K-2nd Grade - Gateway 2

The materials reviewed for Eureka Math² Kindergarten through Grade 2 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 in Grades 1-2, which assess the lesson’s learning, and Practice, which provides additional problems aligned to the lesson’s learning objectives. In Kindergarten through Grade 2, Topic Tickets assess students’ independent understanding of the mathematics within each topic. 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:

• Kindergarten, Module 1, Topic B, Lesson 9, Teach Book, Learn, students develop conceptual understanding as they count objects and reason that the number of objects in a set remains the same when the objects are rearranged. Students connect the quantity of the set to a written numeral and explain how they know the quantity remains unchanged after the arrangement changes. The teaching notes state, “Count and use numeral cards to describe a set before and after it is rearranged. Display 3 Unifix Cubes in a linear configuration and cards 1–5. ‘How many cubes are there?’ (3) ‘Which number card tells how many cubes there are?’ (The 3 card) Move the 3 card below the cubes. Move the cubes slowly, deliberately letting students see you create a scattered configuration. ‘How many are there now? How do you know?’ Some students will know right away that there are still 3 cubes. Others will need to count to make sure. Validate both strategies by calling on students to share how they know there are still 3 cubes. (I know there are still 3 because I counted them again. I know there are still 3 cubes because I saw you move them. If there were 3 before, then that means there are still 3 now.) Which number card tells how many there are?” (K.CC.4)

• Grade 1, Module 5, Topic D, Lesson 19, Teach Book, Problem Set, students demonstrate conceptual understanding of place value as they use tens and ones to represent quantities and determine how adding tens changes the value of a two-digit number. The teaching notes state, “Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex. Directions and word problems may be read aloud. Students may use the Double Place Value Chart removable as they complete the problems.” Problem 1 shows 3 boxes of crayons and 4 additional crayons and states, “Kit has 3 boxes of crayons and 4 more. She gets 2 more boxes. How many crayons does she have now? Write, Kit has _____ crayons.” (1.NBT.4)

• Grade 2, Module 2, Topic D, Lesson 21, Teach Book, Fluency, students demonstrate conceptual understanding of place value as they represent two-digit numbers with place value disks and connect each representation to the number expressed in unit form as tens and ones. The teaching notes state, “Invite students to make a chart on their desks. Distribute a place value disks set to each student. Display the chart and the number 24. ‘Use your place value disks to show the number 24. Arrange them in 5-group formation.’ Give students time to work. Circulate and provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the place value disks on the chart. ‘On my signal, say the number in unit form. Ready?’ (2 tens 4 ones) Display the number in unit form. Repeat the process with the following sequence: 27, 36, 45, 52, 60, 71, 89.” (2.NBT.1)

The materials reviewed for Eureka Math² Kindergarten through Grade 2 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 skills with key mathematical concepts. Students independently demonstrate procedural skill through Exit Tickets, which serve as formative assessments that close the 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 K and 1–2 Implementation Guides state, “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:

• Kindergarten, Module 5, Topic A, Lesson 7, Teach Book, Learn, students develop procedural skill and fluency with addition as they choose and use a strategy to find the total and compare the efficiency of different strategies. The teaching notes state, Write 3+6= ___ and tell students they will find the total in a different way than they did before. Remind students of the available tools, such as cubes, 10-frames, number paths, whiteboards, fingers, or drawing. Invite students to share their thinking with a partner. Then ask them to use their chosen way to find the total. ‘What is the total of 3+6?’ Ask a few students to share their strategy for finding the total. ‘Turn and talk to your partner: Which way was easier for you to solve, the way you solved 4+3 or the way you solved 3+6? Why?’” (K.OA.5)

• Grade 1, Module 2, Topic A, Lesson 2, Teach Book, Land, Problem Set, students independently demonstrate procedural skill and fluency with subtraction as they subtract a number from itself and subtract 0 and use generalizations to solve related problems efficiently. The teaching notes state, “Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex. Sample solutions are at the end of this lesson. Directions may be read aloud. As needed, students may use mental math, an anchor chart, their fingers, or cubes to solve. 1. Subtract. 5-5=__, 9-9=__, 7-7=__, 12-12=__. Write 2 number sentences like these.” The Problem Set also includes four problems in which students subtract 0. (1.OA.6)

• Grade 2, Module 2, Topic A, Lesson 5, Teach Book, Learn, students develop procedural skill and fluency with addition within 100 as they use benchmark numbers to make addition problems easier to solve and independently apply the strategy to related problems. The teaching notes state, “Write 38+15 and have students do the same. ‘Which of these numbers is closer to a ten, a benchmark number?’ ‘Which ten is 38 closest to?’ ‘How much more do we need to add to 38 to make 40?’ ‘Where can we get the 2 from?’ ‘How can we decompose 15 to get 2?’ (We can decompose 15 into 2 and 13.) Direct students to draw a number bond to show how to decompose 15 into 2 and 13. Circle 38 and 2 as you demonstrate. ‘What expression shows how we can make a ten?’ (38+2+13) ‘What expression shows how we can make a simpler problem?’ (40+13) ‘How do we know that 40+13 is equal to 38+15?’ (We took 2 from the 15 and gave it to the other addend. They have the same total. I know 40 is 2 more than 38 and 13 is 2 less than 15.) ‘Did we take anything or add anything to the total?’ ‘What does our work show?’ (We broke 15 into 2 and 13 to make a ten. 38+2+13 is equal to 40+13. 40+13=53 Invite students to turn and talk about why 40+13 is easier to add than 38+15. Write 38+25 and prompt students to find the answer independently on their whiteboards. Invite students to think–pair–share about how making a benchmark number, such as a ten, is a helpful simplifying strategy for addition problems. (It helps me to think about a problem by using numbers that are easier for me to add in my head. I can move part of one addend to the other addend. It’s the same as the original problem but easier to add in my head.)” (2.NBT.5)

