3rd-5th Grade - Gateway 2

The materials reviewed for Eureka Math² Grade 3 through Grade 5 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 3, Module 5, Topic D, Lesson 18, Teach Book, Learn, students develop conceptual understanding as they represent fractions on a number line, reason about fraction magnitude based on position, compare fractions, and identify fractions that are equivalent to whole numbers. Teachers state, “Draw a number line, labeling the starting mark as 1 and the ending mark as 4. Invite students to do the same. Write the following fractions: \frac{2}{2}, \frac{7}{2}. What should we do to our number line to help us locate these fractions? Partition and label the number line to show the whole numbers and then partition again to show halves. Invite students to do the same. Where does 2 halves belong on the number line? How do you know? Label \frac{2}{2} on the number line. Direct students to label \frac{2}{2}, and invite them to think-pair-share about where to label \frac{7}{2}. This number line doesn’t show 0. Where is 0? Which fraction is closer to 0, \frac{2}{2} or \frac{7}{2}? Which fraction is greater, \frac{2}{2} or \frac{7}{2}? How do you know? Write \frac{5}{2} and \frac{8}{2} next to \frac{2}{2} and \frac{7}{2}. Invite students to work with a partner to place them on a number line. Let’s draw a box around the fractions that are equivalent to whole numbers. Which fractions will be in a box? Draw a box around \frac{2}{2} and \frac{8}{2}. Direct students to do the same. 5 halves is not equivalent to a whole number. How did you decide where to place it? Which fraction is less, \frac{5}{2} or \frac{8}{2}? How do you know?” (3.NF.2) 

• Grade 4, Module 5, Topic A, Lesson 3, Teach Book, Land, students demonstrate conceptual understanding as they connect visual representations of tenths to equivalent decimal fraction and decimal notation. Exit Ticket, Problem 1, “Write the decimal fraction as a decimal number. 6 tenths, \frac{6}{10}=___.” An area model is provided. Problem 2, “Write the decimal number as a decimal fraction. 4 tenths, 0.4 = ___.” Four place value disks are provided. (4.NF.6)

• Grade 5, Module 3, Topic A, Lesson 3, Teach Book, Problem Set, students demonstrate conceptual understanding of fraction multiplication as they use a number line and repeated addition to represent and determine the product of a fraction and a whole number. Problem 1 states, “Use the number line to find the product. Then write a repeated addition sentence to check your work. Write your answer as a whole number when possible. \frac{1}{2}\times4=___.” A number line from 0 to 4 is provided. (5.NF.4)

The materials reviewed for Eureka Math² Grades 3 through Grade 5 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.” 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 3, Module 2, Topic C, Lesson 18, Teach Book, Fluency, students develop procedural skill and fluency with multiplication within 100 as they determine products and state multiplication equations using factors represented by the number of fingers shown. The teaching notes state, “Let’s play Ready, Set, Multiply. Have students form pairs and stand facing each other. Model the action: Make a fist, and shake it on each word as you say, ‘Ready, set, multiply.’ At ‘multiply,’ open your fist, and hold up any number of fingers greater than 1. Tell students that they will make the same motion. At ‘multiply,’ they will show their partner any number of fingers other than 1. Consider doing a practice round with students. Clarify the following directions: Only show 2, 3, 4, or 5 using one hand. Try to use different numbers each time to surprise your partner. Each time partners show fingers, have them both say the product. Then have each student say the multiplication equation, starting with the number of fingers on their own hand. See the sample dialogue under the photograph.” (3.OA.7)

• Grade 4, Module 1, Topic D, Lesson 16, Teach Book, Problem Set, students independently demonstrate procedural skill and fluency with adding multi-digit whole numbers as they use the standard algorithm to add numbers with up to six digits. Problems 1–3 state, “Add by using the standard algorithm. Problem 1, 5,212+367; Problem 2, 5,212+1,367; Problem 3, 5,215+1,367.” (4.NBT.4)

• Grade 5, Module 5, Topic D, Lesson 22, Teach Book, Problem Set, students independently demonstrate procedural skill and fluency with multiplying multi-digit whole numbers as they multiply the area of the base by the height to determine the volume of right rectangular prisms. Problems 4–5 state, “Calculate the volume of the right rectangular prism. 4. The area of the base is 48 square inches, and the height is 5 inches. 5. The area of the base is 24 square centimeters, and the height is 4 centimeters.” (5.NBT.5)

The materials reviewed for Eureka Math² Grades 3 through Grade 5 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 Math² 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 3, Module 1, Topic E, Lesson 23, Teach Book, Learn, Problem Set, students apply their understanding of multiplication and division in a non-routine problem as they determine the number of fruit snacks in each equal group and use that information to determine how many fruit snacks remain. Problem 4 states, “Ivan has a bag of 18 fruit snacks. There is an equal number of peach, cherry, and grape fruit snacks. Ivan eats all the grape fruit snacks. How many fruit snacks does Ivan have left?” (3.OA.3)

• Grade 4, Module 4, Topic E, Lesson 28, Teach Book, Learn, Classwork, students apply their understanding of addition and subtraction of mixed numbers in a routine problem as they use the Read–Draw–Write process to represent the situation and determine the number of pizzas eaten. Problem 1 states, “Use the Read–Draw–Write process to solve the problem. There were 6\frac{3}{8} pizzas on a table when lunch started. At the end of lunch, there were 2\frac{6}{8} pizzas left. How many pizzas were eaten during lunch? Guide students to reason about the problem by asking questions such as the following: What information does the problem give us? What does the question ask? What can you draw to represent the problem?” (4.NF.3d)

