6th to 8th Grade - Gateway 2

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

The Learn portion of the lesson presents new learning through instructional segments that develop conceptual understanding of key mathematical concepts. Teachers use prompts provided in the margin notes to facilitate discussion and develop students’ understanding of the concepts. Students independently demonstrate conceptual understanding through Exit Tickets, which assess the lesson’s learning, and Practice, which provides additional problems aligned to the lesson’s learning objectives. In each module, under Module Resources, the Instructional Design and Curriculum document states, “The Eureka Math^2@ lessons strategically balance conceptual understanding, procedural fluency, and real-world application. Conceptual Understanding: Daily lessons include concrete–pictorial–symbolic representations and simple-to-complex progressions to develop understanding, as well as opportunities to identify math relationships and patterns and engage in meaningful discourse to answer Why does this work?”

Examples include:

• Grade 6, Module 3, Topic B, Lesson 9, Learn, Problem 1, students develop conceptual understanding as they reason about the order and absolute value of rational numbers and connect these concepts to quantities in a real-world context. Students use a video and their own sketches to represent elevations above and below sea level, interpret inequalities in context, and connect absolute value to distance and magnitude. In the teacher notes, the material provides suggestions for facilitating discussion around absolute value that helps students interpret the absolute value of a rational number as the magnitude of a positive or negative quantity in a real-world situation. The teacher notes say, “What does the absolute value of 23 represent in this situation? We can think of both of these absolute values as representing either the distance between two points, such as the net and sea level, or the length of the rope between two points. Can you determine how much more rope is below sea level than above? Explain. How long is the rope? How do you know? Why is the length of the rope not −61 feet?” The student's Learn book states, “For parts (a)–(e), consider the situation from the video of the fishing boat and the net. a. Use a vertical number line to sketch the situation. Include sea level, the boat, the top of the rope, and the net in your sketch. b. What is the elevation of the top of the rope? c. What is the elevation of the net? d. Interpret the meaning of −38<23 in this situation. e. Interpret the meaning of \lvert-38\rvert>\lvert23\rvert in this situation. What does the absolute value of −38 represent in this situation?” (6.NS.7)

• Grade 7, Module 2, Topic A, Lesson 5, Learn, Problem 3, students develop conceptual understanding as they identify additive inverses, represent them on a number line, write number sentences that show their relationship, and describe what they notice about the locations of additive inverses on the number line. The teacher notes say, “Display the number line again, this time with the point 3.4 plotted. Direct students’ attention to the Additive Inverse problem. Have students plot 3.4 on their number lines for problem 3.” The student Learn book states, “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?” (7.NS.1)

• Grade 8, Module 4, Topic C, Lesson 13, Learn, Problem 5, students demonstrate conceptual understanding as they connect ordered pairs that satisfy a linear equation to points on its graph and reason about why using 0 for either variable produces the x- and y-intercept points. Students use the intercept points to graph a linear equation in standard form. The student Learn book states, “Use the table to find two ordered pairs that satisfy the equation 4x-3y=12. Then use the ordered pairs to graph the equation in the coordinate plane.” The teacher notes say, “What x- or y-value did you use to find an ordered pair that satisfies the equation? What ordered pair resulted from that value? Why did you use that value? Why would we want to use 0 as a value for x or y to find an ordered pair that satisfies the equation 4x-3y=12? What is special about the locations of these points? These points are called intercept points because the line meets the axes at these points. An x-intercept point of a graph has the coordinates (a,0) and is a point where the graph intersects the x-axis. A y-intercept point of a graph has the coordinates (0,b) and is a point where the graph intersects the y-axis.” (8.EE.6)

The materials reviewed for Eureka Math² Grades 6 through Grade 8 meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

The Learn portion of the lesson presents new learning through instructional segments that develop procedural skill and fluency with key mathematical concepts. Students independently demonstrate procedural skill and fluency through Exit Tickets, which assess the lesson’s learning, and Practice, which provides additional problems aligned to the lesson’s learning objectives. Each lesson also begins with Fluency problems that provide distributed practice with previously learned material and prepare students for new learning. 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.” In each module, under Module Resources, the Instructional Design and Curriculum document states, “The Eureka Math^2@ lessons strategically balance conceptual understanding, procedural fluency, and real-world application. Procedural Fluency: Distributed and interleaved practice through fluency and practice problems provides ample daily opportunities for students to grow in mathematical fluency.”

Examples include:

• Grade 6, Module 2, Topic E, Lesson 20, Learn, students develop procedural skill and fluency with dividing multi-digit decimals as they create and solve division word problems and apply division procedures to problems with different dividends, divisors, and quotients. The teaching notes state, “Use the Numbered Heads routine. Organize students into an even number of groups with 2–3 students per group. Assign each student a number, 1 through 3. Display the following constraints. A. 3-digit or 4-digit dividend, 2-digit divisor with a whole-number quotient. B. 3-digit or 4-digit dividend, 1-digit divisor with a decimal quotient. Assign each group one of the constraints. Give students 5–7 minutes to create a division word problem that meets that constraint and to solve their problem. Have students record their word problem and solution in their books for problem 11. Then give each group an index card. Write the letter that you were assigned on the front of the index card. Write the word problem your group created on the back of the index card. Put your completed card in the correct stack: A or B. Once groups have turned in a card, have each group choose a card from the opposite stack and solve the problem on the card. Have them record the problem and solution in their books for problem 12. Remind students that any one of them could be the spokesperson for the group, so they should be prepared to share.” Problem 5 states, “Kayla makes 175 ounces of lemonade for a lemonade sale. She wants to fill each cup with 8 ounces of lemonade. What is the greatest number of 8-ounce cups Kayla can fill?” (6.NS.3) 

• Grade 7, Module 3, Topic B, Lesson 7, Learn, Problem 3, students demonstrate procedural skill and fluency as they write and solve equations to determine unknown angle measures and check their solutions by substituting values into the equations. Problem 3 states, “In the diagram, \angle DAB is a straight angle. a. Describe the relationship between \angle DAC and \angle CAB. b. Write an equation for the angle relationship and solve for x.” The straight line shown has angle measures x and 23° The teaching notes state, “Have students return to the whole group. Model how to check the solution to an equation by completing part (c) together. Emphasize that substituting the correct solution for the variable creates a true number sentence. If students end up with a false number sentence, then they need to determine whether they made an error when substituting the value or when solving the equation. Tell students that they are expected to check their solutions to equations from now on.” (7.EE.4a)