The materials reviewed for Eureka Math²  Kindergarten through Grade 2 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:

• Kindergarten, Module 2, Topic A, Lesson 5, Teach Book, Land, Debrief, students apply their understanding of shapes and positional words as they identify shapes based on their positions relative to other objects in a visual context. Students examine a picture of birthday candles and toothpicks and respond to prompts such as, “I’m thinking of the shape that is beside the hexagon. What shape am I thinking about?” Students then select shapes in the picture and use position words such as “above, below, beside, in front of, behind, around” to describe their locations. (K.G.1)

• Grade 1, Module 2, Topic C, Lesson 11, Teach Book, Learn, students apply their understanding of subtraction in a routine problem as they use craft sticks to represent quantities, determine an unknown part, and write a subtraction equation to represent a real-world context. The problem states, “I put 9 pencils on the desk. Some fell off. Now there are 3 pencils. How many pencils fell off the desk?” Students model the situation with craft sticks and determine that 6 pencils fell off, then write the equation 9-6=3 to represent the situation. (1.OA.1)

• Grade 2, Module 2, Topic C, Lesson 14, Teach Book, Launch, students apply their understanding of addition and subtraction in a routine problem as they use the Read–Draw–Write process and a tape diagram to represent quantities and determine an unknown value in a real-world context. The problem states, “Mrs. King makes 52 cups of ice cream. She gives 28 cups of ice cream to the students. How many cups are left?” Students use addition or subtraction strategies to solve and write an equation with an unknown, such as “52-28=__” or “28+__=52” (2.OA.1)

The materials reviewed for Eureka Math² Grades 3 through Grade 5 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:

• Kindergarten, Module 4, Topic B, Lesson 5, Teach Book, Learn, students demonstrate conceptual understanding and procedural skill and fluency as they decompose 5 in multiple ways, distinguish between the total and parts, and record each decomposition with numerals in a number bond. Students sort 5 bears by different attributes to produce different partners to 5 and “record the sort with numerals in a number bond.” The lesson states, “We sorted, or broke, the total into two parts,” and guides students to a final decomposition with 0 and 5 as the parts. (K.OA.3)

• Grade 1, Module 5, Topic A, Lesson 6, Teach Book, Learn, students demonstrate conceptual understanding and procedural skill and fluency as they use dimes and pennies to add and subtract 10, connect changes in the quantity to changes in the tens place, and repeatedly determine 10 more or 10 less than a number. Students determine that Ko has 27 cents and then 17 cents after she removes a dime, explaining, “10 cents less than 27 cents is 17 cents. Dimes are the same as ten cents. We can use dimes to show adding 10 or taking 10.” Students then find 10 more than 54 and 10 less than 42 and connect these calculations to place value: “When we add a 10, the digit in the tens place is 1 more” and “When we take a 10, the digit in the tens place is 1 less.” (1.OA.6)

• Grade 2, Module 4, Topic B, Lesson 7, Teach Book, Launch, students demonstrate conceptual understanding and procedural skill and fluency as they use place value disks to represent addition within 1,000, connect the concrete representation to addition expressions, and prepare to represent the addition using vertical form. Students examine a place value model representing 136+285 and identify the hundreds, tens, and ones before writing an addition expression to represent the disks. Students write 136+285; 285+136, and the lesson states, “Today, we will show addition by using a place value model and vertical form.” (2.NBT.7)

The materials reviewed for Eureka Math² Kindergarten through Grade 2 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:

• Kindergarten, Module 3, Topic A, Lesson 3, Teach Book, Learn, students make sense of measurement problems and persevere in solving them as they determine how to create objects about the same length as other objects. In Sort by Length, students develop a strategy to cut a piece of yarn that is about the same length as a bracelet without directly comparing the objects, then test their estimate and determine whether the yarn is longer than, shorter than, or about the same length as the bracelet. Promoting Mathematical Practice margin note states, “When students try to cut a piece of yarn that is the same length as the bracelet, they make sense of problems and persevere in solving them. Since students cannot compare the yarn directly with the bracelet, encourage them to persevere in finding a strategy to develop their best guess. Using the questions in this section to analyze the chart showing all of the students’ pieces of yarn encourages students to evaluate why this task is challenging.” In Make It Long Enough, students independently create a clay or dough bracelet that is long enough to fit around their wrist. During Land, students reflect on how they determined whether their strategies worked and describe how they adjusted their work. Students explain, “I put the bracelet around my wrist to see if it fit” and “I rolled out a long piece of dough and put it on my wrist. Then I cut off the part I didn’t need.” Students also compare approaches across the activities and explain, “We could put the yarn next to the cube stick to see if it was too long or too short” and “It was easy to put more cubes on the stick until it was about the same as the yarn.”

• Grade 1, Module 1, Topic A, Lesson 5, Teach Book, Learn, students make sense of counting problems and persevere in solving them as they organize and represent categorical data. In Sort and Count a Set, students select a set of objects, develop a plan for counting the objects accurately, carry out their plan, and adjust their approach as needed. As students work, they explain why they chose their plan, monitor which objects they have already counted and which still need to be counted, and consider ways to reorganize the objects to make counting easier and more accurate. Promoting Mathematical Practice margin note states, “When students select a set of objects, make a plan about how to count, carry out the plan, and adjust the plan as needed, they make sense of problems and persevere in solving them. Ask the following questions: How could you explain your plan in your own words? Is your plan working? Is there something else you could try?” As students work, teachers ask, “What is your plan? Why did you choose that plan? Show or tell me how you are counting. How are you keeping track of what you already counted and what you still need to count? How can you organize to make counting easier?” Students use strategies such as moving objects as they count, using tools to group or organize objects, or organizing objects into 5-groups. Teachers select student work that demonstrates accurate sorting and different counting strategies for students to share and compare.

