students add integers, “$12+(-15)$.” Students practice fluency of 7.NS.1 (Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram).

The materials reviewed for Eureka Math² Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

The Learn portion of the lesson presents new learning through instructional segments to develop application of mathematical concepts. Students independently demonstrate routine application of the mathematics in Exit Tickets, formative assessments that close the learning, and Practice, additional practice problems aligned to the lesson’s learning objectives.

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Module 2, Topic D, Lesson 22: Multiplication and Division Expressions, Learn, Problem 15, students create true number sentences involving multiplication and division of rational numbers. “The product of two numbers is -0.75. Find two different pairs of rational numbers that have a product of -0.75. Show that your answers make a true number sentence.” In this non-routine problem, students apply the mathematics of 7.NS.2 (Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers).

• Module 3, Topic C, Lesson 12: Solving Problem Algebraically and Arithmetically, Learn, Problem 3, students model a context with an equation and solve the equation. “A seventh grade class and 7 adults go on a field trip. Two-thirds of the people ride on a bus. Everyone else rides in vans. If 54 people ride on the bus, how many seventh graders go on the field trip? Write and solve an equation. Check your solution.” In this routine problem, students apply the mathematics 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 quantities).

• Module 4, Topic C, Lesson 15: Watering a Lawn, Learn, students use technology to model the problem of designing an efficient plan to water a lawn. Students watch videos of how different sprinklers move, design a sprinkler system for a lawn, adjust their model, and assess the effectiveness of their design by using what they know about the area. “How did you find the total area of the lawn that your sprinklers covered? How does the area of the lawn compare to the area of the lawn your sprinklers cover? What does that tell you about your design? What changes would you make to your sprinkler placement? Why? What do you notice about the different models? What do you wonder? How can we figure out the area that is overlapping? Or the area of the lawn that was not watered?” In this non-routine problem, students apply the mathematics of 7.G.4 (Know the formulas for the area and circumference of a circle and use them to solve problems: give an informal derivation of the relationship between the circumference and area of a circle).

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

• Module 1, Topic C, Lesson 14: Extreme Bicycles, Practice, Problems 1-3, students create their own question to compare two bicycles using proportional reasoning. “1. When you modeled the bicycle situation by using a proportional relationship, what assumptions did you make? 2. How was the bicycle situation modeled in class different from a proportional relationship? 3. If you had more time to examine another question related to the bicycle video, what question would you consider? What would your plan be to determine the answer?” In this non-routine problem, students independently apply the mathematics of 7.RP.2 (Recognize and represent proportional relationships between quantities).

• Module 3, Topic C, Lesson 11: Dominos and Dominos, Launch, students watch a video and estimate the number of dominoes used to make a tower before it falls. “What questions do you have? We will not be able to explore all these wonderings today. Let’s first tackle the question of how many dominoes made up the tower before it fell. How many dominoes do you think made up the tower before it fell? What is an unreasonable guess? What is too high or too low?” In this routine problem, students independently apply the mathematics of 7.EE.3 (Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form, using tools strategically).

• Module 5, Topic A, Lesson 2: Racing For Percents, Practice, Problem 8, students identify proportional relationships and write the constant of proportionality as a percent.“Pedro’s car completes 2.2 laps for every 2 laps Noor’s car completes. a. How many laps does Pedro’s car complete when Noor’s car completes 100 laps? b. Pedro’s car completes what percent of the number of laps that Noor’s car completes?” In this routine problem, students independently apply the mathematics of 7.RP.3 (Use proportional relationships to solve multistep ratio and percent problems).

The materials reviewed for Eureka Math² Grade 7 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. 

All three aspects of rigor are present independently throughout the grade. Examples where the materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Module 3, Topic D, Lesson 21: Solving Two-Step Inequalities, Practice, Problems 1-5, students solve two-step inequalities. “For problems 1–5, solve the inequality and graph the solution set. 1. $3x-6\geq-9$, 2. $-2x+8<14$, 3. $21.75\geq7.25-0.5x$, 4. $2-\frac{1}{3}x\geq-4$, 5. $\frac{-18}{4}>4.5x+13.5$.” Students attend to the procedural skill 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).

• Module 4, Topic E, Lesson 26: Designing a Fish Tank, Exit Ticket, Problem 1, students calculate surface area. “Logan plans to paint the entire outside of his doghouse, not including the roof. a. Explain whether Logan needs to find surface area or volume to solve the problem. b. Calculate the amount of wood that Logan needs to paint.” Students are shown a diagram with dimensions of the dog house. Students engage with the application of 7.G.6 (Solve real- world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms).

• Module 5, Topic E, Lesson 22: Making MIxtures, Exit Ticket, students compare mixtures made from percents of two or more liquids. “Consider the following pitchers of fruit punch with a mixture of ingredients as shown. The purple ingredient is grape juice. The orange ingredient is orange juice. a. What percent of the fruit punch in each pitcher is grape juice? b. Which fruit punch do you think will have a stronger grape taste? How do you know?” Pitcher 1 has two parts orange and three parts purple and Pitcher 2 has three parts orange and five parts purple. Students develop conceptual understanding of 7.RP.3 (Use proportional relationships to solve multi-step ratio and percent problems).

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single unit of study throughout the materials. Examples include:

• Module 1, Topic B, Lesson 9: Comparing Proportional Relationships, Practice, Problem 3, students explain how to use the point to find the unit rate of a proportional relationship. “Line 𝒶 shows how many coins machine A sorts per second. Line 𝒷 shows how many coins machine B sorts per second. a. How do you know that these lines represent proportional relationships? b. Write an equation relating the number of seconds t to the number of coins sorted c for machine A. c. Write an equation relating the number of seconds t to the number of coins sorted c for machine B. d. Which machine sorts coins faster? How do you know?” Students engage in procedural skill, conceptual understanding, and application of 7.RP.2 (Recognize and represent proportional relationships between quantities).

