The materials reviewed for Eureka Math² Grade 2 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. 

Materials provide opportunities for students to develop conceptual understanding throughout the grade level. These opportunities are most often found within the Launch and Learn portions of lessons. Examples include:

• Module 2, Topic D, Lesson 21: Use concrete models to decompose a ten with two-digit totals, Fluency, Model Numbers with Place Value Disks, students demonstrate conceptual understanding by using place value disks to model a two-digit number and say the number in unit form to prepare for modeling subtraction and decomposition of a ten. “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)

• Module 3, Topic B, Lesson 8: Create composite shapes by using equal parts and name them as halves, thirds, and fourths, Learn, students develop conceptual understanding of equal parts as they partition circles and rectangles into two, three, or four equal shares and explain their reasoning as they partition. “Have students gather the two smallest triangles and the parallelogram from their tangram pieces. Direct students to use the two triangles to make a polygon. Circulate and observe student work. Select a student who composed one of the following polygons to share their work: a square, a larger triangle, or a parallelogram with no right angles. Repeat with two more students who each composed one of the other two polygons. Consider drawing the polygons on a chart as each student shares. ‘What do you notice about each of these polygons?’ (They are all made up of 2 triangles. The triangles are turned in different directions to make different polygons.) ‘How many parts make up each polygon?’ (2 parts) ‘Are the 2 parts equal?’ (Yes. Each polygon is made up of 2 equal parts, or units, called halves.) ‘How many halves compose, or make up, the whole triangle?’ (2) Direct students to think–pair–share to predict whether they can make halves by using the small triangle and the parallelogram. Encourage students to test their reasoning with the tangram pieces. (I think it might be possible because we will have 2 parts. I don’t think it is possible, because the 2 parts have to be equal, and a parallelogram is not the same as a triangle. No, it won’t be possible because the parts are different sizes. When I put the triangle on top of the parallelogram, they don’t match.) Refer students to the hexagon pattern block. Challenge them to find one pattern block that covers half of the hexagon. ‘Which polygon is half of the hexagon?’ (A trapezoid) ‘How many trapezoids compose a whole hexagon?’ (2 trapezoids) ‘Are they equal parts?’ (Yes.) ‘How do you know?’ (I used 2 trapezoids to make the hexagon, and the trapezoids are the same size and shape, so I know they are equal.) ‘How many halves compose the whole?’ (2 halves compose the whole.) Repeat the process for a rhombus, covering it with 2 equilateral triangles. Invite students to think–pair–share about how the polygons are similar or different. (Each polygon is composed of 2 equal parts. The polygons are different, but they all show halves. Different-sized polygons have different-sized halves, but they all show 2 equal parts.)” (2.G.3)

• Module 6, Topic B, Lesson 7: Distinguish between rows and columns and use math drawings to represent arrays, Learn, Draw Arrays to Show Addition or Subtraction of a Unit, students develop conceptual understanding multiplication as they use arrays and distinguish between columns and rows. “Direct partner A to draw an array with 5 columns of 3 circles and to draw a line between each column. Direct partner B to draw an array of 5 rows of 3 circles and to draw a line between each row. Direct partner A to write a repeated addition equation to match the columns and direct partner B to write a repeated addition equation to match the rows. ‘Do your arrays look the same?’ (No.) ‘Partner A, what repeated addition equation did you write?’ ($3+3+3+3+3=15$) ‘Partner B, what repeated addition equation did you write?’ ($3+3+3+3+3=15$) Invite students to think–pair–share about what would cause the total of the array to change. (The total of the array would change if we added a column or a row of circles. The total of the array would change if we took away a column or a row of circles.) ‘What will happen if we add 1 more column to the array with 5 columns of 3?’ (There will be 6 columns of 3. The total number of tiles will be 18.) Direct students to draw another column of circles and to write a new repeated addition equation to match the new array. ‘What is your new repeated addition equation?’ ($3+3+3+3+3+3=18$) ‘What will happen if we take away 1 row from the new array?’ (The total will be 6 less. The total number of circles will be 12.) Direct students to cross off 1 row. ‘What is the new repeated addition equation to match the columns?’ ($2+2+2+2+ 2+2=12$) Invite students to think–pair–share about why the repeated addition equation is now 6 twos and not 6 threes. (The repeated addition equation is 6 twos because we took away 1 row, so we took away 1 circle from each column. Now there are only 2 circles in each column.)” (2.OA.4) 