• Grade 5, Module 6, Topic D, Lesson 17, Teach Book, Learn, Classwork, students apply their understanding of the coordinate plane in a non-routine problem as they collect data, represent the data as ordered pairs in the coordinate plane, and use the resulting graph to draw conclusions about the relationship between the number of consonants and vowels in words. Problem 3 states, “a. Write a word of each type, the number of consonants in the word, and the number of vowels in the word. Do not write the same words as the words in problem 1. b. Label the x-axis Number of Consonants and the y-axis Number of Vowels. Label the title Word Data. Use the data collected in part (a) to form ordered pairs. Plot points that represent the ordered pairs in the coordinate plane. c. Based on the data, do you think it is true that the more consonants a word has, the more vowels it has? Why?” (5.G.2)

The materials reviewed for Eureka Math² Grade 3 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 3, Module 6, Topic A, Lesson 3, Teach Book, Learn, students demonstrate conceptual understanding and application as they use a number line with 5-minute intervals to reason about elapsed time and determine an unknown finish time in a real-world context. The problem states, “Pablo puts noodles into boiling water at 5:27 p.m. The directions say they need to cook for 16 minutes. What time will the noodles be done?” (3.MD.1)

• Grade 4, Module 4, Topic B, Lesson 11, Teach Book, Learn, students demonstrate conceptual understanding and procedural skill and fluency as they use tape diagrams and number lines to represent equivalent fractions, connect the representations to multiplication, and write equations to represent equivalent fractions. The lesson states, “Write \frac{6}{10}=\frac{6}{10}+\frac{2}{2}=\frac{3}{5}. Invite students to turn and talk about how the tape diagram and number line represent the multiplication.” Students then write equations to represent equivalent fractions. (4.NF.1) 

• Grade 5, Module 5, Topic D, Lesson 23, Teach Book, Learn, Classwork, students demonstrate conceptual understanding and application as they use their understanding of volume formulas to determine and compare the volumes of right rectangular prisms represented in different ways. Problem 1 states, “Which right rectangular prism has the greater volume?” Prism A has dimensions 14 inches, 3 inches, and 6 inches, and Prism B states, “The area of the base is 20 square inches, and the height is 12 inches.” (5.MD.5a, 5.MD.5b)

The materials reviewed for Eureka Math² Grade 3 through Grade 5 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 them 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 3, Module 4, Topic D, Lesson 16, Teach Book, Learn, students make sense of a new problem type as they determine the area of a shaded part within a larger square. Students consider what they know about the area of the whole square and look for an entry point to begin solving the problem. They consider strategies for determining the area of the shaded region and use their understanding of area to evaluate whether a solution is reasonable. Promoting Mathematical Practice margin note states, “Students make sense of problems as they encounter a new problem type, finding the area of a shaded part of a larger shape and look for entry points to its solution. Ask the following questions: What can you figure out about the area of the shaded part by looking at the area of the whole square? What are some strategies you can try to start finding the area?” Problem Set, Problem 4, students determine the area of a quilt square and the area covered by blue fabric and write an equation showing the relationship between the two areas. They then evaluate the statement, “The area covered by the orange fabric is equal to the area covered by the blue fabric. Do you agree? Why?” 

• Grade 4, Module 2, Topic C, Lesson 16, Teach Book, Learn, students make sense of a division word problem as they use the Read–Draw–Write process to determine how many chairs can be made from 304 chair legs when each chair has 4 legs. Students develop a plan for solving the problem, monitor whether their strategy is working, consider another strategy when needed, and determine whether their answer makes sense. Promoting Mathematical Practice margin note states, “Students make sense of problems and persevere in solving them as they use the Read–Draw–Write process to solve a word problem. Ask the following questions: What is your plan to find how many chairs can be made? Is your strategy working? Is there something else you could try? Does your answer make sense? Why?” Problem Set, Problem 5 states, “Think about 308\div4. a. Use two different strategies to find the quotient. b. Draw a star next to the strategy that is more efficient. c. Explain why the strategy you chose is more efficient.” 

• Grade 5, Module 1, Topic C, Lesson 14, Teach Book, Learn, students make sense of division word problems and persevere in solving them as they use the Read–Draw–Write process to solve real-world problems. Students make an estimate, monitor and adjust their estimate as needed, interpret remainders in context, and determine whether their answer makes sense. Promoting Mathematical Practice margin note states, “When students make estimates, adjust their estimates, and interpret remainders to solve real-world problems by using division, they are making sense of problems and persevering in solving them. Ask the following questions: How can you simplify the problem? Does your estimate work? Is there something else you could try? Does your answer make sense? Why?” Problem Set, Problem 10 states, “Use the Read–Draw–Write process to solve the problem. Kayla’s book has 307 pages. She plans to read 45 pages each day. How many days will it take Kayla to finish reading the book?”