• Grade 2, Module 1, Topic D, Lesson 19, Teach Book, Learn, students make sense of comparison word problems and persevere in solving them as they determine how to represent the quantities and find the unknown difference. Students consider different representations, including a number line, measuring tape, tape diagram, or equation, independently select a tool or strategy, record their approach, and consider whether their representation makes sense in the context of the problem. Promoting Mathematical Practice margin note states, “When students solve comparison problems, they make sense of problems and persevere in solving them. In measurement problems, models such as the measuring tape and tape diagram help students represent the problem more concretely. Ask the following questions: What are some things you could try to start solving the problem? Does your drawing make sense with the problem?” During the activity, students independently choose from the available measurement tools to represent and solve the problem and record their strategy. Teachers ask, “How can you represent the information in the problem?” and select student work that demonstrates different strategies and progresses toward more abstract representations of finding the difference. During Land, Debrief, students find a partner who solved Problem 1 in a different way and compare their strategies. Teachers promote reflection by asking, “What worked well?” and “Why did you ___?” Students then consider a different strategy for future problems when the teacher asks, “What strategy did you hear today that you are going to try next time?” Students identify approaches such as, “I am going to draw my own number line and make bigger hops” and “I am going to count back to a benchmark number.” Problem Set, Problem 1 states, “Sal jumps rope 65 times. Hope jumps rope 40 times. How many more times does Sal jump rope than Hope?”

The materials reviewed for Eureka Math² Kindergarten through Grade 2 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:

• Kindergarten, Module 6, Topic B, Lesson 8, Teach Book, Learn, students reason abstractly and quantitatively as they move between the context of a bracelet story problem and concrete, visual, and symbolic representations of the quantities and relationships in the story. After independently solving the problem with self-selected tools and strategies, students connect Ko’s 10 bracelets, Isaac’s 7 bracelets, and the total of 17 bracelets to representations using cubes, Hide Zero cards, a number bond, and an addition sentence, and explain what each part of their representation means in the context of the story. Promoting Mathematical Practice margin note states, “As students work to solve the bracelets problem, they reason abstractly and quantitatively. Students decontextualize by using representations such as cubes, numbers, number bonds, and number sentences to represent the bracelets. They recontextualize by explaining which parts of the story the referents represent. In this instance, students are also asked to reason abstractly by making connections between the different representations, recognizing that the parts and total can be represented in different, but equivalent, ways.” During the activity, selected students share how they represented and solved the same problem using different approaches. One student explains using Hide Zero cards, “I took the 10 to show Ko’s 10 bracelets. Then I took the 7 to show Isaac’s 7 bracelets. I put the cards together to make 17,” while another explains using a number bond, “I put 10 cubes in one part and 7 cubes in another part. Then I counted the cubes, and there were 17.” A third student connects the quantities and operation in the context to the symbolic representation by explaining, “I wrote 10 plus 7 because I knew I needed to add their bracelets to get the answer. 10 plus 7 is 17, so I knew they had 17 bracelets.” Teachers display the different representations and ask students to identify the referents in each drawing or number sentence, including where they see Ko’s bracelets, Isaac’s bracelets, and the total number of bracelets. Students compare and connect the different representations and recognize that the same quantities and part-total relationship can be represented concretely, visually, and symbolically. 

• Grade 1, Module 4, Topic B, Lesson 6, Teach Book, Learn, students reason abstractly and quantitatively as they use numerical measurements to compare and order the heights of objects rather than relying on direct comparison. Students connect the measured heights of plants to the quantities they represent, use those measurements to determine which plant is taller or shorter, and represent the quantitative relationships symbolically with comparison number sentences using > and <. Promoting Mathematical Practice margin note states, “When students use measurements to order objects, they reason abstractly and quantitatively because they use numbers to order the objects instead of direct comparison. This lesson creates a need for measuring by presenting scenarios where it is more convenient to measure objects than to compare them directly.” During the activity, students compare Sam’s 12-centimeter plant and Wes’s 9-centimeter plant and explain how the measurements show which plant is taller or shorter. Teachers connect the quantities in the context to symbolic representations by asking, “Is Sam's plant taller or shorter than Wes's plant? How do we know?” and recording students’ reasoning as 12>9, then asking, “Is Wes's plant taller or shorter than Sam's plant? How do we know?” and recording 9<12. Students then independently select two plants to complete contextual comparison statements, use the corresponding measurements to determine whether the statement is true, and write a comparison number sentence to represent the relationship between the plants’ heights. For example, students determine that Sam’s plant is shorter than Baz’s plant because Sam’s plant is 12 centimeters tall and Baz’s is 14 centimeters tall, represent the relationship as 12<14, and read the number sentence aloud to a partner. Problem Set, Problem 3 states, “Compare the lengths. Write the letters. is shorter than . is longer than .” Students then write the corresponding measurements with the < and > symbols.