• Grade 8, Module 4, Topic A, Lesson 2, Learn, students develop procedural skill and fluency as they solve linear equations with variables on both sides and check their solutions for accuracy. Problems 3- 5 state, “For problems 3–5, solve the equation. Check your solution. 3. 27−3a=8a+5, 4. 9(c−2)=3c−28, 5. 8 −4m=2(m−1)+10”. The teaching notes state, “Direct students’ attention to the equations in problems 3–5 and ask the following questions. How are the equations in problems 3–5 similar to the equation in problem 2? How are some of these equations different from the equation in problem 2? For problems 3–5, solve the equation. Check your solution. 3. 27-3a=8a+5, 4. 9(c-2)=3c-28, 5. 8-4m=2(m-1)+10. Confirm answers as a class. If there are any discrepancies, consider having pairs share their solution steps. Emphasize that checking solutions is an important step to ensure accuracy. If time allows, consider asking students to identify instances in their work where they applied the distributive property and, in more subtle ways, the commutative and associative properties.” (8.EE.7)

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

The Learn portion of the lesson presents new learning through instructional segments that develop students’ application of mathematical concepts. Students independently demonstrate routine application of mathematics through Exit Tickets, which assess the lesson’s learning, and Practice, which provides problems aligned to the lesson’s learning objectives. In each module, under Module Resources, the Instructional Design and Curriculum document states, “The Eureka Math^2@ lessons strategically balance conceptual understanding, procedural fluency, and real-world application. Application: Real-world problems and collaborative tasks consolidate student thinking through why and how questions, inviting students to draw conclusions and make deep connections among math ideas.”

Examples include:

• Grade 6, Module 1, Topic B, Lesson 6, Learn, students apply their understanding of equivalent ratios in a routine problem as they use a double number line to represent the relationship between quantities and determine unknown values in a real-world context. Problem 5 states, “Leo buys fabric at a craft store. Every 2 yards of fabric costs $7.00. a. Create a double number line to show the relationship between possible amounts of fabric in yards and the total cost in dollars. b. If the total cost of the fabric is $21.00, how many yards of fabric does Leo buy? c. If Leo buys 1 yard of fabric, what is the total cost of the fabric?” (6.RP.3)

• Grade 7, Module 5, Topic A, Lesson 2, Practice, Problem 8, students apply their understanding of proportional relationships in a routine problem as they determine an unknown quantity and express one quantity as a percent of another in a real-world context. Problem 8 states, “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?” (7.RP.3)

• Grade 8, Module 6, Topic B, Lesson 9, Practice, students apply their understanding of qualitative features of functions in a non-routine problem as they create a real-world relationship between two quantities and construct a graph that represents the relationship. Problem 5 states, “Create your own graph story. a. Write a story about the relationship between two quantities. Any quantities can be used, such as distance and time, money and hours, or age and growth. Include keywords in your story, such as increase and decrease, to describe the relationship. b. Label each axis with the quantity of your choice. Then sketch a graph of the function that represents the relationship described in your story.” (8.F.5)

The materials reviewed for Eureka Math² Grade 6 through Grade 8 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 as reflected by the standards.

Multiple aspects of rigor are engaged simultaneously across the materials to develop students' mathematical understanding of individual lessons. Each lesson within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way. 

Examples include:

• Grade 6, Module 1, Topic D, Lesson 16, Practice, students demonstrate conceptual understanding and application as they interpret the meaning of a rate in a real-world context and connect the rate to a ratio table and double number line. Problem 1 states, “Karl Benz drove the first car in Mannheim, Germany, in 1886. The car traveled at a top speed of 10 miles per hour. Assume the car kept that constant speed. a. Interpret the meaning of the car’s speed. b. Use your answer from part (a) to complete the ratio table. c. Create a double number line to represent the situation.” (6.RP.3)

• Grade 7, Module 2, Topic E, Lesson 26, Practice, students demonstrate procedural skill and application as they calculate the mean of a set of data and interpret the value in a real-world context. Problem 3 states, “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?” (7.EE.3)

• Grade 8, Module 6, Topic B, Lesson 6, Exit Ticket, students demonstrate conceptual understanding, procedural skill, and application as they determine whether functions represented by tables are linear, calculate rates of change, and interpret the rates of change in a real-world context. The Exit Ticket states, “The tables show values of two functions. The functions represent the number of downloads for two different songs for a given number of days after their release. a. Is the function representing downloads for song B a linear function? Explain. b. If the function representing downloads for song B is a linear function, what is the rate of change? What does the rate of change mean in context? c. Is the function representing downloads for song C a linear function? Explain d. If the function representing downloads for song C is a linear function, what is the rate of change? What does the rate of change mean in context?” (8.F.4)

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP1 throughout the year. Each lesson identifies one or more Standards for Mathematical Practice and names a focus MP for teachers at the beginning of the lesson. Margin notes titled “Promoting the Standards for Mathematical Practice” identify places within lessons where students engage in or build experience with the focus practice. The notes provide teachers with lesson-specific information, ideas, prompts, and questions to promote students’ engagement with the focus MP. In addition, the Implementation Guide explains that while lessons may provide opportunities for students to engage with multiple mathematical practices, the guidance identifies a focus MP within each lesson.

Across the grades, students engage in open-ended tasks that support key components of MP1, including sense-making, strategy development, and perseverance. These tasks prompt students to make sense of mathematical situations, develop and revise strategies, and persist through challenges. They are encouraged to analyze problems by engaging with the information and questions presented, use strategies that make sense to them, monitor and evaluate their progress, determine whether their answers are reasonable, reflect on and revise their approaches, and increasingly devise strategies independently. 

Examples include:

• Grade 6, Module 1, Topic D, Lesson 20, Teach Book, Learn, Station 2, students make sense of multi-step rate problems and persevere in solving them as they determine the speeds of different families in miles per hour from distances traveled over portions of an hour. Students identify steps to begin solving, simplify the problem, create representations to support their reasoning, monitor their progress, and determine whether their calculated values make sense. Promoting Mathematical Practice margin note states, “As students solve multi-step rate problems by finding entry points, monitoring their own progress, and questioning whether the values they calculate make sense, they are making sense of problems and persevering in solving them. Ask the following questions: What steps can you take to start solving the problem? How can you simplify the problem? Does your answer make sense? Why?” Practice, Problem 6 states, “Store A sells a \frac{1}{2}-gallon size of juice for $2.65. This store sells the same juice in a 1-gallon size for $5.50. a. Which size of juice at store A, the \frac{1}{2}-gallon size or the 1-gallon size, has the better price per gallon? Explain. b. Store B sells the same juice in a 1-liter bottle for $1.50. Does the juice from store B have a better price per gallon than the juice from store A? Explain. Use 1 gallon ≈ 3.79 liters.”