The materials reviewed for Eureka Math² Grade 3 through Grade 5 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 3, Module 2, Topic B, Lesson 9, Teach Book, Learn, Classwork, students reason abstractly and quantitatively as they connect a measurement of 73 milliliters in a graduated cylinder to its representation on a vertical number line and use the quantities in the context to round to the nearest ten. During Classwork, students consider a graduated cylinder containing 73 milliliters of water and use a vertical number line to round the measurement. Teachers ask, “How many tens are in 73?” “What is 1 more ten than 7 tens?” and “What number is halfway between 7 tens and 8 tens?” as students connect the quantity of water to the interval from 70 to 80 and use the number line to determine the rounded measurement. Students interpret 73 as 7 tens and 3 ones, identify 70 and 80 as the endpoints of the interval and 75 as the halfway point, and use the number line to reason about the amount of water in the graduated cylinder. Promoting Mathematical Practice margin note states, “Students reason quantitatively and abstractly as they consider rounding in different contexts (such as water in a graduated cylinder) as well as on a number line. Ask the following questions: How does the graduated cylinder help you see what numbers to mark on your number line? What does your number line tell you about the amount of water in the graduated cylinder?” Problem Set, Problem 1 states, “Use the picture to help you answer parts (a)-(e). a. How much water is in the container? milliliters. b. Which two tens is the amount of water between? c. Is the amount of water less than halfway or more than halfway between the tens? Explain your answer. d. Which ten is the amount of water closer to? e. To the nearest ten millimeters, the amount of water in the container is _ millimeters." 

• Grade 4, Module 4, Topic F, Lesson 33, Teach Book, Learn, students reason abstractly and quantitatively as they use the Read–Draw–Write process to make sense of the quantities and relationships in a multiplicative comparison situation. Classwork, Problem 2 states, “A kitten weighs \frac{4}{5} kilograms. A puppy is 6 times as heavy as the kitten. How many kilograms does the puppy weigh?” Students represent the kitten’s weight of \frac{4}{5} kilogram and the puppy’s weight as 6 times as heavy with a tape diagram, decontextualize the situation to multiply a fraction by a whole number, and determine whether the resulting quantity is appropriately expressed as a fraction or mixed number. Promoting Mathematical Practice margin note states, “Students reason abstractly and quantitatively as they use the Read–Draw–Write process to solve a word problem, decontextualize to multiply a fraction by a whole number, and decide to express the product as a fraction or as a mixed number. Ask the following questions: What does the given information in the problem tell you about how to draw a tape diagram to represent the situation? How does your tape diagram represent how the weights of the kitten and puppy are related? Does your answer make sense mathematically? Problem Set, Problem 6 states, “A piece of blue ribbon is \frac{5}{6} yards long. A piece of red ribbon is 11 times as long as the piece of blue ribbon. What is the total length of the two pieces of ribbon?”

• Grade 5, Module 5, Topic B, Lesson 15, Teach Book, Classwork, students reason abstractly and quantitatively as they make sense of real-world quantities and relationships across increasingly challenging multi-step word problems. Students use the Read–Draw–Write process to determine the meaning of each problem, attend to the units involved, estimate a solution, decontextualize the situation by representing it with an expression, and recontextualize their solution to determine whether it makes sense in the situation. Promoting Mathematical Practice margin note states, “When students determine the meaning of each real-world problem, estimate a solution, represent it with an expression, and recontextualize the solution while attending to units, they are reasoning quantitatively and abstractly. Ask the following questions: What does the problem ask you to do? How do the units involved in the situation help you think about the problem? Does your answer make sense in this context?.” During Classwork, students rotate through four stations that increase in complexity and are directed to “Use the Read-Draw-Write process to complete the problem at each station. Estimate before you solve the problem.” Problem Set, Problem 3 states, “In a sprint triathlon, a participant swims a distance of \frac{23}{50} miles, bikes a distance of 12\frac{2}{5} miles, and runs a distance of 3\frac{1}{10} miles. Ryan participated in 4 sprint triathlons this year. a. How many total miles did Ryan swim in the 4 sprint triathlons? b. How many total miles did Ryan swim, bike, and run in all 4 triathlons? c. Ryan completed his first sprint triathlon in 2\frac{1}{4} hours and completed his second sprint triathlon in 1\frac{3}{4} hours. How much faster was Ryan’s second triathlon than his first triathlon?”

The materials reviewed for Eureka Math² Grade 3 through Grade 5 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 3, Module 5, Topic D, Lesson 18, Teach Book, Launch, students construct viable arguments and critique the reasoning of others as they justify the placement of fractions on a number line, listen to their classmates’ reasoning, and determine whether they agree or disagree and why. Students select fraction index cards, place the fractions on an interactive number line from 0 to 4, and justify their placement. The class evaluates each student’s reasoning and discusses whether the placement is correct. Students are asked, “What is the largest number on this number line? How do you know? How does the number line show that 3 fourths is greater than 1 fourth?” Promoting Mathematical Practice margin note states, “Students construct viable arguments as they justify the placement of their index cards on the number line. The class critiques each student’s reasoning as they discuss whether they agree or disagree and why. Ask the following questions: Is the place you put your index card a guess, or do you know for sure that is where it goes? How do you know for sure? What questions can you ask your classmate to make sure you understand their reasoning?” Practice, Problem 1 asks students to use a number line labeled in fourths to compare fractions, including, “c. Which whole number shown on the number line is greater than \frac{12}{4}? How do you know?” Students use the number line as evidence to determine the relationship between the fraction and whole numbers and construct an explanation that justifies their conclusion.

• Grade 4, Module 3, Topic F, Lesson 24, Teach Book, Learn, Classwork, students construct viable arguments and critique the reasoning of others as they solve a multi-step word problem, share and justify their solution paths, and listen to and analyze their peers’ solution paths. Problem 2 states, “A restaurant uses 161 gallons of milk each week. A school uses 483 gallons of milk each week. How many more gallons of milk does the school use than the restaurant in 4 weeks?” Students work in pairs to solve the problem and then share their solution path with another pair, explaining why their method works while the other pair listens to, analyzes, and asks questions about their reasoning. Promoting Mathematical Practice margin note states, “When students share their solution path and listen to and analyze their peers’ solution path for solving a word problem, they are constructing viable arguments and critiquing the reasoning of others. Ask the following questions: Why does your method work? Convince the other pair in your group. What questions can you ask the other pair to make sure you understand their method?” Practice, Problem 1 states, “A newspaper carrier has 4 stacks of newspapers. Each stack has 38 newspapers. The carrier delivers 62 newspapers to houses and 15 newspapers to a school. a. About how many newspapers does the newspaper carrier still have left to deliver? b. What is the exact number of newspapers the newspaper carrier still has left to deliver? c. Is your answer from part (b) reasonable? Why?” 