• Grade 2, Module 3, Topic D, Lesson 17, Teach Book, Learn, students reason abstractly and quantitatively as they connect the quantities of minutes and groups of 5 minutes to their representations on an analog clock. Students use individual cubes to represent minutes and sticks of five cubes to represent groups of 5 minutes, connect these concrete representations to the numbers on the clock, and interpret the meaning of a number differently depending on whether it is referenced by the hour hand or the minute hand. Promoting Mathematical Practice margin note states, “Students reason quantitatively and abstractly when they use cubes to represent minutes and a stick of five cubes to represent every fifth minute as they build a clock. Ask the following questions: How are the cubes and the sticks of five cubes labeled on the clock? How does the number that the minute hand is pointing to relate to the number of 5 minutes that have gone by?” During the activity, students build a clock by placing one cube for each minute and combining the cubes into sticks of five to represent successive 5-minute intervals. Once the unit of five is established, students place a 5-stick at each interval and label the corresponding number of minutes. Teachers connect the quantities to the clock representation by asking, “What do you think each number represents when the minute hand points to them?” When the clock shows 5:05, students consider why the minute hand points to 1 even though 5 minutes have elapsed and explain that “The 1 represents 5 minutes” and “The 1 stands for 1 group of 5 minutes.” Students then connect the quantities of minutes, groups of five, and clock numbers as they count the hour as 5, 10, 15, …, 60 minutes and as 1 five, 2 fives, 3 fives, …, 12 fives while moving the minute hand on their paper clocks. Practice, Problem 2 states, “Plot each time on the number line.” Students take a time represented with hours and minutes and represent that time on a number line from zero to 60 minutes.

The materials reviewed for Eureka Math² Kindergarten through Grade 2 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:

• Kindergarten, Module 5, Topic B, Lesson 13, Teach Book, Land, Debrief, students construct viable arguments and critique the reasoning of others as they determine which subtraction equation matches a picture of baseballs, explain their reasoning, and identify the flaw in the teacher’s argument. Students establish 6 as the total by counting all the baseballs and determine that 6-2=4 matches the picture. The teacher challenges students by arguing that 4-2=2 also matches because 4 baseballs and 2 baseballs can be seen in the picture. Students critique the teacher’s reasoning and explain why the equation does not match, using the relationship between the parts and total as evidence to support their argument. Students are asked, “Why doesn’t this match?” and “Use the words part and total to explain why this number sentence matches.” Promoting Mathematical Practice margin note states, “As students explain why one equation matches the baseballs and the other doesn’t, they construct viable arguments and critique the reasoning of others. The argument presented by the teacher, that the baseballs match 4-2=2, gives students an opportunity to find the flaw in someone else’s reasoning. Encouraging students to use precise language, such as part and total, helps them to see how their specific mathematical knowledge can be used to explain the teacher’s mistake.”

• Grade 1, Module 2, Topic D, Lesson 16, Teach Book, Land, Debrief, students construct viable arguments and critique the reasoning of others as they discuss, defend, and evaluate the efficiency of counting on and counting back strategies for subtraction. Students consider the problem 11-9 and evaluate the teacher’s claim that counting back is the best strategy because the problem involves subtraction. Students indicate whether they agree or disagree with the strategy and explain their reasoning. Students who disagree argue that counting on from 9 is more efficient because 9 is close to 11 and requires fewer counts, and the class discusses when counting on or counting back is the more efficient strategy. Students are asked, “Why is counting on more efficient for this problem? Convince me,” and use their reasoning to establish when each strategy is efficient. Promoting Mathematical Practice margin note states, “Students construct viable arguments and critique the reasoning of others when they discuss and defend the efficiency of strategies. The following questions provided in the sample dialogue promote this practice: Why is counting on or counting back more efficient for this problem? Why is the other counting strategy not as efficient for this problem? What can you do to figure out if counting on or counting back will be more efficient?” 

• Grade 2, Module 2, Topic B, Lesson 12, Teach Book, Learn, students construct viable arguments and critique the reasoning of others as they evaluate whether tens can be added first when adding any two-digit numbers, justify their conclusions, listen to their classmates’ reasoning, and determine whether they agree or disagree and why. Students work with 84+47 and compare two written methods for finding the sum, discussing how the recordings are similar and different. Students then evaluate the statement, “When I add two-digit numbers, I can add the tens first,” as always, sometimes, or never true and justify their answers. Students discuss their reasoning with a partner and then share with the class, using examples and nonexamples to support their claims and evaluating their classmates’ arguments before the class reaches a consensus. Promoting Mathematical Practice margin note states, “When students listen to and evaluate their classmates’ analysis of whether they can add the tens first in any two-digit number, they are constructing viable arguments and critiquing the reasoning of others. Students are explicitly asked to evaluate a statement as always, sometimes, or never true and to justify their answers. They can also use this as an opportunity to critique their classmates’ arguments if they disagree. Ask the following questions: When is your classmate correct? Is there a general statement you can make that the class can agree to?” Practice directs students to, “Add. Show how you know.” Practice, Problem 1 states, “75+27=” and Problem 2 states, “___=96+68.” Students solve each addition problem and show how they know, providing mathematical reasoning to support their solutions.

The materials reviewed for Eureka Math² Kindergarten through Grade 2 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:

• Kindergarten, Module 1, Topic A, Lesson 4, Teach Book, Learn, Problem Set, students model with mathematics as they develop their own rule to sort a collection of physical objects into three groups, count the objects in each group, and create a drawing to represent how they sorted the objects. Students sort the objects based on attributes they identify and explain how their sorting rule determines where each object belongs. Teacher directions state, “Distribute sorting bags to individual students or pairs. Make number paths available for students who want to use them for counting. Invite students to sort the objects in their bags into three groups. Allow time for them to consider the differences between the objects in a bag and develop their own sorting rule. Support students by providing a way to sort only if necessary.” Students describe how they sorted their objects and connect the characteristics of individual objects to their groups as teachers ask, “How did you sort into groups?” and “How did you decide where to put this? (Holding object.) Why didn’t it fit into the other group(s)?” Students then count the objects in their groups and, if time permits, draw the groups on the Problem Set page, creating a mathematical representation of their physical sort that they can compare to the original objects. Promoting Mathematical Practice margin note states, “When students draw their sort and compare their drawing to the physical objects, they are modeling with mathematics.” Problem Set extends students’ use of mathematical models by stating, “I can sort things into three groups. I can count how many in each group” and directing students to “Draw your groups to show how you sorted.” Students use their drawing to represent the groups they created with the physical objects, connect the representation to their self-selected sorting rule, and compare the drawing with the original sort to determine whether their model accurately represents how the objects were classified.