• Grade 7, Module 4, Topic D, Lesson 16, Teach Book, Learn, Problem 4, students make sense of an area problem and persevere in solving it as they use different strategies to determine the area of a figure. Students determine the information needed to solve the problem, consider strategies for finding an entry point, use both decomposition and composition to calculate the area, compare the strategies, and monitor whether a selected strategy is effective or whether they need to change their approach. Promoting Mathematical Practice margin note states, “Students make sense of problems and persevere in solving them as they use different strategies to calculate area. They find entry points into problems by selecting a strategy, and they continually monitor the effectiveness of their strategy and change course if necessary. Ask the following questions: 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?” Practice, Problem 2 states, “Consider the figure. a. Find the area of the figure by using any strategy. b. Identify the strategy you used and explain why you picked that strategy.”

• Grade 8, Module 2, Topic C, Lesson 16, Teach Book, Learn, Problem 2, students make sense of problems and persevere in solving them as they consider the possible angle relationships in a diagram and develop a pathway for determining an unknown value. Students identify relevant angle relationships, develop a plan for approaching the problem, use those relationships to solve, and consider whether the value they find makes sense. Promoting Mathematical Practice margin note states, “When students consider all the possible angle relationships in a diagram and develop a pathway to solve for the indicated variable, they are making sense of problems and persevering in solving them. Ask the following questions: What can you figure out about the angle relationships by looking at the diagram? What is your plan for approaching each problem? Does the value you found for x make sense?” Practice, Problem 9 states, “Ava solves this problem correctly. Analyze Ava’s work. Then find the measure of \angle IFO by using a different strategy. Explain your work.”

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP2 throughout the year. Each lesson identifies one or more Standards for Mathematical Practice and names a focus MP for teachers at the beginning of the lesson. Margin notes titled “Promoting the Standards for Mathematical Practice” identify places within lessons where students engage in or build experience with the focus practice. The notes provide teachers with lesson-specific information, ideas, prompts, and questions to promote students’ engagement with the focus MP. In addition, the Implementation Guide explains that while lessons may provide opportunities for students to engage with multiple mathematical practices, the guidance identifies a focus MP within each lesson.

Across the grades, students participate in tasks that support key components of MP2, including reasoning with quantities, representing situations symbolically, and interpreting the meaning of numbers and symbols in context. These tasks encourage students to consider the units involved in a problem, analyze the relationships between quantities, and connect real-world scenarios to mathematical representations. Teachers are guided to support this development by modeling the use of mathematical notation, asking clarifying and probing questions, and facilitating conversations that help students make connections between multiple representations. 

Examples include:

• Grade 6, Module 4, Topic B, Lesson 10, Teach Book, Learn, students reason abstractly and quantitatively as they represent, write, and interpret algebraic expressions that describe relationships between quantities in real-world situations. Students identify the relationship between the number of cups of lemonade and seltzer water in a punch recipe, decontextualize the situation by representing the relationship symbolically with an expression, interpret the coefficient 1\frac{1}{2} in relation to the quantities, and consider whether their expression makes sense in the context. Promoting Mathematical Practice margin note states, “Students reason quantitatively and abstractly as they write and interpret algebraic expressions that represent real-world situations. Ask the following questions: How does this expression represent this situation? What does this coefficient mean in this situation? Does the expression you wrote make sense mathematically?” Problem 1 states, “The number of cups of seltzer water in a punch recipe is 1\frac{1}{2} times the number of cups of lemonade. Complete the table.” Students use the table to represent the corresponding quantities of lemonade and seltzer water and connect the numerical relationship to an algebraic expression. Practice, Problem 4 states, “A can of mixed nuts has 4.5 times as many grams of peanuts as grams of walnuts. a. What is the ratio of the number of grams of peanuts to the number of grams of walnuts? b. Complete the table. c. Let w represent the number of grams of walnuts. Write an expression that represents the number of grams of peanuts for w grams of walnuts.”

• Grade 7, Module 1, Topic A, Lesson 5, Teach Book, Learn, Discussion, students reason abstractly and quantitatively as they connect graphs, equations, ratios, and rates to the quantities and relationships in real-world situations. Students interpret the constant of proportionality within different representations, connect the point (1,k) on a graph to the unit rate and the quantities in the context, and use k symbolically to represent the constant of proportionality in an equation. Students compare representations to determine how each conveys the same proportional relationship and explain the meaning of the mathematical representations in context. Promoting Mathematical Practice margin note states, “Students reason abstractly and quantitatively when they make sense of graphs, equations, ratios, and rates that represent real-world contexts, such as exchanging currency or constant price per unit. Ask the following questions: 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?” During Discussion, students analyze two proportional relationships and consider, “What do you notice about where the constant of proportionality is determined on both graphs?” and “Can the constant of proportionality always be determined from the point (1,k) on the graph of a proportional relationship? How do you know?” Students also consider which representation was most helpful for writing the equation and compare what proportional and nonproportional relationships have in common. Practice, Problem 7 states, “The table shows the widths and heights of different logos for a local school. a. Graph the pairs of values from the table. b. Explain how you can determine that the width of the logo in inches is proportional to the height of the logo in inches. c. Explain where you can find the constant of proportionality in both the table and the graph. What does the constant of proportionality represent in this situation? d. Write an equation to describe the relationship between the width w of the logo in inches and the height h of the logo in inches.”

• Grade 8, Module 3, Topic D, Lesson 15, Teach Book, Learn, students reason abstractly and quantitatively as they decontextualize real-world situations by drawing diagrams to represent quantities and relationships and use facts about similar triangles to determine unknown measurements. Students identify relevant quantities from each context, represent the relationships mathematically in a diagram, use proportional relationships within similar triangles to solve, and recontextualize their solutions by expressing the measurements with the appropriate level of precision for the situation. Promoting Mathematical Practice margin note states, “When students decontextualize problems by drawing a diagram to represent real-world scenarios and use facts about similar triangles to solve these problems, they are reasoning quantitatively and abstractly.” Learn, Problems 3 and 4 direct students to “use the space provided to take notes and draw a diagram from the video. Ask the following questions: What does this problem ask you to do? How do similar triangles represent the problem posed in this video?” Problem 3 asks, “What is the height of the building to the nearest foot?” and Problem 4 asks, “What is the length of the flagpole to the nearest tenth of a foot?” Practice, Problem 6 states, “Create a context for the diagram. Provide a question based on the unknown length in the diagram, and then answer that question.”