• Grade 5, Module 2, Topic D, Lesson 16, Teach Book, Learn, Classwork, students construct viable arguments and critique the reasoning of others as they use data from a line plot to solve problems, evaluate a claim about the data, and justify whether they agree or disagree with the claim. Problem 1 states, “Mr. Sharma weighs each pumpkin he sells at his pumpkin farm. He records the data on a line plot. a. How many pumpkins did Mr. Sharma sell? b. What is the weight of the heaviest pumpkin? c. What is the most frequent weight of the pumpkins sold? d. What is the total weight of the two lightest pumpkins? e. How many pumpkins weigh at least 12\frac{1}{4} pounds? f. Eddie bought the two heaviest pumpkins. Jada bought the lightest pumpkin. What is the difference in weight between Jada’s pumpkin and the total weight of Eddie’s pumpkins? g. Julie bought two pumpkins that have a total weight of 25 pounds. Based on the data on the line plot, what could be the weights of Julie’s pumpkins?” Students compare their work, identify errors, and discuss with a partner why an error was made and what they would do differently when solving a similar problem. Promoting Mathematical Practice margin note states, “When students turn and talk about whether the claim that most of the pumpkins sold weighed more than 12\frac{1}{4} pounds is true based on the data on the line plot, they are constructing viable arguments and critiquing the reasoning of others. Ask the following questions: Is the claim that most of the pumpkins sold weighed more than 12\frac{1}{4} pounds true? How do you know? Which parts of Mr. Sharma’s claim do you question?” Practice presents a line plot showing the weights of backpacks of Miss Song’s students, and Problem 5 states, “Miss Song says that most of the backpacks weigh less than 9 pounds. Do you agree with Miss Song? Why?” Students analyze the data on the line plot, evaluate Miss Song’s claim, and construct an explanation using the data as evidence to justify whether they agree or disagree with her reasoning.

The materials reviewed for Eureka Math² Grade 3 through Grade 5 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 3, Module 3, Topic D, Lesson 22, Teach Book, Learn, students model with mathematics as they use the Read–Draw–Write process to represent and solve a real-world two-step problem involving multiplication of single-digit factors and multiples of 10 and use their model to determine the total number of carpet squares. Solve a Two-Step Array Problem, Classwork, Problem 1, states, “Use the Read–Draw–Write process to solve the problem. Use a letter to represent each unknown. There are 8 rows of 10 carpet squares in each classroom. How many carpet squares are in 4 classrooms?” Students identify the important quantities and relationships in the situation, create a tape diagram, place value chart, or other pictorial model that represents the number of carpet squares in each classroom and across 4 classrooms, and use the model to determine the unknown total and write a solution in context. Students consider what they can draw to help them understand the situation and what key ideas from the problem need to be represented in their model. Promoting Mathematical Practice margin note states, “Students model with mathematics as they use tape diagrams, place value charts, and other pictorial models to represent and make sense of word problems. The RDW process scaffolds students’ movement between the concrete problem and their abstract representation of it. Ask the following questions: What can you draw to help you understand problem 1? What key ideas in problem 1 do you need to make sure are in your model?” During Predict the Next Step, students analyze and compare different mathematical models for the contextual problems, including arrays, tape diagrams, and open number lines, and explain how quantities and operations in each representation correspond to the situation. In Land, students evaluate the usefulness of their models as the teacher asks, “How do you decide which model to use when solving a two-step word problem?” Students explain, “I want to use a model that helps me understand what is happening in the problem. If I start to draw a model and it doesn’t make sense, I may switch to a different model.” Practice extends students’ use of mathematical models by directing students to “Use the Read–Draw–Write process to solve each problem. Use a letter to represent each unknown.” Practice, Problem 1 states, “There are 20 apple trees in a garden. Luke picks 6 apples from each tree. a. What is the total number of apples Luke picks? b. Eva picks 165 apples from the apple trees. How many more apples does Eva pick than Luke?” Students use the Read–Draw–Write process to create a mathematical representation of the quantities and relationships in the real-world situation, use the model to determine the total number of apples Luke picks and then compare that amount with the number of apples Eva picks, and communicate the resulting difference in the context of the problem.