• Module 2, Topic E, Lesson 26: Writing and Evaluating Expressions with Rational Numbers, Part 2, Practice, Problem 3, students write and evaluate numerical expressions and interpret their value in context. “The diagram shows the annual operating income in millions of dollars for the Arizona Diamondbacks from 2002 to 2011. Over this 10-year period, what was the Arizona Diamondbacks’ mean operating income? What does this number tell you?” Students engage in procedural skill and application of 7.EE.3 (Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form, using tools strategically).

• Module 4, Topic C, Lesson 14: Composite Figures with Circular Regions, Debrief, students solve problems involving area and perimeter of composite figures.“Facilitate a class discussion by using the following prompts. Encourage students to restate or build upon their classmates’ responses. What makes a composite figure different from other figures for which we have found the perimeter and the area? How can we find the perimeter of a composite figure? How can we find the area of a composite figure? How can we determine lengths when they are not given in a diagram?” Students engage in conceptual understanding and procedural skill of 7.G.4 (Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle).

The materials reviewed for Eureka Math² Grade 7 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

Materials provide intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the units. Examples include:

• Module 2, Topic A, Lesson 4: KAKOOMA, Learn, students analyze and make sense of problems as they add integers to create a KAKOOMA puzzle. “Create your own KAKOOMA puzzle by using the blank puzzle shown. Refer to problem 1, and make sure that your KAKOOMA puzzle follows the structure. Use integer values of -9 to 9. The same integer cannot be used more than once in a pentagon.” Teacher margin note states, “Ask the following questions to promote MP1: What are some things you could try to start solving the problem? What facts do you need to determine the five numbers within one of the pentagon sections of the puzzle? Is starting with the outer pentagons working? Is there something else you could try?” 

• Module 3, Topic B, Lesson 10: Problem Solving with Unknown Angle Measures, Learn, Problem 2, students monitor and evaluate their progress to solve multi-step unknown angle problems. “Sketch a picture that represents the angle relationship described. Then determine all unknown angle measures. Two supplementary angles have measures that are in a ratio of 3 : 2. What are the measures of these angles?” Teacher margin note states, “Ask the following questions to promote MP1: What can you figure out about the relationships among the angles by looking at the diagram? What is your plan to find the unknown angle measure? Does your answer make sense? Why?”

• Module 4, Topic D, Lesson 16: Solving Area Problems by Composition and Decomposition”, Learn, Problem 4, students use a variety of strategies to calculate area. “Consider the figure. a. Find the area of the figure by using decomposition. b. Find the area of the figure by using composition. c. Which strategy do you prefer to use to find the area of this figure? Why?” Teacher margin note states, “Ask the following questions to promote MP1: What information or facts do you need to find the area of the given figure? What are some strategies you can try to start determining the area of the given figure? Is decomposing or composing working? Is there something else you can try?” 

Materials provide intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the units. Examples include:

• Module 1, Topic A, Lesson 5: Analyzing Graphs of Proportional Relationships, Learn, Discussion, students represent situations symbolically using graphs, equations, ratios, and rates that represent real-world contexts. “Once students have shared their responses, draw students’ attention to the two posters that represent the proportional relationships. Have students think–pair–share with a partner about the following questions. After each question, select a few students to share their ideas with the class. As students respond, point to where the constant of proportionality is conveyed on each displayed graph by circling or highlighting the point (1,k). What do you notice about where the constant of proportionality is determined on both graphs? We use the variable k to represent the constant of proportionality. Can the constant of proportionality always be determined from the point (1,k) on the graph of a proportional relationship? How do you know? What representation was most helpful for you in writing the equation, and why? What do the proportional relationships have in common? What do the relationships that are not proportional have in common?” Teacher margin note states, “Ask the following questions to promote MP2: What does this point on the graph mean in this situation? What does this ratio (or unit rate) mean in this situation? What does this context tell you about the relationship between these two quantities?”

• Module 4, Topic A, Lesson 3: Side Lengths of a Triangle, Learn, Problem 9, students analyze the relationship among distances and relate these to the side lengths of a triangle. “The diagram shows the location of your school, your home, and Maya’s home. a. What is the shortest path that you can take to walk home from school? Draw this path on the diagram and explain why you think it is the shortest. b. Maya missed school today. You walk to her home to drop off homework. What is the shortest path you can take from school to Maya’s home and then to your home? Draw this path on the diagram and explain why you think it is the shortest. c. From school, is the path home shorter or longer if you stop at Maya’s home? Explain. d. What if we could change the location of Maya’s home? Could we make the path from school to Maya’s home and then home shorter than the path straight home? e. Imagine that Maya’s home is on the path you drew from school to your home. How does the distance of the path from school to home compare to the distance of the path from school to Maya’s home and then to your home?” Teacher margin note states, “Ask the following questions to promote MP2: How do the lengths you drew represent the distances walked? What do the distances walked in this context tell you about the side lengths of the triangle? What real-world situations are modeled by distances that form a triangle?”

• Module 6, Topic B, Lesson 8: Picking Blue, Launch, students attend to the meaning of quantities as they analyze observations from a chance experiment to estimate the theoretical probability of pulling a blue chip from a bucket. “You are going to be a contestant on a game show called Picking Blue. Let’s see how the game is played. How would you explain the game to a classmate?” Teacher margin note states, “What does the relative frequency of blue chips pulled from a bucket tell you about the contents of that bucket? What do the results of the activity tell you about the theoretical probability of pulling a blue chip from that bucket?”