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP3 throughout the year. Each lesson identifies one or more Standards for Mathematical Practice and names a focus MP for teachers at the beginning of the lesson. Margin notes titled “Promoting the Standards for Mathematical Practice” identify places within lessons where students engage in or build experience with the focus practice. The notes provide teachers with lesson-specific information, ideas, prompts, and questions to promote students’ engagement with the focus MP. In addition, the Implementation Guide explains that while lessons may provide opportunities for students to engage with multiple mathematical practices, the guidance identifies a focus MP within each lesson.

Across the grades, students participate in tasks that support key components of MP3. These include constructing mathematical arguments, analyzing errors in sample student work, and explaining or justifying their thinking orally or in writing using concrete models, drawings, numbers, or actions. Students are encouraged to listen to or read the arguments of others, evaluate their reasoning, and ask clarifying questions to strengthen or improve the argument. They also have opportunities to make and test conjectures as they solve problems. Teachers are guided to support this development by providing opportunities for students to engage in mathematical discourse, set clear expectations for explanation and justification, and compare different strategies or solutions. Teachers are prompted to ask clarifying and probing questions, support students in presenting their solutions as arguments, and facilitate discussions that help students reflect on and refine their reasoning.

Examples include:

• Grade 6, Module 1, Topic E, Lesson 24, Teach Book, Learn, students construct viable arguments and critique the reasoning of others as they analyze, compare, question, and justify different methods for calculating a percent of a number. Problem 2 states, “Lisa, Julie, and Toby each calculated 16% of 40 but in different ways. Use their work to answer parts (a)–(g).” Students explain how Lisa and Julie calculated 16% of 40, apply Lisa’s method to a new calculation, determine another percent for which her method can be used, and compare Lisa’s and Toby’s methods by explaining how their reasoning is similar and different. Promoting Mathematical Practice margin note states, “When students analyze methods for calculating percents and justify conclusions, they are constructing viable arguments and critiquing the reasoning of others. Ask the following questions: What parts of Toby’s method do you question? When do you think Lisa’s method works? Can you find a situation other than 10% of 40 where Noah’s method in problem 3 works?” Practice, Problem 14 states, “Which amount is greater, 15% of 20 or 20% of 15? Explain.”

• Grade 7, Module 5, Topic B, Lesson 8, Teach Book, Learn, students construct viable arguments and critique the reasoning of others as they justify, question, and evaluate strategies for calculating percent-based fees. Problem 1 states, “Henry wants to transfer $40 to Kabir through one of three apps on his phone.” Students determine the percent-based fees charged by three different apps, determine the unknown percent charged by one app, and compare the fees to determine which app will charge Henry the least. Students justify their strategies to a partner, listen to their partner’s reasoning, question parts of the justification, and consider how the argument could be made more mathematically accurate. Promoting Mathematical Practice margin note states, “When students justify each step of solving percent problems and listen to their peers’ explanations, they are constructing viable arguments and critiquing the reasoning of others. Ask the following questions: 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.” Practice, Problem 4 states, “Maya wants to transfer $15 to Noor through one of two apps on her phone. App A charges a fee of 2.5% of the transferred amount, while app B charges a fee of $0.20 plus 1.5% of the transferred amount. a. Determine the fee app A charges to transfer 15. b. Determine the fee app B charges to transfer $15. c. Which app will charge Maya less in fees?”

• Grade 8, Module 4, Topic D, Lesson 18, Teach Book, Learn, students construct viable arguments and critique the reasoning of others as they analyze, question, and identify errors in different methods for using slope triangles to graph a falling line. Problem 6 states, “A line with slope -\frac{4}{3} passes through the point (−6,1). Four students use the slope and the given point to locate a second point and graph the line. Their work is shown. Three of the students made an error. a. Who graphed the line correctly? b. Describe the error made by each of the other three students.” Students evaluate four students’ representations, determine which reasoning correctly represents the given negative slope, and critique the other three students’ reasoning by identifying and explaining the mathematical error in each approach. Promoting Mathematical Practice margin note states, “When students analyze other students’ use of slope triangles to graph a line and identify the underlying misconceptions, they are critiquing the reasoning of others. Ask the following questions: What parts of this student’s work do you question? Why? How would you change this student’s work to make it more accurate?” Practice, Problem 9 states, “The slope of a line is -\frac{3}{2}. Which statements about the line are true? Choose all that apply.” Students evaluate multiple mathematical claims about the direction of the line and possible slope triangles, determining which claims are consistent with a slope of -\frac{3}{2}.

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP4: Model with Mathematics, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP4 throughout the year. Each lesson identifies one or more Standards for Mathematical Practice and names a focus MP for teachers at the beginning of the lesson. Margin notes titled “Promoting the Standards for Mathematical Practice” identify places within lessons where students engage in or build experience with the focus practice. The notes provide teachers with lesson-specific information, ideas, prompts, and questions to promote students’ engagement with the focus MP. In addition, the Implementation Guide explains that while lessons may provide opportunities for students to engage with multiple mathematical practices, the guidance identifies a focus MP within each lesson.

Across the grades, students participate in tasks that support the intentional development of MP4, including putting problems or situations in their own words and identifying important information, using the math they know to solve problems and everyday situations, modeling the situation with appropriate representations and strategies, describing how their model relates to the problem situation, and checking whether their answer makes sense, revising the model when necessary.

Examples include:

• Grade 6, Module 3, Topic C, Lesson 14, Teach Book, Learn, students model with mathematics as they use data, a table, and a graph to represent the elevation of a deep-diving vehicle over time and use their model to approximate how long it would take the vehicle to descend from sea level to the bottom of the Mariana Trench. Problem 2 states, “The Mariana Trench is in the Pacific Ocean and is the deepest oceanic trench on Earth. An oceanic trench is like a valley on the ocean floor. The deepest place in the bottom of the Mariana Trench is called Challenger Deep. It is farther from sea level than the summit of Mt. Everest is from sea level. How long do you think it would take a vehicle to descend from sea level to Challenger Deep? Record your data in the table. Then graph your time and elevation data for the dive.” Students identify and make assumptions about information needed to represent the situation, record time and elevation data in a table, and graph the data to create a mathematical model of the vehicle’s descent. Students use the model to approximate the time required to reach Challenger Deep and consider how additional information or changes to their assumptions could improve the accuracy of their model. Promoting Mathematical Practice margin note states, “When students record and analyze data to answer a question about the time it takes a deep-diving vehicle to reach the bottom of the Mariana Trench, they are modeling with mathematics. Ask the following questions: How could you improve your graph to better represent the time it takes the vehicle to dive to the bottom of the trench? What assumptions could you make to help you approximate the time it takes the vehicle to dive to the bottom of the trench? What do you wish you knew in order to determine how long it would take the vehicle to reach the bottom of the trench?” Practice, Problem 2 extends students’ use of mathematical models by providing daily high temperatures over 15 days and asking students to represent the contextual data with a time graph. Students “a. Create a time graph of the temperature data on the grid. Draw axes, choose appropriate interval lengths for the axes, plot the data points, and connect the points with line segments.” Students then use and interpret their model to determine the warmest high temperature, identify when the temperature decreased the most, and “Summarize how the high temperature in this city changed over the 15 days.”