• Grade 4, Module 5, Topic D, Lesson 14, Teach Book, Learn, students model with mathematics as they use the Read–Draw–Write process to represent and solve a real-world comparison problem involving decimal measurements and use their model to determine the length of a wall. Solve a Comparison Word Problem, Classwork, states, “Use the Read–Draw–Write process to solve the problem. Write the solution statement by using a decimal number. A wall in Mrs. Smith’s living room is 0.78 meters longer than a wall in her bedroom. The wall in Mrs. Smith’s bedroom is 4.32 meters long. How long is the wall in her living room?” Students identify the important quantities and relationships in the situation, create a diagram that represents the comparison between the two wall lengths, and use the model to determine the unknown length and write a solution statement in context. Students consider whether their diagram includes the key ideas from the situation, how they could improve the diagram to better represent the context, and how simplifying the problem can help them estimate and evaluate their solution. Promoting Mathematical Practice margin note states, “When students use the Read–Draw–Write process to solve problems with decimal numbers, they are modeling with mathematics. Ask the following questions: What key ideas in the problem do you need to make sure are in your diagram? How can you improve your diagram to better represent the context? How can you simplify the problem to help estimate the answer to the problem?” Practice extends students’ use of mathematical models by directing students to “Use the Read–Draw–Write process to solve each problem. Write the solution statement by using a decimal number.” Practice, Problem 2 states, “A restaurant sells 6.14 liters of apple juice. It sells 7.42 more liters of orange juice than apple juice. How many liters of orange juice does the restaurant sell?” Students use the Read–Draw–Write process to create a mathematical representation of the quantities and comparison relationship in the real-world situation, use the model to determine the amount of orange juice sold, and communicate the result with a decimal number and appropriate unit in the context of the problem.

• Grade 5, Module 6, Topic D, Lesson 20, Teach Book, Learn, students model with mathematics as they collect, organize, and analyze data from a real-world coin drive, use patterns in the data to make predictions, and develop a method for determining the number of nickels put into a jar on the last day of the fundraiser. In the Coin Drive Challenge, Classwork, Problem 2, students consider how many nickels may have been collected on the final day and evaluate the reasonableness of possible values. Students discuss “how many nickels they think were put into the jar on the last day of the coin drive” and consider, “What is an unreasonable estimate for the number of nickels put into the jar on the last day? What is a number that is too low? What is a number that is too high?” Students use the collected data to create mathematical representations of the situation, analyze patterns, make and evaluate predictions, and develop a method for answering the contextual question. Students also consider assumptions they can make and additional information that would help them improve their model and determine the number of nickels collected on the final day. Promoting Mathematical Practice margin note states, “When students collect, organize, and analyze data, make predictions, and develop a method for determining the number of nickels put into the jar on the last day of the fundraiser, they are modeling with mathematics. Ask the following questions: What math can you write or draw to represent the coin drive problem? What assumptions can you make to help you determine the number of nickels put into the jar on the last day? What do you wish you knew that would help you find the answer?” Practice Problem 1 extends students’ use of mathematical representations by asking students to “Use the table and coordinate plane to complete parts (a)–(c).” Students complete a table according to given rules, plot the four resulting ordered pairs on a coordinate plane, and then use the relationship represented by the pattern to determine the corresponding y-coordinate when the x-coordinate is 20. Students connect the numerical pattern, table, and coordinate-plane representations and use the mathematical relationship they identify to determine an unknown value.

The materials reviewed for Eureka Math² Grade 3 through Grade 5 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 3, Module 6, Topic B, Lesson 10, Teach Book, Learn, students use appropriate tools strategically as they choose between a ruler and a right-angle tool to draw polygons with specified attributes and consider which tool is useful for creating particular attributes. Students draw polygons with attributes such as an angle larger than a right angle and a pair of parallel sides and determine whether a ruler or right-angle tool can help them construct the required figure. Promoting Mathematical Practice margin note states, “Students use appropriate tools strategically as they use their ruler or right-angle tool to draw polygons with specific attributes. In particular, students select appropriate tools as they recognize which tool helps them draw which attributes. Ask the following questions: What tool can help you draw a pair of parallel sides? Can you use your right-angle tool to help you draw your polygon? Why did you choose to use your ruler? Did that work well?” Practice, Problem 1 states, “Use a ruler and a right-angle tool to draw a polygon with at least 1 pair of parallel sides and all sides measuring 2 centimeters long.” Students then label the equal side lengths, highlight each pair of parallel sides, identify the polygon, and explain how they know.

• Grade 4, Module 5, Topic C, Lesson 10, Teach Book, Launch, students use appropriate tools strategically as they choose among an area model, a tape diagram, a number line, or place value disks to represent a decimal number and explain the reason for their selection. During the Take a Stand routine, students consider the decimal number 0.5, select the pictorial representation they would prefer to use, and discuss with classmates why they chose that representation. Students consider the usefulness of different representations and determine which would be most helpful for comparing numbers in decimal form. Promoting Mathematical Practice margin note states, “When students choose to represent a decimal number with an area model, a tape diagram, a number line, or place value disks and explain the reason for their selection, they are using appropriate tools strategically. Ask the following questions: Why did you choose to use the pictorial representation you did? Which pictorial representation would be the most helpful to you to compare numbers in decimal form? Why?” Practice provides students with opportunities to use different pictorial representations to compare decimal numbers. Problem 1 states, “Use >, =, or < to compare the numbers. Shade the area models to help you. Each square represents 1.” Students use area models to compare 0.4 and 0.47. Problems 4 and 5 then direct students to “Locate the numbers on the number line to help you” as they compare 5.43 and 5.6 and compare 8.76 and 8.67. 

• Grade 5, Module 5, Topic B, Lesson 12, Teach Book, Learn, Multiply Two Mixed Numbers, Classwork, Problem 2, students use appropriate tools strategically as they choose among an area model, rewriting mixed numbers as fractions greater than 1, or using the break apart and distribute strategy to multiply mixed numbers and provide reasoning for their choice. Students use two different methods to evaluate 2\frac{3}{5}\times3\frac{1}{8} compare available approaches, estimate the product to determine whether their result is reasonable, and consider which method or strategy is most efficient for multiplying mixed numbers. Promoting Mathematical Practice margin note states, “When students choose between using the area model, writing mixed numbers as fractions greater than 1, and using the break apart and distribute strategy to multiply mixed numbers and provide reasoning for their choice, they are using appropriate tools strategically. Ask the following questions: How can you estimate the product? Does your estimate sound reasonable? What kind of model or strategy would be helpful? Which method or strategy would be the most efficient to use to multiply mixed numbers? Why?” Practice provides additional opportunities for students to use different representations and strategies to multiply mixed numbers. Problems 1 and 2 ask students to interpret area models and write the multiplication expressions they represent, while Problem 3 asks students to “Multiply. Show your work,” allowing them to apply a strategy to 6\times5\frac{1}{2}. Problem 4 specifically requires students to “Draw an area model to find 3\frac{5}{8}\times4\frac{2}{3}.” 