The materials reviewed for Eureka Math² Grade 7 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

Materials provide support for the intentional development of MP3 by providing opportunities for students to construct viable arguments in connection to grade-level content. Examples include:

• Module 2, Topic C, Lesson 14: Understand the Product of Two Negative Numbers, Learn, Problems 6-10 students create conjectures as they determine the sign of the product based on the number of negative factors. “Have students examine the expressions in problems 6–10 for about a minute before asking the following questions. What is the same about these expressions? What is different? Have students turn and talk about the following prompt. Without evaluating the expressions, predict which expressions have a negative product and which expressions have a positive product. Explain your reasoning. For problems 6–10, evaluate the expression. 6. $1(3)(5)(2)$ 7. $1(3)(5)(-2)$ 8. $1(3)(-5)(-2)$ 9. $1(-3)(-5)(-2)$ 10. $-1(-3)(-5)(-2)$ What patterns do you notice? Create a conjecture about how you can predict the sign of the product.Is your conjecture accurate? Why does an even number of negative factors result in a positive product?” 

• Module 4, Topic D, Lesson 17: Surface Area of Right Rectangular and Right Triangular Prisms, Exit Ticket, students justify their thinking as they calculate the surface area of right rectangular and triangular prisms. “Find the surface area of the prism shown. Explain how you got your answer.” The prism shown has a right triangular base with a height of 15 feet, width of 8 feet and hypotenuse of 17 feet. The height of the prism is 20 feet. 

• Module 5, Topic B, Lesson 9: Tax as a Fee, Learn, Problem 4, students construct viable arguments as they find the tax owed on a purchase. “Abdul found two receipts from his road trip and wondered how the tax rates compared in Louisiana and Texas. a. Determine the sales tax rate Abdul paid in Louisiana. Round to the nearest tenth of a percent. b. Determine the sales tax rate Abdul paid in Texas. Round to the nearest tenth of a percent. c. How does the sales tax rate that Abdul paid in Louisiana compare to the sales tax rate he paid in Texas? How does the sales tax rate paid by Abdul in Louisiana compare to the sales tax rate he paid in Texas? How do you know?”

Materials provide support for the intentional development of MP3 by providing opportunities for students to critique the reasoning of others in connection to grade-level content. Examples include:

• Module 2, Topic A, Lesson 6: Adding Rational Numbers, Learn, Problem 3, students critique the reasoning of others as they add rational numbers. “For Problems 3 and 4, refer to Shawn’s way which follows.$10.9+(-6.85)$  The addends have opposite signs. $10.9-6.85=10+0.9-6-0.85=10-6-0.9-0.85=4+0.05=4.05$ The positive addend has the greater absolute value so the sum is 4.05. Did Shawn get the right answer? Explain why.”

• Module 3, Topic A, Lesson 1: Equivalent Expressions, Learn, Problems 10, students perform error analysis as they verify equivalent expressions. “Logan said that $2(x + 4)$ must be equivalent to $5x + 8$ because he tested 0 and it worked. Is Logan correct? Explain your reasoning.” 

• Module 5, Topic B, Lesson 8: Determining Fees, Learn, Problem 1, students critique the reasoning of others as they calculate fees based on percentages. “Henry wants to transfer $40 to Kabir through one of three apps on his phone. a. App A charges a fee of 3% of the transferred amount. Determine the fee charged by app A. b. Kabir knows about another app, app B, which also charges a percent-based fee. He transferred $40 to a different friend by using app B, and the fee was $1. What percent did app B charge as a fee? c. App C charges a fee of $0.95 plus 1.25% of the transferred amount. Determine the fee charged by app C to transfer 40. d. Which app will charge Henry the least in fees for transferring $40?” Teachers are prompted to ask, “What parts of your partner’s justification do you question? Why? How would you change your partner’s justification to make it more accurate? Why does your strategy work? Convince your partner.”

The materials reviewed for Eureka Math² Grade 7 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

Materials provide intentional development of MP4 to meet its full intent in connection to grade-level content. Students model with mathematics to solve real-world problems, identify important quantities to make sense of relationships, and represent them mathematically as they work with the support of the teacher and independently throughout the units. Examples include:

• Module 1, Topic B, Lesson 7: Handstand Sprint, Launch, students collect and analyze data to determine how long it takes a person to reach the finish line. Students watch part 1 of a video to determine, “How long will it take to reach the finish line?” Teacher prompts include, “What math can you write or draw to represent the hand-walking situation? What assumptions can you make to help estimate the time it takes to reach the finish line? What do you wish you knew that would help you find the answer?”

• Module 2, Topic E, Lesson 26: Writing and Evaluating Expressions with Rational Numbers, Part 2, Learn, Problem 4, students write and evaluate rational number expressions that model a real-world context. “Each year, Major League Baseball teams report their operating income. This represents how much money a team earns before it pays interest or taxes. The diagram shows the operating income in millions of dollars for the Toronto Blue Jays from 2009 to 2018. Over this 10-year period, what was the Toronto Blue Jays’ mean operating income? What does this number tell you?” Teacher prompts include, “How could you improve your expression to better represent the context? What assumptions could you make to help solve the problem? What mathematical models could you draw to represent this situation?”

• Module 5, Topic E, Lesson 20: Making Money, Day 1, students model with mathematics as they consider how a pet store makes money, make decisions and assumptions about markups, and the number of goods and services they will sell. “Task: Today, we will determine how much money our pet store can make in one month from selling, at most, five types of goods and services.” Teachers are prompted to ask, “How can you use expressions or equations to express the amount of money made? What assumptions can you make to help you better estimate how many of an item you will sell? What do you wish you knew about how the store makes money? Can you make a reasonable estimate or assumption about that?”