• Grade 1, Module 2, Topic B, Lesson 6, Teach Book, Learn, students model with mathematics as they use the Read–Draw–Write process to represent and solve addition and subtraction result-unknown situations and connect their drawings and number sentences to the actions in the stories. Related Subtraction Word Problem states, “9 bugs are on a leaf. 3 bugs fly off the leaf. How many bugs are on the leaf now?” Students retell the story, draw and label a representation showing the starting number of bugs and the bugs that fly away, and use their drawing to determine how many bugs remain. Students connect their mathematical representations to the context by identifying what each number and symbol in their number sentence represents and where those quantities and actions appear in their drawings. Students compare their work with classmates’ representations and consider whether their drawings and number sentences accurately represent what happened in the story. Promoting Mathematical Practice margin note states, “When students use a drawing to represent the story, label their drawing, and use their drawing to solve the problem, they model with mathematics. Ask the following questions: How did you show the bugs that were on the leaf at the start? How did you show the bugs that flew on or off the leaf? How did you show the bugs that are on the leaf now?” Practice extends students’ use of mathematical models by providing related addition and subtraction situations and asking students to use the Read–Draw–Write process to represent and solve each problem. Practice, Problem 1 states, “5 ducks are in a pond. 3 more fly into the pond. How many ducks are in the pond now?” Students draw the situation, write a number sentence, and complete a contextual solution statement.

• Grade 2, Module 3, Topic C, Lesson 13, Teach Book, Launch, students model with mathematics as they choose and draw a mathematical representation to solve a real-world equal-sharing problem and use their models to determine how five granola bars can be shared equally among four friends. Students are presented with the problem, “4 friends have 5 granola bars. They want to share them equally. How many granola bars will each friend get?” Students identify the important quantities and equal-sharing relationship in the situation, choose how to represent the problem, and partition the granola bars into fourths to determine each friend’s share. Students consider different ways the same-size granola bars can be partitioned into fourths, compare their models with classmates’ approaches, and make connections among representations that use differently shaped equal parts while still accurately modeling the same situation. Promoting Mathematical Practice margin note states, “When students choose and draw a model to solve a real-world problem, they are modeling with mathematics. Ask the following questions: What key pieces of information should be in your model? Do you agree or disagree with a different approach? Why?” Practice, Problem 4 extends students’ use of mathematical models through another real-world situation and provides spaces for students to draw and write their solution. Problem 4 states, “Miss Wells picks 65 apples. She uses some of the apples to bake pies. She has 27 apples left. How many apples does Miss Wells use?” Students create a drawing to represent the known and unknown quantities and their relationship, use the model to determine the number of apples Miss Wells uses, and interpret and communicate the result in the context of the situation.

The materials reviewed for Eureka Math² Grade Kindergarten through Grade 2 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 providing a variety of tools, modeling their effective use, and encouraging students to make strategic decisions about when and how to use them. 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:

• Kindergarten, Module 6, Topic C, Lesson 15, Teach Book, Learn, Scavenger Hunt, students use appropriate tools strategically as they search among assorted math tools for a tool that makes it easy to see 10 and use their selected tool to count by tens. Students consider the mathematical purpose of the available tools, select a tool based on the specific attribute that makes it useful for counting by tens, and justify why their chosen tool is appropriate. Promoting the Standards for Mathematical Practice Promoting Mathematical Practice margin note states, “Students choose appropriate tools strategically when they search for a tool that makes it easy to see 10. Specifically instructing students to look for a tool with this property will help them choose tools more strategically when problem solving in the future.” During the scavenger hunt, students are directed to “look for tools that help us count by tens” and find “a certain kind of math tool: One that makes it easy to see 10.” Students may select a tool that shows exactly 10 or multiple groups of 10, and unexpected tool choices are accepted when students provide a solid justification for their selection. Students then demonstrate to a partner how their chosen tool can be used to count by tens and explain what makes the tool useful. For example, students consider how a two-hands mat makes 10 easy to see because two groups of 5 make 10 and use multiple mats to count 10, 20, 30, 40. The discussion also provides students an opportunity to consider the limitations of a tool when the teacher presents a dot card on which 10 is not easily recognizable and asks, “Is it easy to see 10 on this tool?” Students distinguish between tools that make the mathematical structure useful and visible and those that do not, providing experience selecting, using, justifying, and evaluating mathematical tools based on their effectiveness for counting by tens.

• Grade 1, Module 5, Topic D, Lesson 15, Teach Book, Launch, students use appropriate tools strategically as they independently choose among strategies and tools from their mathematical toolbox to solve a subtraction word problem and justify their selection. Students consider the situation, “Miss Lin had a roll of 90 stickers. She gave 30 stickers to students. How many stickers are left on the roll?” and use the Read–Draw–Write process to solve the problem. Students self-select strategies and tools such as base 10 rods, fingers, personal whiteboards, number paths, or mental math and use a representation to justify their solution. Students then compare the different ways the problem was solved and consider whether their selected approach worked well and whether another approach might be helpful. Promoting Mathematical Practice margin note states, “As students decide which strategy to use to solve the sticker problem, they use appropriate tools strategically. Throughout the year, students have built up their mathematical “toolbox” of strategies, which they can now apply to this new problem type. Encourage students to think strategically by asking the following questions: Why did you choose to solve the problem that way? Did it work well? What other way could you use to solve the problem? Why might that way be helpful?” Practice, Problem 1 provides additional opportunities for students to apply strategies and representations as they add and subtract by tens. Students are directed to “Add or subtract. Show how you know” as they solve 20+40, 70-40, 40+30, 80-30, 60+20, 90-20. 