The materials reviewed for Eureka Math² Grade 3 through Grade 5 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 3, Module 5, Topic A, Lesson 3, Teach Book, Learn, students attend to precision as they precisely measure and partition fraction strips into equal-sized parts and communicate carefully using accurate fraction names and fractional-unit language. Students use a ruler to determine how to partition a 6-inch strip into 3 equal units, place tick marks at precise measurements, fold at each mark, and label the resulting fractional units. Students then create a fraction strip showing sixths and compare the thirds and sixths strips. Teachers guide students to use precise mathematical language as they complete the statements, “There are 3 equal parts in all. The fractional unit is thirds. One unit is called 1 third.” Promoting Mathematical Practice margin note states, “Students attend to precision as they communicate carefully to name fractions and fractional units, and as they make sure the parts of their fraction strips are equally sized. Ask the following questions: How can you describe your fraction strip by using the term fractional unit? When folding your fraction strip, what steps do you need to be precise with? Why?” Practice provides additional opportunities for students to precisely identify equal parts and use accurate fractional-unit language. Problems 1-4 ask students to identify strips partitioned into equal parts, describe and label fractional units, and identify an error in the labeling of a fraction strip. Problem 1 states, “Circle the paper strips that are folded into equal parts.” Problems 2–3 state, “Complete the sentences to describe each fraction strip. Then label each equal part,” including, “There are ___ equal parts in all. The fractional unit is ___. One unit is called ___.” Problem 4 presents a strip partitioned into four equal parts, each labeled “1 third,” and asks, “What mistake does Robin make?”

• Grade 4, Module 6, Topic A, Lesson 4, Teach Book, Learn, students attend to precision as they identify perpendicular line segments, use right-angle tools to verify perpendicularity, and use proper mathematical notation and symbols to name perpendicular sides. Students trace pairs of sides in polygons, use a right-angle tool to determine whether the sides form a right angle, mark right angles with a small square, and record perpendicular relationships using the symbol ⊥, such as \overline{AD}\perp\overline{AB}. Students work with partners to identify and precisely name additional pairs of perpendicular sides and explain how they know the sides are perpendicular. Promoting Mathematical Practice margin note states, “When students identify or draw and use proper notation for perpendicular lines, rays, and line segments, they are attending to precision. Ask the following questions: How can we write \overline{BC} is perpendicular to \overline{CD} using the new symbol we learned? When drawing perpendicular lines, what steps do you need to be careful with? Why?” Practice, Problems 1–2 state, “Trace at least one pair of line segments that appear to be perpendicular on each object. Mark their shared right angle with a small square.” Problem 3 states, “Draw a line segment perpendicular to each given line segment. Use a straightedge and a right-angle tool. Mark the right angle with a small square.” Problems 4–5 direct students, “Use your right-angle tool to find and name the perpendicular sides. If there are no perpendicular sides, write the word None. Mark each right angle with a small square.”

• Grade 5, Module 2, Topic C, Lesson 12, Teach Book, Learn, students attend to precision as they use precise mathematical vocabulary to describe the quantities, parts, and operations involved in subtracting a mixed number from a whole number and explain the subtraction method used to find 5−2\frac{3}{4}. Students use terms such as minuend, subtrahend, part, total, difference, decompose, whole number, ones, and fourths as they decompose the minuend, subtract the whole-number part, and rewrite 1 as \frac{4}{4} to subtract the fractional part. Students connect the vocabulary to the mathematical representations and explain the meaning of the quantities as they record \frac{4}{4}-\frac{3}{4}=\frac{1}{4}, 2+\frac{1}{4}=2\frac{1}{4}, 5-2\frac{3}{4}=3-\frac{3}{4}. Promoting Mathematical Practice margin note states, “As students access their mathematical vocabulary and use terms such as minuend, subtrahend, part, total, difference, decompose, whole number, ones, and fourths to explain the subtraction method used to find 5-2\frac{3}{4}, they are attending to precision. Ask the following questions: How can we describe the subtraction expression 5-2\frac{3}{4} by using the words minuend and subtrahend? What does the word part mean in the number bond? Total? Subtrahend? Minuend? Difference?” Practice, Problem 1 states, “Subtract like units to find the difference,” for 7\frac{3}{8}-4. Problem 2 states, “Decompose the part to take away from the total. Use the number line to show your work,” for 6-2\frac{4}{9}. Problem 3 states, “Use the arrow way to add on to find the unknown part,” for 3-1\frac{2}{5}. Problem 4 states, “Subtract the whole number parts. Then decompose the whole number to take from 1 to subtract,” for 9-5\frac{6}{7}.