• Grade 7, Module 2, Topic E, Lesson 26, Teach Book, Learn, students model with mathematics as they use rational number expressions to represent the operating income of a Major League Baseball team over a 10-year period and use their model to determine and interpret the team’s mean operating income. Problem 4 states, “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?” Students use the operating income data to write and evaluate a rational number expression that models the situation, determine the mean operating income, and interpret the resulting value in the context of the team’s earnings. Students consider assumptions that could help them solve the problem, evaluate how well their expression represents the context, and consider other mathematical models that could represent the situation. Promoting Mathematical Practice margin note states, “When students use rational expressions to represent and solve problems in real-world situations such as decorating, fundraising, baseball, or golf, they are modeling with mathematics. Ask the following questions: 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?” Practice, Problem 3 extends students’ use of mathematical models by providing the annual operating income of the Arizona Diamondbacks from 2002 to 2011 in a diagram and asking students to use the data to determine and interpret the team’s mean operating income. Problem 3 states, “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 use the values represented in the diagram to create and evaluate a mathematical expression for the mean and then interpret what the result means in the real-world context of the team’s operating income.

• Grade 8, Module 4, Topic B, Lesson 11, Teach Book, Learn, students model with mathematics as they write and solve linear equations to represent the time it takes to travel to a destination by car or plane and use their models to determine the distance for which the travel times are the same. Students use speeds of 65 miles per hour for the car and 575 miles per hour for the plane and an airport waiting time of 2 hours and 30 minutes, define a variable to represent the distance of the trip, and write and solve an equation that models the travel situation. Students consider assumptions and additional information needed to represent the travel times, determine how well their equations represent the context, and adjust their equations based on changes to the scenario. Promoting Mathematical Practice margin note states, “When students write linear equations to represent the time it takes to drive or fly to a destination, when they acknowledge the assumptions they make to write the linear equation, and when they adjust the equation based on changes to the scenario they are given, they are modeling with mathematics. Ask the following questions: What math can you write to represent the total time it takes to travel to your destination by car? By plane? What assumptions can you make to help write an expression that represents the total time it takes to travel to your destination by car? By plane? What do you wish you knew to find the total time it takes to travel to your destination by car? By plane? Can you make a reasonable estimation or assumption about that?” Practice, Problem 3 extends students’ use of mathematical models by providing a situation in which two people leave home at different times and travel along the same path at different speeds. Problem 3 states, “Sara leaves home at 8:00 a.m. and drives at an average speed of 45 miles per hour. Her brother Pedro leaves home at 9:00 a.m. and drives along the same path at an average speed of 65 miles per hour. a. How many hours will Sara have driven when Pedro catches up with her? b. How far from home are they when Pedro catches up with Sara?” Students use the quantities and relationships in the travel situation to create and solve a linear equation that models when Pedro catches up with Sara and then use the solution to determine and interpret how far from home they are when the two travel distances are equal.

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP5: Use appropriate tools strategically, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP5 throughout the year. Each lesson identifies one or more Standards for Mathematical Practice and names a focus MP for teachers at the beginning of the lesson. Margin notes titled “Promoting the Standards for Mathematical Practice” identify places within lessons where students engage in or build experience with the focus practice. The notes provide teachers with lesson-specific information, ideas, prompts, and questions to promote students’ engagement with the focus MP. In addition, the Implementation Guide explains that while lessons may provide opportunities for students to engage with multiple mathematical practices, the guidance identifies a focus MP within each lesson.

Across the grades, students participate in tasks that support key components of MP5. These include selecting and using appropriate tools and strategies to explore mathematical ideas, solve problems, and communicate their thinking. Students are encouraged to consider the advantages and limitations of various tools such as manipulatives, drawings, measuring devices, and digital technologies, and to choose tools that best support their understanding and reasoning. They have opportunities to use tools flexibly for investigation, calculation, representation, and sense-making. Teachers are guided to support this development by making a variety of tools available, modeling their effective use, and encouraging students to make strategic decisions about when and how to use tools. Teacher materials prompt opportunities for student choice in tool selection, promote discussion around tool effectiveness, and support the comparison of multiple tools or representations. Teachers are also encouraged to highlight how different tools can yield different insights and to help students reflect on their tool choices as part of the problem-solving process.

Examples include:

• Grade 6, Module 4, Topic E, Lesson 23, Teach Book, Learn, students use appropriate tools strategically as they choose from a table, a graph, or an equation to help solve a real-world problem and compare the effectiveness of the tools. Students consider a situation involving gasoline purchased at a rate of $2.50 per gallon and complete a table showing the number of gallons of gasoline, the calculation, and the total cost. Students consider which available tools could help them solve the problem, use estimation to determine whether their results are reasonable, and determine which tool would be most helpful for finding an exact value. Promoting Mathematical Practice margin note states, “Students use appropriate tools strategically when they choose from a table, a graph, or an equation to help them find total earnings in a real-world situation and then compare the effectiveness of the tools. Ask the following questions: What tools could help you solve this problem? How can you estimate your total earnings? Does your estimate sound reasonable? Which tool would be the most helpful to find your exact earnings? Why?” In Practice, Problem 3 provides a real-world earnings situation that students represent with a table and an equation. Problem 3 states, “Tara earns $11.45 per hour at her job. Let d represent Tara’s total earnings in dollars. Let h represent the number of hours Tara works.” Students determine the independent and dependent variables, complete a table showing Tara’s earnings for the first 6 hours she works, determine the value of the ratio of total earnings to hours worked, and write an equation representing Tara’s earnings for a given number of hours. Students then use the equation to “Determine Tara’s total earnings when she works 20 hours.” Students use multiple mathematical representations of the same ratio relationship and use the equation to determine an unknown value in the real-world context.