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. The Learn (Problem Set) and Land (Exit Tickets) portions of lessons consistently include these opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of conceptual understanding. Examples include:

• Module 1, Topic I, Lesson 35: Compare three-digit numbers using >, =, and <, Learn, Problem Set, students demonstrate conceptual understanding of place value as they compare numbers based on the value of digits. “Differentiate the set by selecting problems for students to finish independently within the timeframe. Help students recognize the words compare, greater, and equal in print. Invite students to underline the words as you read them aloud. (Student page has two place value charts, one for 97 and the other for 200, for the students to show their work.) Draw each number on the place value chart. Then circle >, =, or < to compare.” Additional problems include comparison of 227 and 127, 241 and 251, 245 and 99, 899 and 900, 181 and 159, 419 and four hundred nineteen. (2.NBT.4)

• Module 4, Topic D, Lesson 17: Use place value drawings to represent subtraction with one decomposition and relate them to written recordings, Land, Exit Ticket, students independently demonstrate conceptual understanding of addition and subtraction of three-digit numbers, including composing or decomposing tens or hundreds, as needed. “Jack finds $849-374$  two ways. Look at his work. What mistake did Jack make? Show the correct work.” Students are shown a problem presented vertically with an answer of 375 and a place value chart showing decomposition with an answer of 475. (2.NBT.7) 

• Module 6, Topic C, Lesson 10: Use math drawings to compose a rectangle, Learn, Use Arrays to Solve Problems, students demonstrate conceptual understanding as they use addition to find the total number of objects arranged in rectangular arrays. “Display the word problem. ‘Alex bakes two pans of brownies. In the first pan, he cuts 2 rows of 8. In the second pan, he cuts 4 rows of 4. How many brownies did Alex bake altogether?’ Read the problem chorally with the class. Invite students to use Read–Draw–Write to solve the problem and answer the question. Give students 5 minutes of independent work time. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose work that allows for rich discussion. Use the Math Chat routine to engage students in mathematical discourse about their problem-solving process and their array drawings. ‘What do you notice about the 2 arrays?’ (They both have 16 brownies. One array is a rectangle and one is a square. If you moved the bottom 2 rows of the square array next to the top two rows, you would have 2 rows of 8.) ‘What do you notice about the 2 repeated addition equations?’ (They both equal 16. You can group the twos together to make the other repeated addition equation. I notice a lot of doubles in the repeated addition equations and the equation I wrote to find my answer.)” (2.OA.4)

The materials reviewed for Eureka Math² Grade 2 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Examples can be found within various portions of lessons, including Fluency, Launch, and Learn. They include:

The materials develop procedural skill and fluency throughout the grade level, within various portions of lessons, including Fluency, Launch, and Learn. There are also opportunities for students to independently demonstrate procedural skill and fluency. Examples include:

• Module 2, Topic A, Lesson 5: Make a ten to add within 100, Learn, Make a Ten When One Addend Ends in 8, students develop procedural skill and fluency with addition as they use benchmark numbers to make problems easier. “Write $38+15$ and have students do the same. ‘Which of these numbers is closer to a ten, a benchmark number?’ (38) ‘Which ten is 38 closest to?’ (40) ‘How much more do we need to add to 38 to make 40?’ (2) ‘Where can we get the 2 from?’ (15) ‘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?’ (No.) ‘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)

• Module 6, Topic A, Lesson 2: Organize, count, and represent a collection of objects, Fluency, Counting the Math Way by Fives, students develop procedural skill and fluency as the practice counting in both directions by 5s. “‘Let’s count the math way. Each finger represents 5, just like the 5 beads in a row on the rekenrek.’  Face the students and direct them to mirror you. For each skip-count, show the math way on your own fingers while students count, but do not count aloud. ‘Show me 0. Now raise your left pinkie. That’s 5. Put up your very next finger. That’s 10. Keep counting by fives to 50. Stay here at 50. Now count by fives back down to 0. Ready?’ Have students count the math way by fives from 50 to 0. Offer more practice counting the math way by fives from 0 to 50 and then back down to 0.”  (2.NBT.2)

Materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. These are often found within the Problem Set or within Topic Tickets, Learn and Land lesson sections respectively. When appropriate, teachers may use other portions of lessons for independent demonstration of procedural skill and fluency. Examples include:

• Module 2, Topic D, Lesson 27: Solve two-step word problems within 100, Topic Ticket, Item 1, students independently demonstrate procedural skill and fluency with adding and subtracting within 100. “There are 67 books in the red bin. There are 48 fewer books in the green bin. How many books are in the green bin? How many books are in both bins?” (2.NBT.5) 

• Module 4, Topic A, Lesson 2: Mentally add and subtract multiples of 10 and 100 with unknowns in various positions, Learn, Problem Set, Problem 1, students demonstrate procedural skill and fluency as they mentally add or subtract multiples of 10 or 100. “Find the unknown. $40+40=$__; $400+200=$__; $440+240=$__.” (2.NBT.8)

• Module 6, Topic D, Lesson 18: Use various strategies to fluently add and subtract within 100 and know all sums and differences within 20 from memory, Learn, Number Line Hop: Race to 100, students demonstrate procedural skill and fluency with addition as they play a game to see who can get closest to 100. “Pair students and distribute a measuring tape, and have students play according to the following rules: Partners place their counters at 0 on the measuring tape. Partner A rolls the dice and finds the sum of the roll, for example, “3 and 4 is 7.” Partner A says an equation beginning with the location of their counter and adding the sum of the roll, for example, “$0+7=7$.” Partner A moves their counter to the sum on the measuring tape. Partner B rolls the dice, repeating the procedure. The first person to get closest to 100 is the winner. For example, if partner A gets to 98 and partner B gets to 103, partner A wins.” (2.OA.2)

The materials reviewed for Eureka Math² Grade 2 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. 

There are multiple routine and non-routine application problems throughout the grade level, including opportunities for students to work with support of the teacher and independently. While single and multi-step application problems are included across various portions of lessons, independent application opportunities are most often found within Problem Sets or the Lesson Debrief, Learn and Land sections respectively.

Examples of routine applications of the math include:

• Module 1, Topic A, Lesson 2: Draw and label a bar graph to represent data, Land, Exit Ticket, students independently solve a routine problem where they complete a bar graph. Teacher directions state, “Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.” Students see data, “Games We Like” in a table: Tag 4, Kickball 7, Jump Rope 10, Hide and Seek 3. “Make a bar graph.” (2.MD.10)

• Module 2, Topic C, Lesson 14: Use addition and subtraction strategies to find an unknown part, Launch, students solve routine addition and subtraction application problems with teacher support. “Gather the class. Read the following problem aloud and invite students to picture it in their minds. ‘Mrs. King makes 52 cups of ice cream. She gives 28 cups of ice cream to the students. How many cups are left? Let’s use the Read–Draw–Write process to help us solve this problem.’ Read the first sentence aloud. ‘What do we know?’ (We know there are 52 cups of ice cream.) ‘What can we draw?’ (We can draw a tape diagram to represent the 52 cups of ice cream. We can draw a tape diagram and label it with 52 as the total. We know 52 is the total cups of ice cream.) Draw a tape diagram and label the total 52, as students do the same. Read the next sentence aloud. ‘What do we know?’ (Mrs. King gives 28 cups to students.) ‘What can we draw?’ (We can split the tape into two parts and label one part 28, since that is the part Mrs. King gives away.) Partition and label the tape diagram, as students do the same. Read the next sentence aloud. ‘What do we know?’ (We know we need to find how many cups are left.) ‘What can we draw?’ (We can draw a question mark in the other part to represent the unknown part.) Write a question mark in the tape diagram as students do the same. Invite students to turn and talk about what the tape diagram shows. Direct students to solve the problem. Provide students time to work. As students work, circulate and select student work to share. Invite students to think–pair–share about their solution strategies. (I counted back by tens and ones on the number line and got to 24. First I counted back by ones to get to 50. Then I counted back from 50 to 30 by tens. Then I counted back 6 more and got to 24. I subtracted in parts. I know $52-20=32$. I subtracted 2 of the 8 ones to get to 30. Then I still needed to subtract 6, and $30-6=24$. I used addition. I counted on from 28 to 52. I counted on 2 to get to 30 and then 22 more to get to 52. I counted 24 all together.) ‘What equation, with an unknown, did you write to solve this problem?’ ($52-28=$__ ) ‘Let’s answer the question in a complete sentence. How many cups are left?’ (24 cups are left). Transition to the next segment by framing the work. ‘Today, we will use addition and subtraction strategies to solve problems with an unknown part.’” (2.OA.1)