Materials provide intentional development of MP5 to meet its full intent in connection to grade-level content. Students use appropriate tools strategically as they work with the support of the teacher and independently. Examples include:

• Module 1, Topic B, Lesson 13: Multi-Step Ratio Problems, Part 2, Learn, students identify and use proportional relationships in multi-part ratio situations. “A restaurant called The Irrational Pie provides the pizza toppings shown in the list. a. Create a recipe for a small pizza from the list of toppings. Your recipe must total exactly 5 cups of toppings, include at least 3 different toppings, include fractional amounts of at least 2 toppings, and include some amount of cheese. b. A large pizza calls for a total of 8 cups of toppings. Calculate the amount of each topping you need to create a large pizza if the ratios of ingredients remain the same as in the recipe you created for your small pizza. c. A medium pizza calls for a total of 612 cups of toppings. Calculate how much of each topping you will need to create a medium pizza if the ratios of ingredients remain the same as in the recipe you created for your small pizza. d. A customer wants to order a pizza made from the recipe you created for your small pizza, but the customer asks for 412 cups of cheese. Determine the amounts of all the other ingredients the customer’s pizza will have if the ratios of ingredients remain the same as in your recipe. What will the total amount of toppings be? e. You are running out of cheese and only have 12 cups left for a pizza. Determine the amounts of all the other ingredients the pizza will have if you keep the ratios of ingredients the same. What will the total amount of toppings be?” Teachers are prompted to ask, “What kind of diagram or strategy would be helpful in determining the amounts of pizza toppings? How can you estimate the amount of pizza toppings in each part of the problem? Do your estimates sound reasonable?”

• Module 2, Topic D, Lesson 21: Comparing and Ordering Rational Numbers, Learn, Problem 10, students compare numbers written in different forms. “Three students recorded their distances in a standing long jump competition. Logan jumped $2\frac{3}{8}$ meters, Abdul jumped 2.36 meters, and Shawn jumped $2\frac{4}{9}$ meters. a. Who jumped the farthest, b. Historically, the average distance of a jump in this competition is 2.3 meters. Who was closest to the historical average? Explain your thinking.” Teachers are prompted to ask, “What tool could help you model this problem? Why did you choose to use this tool? Did it work well?”

• Module 6, Topic B, Lesson 10: Simulations with Random Number Tables, Learn, Problem 7, students conduct a simulation to find an empirical probability. “You and your friend are both going to the park. You will each arrive between 1:00 p.m. and 2:00 p.m. and remain for 15 minutes. What is the theoretical probability that you and your friend will be at the park at the same time, for at least a minute? a. Describe a trial for your simulation. b. Describe a successful trial for your simulation.” Teachers are prompted to ask, “What tool would be the most efficient to simulate the theoretical probability that two people are at the park at the same time for at least a minute? Why? How can you estimate the theoretical probability that two people are at the park at the same time for at least a minute?”

The materials reviewed for Eureka Math² Grade 7 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. Margin Notes, Language Support, provide suggestions for student-to-student discourse, support of new and familiar content-specific terminology or academic language, or support of multiple-meaning words.

Materials provide intentional development of MP6 to meet its full intent in connection to grade-level content. Examples include:

• Module 1, Lesson 13: Multi-Step Ratio Problems, Part 2, Learn, Teacher Note, “Based on the fractions that students chose for their original recipe, some students may get unrealistic measurements of toppings in their answers to different parts of the problem. If needed, allow students to go back and change their original recipe to include friendlier numbers in their measurements. Prompt student thinking about real-world connections by using the following discussion questions: Is it reasonable to see a measurement like $\frac{13}{20}$ cups of olives in a recipe? What kinds of measurements are reasonable? How might a person making pizza in real life estimate a quantity like cups of olives?”

• Module 3, Topic D, Lesson 18: Understanding Inequalities and Their Solutions. Learn, students attend to precision as they write inequalities from graphs and context. “Write an inequality to represent the solutions shown on the graph.” Teachers are prompted to ask, “How can we show that the boundary number is a solution when writing the inequality? Is it exactly correct to say that the solutions are less than 5? What can we add or change to be more precise? Where is it easy to make mistakes when graphing solutions?”

• Module 4, Topic A, Lesson 2: Constructing Parallelograms and Other Quadrilaterals, Learn, Problem 4, students attend to precision as they construct quadrilaterals with given conditions. “For problems 4–8, use tools to construct the quadrilateral with the given conditions. Mark the figure to indicate parallel sides and equal lengths when appropriate. Construct a quadrilateral that has four different side lengths and four different angle measures. Label the side lengths and the angle measures.” Teachers are prompted to ask, “What details are important to think about when constructing a parallelogram? Where is it easy to make mistakes when constructing a rhombus? How precise do you need to be?”

The instructional materials attend to the specialized language of mathematics. Examples include:

• Module 2, Topic B, Lesson 7: What Subtraction Means, Learn, Teacher Note, “Some students might offer a comparison interpretation for $8-10$. They could interpret $8-10$ as ‘10 is how much greater than 8?’ This interpretation is accurate in this case, but interpreting $8-10$ the same way could introduce confusion. Following suit, students may interpret $8-10$ as ‘8 is how much greater than 10?’ However, that does not produce an accurate response. Should the conversation arise, encourage students to make conjectures as to why this interpretation does not translate to all subtraction expressions.”

• Module 2, Topic B, Lesson 12: The Integer Game, Learn, students play a game with integers finding sums closest to zero. “Pass out an Integer Game card deck to each group. Direct students to finish as many hands as possible in the next 15 minutes. Remind students that each hand ends after everyone has had a chance to draw and discard four times.” Teachers are prompted to ask, “How are you using strategies for adding and subtracting integers when playing the Integer Game? What details are important to think about when playing the Integer Game?“

• Module 5, Topic A, Lesson 4: Proportion and Percent, Learn, Teacher Note, “Encourage students to continue identifying the part, whole, and percent in equations to make sense of the problem and to further make connections between equations.”