• Grade 7, Module 6, Topic B, Lesson 10, Teach Book, Learn, students use appropriate tools strategically as they determine which physical tool would be most efficient for conducting a simulation and use the simulation to determine an empirical probability. Problem 7 states, “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.” Students determine which available tool would most efficiently simulate the possible arrival times, describe a trial and a successful outcome, and conduct the simulation to estimate the probability of the event. Promoting Mathematical Practice margin note states, “Students use appropriate tools strategically when they decide which physical tools are appropriate for a simulation that determines empirical probabilities. Ask the following questions: 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?” In Practice, Problem 1 extends students’ strategic use of tools by requiring them to determine how a random number table can be used to model a probability situation and then use the tool to conduct repeated trials. Problem 1 states, “A model airplane has two engines. It can still fly if one engine fails, but not if both engines fail. Any time the model airplane flies, the probability of a failure is 0.10 for each engine. Describe how to use a random number table to estimate the theoretical probability that both engines fail the next time the model airplane flies.” Students determine which numbers in the random number table could represent failed and working engines, describe a trial and a successful trial, and then “d. Conduct 50 trials, record your results, and find the empirical probability that both engines fail the next time the model airplane flies.” Students use the random number table to represent the possible outcomes, establish which numbers represent failed and working engines, conduct repeated trials, and use the results to determine the empirical probability of the event.

• Grade 8, Module 2, Topic A, Lesson 1, Teach Book, Launch, students use appropriate tools strategically as they choose among several available tools to recreate a geometric pattern and evaluate the effectiveness of their choices. Problem 1 states, “Study the pattern. a. Use any of the given tools and only Figure A to create the pattern. b. What tools did you use? c. What strategy did you use?” Students are shown a four-petal flower pattern with petals labeled A, B, C, and D and determine how to reproduce the pattern using only Figure A. Teachers are instructed,  “Present the available tools: a dry-erase marker, a piece of paper, a protractor, a ruler, a straightedge, scissors, and a transparency. Allow students time to choose their tools, discuss their strategy, and try to create the pattern shown in problem 1.” Promoting Mathematical Practice margin note states, “Students use appropriate tools strategically when they choose among a ruler, a protractor, a piece of paper, a straightedge, scissors, a transparency, and a dry-erase marker to recreate the pattern in problem 1. Ask the following questions: Why did you choose the tools you chose? Did they work well? Which tool would be the most helpful to recreate the pattern? Why?” In Practice, Problem 21 states, “Rectangle ABCD has a perimeter of 32 units and an area of 48 square units. a. If the coordinates of point A are (-8, 2), what could be the coordinates of the other three vertices? Use the coordinate plane if needed. b. Explain how you determined the coordinates of the other three vertices.” Students determine possible coordinates for the remaining vertices and decide whether to use the provided coordinate plane to support their reasoning. The direction, “Use the coordinate plane if needed,” allows students to choose whether to use the coordinate plane rather than requiring its use.

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP6: Attend to precision, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP6 throughout the year. Each lesson identifies one or more Standards for Mathematical Practice and names a focus MP for teachers at the beginning of the lesson. Margin notes titled “Promoting the Standards for Mathematical Practice” identify places within lessons where students engage in or build experience with the focus practice. The notes provide teachers with lesson-specific information, ideas, prompts, and questions to promote students’ engagement with the focus MP. In addition, the Implementation Guide explains that while lessons may provide opportunities for students to engage with multiple mathematical practices, the guidance identifies a focus MP within each lesson.

Across the grades, students engage in tasks that support key components of MP6. These include formulating clear explanations, using grade-level appropriate vocabulary and conventions, applying definitions and symbols accurately, calculating efficiently, and specifying units of measure. Students are also expected to label tables, graphs, and other representations appropriately, and to use precise language and notation when presenting mathematical ideas. Teachers support this development by modeling accurate mathematical language, ensuring students understand and apply precise definitions, and providing feedback on usage. Lessons include opportunities for students to clarify the meaning of symbols, evaluate the accuracy of their own and others’ work, and refine their communication. Teachers are encouraged to facilitate discussions and structure tasks that promote attention to detail in both reasoning and representation.

Examples include:

• Grade 6, Module 1, Topic A, Lesson 3, Teach Book, students attend to precision as they consider the order of quantities in ratios and use precise ratio language and notation to describe relationships between quantities. Exit Ticket states, “Tyler has 24 quarters and 6 dimes. a. Write and explain the meaning of two ratios that could represent this situation. b. Tyler puts all the coins in bags. Every bag has the same number of quarters and the same number of dimes. How many bags of coins can Tyler make? How many quarters and how many dimes are in each bag? c. Use ratio language to describe the relationship between the number of quarters and the number of dimes in each bag from part (b).” Students write ratios to represent the quantities of quarters and dimes, explain the meaning of the ratios they write, and use ratio language to precisely describe the relationship between the two quantities. Students distinguish the quantities represented by each number in a ratio and maintain the correct order of the quantities when expressing the relationship. Promoting Mathematical Practice margin note states, “Students attend to precision when they consider the order of the numbers in ratios and use correct ratio language and notation to describe the relationship between quantities. Ask the following questions: Where is it easy to make mistakes when writing ratios? Is it exactly right to say that the ratio of pens to pencils is 3 to 6? What can we add or change to be more precise? Is it exactly right to say that the ratio is 3 pens to 6 pencils? What can we add or change to be more precise?” Practice, Problem 3 states, “A pet store has 10 cockatiels and 15 parakeets. a. Write a ratio that relates the number of cockatiels to the number of parakeets. b. Students attend to precision when they consider the order of the numbers in ratios and use correct ratio language and notation to describe the relationship between quantities. c. Use the groups from part (b) to write the ratio of the number of groups of cockatiels to the number of groups of parakeets. d. Use ratio language to describe the relationship between the number of cockatiels and the number of parakeets. e. Explain how the ratios you wrote in part (a) and part (c) represent the same situation.”