• Module 5, Topic C, Lesson 14: Solve addition and subtraction two-step word problems that involve length, Learn, Solve Two-Step Comparison and Total Unknown Word Problems. students solve routine problems, involving length measurements, as they add and subtract within 100. “Direct students’ attention to problem 1 in their books. Read the problem aloud. Reread the first sentence. ‘What can we draw?’ (We can draw two tapes, one longer than the other. The longer tape represents how far the yellow rocket travels and the shorter tape represents how far the blue rocket travels. We can draw two tapes: one to represent the yellow rocket and one to represent the blue rocket. We know the difference between how far the yellow and blue rocket travel is 16 feet because the problem says the blue rocket travels 16 feet less than the yellow rocket.) Draw a tape diagram as students do the same. Reread the second sentence. ‘What can we draw?’ (We can label the tape that represents the yellow rocket with 35 feet.) Label the yellow and blue rockets in the tape diagram as students do the same. Invite students to think–pair–share to restate the problem in their own words. (There is one yellow rocket and one blue rocket. The yellow rocket travels 35 feet, which is 16 feet farther than the blue rocket. We have to find the total number of feet the two rocket ships travel.)  ‘Do you have enough information right now to find how far they both traveled?’ (No.) ‘What other information do you need?’ (We need to know how far the blue rocket travels.) ‘Step one is to find how far the blue rocket travels and step two is to find the total distance both rockets travel in feet.’ Label both unknowns in the tape diagram and have students solve for the distance the blue rocket travels. Invite students to think–pair–share about what equation they can write to find the distance the blue rocket travels.’ (__$+16=35$ ; $16+$__$=35$ ; $35-16=$__) Give students 2 or 3 minutes to solve for the distance the blue rocket travels in feet, as well as the total distance for both rockets in feet, and write an answer statement. ‘How many feet does the blue rocket travel?’ (The blue rocket travels 19 feet.) ‘Let’s go back and make sure we answered the question.’ Have students read the question as you read it aloud. (How many total feet do the two rockets travel?) Have we answered this question?’ (No.) ‘What equation can we write to find the total distance?’ ($19+35=$__) Have students write the equation, solve, and write a statement to answer the question. Confirm that the total distance both rockets travel is 54 feet.” (2.MD.5)

Examples of non-routine applications of the math include:

• Module 1, Topic A, Lesson 1: Draw and label a picture graph to represent data, Launch, students solve non-routine problems as they apply strategies to generate measurement data. “Gather students and invite them to participate in a fun getting-to-know-you activity. ’One way we can get to know each other is to ask questions and record the answers. For example, I could ask you to tell me your favorite subject–reading, writing, math, or science.’ Display the Favorite Subject table. Introduce the terms table and category. ‘This is a table. It lists the four subjects you can choose. Each subject is a category, or type of group. Vote by raising your hand when I call out your favorite subject. I’ll record the number of votes for each category on the table.’ Add the new terms table and category to the terminology chart you prepared in advance. Conduct the survey and record the counts for each category. Then introduce the new term data. ‘The information we just recorded about our favorite subjects is called data.’” (2.MD.10)

• Module 3, Topic B, Lesson 6: Recognize that a whole polygon can be decomposed into smaller parts and the parts can be composed to make a whole, Problem Set, Problem 1, students solve a non-routine application problem independently as they recognize and draw shapes having specified attributes. Students see a hexagon with lines separating the shape into rectangles. “Name one shape you see in the hexagon: I see a __.” There are five additional problems where students apply their understanding of composing and decomposing shapes and demonstrating understanding of attributes. (2.G.1)

• Module 5, Topic A, Lesson 2: Use the fewest number of coins to make a value, Launch, students solve a non-routine problem with the teacher’s guidance when they decide which coins can be used to make one dollar. Teacher displays: “Jill has 100 cents in her pocket. What coins might Jill have in her pocket?” Teacher directions state, “Display the problem. Direct students to solve the problem on their whiteboards. ‘What coins might Jill have in her pocket?’ (Jill might have 4 quarters. She could have 100 pennies. Jill might have 10 dimes. She might have 2 quarters and 10 nickels. Jill might have 2 dimes, 1 nickel, and 3 quarters.) As students share, record possible coin combinations. ‘Are there any coin combinations that Jill cannot have in her pocket?’ (Jill cannot have 5 quarters. Jill cannot have more than 10 dimes. Jill cannot have a group of coins that is worth more than 100 cents.) Direct students to look at all the possible combinations of coins that make 100 cents. Invite them to think–pair–share about which combination of coins is the most efficient or has the fewest number of coins. (Having 10 dimes is more efficient than having 20 nickels. 4 quarters is the most efficient combination because it’s the fewest number of coins.) Transition to the next segment by framing the work. ‘Today, we will look at how a given value can be represented by using the fewest number of coins.’” (2.MD.8)