The materials reviewed for Eureka Math² Grade 7 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

Materials provide intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and make use of structure as they work with the support of the teacher and independently throughout the units. Examples include:

• Module 1, Topic C, Lesson 15: Scale Drawings, Learn, students look for patterns or structures to make generalizations and solve problems as they recognize that corresponding lengths in scale drawings are in a proportional relationship with a scale factor. “Select the figures that are a scale factor of the original figure.” Students are shown a picture of an elephant head and are shown four other elephant head figures. Teachers are prompted to ask, “How are proportional relationships and scale drawings related? How could that help you create scale drawings? How can what you know about proportional relationships help you find unknown lengths in a scale drawing?“

• Module 4, Topic E, Lesson 24: Volume of Other Right Prisms, Learn, Problem 3, students look for and explain the structure within mathematical representations as they find the volume of other right prisms that are composed of right triangular prisms and right rectangular prisms. “Jonas’s work to find the volume of this solid is shown. All measurements are in feet. The volume of the solid is 312 cubic feet. a. Explain how Jonas found the volume of the solid. b. Nora says,’This is still a prism, can we just use $V=Bh$ to find the volume?’ How would you respond to Nora? c. Find the volume of the prism by using Nora’s strategy. How does your answer compare to Jonas’s answer?” Teachers are prompted to ask, “How are rectangular prisms and other types of prisms related? How can that help you find the volume of any prism? How can you use what you know about the base shape of a prism to help you find its volume?”

• Module 6, Topic A, Lesson 5: Multistage Experiments, Learn, Problem 6, students look for patterns or structures to make generalizations and solve problems as they use tree diagrams. “The ancient Egyptian game called Hounds and Jackals is a game of chance using knucklebones. Knucklebones are two-sided throwing sticks with a rounded side and a flat side. A knucklebone is equally likely to land on the rounded side or the flat side. Players move around the board by throwing four knucklebones at the same time to determine their score. The score for each throw is determined by how the knucklebones land. If one knucklebone lands on the flat side and the other three land on the rounded side, the score is 1. If two knucklebones land on the flat side and the other two land on the rounded side, the score is 2. If three knucklebones land on the flat side and the other one lands on the rounded side, the score is 3. If all four knucklebones land on the flat side, the score is 4. If all four knucklebones land on the rounded side, the score is 5. a. Create a tree diagram that represents all the outcomes when throwing knucklebones. b. Use your tree diagram to find the theoretical probability of each score. c. What is the theoretical probability of scoring at least 3? d. At the start of the game, a player’s pieces all start in the home box. The player must score at least 4 to get a piece out of the home box. What is the theoretical probability a player moves a piece out of the home box on the first throw? e. A player wins when they move all their pieces into the opposite home box. The board below shows a player’s pieces that started in the red box and are almost all in the opposite home box. A score of 3 is needed to win. What is the theoretical probability that the player wins on their next throw? Teachers are prompted to ask, “How is this problem similar to the sandwich problem? How can you break the multistage experiment into simpler parts by using a tree diagram? What is another way you can label your tree diagram to help you find the theoretical probability of each score?”

Materials provide intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning to make generalizations and build a deeper understanding of grade-level math concepts. Examples include:

• Module 1, Topic C, Lesson 18: Relating Areas of Scale Drawings, Learn, Problem 14, students evaluate the reasonableness of their answers and thinking as they explore whether their conjecture is also true for rectangles. “Consider the rectangle. a. What is the area of the rectangle? b. Suppose the conjecture you generated for squares also works for rectangles. What should the area of a scale drawing be if it is produced with a scale factor of $\frac{1}{2}$? c. Create the scale drawing by using a scale factor of $\frac{1}{2}$. Find the area of the scale drawing. Does the area you found confirm that your conjecture works for rectangles?” Students are shown a rectangle that is two units by three units. Teachers are prompted to ask, “What patterns did you notice when comparing areas of corresponding regions in the scale drawing and original figure? Will this pattern always work?” 

• Module 2, Topic D, Lesson 19: Rational Numbers as Decimals, Part I, Learn, Problem 4, students notice repeated calculations to understand algorithms as they write rational numbers, given in fraction form, as decimal fractions. “For problems 4–7, if possible, write each number as a decimal fraction. Then write the decimal fraction as an equivalent decimal. $-\frac{6}{10}$” Teachers are prompted to ask, “What patterns did you notice when you found the prime factorizations of different powers of 10? Will this pattern always work?” 

• Module 4, Topic D, Lesson 19: Surface Area of Cylinders, Learn, Problem 2, students determine the exact and approximate surface area of right circular cylinders by creating nets, “Consider the right circular cylinder. a. Sketch a net for the cylinder and label its measurements in terms of $\pi$ as necessary. b. Calculate the surface area of the cylinder.” Students are shown a cylinder with a height of 9 inches and a radius of 2 inches. Teachers are prompted to ask, “What patterns do you notice when you draw the net for a cylinder? What pattern do you notice when you calculate the area of the lateral surface? Will the rectangular region that represents the lateral surface always have an area equal to the cylinder’s height times the circumference of its base?“

The materials reviewed for Eureka Math² Grade 7 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. These are found in a variety of sections within the Implementation Guide, including the Overview, Why, Achievement Descriptors Overview, and Lesson Structure. Examples include:

• Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Overview, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.”

• Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Why, “The Why section gives insight into the decisions made during the writing of the module. This insight helps you understand the underlying structure of the module, flow of the content, and coherence of the different parts of the curriculum.”

• Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Achievement Descriptors, “The Achievement Descriptors: Overview section is a helpful guide that describes what Achievement Descriptors (ADs) are and briefly explains how to use them. It identifies specific ADs for the module, with more guidance provided in the Achievement Descriptors: Proficiency Indicators resource at the end of each Teach book.”

• Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Lesson Structure, “Each lesson is structured in four sections: Fluency, Launch, Learn, and Land. Lessons are designed for one 45-minute instructional period. Fluency provides distributed practice with previously learned material. Launch creates an accessible entry point to the day’s learning through activities that build context and often create productive struggle that leads to a need for the learning that follows. Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Land helps you facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific lessons. This guidance can be found for teachers within boxes called Differentiation, UDL, and Teacher Notes. The Implementation Guide states, “There are six types of instructional guidance that appear in the margins. These notes provide information about facilitation, differentiation, and coherence. Teacher Notes communicate information that helps with implementing the lesson. Teacher Notes may enhance mathematical understanding, explain pedagogical choices, five background information, or help identify common misconceptions. Universal Design for Learning (UDL) suggestions offer strategies and scaffolds that address learner variance. These suggestions promote flexibility with engagement, representation, and action and expression, the three UDL principles described by CAST. These strategies and scaffolds are additional suggestions to complement the curriculum’s overall alignment with the UDL Guidelines.” Examples include: 

• Module 2, Topic A, Lesson 3: Adding Integers Efficiently, Learn, Teacher Note, “Do not give students more than 2–3 minutes to solve the Making a Purchase problem. If students run out of time to solve the problem because they were creating a number line with values up to 54, this is a great opportunity to talk about efficiency. Consider asking a student to share why the time given was not enough to complete the problem. Use this to initiate a class discussion about the need for a more efficient strategy.”

• Module 4, Topic C, Lesson 9: Constructing a Circle, Launch, Differentiation: Challenge, “Consider deepening comprehension by discussing the couple of places where the interactive plots a point but does not form a triangle. Where are these points? What is happening at these points?”

• Module 6, Topic D, Lesson 18: Comparing Population Means, Learn, UDL: Action and Expression, “To support students in monitoring their own progress, consider providing questions that guide self-monitoring and reflection. For example, post the following for students to refer to as they work independently: How is this problem like the jigsaw puzzle problem? How have we used mean absolute deviation in previous problems similar to this?”

The materials reviewed for Eureka Math² Grade 7 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Materials consistently contain adult-level explanations, examples of the more complex grade/ course-level concepts, and concepts beyond the course within Topic Overviews and/or Module Overviews. According to page 7 of the Grade 6-9 Implementation Guide, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.” Page 9 outlines the purpose of the Topic Overview, “Each topic begins with a Topic Overview that is a summary of the development of learning in that topic. It typically includes information about how learning connects to previous or upcoming content.” Examples include:

• Module 2: Operations with Rational Numbers, Module Overview, Why, “I notice that this module does not encourage the use of manipulatives. Why not? The use of manipulatives can support student engagement and provide differentiation and equity. Manipulatives can promote student thinking and aid in communicating about the mathematics being learned. Manipulatives often bridge learning from the conceptual stage to the pictorial or abstract stages of learning. However, students may lose the chance to deepen their understanding of concepts if manipulatives are used in isolation of mathematical connection. Colored chips are a widely used manipulative to engage students in understanding the ‘rules’ for integer arithmetic. Although the chips can be helpful in representing opposites and communicating the associative and commutative properties of addition, some difficulties may occur when students employ these manipulatives: 1. Students are tasked with assigning a number to a set of colored chips and remembering which color represents a positive number and which color represents a negative number. 2. Use of colored chips connects easily to addition, but some learners struggle to conceptualize when subtracting a negative number or multiplying when the first factor is negative. Some interpretations of division cannot be modeled by using the chips. 3. Chips do not represent non-integer rational numbers and cannot be used to model arithmetic with rational numbers. The number line is a coherent model in Eureka Math², and it is a representation that builds understanding of sums and differences of rational numbers. Addition is represented by concatenation of lengths on the number line, beginning in grade 2. This extends through grade 4, where students add fractions with the same unit on the number line, and it further extends to adding, subtracting, multiplying, and dividing fractions in grade 6. Students continue building fluency with addition and subtraction of positive rational numbers on the number line in grade 7, extending this understanding to addition and subtraction of negative rational numbers. In grade 8 and in high school, students work with vectors, and their work with concatenation allows them to understand this as one-dimensional vector addition. For this reason, we have chosen to use the number line instead of colored chips.”

• Module 3: Expressions, Equations, and Inequalities, Module Overview, Why, “I notice that this module includes standards for geometry. Why are these standards addressed in this module? Students understand and apply angle relationships to determine unknown angle measures. These relationships necessitate equivalence. To determine whether angles are complementary, students understand that the two angle measures must sum to 90°. To determine whether angles are supplementary, students understand that the two angle measures must sum to 180°. A natural approach to determine unknown angle measures in these and other cases is to solve for the unknown by using an equation. Determining unknown angle measures drives the need to solve equations. Students use equations to show why angles are equal in measure.”

• Module 5: Percent and Applications of Percent, Module Overview, Why, “Students link prior knowledge of part–whole relationships to the learning of percents. Using a proportion is helpful in determining unknown values in proportional relationships. In this module, students connect percents to proportional relationships and represent them by using equations in the forms $y=k⁢x$ and $ab=cd$. A common error that students make is placing numbers in incorrect locations in the proportion because they lack understanding of what each value represents. Using the procedure of cross multiplication to solve for an unknown value in a proportion also compromises that understanding. This module’s Math Past examines methods of solving proportions, starting with the early Egyptians and continuing through modern times. In this Math Past, cross multiplication is discussed, along with why it works.”