• Grade 7, Module 3, Topic D, Lesson 18, Teach Book, students attend to precision as they determine whether a boundary number is included in the solution set and use inequality symbols and graphs to precisely represent solutions. Learn states, “Write an inequality to represent the solutions shown on the graph.” Students identify the boundary number represented on the graph, determine whether the boundary number is a solution, and use the appropriate inequality symbol to represent the solution set. Promoting Mathematical Practice margin note states, “When students determine whether the boundary number is a solution to an inequality and how that will affect the graph, they are attending to precision. Ask the following questions: How can we show that the boundary number is a solution when writing the inequality? Is it 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?” Practice, Problems 14–17 state, “Write an inequality to represent the solutions shown on the graph.” Students interpret number-line graphs with open or closed boundary points and write inequalities that precisely represent the solution sets shown. Practice, Problems 21–24 state, “Define a variable and write an inequality that represents the situation,” and provide contexts including, “21. Only children 12 years or younger can order from the kids’ menu at Vic’s Diner,” “22. The planning committee wants at least 40 students to attend the school dance,” “23. Ava may play video games for no more than one and a half hours a day,” and “24. The chemical element helium remains a liquid when it is below −452°F.”

• Grade 8, Module 6, Topic E, Lesson 21, Teach Book, Learn, students attend to precision as they use volume formulas, identify the quantities represented by variables, substitute values accurately, and include appropriate units when stating volume. Problem 3 states, “For problems 3–5, find the volume of the solid.” Students are shown a pyramid with a height of 4.5 cm and a base area of 22cm^2. Students use the given base area and height with the volume formula to calculate the volume and state the resulting value with cubic units. Promoting Mathematical Practice margin note states, “When students use a volume formula to solve for an unknown quantity, they attend to precision. Ask the following questions: In problem 3, is it exactly correct to say the volume is 33? What can we add or change to be more precise? What does the variable B represent in the volume formula? What details are important to think about when substituting values in the volume formula?” In Practice, Problem 7 states, “The volume of the pyramid shown is 20in^3. The area of the base of the pyramid is 15in^2. What is the height of the pyramid?” In Practice, Problem 8 states, “The volume of the pyramid shown is 14.75 cubic meters. The height of the pyramid is 7.5 meters. What is the area of the base of the pyramid?”

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP7: look for and make use of structure, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP7 throughout the year. Each lesson identifies one or more Standards for Mathematical Practice and names a focus MP for teachers at the beginning of the lesson. Margin notes titled “Promoting the Standards for Mathematical Practice” identify places within lessons where students engage in or build experience with the focus practice. The notes provide teachers with lesson-specific information, ideas, prompts, and questions to promote students’ engagement with the focus MP. In addition, the Implementation Guide explains that while lessons may provide opportunities for students to engage with multiple mathematical practices, the guidance identifies a focus MP within each lesson.

Across the grades, students engage in tasks that support key components of MP7. These include looking for and describing patterns, identifying structure within mathematical representations, and decomposing complex problems into simpler, more manageable parts. Students are encouraged to analyze problems for underlying structure and to consider multiple solution strategies. They make generalizations based on repeated reasoning and use those generalizations to solve problems efficiently. Teachers support this development by selecting tasks that highlight mathematical structure and by prompting students to attend to and describe patterns they notice. Lessons provide opportunities for students to compare approaches, justify their reasoning, and reflect on how structure helps deepen their understanding. Teachers are encouraged to facilitate discussions and structure lessons in ways that promote the recognition and use of patterns and structure in problem-solving.

Examples include:

• Grade 6, Module 1, Topic B, Lesson 8, Teach Book, Learn, students look for and make use of structure as they recognize addition patterns in tables and graphs of ratio relationships and use the patterns to find equivalent ratios. Problem 1 states, “The graph represents the ratio relationship between the number of cups of orange juice and the number of cups of pineapple juice in batches of a citrus punch. Use the graph to complete the ratio table.” Students use values represented by points on the graph to complete the ratio table, identify addition patterns in the ratio relationship, and use the patterns to determine equivalent ratios. Promoting Mathematical Practice margin note states, “Students look for and make use of structure when they recognize addition patterns in ratio tables and graphs of ratio relationships and use them to find equivalent ratios. Ask the following questions: How are the ratio table and graph of the ratio relationship related? How can that help you find equivalent ratios? What is another way you can complete a ratio table to help you find equivalent ratios? How can what you know about equivalent ratios help you find points on the graph of a ratio relationship?” Practice, Problem 3 states, “A homemade modeling clay recipe calls for salt and flour. a. Use addition patterns to complete the ratio table. b. What is a ratio of the number of cups of salt to the number of cups of flour? c. Describe the ratio relationship by using the addition patterns in the ratio table.”

• Grade 7, Module 4, Topic E, Lesson 24, Teach Book, Learn, students look for and make use of structure as they relate finding the volume of rectangular prisms to finding the volume of other types of right prisms. Problem 3 states, “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?” Students explain Jonas’s method for finding the volume, respond to whether the volume formula V=Bh can be used for the prism, use Nora’s strategy to determine the volume, and compare their result to Jonas’s result. Promoting Mathematical Practice margin note states, “Students look for and make use of structure as they relate finding the volume of rectangular prisms to finding the volume of other types of prisms. Ask the following questions: 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?” Practice, Problem 3 states, “Find the volume of the right trapezoidal prism.” Students are shown a right trapezoidal prism with measurements of 3 cm, 6 cm, 7 cm, 5 cm, and 11 cm. In Practice, Problem 4 states, “Find the volume of the right regular hexagonal prism. The line segment that has a length of 7 m begins at the center of the hexagon and ends at one side of the hexagon.”

• Grade 8, Module 4, Topic B, Lesson 8, Teach Book, Learn, students look for and make use of structure as they compare the linear and constant terms on both sides of equations to determine whether equations have one solution, infinitely many solutions, or no solution. Problem 3 states, “9m-6=6m-9, 3(4n+3)-3n=2+9n-11, and -4x+5=5-4x Complete the table.” The table contains columns for Number of Solutions, Equations, and What I Notice About the Structure. Students compare the structure of the expressions on both sides of each equation, identify similarities and differences in the linear and constant terms, and use those structures to classify the equations according to their number of solutions. Promoting Mathematical Practice margin note states, “When students compare the linear and constant terms on both sides of an equation to distinguish whether the equation has infinitely many solutions or no solution, they are looking for and making use of structure. Ask the following questions: How can you use what -4x=5 and 5-4x have in common to determine the number of solutions to the equation -4x+5=5-4x? How can you use what 10r-2 and 10r+1 have in common to determine the number of solutions to the equation 10r-2=10r+1?” Practice, Problem 1 states, “Match each equation with its number of solutions,” and provides the equations 9w-2=9w+2, 9w-2=8w+2, and 9w-2=9w-2 to match with “Only one solution,” “No solution,” and “Infinitely many solutions.” In Practice, Problem 13 states, “Kabir says the equation 19x+4-(x+1)=19x+4 has infinitely many solutions because the expressions on both sides of the equation have the same linear term 19x and the same constant term 4. Do you agree with Kabir that the equation has infinitely many solutions? Explain.”