The materials reviewed for Eureka Math² Grade 2 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. 

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

• Module 2, Topic A, Lesson 3: Use compensation to add within 100, Learn, Use Compensation with a Number Line Diagram, students develop procedural skill and fluency as they work with the teacher and use a number line to reason about why compensation works. “Gather students and write $57+39$. Have students find the total and use the Math Chat routine to engage them in mathematical discourse. Give students 2 minutes to independently think and write at least one solution strategy. Have students give a silent signal to indicate they are finished. Have students discuss their thinking and strategy with a partner. Circulate and listen as they talk. Identify a few pairs of students to share their strategies. Purposefully choose work that allows for a rich discussion about connections between strategies. As students discuss, highlight thinking that shows the use of benchmark numbers. Then facilitate a class discussion. Invite students to share their thinking and record their reasoning. ‘What strategy did you use to find the total?’ (I added like units. First I added 50 and 30 to get 80. Then I added 7 and 9 to get 16. Then I found $80+16=96$. First I broke apart 57 into 56 and 1. I gave the 1 to 39 to make 40, then I added 56 and 40 to get 96.) ‘Many of you used a benchmark number to help you add. Let’s use benchmark numbers and record our thinking by using the open number line and the arrow way. We will start with the open number line. Which addend is closer to a benchmark number, 57 or 39?’ (39) ‘What benchmark number is close to 39?’ (40) Draw a line, make a tick mark, and label the tick mark 57. Then have students do the same on their personal whiteboards. ‘Add 40 to 57’ Draw a large jump from 57 on the open number line and write +40 above it. Ask students to do the same. ‘That gets us to what number?’ (97) Draw another tick mark and label it 97, then ask students to do the same. ‘We need to add 39 to 57, and 39 is 1 less than 40. So we hop back 1.’ Draw a small hop from 97 and write –1 above it. Draw a tick mark and label it 96. ‘Now let’s record this thinking by using the arrow way.’ (Write 57.) ‘First add 40 to 57.’ (Write +40 and draw an arrow beneath it.) ’What gets us to 97.’ (Write 97.) ‘What do we do next?’ (Subtract 1). Write $-1$ and draw an arrow underneath it. ‘That gets us to 96.’ (Write 96.) Invite students to think-pair-share about how the open number line and the arrow way are similar and different. (They both use lines and arrows. They both show how to add 39 as $+40$ and $-1$. The jumps on the open number line are different sizes, but the lines in arrow way are the same size. The arrow way moves straight across from left to right, but the jumps on the open number line go forward if we add and backward if we subtract.) Distribute measuring tapes. ‘Now let’s use a measuring tape as a number line.’ Use the number line to help students model each part of the process. ‘We start at 57.’ Direct students to slide their fingers from 0 to 57 and help them recall that 57 represents a distance from 0. Have them make a hop of 40 to land at 97. ‘We made a hop of 40. Are we adding 40?’ (No. We’re adding 39.) ‘Why do you think we made a hop of 40?’ (40 is 1 away from 39. It’s easier for me to add 40 because the ones don’t change. You just add 4 tens.) ‘So why do we take away 1?’ (40 is 1 more than 39, so now we have to take away 1.)” (2.NBT.7)

• Module 3, Topic C, Lesson 10: Partition circles and rectangles into equal parts and describe those parts as halves, Learn, Partition a Rectangle to Show Halves, students develop conceptual understanding as they investigate equal shares of identical wholes with varying shapes. ”Direct one partner from each pair to fold their paper in half. Direct the other partner to fold their paper in half a different way. Circulate and assist as needed. ‘We can also partition a rectangle without folding the paper.’ Model partitioning the rectangle on the diagonal by drawing a line from one corner to the opposite corner. Invite students to think–pair–share about how they can tell the rectangles are partitioned into halves. (We made 2 equal parts. The 2 parts are equal shares because the 2 halves match.) Direct students to shade 1 half and label it in unit form. Invite students to think–pair–share about how their papers are similar and different. (We both have 2 equal parts. Our halves look different. We both still have a whole piece of paper.) ‘I hear you say that halves from the same polygon can be different shapes.’” (2.G.3) 