The materials reviewed for Eureka Math² Grade 7 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

The Achievement Descriptors, found in the Overview section, identify, describe, and explain how to use the standards. The lesson overview includes content standards addressed in the lesson. Additionally, a Proficiency Indicators resource at the end of each Teach book, helps assess student proficiency. Correlation information and explanations are present for the mathematics standards addressed throughout the grade level in the context of the series. Examples include:

• Module 1: Ratios and Proportional Relationships, Achievement Descriptors and Standards, “7.Mod1.AD1 Compute unit rates associated with ratios of fractions given within contexts. (7.RP.A.1)”

• Module 3, Topic B, Lesson 7: Angle Relationships and Unknown Angle Measures, Achievement Descriptors and Standards, “7.Mod3.AD12 Write and solve equations to find unknown angle measures by using known facts about angle relationships. (7.G.B.5)”

• Module 4: Percents and Application of Percent, “In module 5, connection to the learning from previous modules drives the need for students to use percents. Students realize the equation $\frac{a}{b}=\frac{c}{d}$ represents proportional relationships and use proportions and rate language to examine percent as a rate per 100. They identify part, whole, and percent and use proportional reasoning to solve percent problems in real-world contexts, understanding that the unknown could either be part of 100 or more or less than 100%.”

The materials reviewed for Eureka Math² Grade 7 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

Materials explain the instructional approaches of the program. According to the Grades 6-9 Implementation Guide, “Eureka Math² features a set of instructional routines that optimize equity by increasing access, engagement, confidence, and students’ sense of belonging. The following is true about Eureka Math² instructional routines: Each routine presents a set of teachable steps so students can develop as much ownership over the routine as the teacher. The routines are flexible and may be used in additional math lessons or in other subject areas. Each routine aligns to the Stanford Language Design Principles (see Works Cited): support sense-making, optimize output, cultivate conversation, maximize linguistic and cognitive meta-awareness.” Examples of instructional routines include:

• Instructional Routine: Always Sometimes Never, students make justifications and support their claims with examples and nonexamples. Implementation Guide states, “Present a mathematical statement to students. This statement may hold true in some, all, or no contexts, but the goal of the discussion is to invite students to explore mathematical conditions that affect the truth of the statement. Give students an appropriate amount of silent think time to evaluate whether the statement is always, sometimes, or never true. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Then facilitate a class discussion. Invite students to share their thinking with the whole group. Encourage them to provide examples and nonexamples to support their claim. Encourage use of the Talking Tool. Conclude by bringing the class to consensus that the statement is [always/sometimes/never] true [because …].”

• Instructional Routine: Critique a Flawed Response, students communicate with one another to critique others’ work, correct errors, and clarify meanings. Implementation Guide states, “Present a prompt that has a partial or broken argument, incomplete or incorrect explanation, common calculation error, or flawed strategy. The work presented may either be authentic student work or fabricated work. Give students an appropriate amount of time to identify the error or ambiguity. Invite students to share their thinking with the class. Then provide an appropriate amount of time for students to solve the problem based on their own understanding. Circulate and identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about the prompt given. Then facilitate a class discussion by inviting students to share their solutions with the whole group. Encourage use of the Talking Tool. Lead the class to a consensus about how best to correct the flawed response.”

• Instructional Routine: Stronger, Clearer Each Time, students revise and refine their written responses. Implementation Guide states, “Present a problem, a claim, or a solution path and prompt students to write an explanation or justification for their solution path, response to the claim, or argument for or against the solution path. Give students an appropriate amount of time to work independently. Then pair students and have them exchange their written explanations. Provide time for students to read silently. Invite pairs to ask clarifying questions and to critique one another’s response. Circulate and listen as students discuss. Ask targeted questions to advance their thinking. Direct students to give specific verbal feedback about what is or is not convincing about their partner’s argument. Finally, invite students to revise their work based on their partner’s feedback. Encourage them to use evidence to improve the justification for their argument.”

Materials include and reference research-based strategies. The Grades 6-9 Implementation Guide states, “In Eureka Math² we’ve put into practice the latest research on supporting multilanguage learners, leveraging Universal Design for Learning principles, and promoting social-emotional learning. The instructional design, instructional routines, and lesson-specific strategies support teachers as they address learner variance and support students with understanding, speaking, and writing English in mathematical contexts. A robust knowledge base underpins the structure and content framework of Eureka Math². A listing of the key research appears in the Works Cited for each module.”

The materials reviewed for Eureka Math² Grade 7 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Each module and individual lesson contains a materials list for the teacher and student. The lesson preparation identifies materials teachers need to create or assemble in advance. Examples include:

• Module 1, Topic B, Lesson 11: Constant Rates, Materials, “Teacher: None. Students: None. Lesson Preparation: Review the Math Past resource to support the delivery of Launch.”

• Module 2: Operations with Rational Numbers, Module Overview, Materials, “The following materials are needed to implement this module. The suggested quantities are based on a class of 24 students and one teacher. Chart paper, tablet (1), Personal whiteboards (24), Colored pencils (24), Personal whiteboard erasers (24), Dry-erase markers (24), Projection device (1), Highlighters (24), Scientific calculators (24), Eureka Math²™ Integer Cards, set of 12 decks, Sticky notes, pads (8), Learn books (24), Student computers or devices (12), Markers (9), Teach book (1), Paper, blank sheets (15), Teacher computer or device (1), Pencils (24).”

• Module 5, Topic A, Lesson 2: Racing for Percents, Materials, “Teacher: None. Students: Computers or devices (1 per student pair). Lesson Preparation: None.”