The materials reviewed for Eureka Math² Grade 6 through Grade 8 meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP8 throughout the year. Each lesson identifies one or more Standards for Mathematical Practice and names a focus MP for teachers at the beginning of the lesson. Margin notes titled “Promoting the Standards for Mathematical Practice” identify places within lessons where students engage in or build experience with the focus practice. The notes provide teachers with lesson-specific information, ideas, prompts, and questions to promote students’ engagement with the focus MP. In addition, the Implementation Guide explains that while lessons may provide opportunities for students to engage with multiple mathematical practices, the guidance identifies a focus MP within each lesson.

Across the grades, students engage in tasks that support key components of MP8. These include noticing and using repeated reasoning to make sense of problems, recognizing patterns, and developing efficient, mathematically sound strategies. Students are encouraged to create, describe, and explain general methods, formulas, or processes based on patterns they identify. Lessons provide opportunities for students to evaluate the reasonableness of their answers and refine their approaches through discussion and reflection. Teachers support this development by structuring tasks that highlight repeated reasoning, prompting students to make generalizations, and guiding them to build conceptual understanding, distinct from relying on memorized tricks. Instructional guidance also encourages teachers to model and elicit strategies that build toward formal algorithms or representations through consistent reasoning.

Examples include:

• Grade 6, Module 5, Topic D, Lesson 17, Teach Book, Learn, students look for and express regularity in repeated reasoning as they compare how changes to one or more edge lengths of right rectangular prisms affect volume and use the repeated relationship to generalize how multiplying an edge length by a number affects the volume. Problem 3 states, “Prism P is a right rectangular prism with edge lengths l, w, and h. a. Write an algebraic expression to represent the volume of prism P. b. Prism S is a right rectangular prism. The volume of prism S is 3 times the volume of prism P. Which of the following could represent the edge lengths of prism S? Choose all that apply. A. 3l, 3w,3h B. 3l,w,h C. 3l,w,3h D. l,w,3h E. l,3w,h F. l,\frac{1}{3}w,9h c. Prism T is a right rectangular prism. Each edge length of prism T is 3 times the corresponding edge length of prism P. Write an algebraic expression to represent the volume of prism T by using as few factors as possible.” Students write an expression for the volume of a right rectangular prism, compare how multiplying different edge lengths changes the volume, identify combinations of edge lengths that produce three times the original volume, and use the repeated relationship to write an expression for the volume when all three edge lengths are multiplied by 3. Promoting Mathematical Practice margin note states, “Students look for and express regularity in repeated reasoning when they notice that when an edge length of a right rectangular prism is multiplied by a number, the volume of the prism is multiplied by the same number. Ask the following questions: What patterns did you notice when you compared the edge lengths and volumes of the prisms to the edge lengths and volumes of prisms A and B? What is the same about how the volume was affected when edge lengths were each multiplied by 2 and when edge lengths were each multiplied by \frac{1}{2}?” Practice, Problem 4 states, “Prism X and prism Y are both right rectangular prisms. Prism X has edge lengths l, w, and h. The volume of prism Y is 4 times the volume of prism X. Which of the following could represent the edge lengths of prism Y? Choose all that apply.” Students choose among combinations including A. 4l, w, h. B. l, 4w, h. C. l, 2w, 2h. D. 4l, 4w, 4h. E. l, 4w, 4h. F. \frac{1}{2}l, \frac{1}{2}w, 16h.”

• Grade 7, Module 1, Topic C, Lesson 18, Teach Book, Learn, students look for and express regularity in repeated reasoning as they identify a pattern between the scale factor and the areas of corresponding figures, make a conjecture based on the pattern, and test whether the conjecture applies to rectangles. Problem 14 states, “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 use their conjecture to predict the area of a scale drawing, create the scale drawing, calculate its area, and compare the calculated area to their prediction to determine whether the conjecture applies to rectangles. Promoting Mathematical Practice margin note states, “When students notice that the area of a region in a scale drawing is always related to the area of the corresponding region in the original figure by a factor of the scale factor squared, they are looking for and expressing regularity in repeated reasoning. Ask the following questions: What patterns did you notice when comparing areas of corresponding regions in the scale drawing and original figure? Will this pattern always work?” In Practice, Problem 1 asks students to determine the scale factor relating two triangles, calculate the areas of both triangles, and then asks, “c. What factor should you apply to the area of \triangle ABC to find the area of \triangle DEF? How does the area of \triangle DEF compare to the area of \triangle ABC? How does this relationship compare to the scale factor?” In Practice, Problem 3 states, “Figure 2 is a scale drawing of figure 1. The area of figure 1 is 15 square units, and the scale factor is 5. What is the area of figure 2? Explain how you know.” 

• Grade 8, Module 3, Topic B, Lesson 8, Teach Book, Learn, students look for and express regularity in repeated reasoning as they compare multiple dilations to identify the relationship between a scale factor and the coordinates of a point and its image. Problem 2 states, “Consider points O and B in the coordinate plane. a. Choose a Scale Factor card. Then plot and label the image of point B under a dilation with center O and scale factor r from the card. b. Check one another’s images. Then complete the table for the group’s four images. c. Look for a pattern within each row of the table. What do you notice about the relationship between the scale factor and the coordinates of points B and B′?” Students plot images of the same point using different scale factors, record the coordinates of the resulting images, compare the repeated calculations across the table, and identify a pattern relating the scale factor to the coordinates of a point and its image. Promoting Mathematical Practice margin note states, “When students look for a relationship between the scale factor of the dilation and the coordinates of a point and its image, they are looking for and expressing regularity in repeated reasoning. Ask the following questions: What is the same about how each group member located their point B′? What arithmetic patterns do you notice when you compare the coordinates of the point and its image? Will the pattern always work?” Practice, Problem 1 states, “Each point A is mapped to point A′ by a dilation centered at the origin with the given scale factor. Complete the table.” Students are given the coordinates (-4,-2),(6,-4), and (-5,3) with scale factors 3, \frac{1}{2}, and 4, respectively, and determine the coordinate of each image. In Practice, Problem 6 states, “Consider \triangle PQR in the coordinate plane. a. What are the coordinates of the images of vertices P, Q, and R under a dilation centered at the origin with scale factor 1.5? b. Graph and label the image of \triangle PQR under the same dilation.”