• Grade 2, Module 5, Topic B, Lesson 11, Teach Book, Launch, students use appropriate tools strategically as they select among available measurement tools to measure and compare the lengths of space creatures and consider which tools and measurement techniques are most effective for different measurement needs. Students first compare the heights, arms, and legs of three space creatures and consider how measurement tools can help them make more precise comparisons. Students reason about how to measure straight and bent lengths, considering a ruler with a mark-and-move-forward strategy as well as using string to trace a bent length and then measuring the string with a ruler. Students then select appropriate measurement tools to create their own space creature, considering which tool would be most helpful for measuring different body parts and how to use the selected tool accurately. Promoting Mathematical Practice margin note states, “When students select among available measurement tools to create a space creature, they use appropriate tools strategically. Ask the following questions: Which tool would be the most helpful to draw the creature’s body? The legs? The arms? What side of the double-sided meter stick should you use?” Practice, Problem 1 students are directed to “Cut out the ruler. Measure and compare.” They use the ruler to determine the approximate lengths of Crayon A and Crayon B and then determine how many inches shorter or longer Crayon A is than Crayon B. 

The materials reviewed for Eureka Math² Kindergarten through Grade 2 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:

Kindergarten, Module 2, Topic A, Lesson 1, Teach Book, Learn, students attend to precision as they use mathematically relevant attributes to sort and describe flat shapes and accurately use the terms straight sides, curves, and corners. Students examine shapes by looking at, feeling, or tracing them to determine whether they have straight sides, curves, or both, and use these attributes to determine where each shape belongs on a sorting chart. Teachers guide students to use precise mathematical language as they describe the attributes they observe and explain whether shapes have been sorted correctly. Students evaluate Puppet’s incorrect placement of a shape and use the sentence frame, “I think (we should move the shape or the shape should stay) because ___,” to explain their reasoning based on the shape’s attributes. Students then sort their own shapes and determine whether they agree or disagree with their classmates’ placements while teachers “encourage students to use the words straight sides, curves, and corners as much as possible.” Promoting Mathematical Practice margin note states, “By correctly using mathematical attributes, such as straight side, corners, and closed to sort and describe shapes, students attend to precision. Eventually students will understand that these attributes are mathematically relevant because they can be used to categorize shapes, in contrast with nonmathematical attributes such as color. For a shape to be a triangle, it must have 3 sides and 3 corners, but the shape’s color does not affect whether a shape is a triangle.”

Grade 1, Module 4, Topic A, Lesson 1, Teach Book, Learn, students attend to precision as they accurately compare and order objects by length, align endpoints before making comparisons, and use precise measurement language such as longer, taller, and shorter. Teachers define length as “how long something is from one end to the other end” and guide students to recognize that the endpoints, or places where objects start and end, must be aligned to make an accurate comparison. Students work with partners to select and align two classroom objects and determine which is longer, then use an Ordering Mat to align the endpoints of five objects and order them from shortest to longest. Teachers ask students to explain and check the precision of their comparisons by asking, “How did you order your objects? How can you check to make sure the order is correct? What can you do if you see an object is out of order? Which object is shortest? Which is longest? How do you know? Why is it important to line up the endpoints?” Students compare their work to an accurate sample and revise their ordering as needed. Promoting Mathematical Practice margin note states, “As students order objects by height, they attend to precision. They do this by using appropriate words like longer, taller, and shorter, and in practice as they align the endpoints of objects before comparing their lengths. The suggested questions in this segment are designed to promote this practice and help students notice the ways they are attending to precision.” Problem Set, Problem 1 presents five objects aligned at a common endpoint and states, “Order from shortest to longest.” Problem 2 asks students to use the letters representing the objects to complete the comparative statements, “___ is shorter than ” and “ is longer than ___.”

Grade 2, Module 5, Topic B, Lesson 10, Teach Book, Land, Debrief, students attend to precision as they express the length of an object using different measurement units, accurately include both the number and unit when reporting measurements, and explain how the size of the unit affects the number of units needed to measure an object. Students revisit Problem Set work in which a fish tank sticker is measured as 4 inches and 10 centimeters and determine that both measurements are correct. Students explain that the measurements are different because centimeters are smaller than inches, so more centimeters than inches are needed to measure the same length. Teachers guide students to precisely communicate the relationship between unit size and the number of units needed and connect this relationship to other units, including hundreds, tens, and ones; pennies, nickels, dimes, and quarters; and units of time. Teachers also ask, “When we record measurements, why is it important to include the number and the unit?” Students explain that “the unit size affects how many units it takes to measure the length” and recognize that a measurement is not precise without identifying its unit. Promoting Mathematical Practice margin note states, “Students communicate with precision when they express the length of an object by using different units and can justify why the unit makes a difference to the reported length. Ask the following questions: Which units, inches or centimeters, are the most efficient to express the length of larger objects? Why might you choose to use centimeters to measure an object if it would take fewer inches?” Practice, Problems 1-2 ask students to measure lines in both inches and centimeters and record each measurement with its appropriate unit. Students then answer, “Which unit did you use more of? inches? centimeters? Why?” to explain the relationship between unit size and the number of units needed to measure the same length.