The materials reviewed for Eureka Math² Grade 7 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

Materials provide strategies, supports, and resources for students in special populations to support their regular and active participation in grade-level mathematics. According to the Implementation Guide, “There are six types of instructional guidance that appear in the margins. These notes provide information about facilitation, differentiation, and coherence.” Additionally, “Universal Design for Learning (UDL) is a framework based on current research from cognitive neuroscience that recognizes learner variance as the norm rather than the exception. The guiding principles of the UDL framework are based on the three primary networks of the brain. Although the concept of UDL has roots in special education, UDL is for all students. When instruction is designed to meet the needs of the widest range of learners, all students benefit. Eureka Math2 lessons are designed with these principles in mind.” Examples include:

• Module 1, Topic C, Lesson 16: Using a Scale Factor, Learn, UDL: Representation, “Have students color-code the sides with the colors of the measurements written in the table. This will help them determine which measurements correspond when they are creating the scale drawing.”

• Module 3, Topic C, Lesson 17: Using Equations to Solve Problems, Learn, Differentiation Support, “If students need more support with solving the equation, suggest that students create a double number line. This grade 6 strategy could help them visualize and make sense of the problem.”

• Module 4, Topic D, Lesson 20: Surface Area of Right Pyramids, Learn, Language Support, “Consider having students highlight the line segments that are 5 cm in one color and write Slant Height in that same color. Then, in a different color, have students highlight the height of 3 cm and in that color write Height of Pyramid.”

The materials reviewed for Eureka Math² Grade 7 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity. 

Materials do not require advanced students to do more assignments than their classmates. Instead, students have opportunities to think differently about learning with alternative questioning, or extension activities. Specific recommendations are routinely highlighted as Teacher Notes within parts of each lesson, as noted in the following examples: 

• Module 1, Topic C, Lesson 20: Creating Multiple Scale Drawings, Differentiation: Challenge, “Challenge groups of students to predict whether a second drawing will be the same if they complete their scale drawings in reverse order. Have students verify their predictions by going back to the original figure their group was assigned and completing the scale drawings again, this time using the scale factors in reverse order. Students should determine that when the scale drawings are completed in reverse order, the final scale drawing still looks the same.”

• Module 3, Topic B, Lesson 9: Solving Equations to Determine Unknown Angle Measures,  Launch, Differentiation: Challenge, “If time permits, challenge groups to solve the equations and determine each angle measure. Print the Challenge answer key and allow students to check their answers after they determine all the angle measures.”

• Module 4, Topic D, Lesson 21: Surface Area of Other Solids, Learn, Differentiation: Challenge, “Challenge students by having them sketch a composite figure that has a given surface area, such as 100 square units.”

The materials reviewed for Eureka Math² Grade 7 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Support for active participation in grade-level mathematics is consistently included within a Language Support Box embedded within parts of lessons. The Implementation Guide explains supports for language learners, “Multilingual learners, or learners who speak a language other than English at home, require specific learning supports for gaining proficiency with the English needed to access the mathematics. Research suggests that best practices for these learners include opportunities and supports for student discourse and for using precise terminology. In addition to precise domain-specific terminology, high-impact academic terminology that supports learners across learning domains is explicitly introduced and used repeatedly in various contexts to build familiarity and fluency across the grade levels. Eureka Math² is designed to promote student discourse through classroom discussions, partner or group talk, and rich questions in every lesson.” 

Examples include:

• Module 1, Topic C, Lesson 16: Using a Scale Factor, Learn, Language Support, “As the class discussion unfolds, consider providing students with sentence frames and key terminology as a scaffold for peer conversations. For example, post the key terms scale drawing, scale factor, enlargement, and reduction on the board along with a sentence frame such as ‘The scale drawing is a _____ because _____.’”

• Module 2, Topic B, Lesson 8: Subtracting Integers Part 1, Launch, Language Support, “In this lesson and throughout topic B, model correct and consistent use of the subtraction terms minuend, subtrahend, and difference. To support student use of the subtraction terminology, consider displaying a model equation with each part labeled and color-coded with the correct term.” An example is shown, “$8-5=3$, Minuend - Subtrahend = Difference.”

• Module 3, Topic D, Lesson 19: Using Equations to Solve Inequalities, Learn, Language Support, provides teachers with guidance to understand a key term for engaging in content. “Students may need support with understanding the word restrictions. Pose the following questions to help students make connections between how the word is used in real life and how the word might be used in a math context: How have you heard the word restrictions used in daily life? What do you think the word restrictions means here? Highlight student responses that identify the restrictions featured in the sample student response.”

The materials reviewed for Eureka Math² Grade 7 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Manipulatives provide accurate representations of mathematical objects. Examples Include:

• Module 2, Topic B, Lesson 10: Subtracting Rational Numbers Part 1, Launch, UDL: Representation, “It also may be helpful to encourage students to draw a number line numbered −5 to 5, scaled by 0.5. Have them use this number line to model the balance of the account after one charge. Then have them use the number line to model the balance of the account after the second charge.”

• Module 4, Topic A, Lesson 3: Side Lengths of a Triangle, Launch, students use pieces of spaghetti to investigate if a triangle can be formed with given side lengths. “Break apart the pieces of spaghetti to form the specific lengths shown. Then manipulate the pieces of spaghetti and determine whether a triangle can be formed with those side lengths. Record your observations in the table.” 

• Module 6, Topic A, Lesson 2: Empirical Probability, Learn, Problem 2, students are given a bag containing 20 cubes of varying colors to identify empirical probability. “You and your partner have 20 cubes, which were randomly pulled from a large bucket of cubes. These 20 cubes represent the outcomes of 20 trials where a cube was randomly pulled from the large bucket and then replaced.” Using a data table, students record how many cubes of each color are drawn. After conducting multiple trials of this game of chance and collecting data, students calculate the empirical probability for an event.