The materials reviewed for Eureka Math² Grade 3 through Grade 5 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 3, Module 4, Topic C, Lesson 10, Teach Book, Learn, students look for and make use of structure as they find and combine the areas of smaller rectangles to find the area of a larger rectangle. Classwork states, “Use the Read–Draw–Write process to solve the problem. Liz and Ray have a picnic. They put blankets together to sit on. Liz’s rectangular blanket is 5 feet in length and 7 feet in width. Ray’s blanket is a rectangle with a length of 8 feet and a width of 7 feet. a. Shade the rectangle to show Liz’s blanket. b. Label the length and width of the shaded and unshaded rectangles. c. What is the area of Liz’s blanket? d. What is the area of Ray’s blanket? e. What is the total area of Liz’s and Ray’s blankets?” Students use the structure of the composed rectangle to identify the dimensions and areas of the two smaller rectangles and combine their areas to determine the area of the larger rectangle. Promoting Mathematical Practice margin note states, “Students look for and make use of structure as they find and combine the areas of smaller rectangles to find the area of a larger rectangle. Ask the following questions: How are the smaller rectangles and the larger rectangle related? How can that help you find the area of the larger rectangle? How is finding the area of the larger rectangle similar to a multiplication problem you’ve solved before?” Practice, Problem 1 states, “Gabe uses square units to make two rectangles. a. Label the length and width of each rectangle. Each represents 1 square unit. b. Find the area of each rectangle. c. Gabe pushes rectangles A and B together to make a larger rectangle. Complete the number bond to show the relationship between the areas of the smaller rectangles and the area of the larger rectangle. d. Explain how you found the area of the new, larger rectangle.”

• Grade 4, Module 4, Topic A, Lesson 1, Teach Book, Learn, students look for and make use of structure as they decompose whole numbers into unit fractions by using tape diagrams and number lines and write equations representing the decompositions as sums of unit fractions. Students first analyze a tape diagram representing 1, determine how 1 is decomposed, and write an equation to represent the decomposition. Students then analyze a tape diagram representing 2, identify where 1 is represented within the diagram, explain how 2 is decomposed, and write an equation with parentheses to represent its structure. Students connect this structure to a number line by partitioning both a tape diagram and number line into fourths, identifying each part as one-fourth, and writing an equation showing 1 as a sum of fourths. Students extend the representations to show 2 decomposed into fourths, revise the equation accordingly, and discuss how the number line and tape diagram represent the same equation. Promoting Mathematical Practice margin note states, “As students decompose a whole number into unit fractions by using a tape diagram and a number line and write an equation showing the decomposition as a sum of unit fractions, they are looking for and making use of structure. Ask the following questions: How is the decomposition of 1 by using a tape diagram and by using a number line related? How does this help you write an equation showing 1 as a sum of unit fractions? How is decomposing 2 into thirds similar to decomposing 1 into thirds?” Practice, Problem 1 states, “Complete the equations to match the partitioned circle. The circle represents 1.” Students use the structure of a circle partitioned into fourths to represent 1 as a sum of four unit fractions. Practice, Problem 2 states, “Complete the number bond and equation to match the tape diagram.” Students are shown a tape diagram representing 2 partitioned into six equal parts and connect its structure to a number bond and equation. Practice, Problem 3 states, “Draw and label a tape diagram to represent the equation,” and students represent the equation 1=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6} with a tape diagram.

• Grade 5, Module 4, Topic B, Lesson 11, Teach Book, Learn, students look for and make use of structure as they apply the relationship between addition and subtraction, decompose decimal numbers into simpler parts, and compare different representations and methods to solve decimal-number subtraction problems. Teachers state, “We can use the relationship between addition and subtraction to help us subtract whole numbers and decimal numbers. Write 12.3-4.8= ___What related addition equation can we write to help us find 12.3-4.8? Let a represent the unknown value.” Students use the structure of addition and subtraction as inverse operations to rewrite a decimal subtraction problem as a related addition equation and use the relationship to determine the unknown difference. Promoting Mathematical Practice margin note states, “When students apply their understanding of the relationship between addition and subtraction to solve decimal-number subtraction problems, they are making use of structure. Ask the following questions: How are addition and subtraction related? How can that help you subtract two decimal numbers? What is another way you can write the subtraction problem to help you find the difference?” Learn, Subtract Decimal Numbers by Decomposing, students decompose decimal numbers to make subtraction problems simpler and connect the structure shown with place value disks to a number bond and written solution. Students analyze 9.4-3.7 and explain how the written solution represents the steps modeled with place value disks. Students then apply the method to 8.62-1.09 as teachers ask, “How might you decompose to take from the next unit?” and “How could you make the problem simpler?” Students also discuss when the decomposition method is useful. Learn, Subtract Decimal Numbers by Using a Number Line, students analyze and compare different approaches to subtraction. Students compare two number lines representing 3.4−0.9, including one that subtracts 0.9 at once and another that decomposes 0.9 into 0.4 and 0.5 to reach a whole-number benchmark. Students then solve 2.8−0.03 and 2.8−0.63 and discuss “whether they partitioned the number lines the same way, whether they used the same way to find the differences, or whether they used different ways and why.” Land, Debrief, students generalize the structures across the methods they used by discussing, “How are the methods we use to subtract decimal numbers similar to and different from methods we use to subtract whole numbers?” Students connect unit form, place value, regrouping and renaming, and number lines across whole-number and decimal subtraction. Students also share which method they used for Problem Set, Problems 5–18 and explain why they selected that method. Problem Set, Problems 5 and 6 direct students to “Use a related addition equation to subtract. Let the letter represent the unknown part. Fill in the blanks and use the arrow way to record your thinking.” Problems 7–10 direct students to subtract by using a number bond, and Problems 11–14 direct students to subtract by using a number line. Problems 15–18 state, “Subtract,” and students select a method to determine each difference.