The materials reviewed for Eureka Math² Kindergarten through Grade 2 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:

• Kindergarten, Module 4, Topic C, Lesson 16, Teach Book, Learn and Land, students look for and make use of structure as they compose and decompose numbers and shapes, identify multiple ways to decompose the same whole, and connect the part-total relationship in numbers to the part-whole relationship in shapes. Learn, Story Problems, students choose from math tools to model and solve put-together and take-apart story problems. Take-apart problems engage students with multiple decompositions of the same total. For example, students consider, “There are 8 ladybugs. Some are flying. Some are on a leaf.” Teacher guidance states, “As with all take apart with both addends unknown stories, there are multiple correct solutions. Learn, Number Bond Hunt, students analyze pictures, identify parts and a total, and write number bonds to represent the structure they identify. Students also consider alternative ways to decompose the same whole. Teacher directions state, “If students finish before it is time to rotate to the next station, ask them to find other ways to sort the pictures into parts.”Land, Debrief, students connect the structures they encountered across the lesson. Teacher directions state, “Where did you see parts and totals, or wholes, today?” Students identify how parts and wholes or totals appeared across pictures, number bonds, groups of bears, and shape puzzles. Students then think–pair–share about the questions, “What have you learned about parts and totals for numbers?” and “What have you learned about parts and wholes for shapes?” Students recognize the common structure across numbers and shapes by explaining that parts combine to make a total in a number bond and smaller shapes combine to make a whole shape puzzle. Students also consider, “What math tools can you use to show your thinking about parts and totals, or wholes?” Promoting Mathematical Practice margin note states, “Students look for and make use of structure when they connect the different ways parts and wholes/totals were seen in the different stations. Seeing the relationship between the parts/whole in a shape puzzle and the parts/total in a number bond can help students gain a deeper understanding of the part-total relationship going forward.”

• Grade 1, Module 2, Topic B, Lesson 7, Teach Book, Learn, students look for and make use of structure as they recognize the relationship between related addition and subtraction problems and use the same parts and total to solve both problems. Students create and solve an addition problem about pennies, write a number sentence to represent their work, and then use the total and the number of pennies found in the first problem to create a related subtraction problem. Students solve and represent both problems by counting up and back on a number path and draw a single number bond to represent both problems. Teachers ask, “Let’s look at this work. What do you all notice?” Students recognize that the problems start and end with the same number of pennies, involve adding and subtracting the same amount, and contain the same numbers. Teachers ask, “How does this number bond represent your addition word problem?” and “How does this number bond represent the subtraction word problem?” Students explain that in the addition problem they combine the parts to find the total, while in the subtraction problem they start with the total, take away one part, and determine the remaining part. Students explain that one number bond represents both problems because “They have the same parts and total.” Promoting Mathematical Practice margin note states, “Students look for and make use of structure when they notice that they start and end with the same number of pennies and when they use that idea to help them solve the problem. In particular, students should see that the equations are related because they include the same numbers but in a different order and with different operations. Ask the following questions: Why is it helpful to notice that you find and lose the same number of pennies? How do the equations show that you found and lost the same number of pennies?” Practice, Problem 1 states, “Add or subtract. Show how you know. You have 7 pennies. You find 3 pennies. How many pennies do you have now?” Students then use the same numbers in a related subtraction problem: “You had 10 pennies. You lost 3 pennies. How many pennies do you have now?”

• Grade 2, Module 3, Topic A, Lesson 5, Teach Book, Learn, students look for and make use of structure as they apply their understanding of the relationship between two-dimensional and three-dimensional shapes to identify and describe the attributes of a cube. Teachers show students a cube and ask, “Which two-dimensional shape makes up the face, or the flat part, of this three-dimensional shape?” Students identify a square and determine that a cube is composed of six square faces. Students then use what they know about the attributes of a square to construct one face of a cube with toothpicks and marshmallows before extending the structure to construct the entire cube. Students count the edges of the cube to determine that 12 toothpicks are needed and identify the marshmallows as representing the vertices. Students think–pair–share to describe the cube’s attributes using the terms faces, edges, and vertices and determine that a cube is composed of 6 square faces, 12 edges, and 8 vertices. Promoting Mathematical Practice margin note states, “When students apply their understanding of the relationship between two-dimensional shapes and three-dimensional shapes, they are making use of structure. Ask the following questions: How are a square and a cube alike? How are they different? How can you use what you know about a square to find the attributes of a cube?” Practice, Problem 1 states, “Circle the shapes that can make a cube,” and students distinguish squares from other two-dimensional shapes based on the structure of a cube’s faces. Practice, Problem 2 states, “Circle the cubes,” and students identify cubes from a set of three-dimensional shapes. Practice, Problem 3 states, “Pam says this rectangle can be 1 face of a cube. Is she correct? Tell how you know.” Students use their understanding that the faces of a cube are squares to evaluate the claim and explain their reasoning.

The materials reviewed for Eureka Math² Kindergarten through Grade 2 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:

• Kindergarten, Module 5, Topic B, Lesson 8, Teach Book, Launch, students look for and express regularity in repeated reasoning as they examine take-away situations, notice that taking away an amount greater than zero repeatedly results in fewer objects, and generalize that subtraction results in a smaller number. Students compare two pictures of an apple tree and are asked, “One thing is different in these pictures. Can you find it?” Students notice that “One tree has more apples than the other.” Teachers reveal that the boy took an apple and state, “There are fewer apples because the boy took 1 of the apples. 6 take away 1 is 5. When something gets taken away, and we figure out how many are left, we call it subtraction.” Promoting Mathematical Practice margin note states, “Students look for and express regularity in repeated reasoning when they recognize that take away stories such as this one, where the amount taken away is greater than zero, result in the number of objects getting smaller. The context of Farmer Brown’s apples gives students an accessible entry point to this line of reasoning. Understanding this relationship becomes increasingly important through the rest of kindergarten and into grade 1, as the scenarios students consider become increasingly complex.” Learn, Farmer Brown, provides repeated examples of this relationship as students begin with 10 apples and repeatedly cross out an apple each time Farmer Brown eats one. After the third apple is crossed off, teachers ask, “What happens to the number every time Farmer Brown eats an apple?” Students respond, “The number gets smaller,” and continue the repeated process until all the apples are crossed off and 0 remains. Land, Debrief, prompts students to generalize the relationship beyond the repeated apple examples. Teachers ask, “If I take away blueberries by eating some, will there be more blueberries or fewer?” and students respond, “Fewer.” Teachers then ask, “If there are 7 blueberries and I eat some, how many could be left?” Students identify possible amounts from one through six. Teachers ask, “Why can’t there be 10 blueberries left?” Students explain, “Because you ate some. 10 is more than 7. You can’t have more than when you started,” using the repeated relationship between taking away and a smaller quantity to evaluate whether a result makes sense. Teachers then ask, “What happens when we take things away, or subtract?” Students generalize, “You take away, and then you get a smaller number. You have fewer when you subtract.”

• Grade 1, Module 1, Topic D, Lesson 21, Teach Book, Launch, students look for and express regularity in repeated reasoning as they examine the number of two-addend expressions for successive totals, identify the repeated pattern that each new total has 1 more expression than the previous total, and generalize the pattern to predict the number of expressions for totals of 7 and 8. Teachers display the Addition Totals chart and ask students to determine the number of expressions for each total from 0 through 6. Teachers then ask, “Think about the number of expressions for each total. What patterns do you notice?” Students notice, “Every time the total is 1 more, there is 1 more expression. There is 1 more expression than the total.” Students use this repeated relationship to make and justify predictions for new totals. Teachers ask, “How many expressions do you think we can make for 7? Why?” Students reason, “8 expressions. There are 7 expressions for 6, so there are probably 8 expressions for 7,” or “8 expressions. The total is 7, and there is usually 1 more expression than the total.” Students then predict how many expressions can be made for 8, and teachers record their predictions to validate later in the lesson. Promoting Mathematical Practice margin note states, “When students predict how many two-addend expressions there will be with totals of 7 and 8, they look for and express regularity in repeated reasoning. Students make use of their previous work with finding all the expressions with totals of 0–5 to make this prediction. They notice the repeated pattern that each total has 1 more expression than the previous total and express regularity to predict that the pattern will continue.” Learn, Two-Part Expressions of 7, students use a systematic process with two-color counters to find and record all the two-part expressions of 7 and then evaluate their prediction. Teachers ask, “How many expressions are there for 7? Is our prediction correct?” Students determine, “There are 8 expressions. Our prediction was right.” Students identify patterns in the repeated process, including, “Each time yellow gets 1 less, and red gets 1 more.” Students repeat the process for 8 and again evaluate their prediction. Teachers ask, “How many expressions are there for 8? Is our prediction correct?” Students determine, “There are 9 expressions. Our prediction was right.” Land, Debrief, students use the established patterns to evaluate whether they found every possible expression. Teachers ask, “How do we know these are all the ways to make 7 with two parts?” Students explain, “I see all the matches 0 + 7 and 7 + 0, …, 3 + 4 and 4 + 3,” and “The total of 6 has 7 expression cards, and the total of 7 has 1 more.” Teachers repeat the process with expressions of 8 and have students extend the repeated relationship to predict the number of expressions for totals of 9 and 10. Practice, Problems 1–2 state, “1. Show three ways to make 7,” and “2. Show two ways to make 8.” Students apply the established relationships by generating multiple two-part decompositions for the totals of 7 and 8, using number bonds and representations of 7 or 8 objects to show different expressions with each total.

• Grade 2, Module 6, Topic A, Lesson 3, Teach Book, Learn, students look for and express regularity in repeated reasoning as they relate equal groups to repeated addition equations, repeatedly compose smaller equal groups into larger groups, and use the resulting pattern to develop a more efficient way to add. Students use tiles to make 4 groups of 5 and connect the groups to the repeated addition equation 5+5+5+5=20. Teachers ask, “How many fives should we write? Why?” Students explain, “We need to write 4 fives because there are 4 groups of 5.” After finding the total through repeated addition, students think–pair–share about a more efficient way to add the 4 fives. Students recognize that they can compose the groups, explaining, “We can combine 2 groups of 5 to make 1 group of 10. Then we would have 2 tens, which I can easily add in my head. 10 and 10 make 20.” Promoting Mathematical Practice margin note states, “Students look for and express regularity in repeated reasoning when they relate repeated addition equations to equal groups and then compose equal groups to reduce the number of addends so they can add more efficiently. Ask the following questions: What is the same about equal groups and repeated addition equations? How can composing 2 smaller equal groups into 1 larger group help you add more efficiently? Will this always work?” Practice, Problems 1–2 state, “Write the equation that matches the picture. Then group the addends to show a more efficient way to add.” In Problem 1, students examine a representation of 4 equal groups with 3 objects in each group, write the corresponding repeated addition equation, regroup the four equal addends into two larger equal addends, and complete the generalization, “4 groups of ___ is equal to 2 groups of ___.” Problem 2 provides another repeated case with 5 equal groups of 3 objects and asks students to write the repeated addition equation and group the addends to find a more efficient way to add. Across the problems, students apply the repeated relationship between equal groups and repeated addition and use the general strategy of composing smaller equal groups into larger groups to reduce the number of addends and calculate more efficiently.