The materials reviewed for Eureka Math² Grade 3 through Grade 5 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 notice and use repeated reasoning to make sense of problems, recognize patterns, and develop 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 3, Module 3, Topic C, Lesson 15, Teach Book, Launch, students look for and express regularity in repeated reasoning as they examine multiplication and division equations with units of 1 and 0, identify what repeats across the equations and representations, and generalize the patterns to reason more efficiently about multiplication and division facts. Classwork, Problem 5 states, “___ divided into ___ equal groups is ___ in each group. ___ \div ___ = ___.” Students work with partners to identify what is known and unknown in the division situation and represent the relationship with an equation. Students then “turn and talk about what patterns they notice in their equations” before using the equations in the next segment to examine patterns across multiplication and division facts. Teachers frame the work by stating, “Today, we will look at patterns in multiplication and division equations to help us learn more facts.” Promoting Mathematical Practice margin note states, “Students look for and express regularity in repeated reasoning as they generalize the patterns they see when looking at equations involving multiplication and division with units of 1 and 0. Ask the following questions: When you look at the answers to problem 1, is anything repeating? How could that help you multiply with 1 more efficiently? What is the same about the class’s drawings for problem 5?” Practice, Problem 1-3 state, “Use the equal groups pictures to fill in the blanks. 1. 3 groups of 1 is . 3\times1= _. 3 divided into groups of 1 is ___ groups. 3\div1= ___. 2. 8 groups of 1 is . 8\times1= _. 8 divided into 8 equal groups is in each group. 8\div8= _. 3. 1 group of 6 is . 1\times6= _. 6 divided into 1 group is ___ in each group. 6\div1= ___.” Students use the repeated examples and equal-groups representations to recognize and apply general relationships involving multiplication and division with 1, including that multiplying or dividing a number by 1 results in that number and that dividing a number by itself results in 1. 

• Grade 4, Module 1, Topic B, Lesson 6, Teach Book, Learn, students look for and express regularity in repeated reasoning as they repeatedly multiply and divide place value units by ten, identify what repeats, generalize the relationship between adjacent place value units, and use the pattern to reason more efficiently. Students repeatedly draw 10 of one place value unit, bundle the units to make one of the next larger unit, and extend the process across the place value chart. Students Think–Pair–Share about what happens and determine, “Every time we have 10 times as much as a unit, we can rename it as 1 of the next larger unit. We shift one place to the left on the place value chart every time.” Teachers then explain, “Representing numbers to show 10 times as many by drawing dots and then bundling and renaming may take a long time. We can represent the process more efficiently by drawing 1 dot and then labeling the arrow to represent the multiplication.” Promoting Mathematical Practice margin note states, “When students repeatedly multiply by 10 and recognize a digit in one place represents 10 times the value it represents in the place to its right, they are looking for and expressing regularity in repeated reasoning. Ask the following questions: What patterns do you notice when you multiply a number of units by 10? How can that help you find 10\times6 thousands or 10\times6 ten thousands? What is the same about the relationship between the value in any one place and the value of the place to its right?” Students extend the repeated reasoning to division by ten and determine, “Every time we divided by ten, the 9 shifted to the right one place value.” In Land, students use the pattern to reason about new units and explain, “We can use the pattern and shift 1 million one place to the left to get 1 ten million.” Exit Ticket states, “a. 1 ten thousand is _ times as much as 1 thousand. b. Explain how you know your answer is correct.”

• Grade 5, Module 4, Topic A, Lesson 5, Learn, Teach Book, students look for and express regularity in repeated reasoning as they repeatedly divide and multiply decimal numbers by powers of 10, identify the relationship between the exponent in the power of 10 and the change in the value of each digit, and apply the pattern to determine products and quotients more efficiently. Classwork, Problems 7–10 state, “Find the quotient and write it in standard form. Then write a related multiplication equation with the power of 10 expressed as a fraction. 7. 4\div10=___; 8. 0.3\div10^2=___; 9. 72.6\div10^3=___; 10. Determine the power of 10 that makes a true statement. 43.2\div___$$=0.432$$.” Students repeatedly divide decimal numbers by different powers of 10, connect each quotient to a related multiplication equation, and use the repeated relationship to determine an unknown power of 10. Promoting Mathematical Practice margin note states, “When students repeatedly divide by powers of 10 to notice the relationship between the power of 10 in the divisor, the decimal number dividend, and the quotient; multiply by powers of 10 to notice the relationship between the exponent and the product; and apply this understanding to divide and multiply decimal numbers by powers of 10 more efficiently, they are looking for and expressing regularity in repeated reasoning. Ask the following questions: What patterns do you notice when you divide a decimal number by a power of 10? How can that help you more efficiently determine the quotient? What patterns do you notice when you multiply a decimal number by a power of 10? How can that help you more efficiently determine the product? Does the exponent of the power of 10 always represent the number of places that each digit shifts from the decimal number dividend to the quotient? Explain.” In Practice, Problems 1–3 state, “Use the place value chart to complete the equations,” and have students work with repeated examples including 0.8\times10\times10, 0.8\times100, 0.6\div100, and 30\div1,000 while using place value charts to make the repeated changes in the values of the digits visible. Students then apply the generalized relationship without the place value chart in Practice, Problems 4–7, which state, “Find the product or quotient and write it in standard form,” including “$$3.04\div10=$$ , 0.205\div10= , 3.205\times10^3= , and 0.9\times10^2= .” Students use the patterns established through repeated reasoning with the place value charts to efficiently multiply and divide new decimal numbers by powers of 10.