• Module 5, Topic C, Lesson 14: Solve addition and subtraction two-step word problems that involve length, Learn, Critique a Flawed Response, students solve non-routine application problems involving addition and subtraction. “Refer students to the next problem. Display the sample student work. Introduce the Critique a Flawed Response routine and present the following prompt. ‘Imani says the green rocket is 32 inches longer than the blue rocket. Is she correct? How do you know?’ Give students 2 minutes to identify the error. Invite students to share. (Imani is incorrect because she found the length of the green rocket, not the difference between the two rockets. The green rocket can’t be 32 inches longer than the blue rocket because that would make the green rocket 57 inches long and the two rockets together would be 82 inches. That is not possible because the problem tells us that the two rockets are 57 inches in all.) Give students 1 minute to check their work and correct any errors or add to their drawings. Circulate and identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about the steps needed to solve the problem.” (2.MD.5)

Multiple aspects of rigor are engaged simultaneously throughout the materials in order to develop students’ mathematical understanding of grade-level topics. Examples include:

• Module 1, Topic I, Lesson 35: Compare three-digit numbers by using >, =, and <, Learn, Compare with Drawings and Symbols, students develop procedural skill and fluency alongside application as they compare three-digit numbers by using what they know about place value. “Direct students to the first two place value charts in their books and have them draw to represent 349 and 329. Invite students to think–pair–share about which number is greater than and which number is less than the other number. (349 is greater than 329 because 349 has 2 more tens than 329. They both have 3 hundreds, but 329 is smaller because it has fewer tens.) ‘Instead of writing the terms greater than and less than, we can use symbols that represent the words.’ Display the symbols. ‘This is the greater than symbol. (Point to the symbol.) This is the less than symbol. (Point to the symbol.) You read the statement the same way with the symbol as you would if the words were there.’ Have students compare 349 and 329 and write a comparison statement by using symbols. Invite students to share their answers with the class. (349 is greater than 329. We can write 349 > 329. 329 is less than 349. We can write 329 < 349.) Direct students to the next two place value charts and have them draw to represent 932 and 934. ‘Both numbers have the same digit in the hundreds and tens place, so where can you look to compare them?’ (In the ones place) ‘Which number is less?’ (932) ‘Compare 932 and 934 and write a comparison statement by using symbols.’ Invite students to turn and talk about the meaning of the two comparison symbols.” (2.NBT.4)

• Module 2, Topic A, Lesson 2: Break apart and add like units, Land, Debrief, students develop conceptual understanding alongside application as they use place value to add and subtract three-digit numbers. “Initiate a class discussion by using the prompts below. Encourage students to restate their classmates’ responses in their own words. ‘What did we do today to help simplify math problems, or make them easier to solve?’ (We used number bonds to break apart numbers in different forms and then added like units.) ‘How can standard form, expanded form, and unit form help add like units?’ (The different forms help us see the place value units, so we know which units to add together.) ‘Look at problem 8. Are we able to break apart and add like units? How do you know?’ (Yes. I know 1 hundred and 0 hundreds is 1 hundred, 2 tens and 1 ten is 3 tens, and 5 ones and 6 ones is 11 ones. That makes another ten, so now I have 4 tens. So my answer is 1 hundred, 4 tens, and 1 one, or 141. Yes. But when you add the ones to the ones, you get 11. You have to add another ten to the 3 tens you got when you added the tens to the tens. You will have 4 tens and 1 one. The answer is 141.) ‘What mental math facts did you use to help you solve this problem?’ (I used $1+0=1$ when I added 1 hundred and 0 hundreds. I used $2+1=3$ when I added 2 tens and 1 ten. I used $5+6=11$ when I added the ones. Then I used $3+1$ to add the ten from 11.) ‘How does place value understanding help in adding two- and three-digit numbers?’ (Place value helps me know which units to add together. I know I have to add tens to tens and ones to ones. When I rename large numbers as smaller units, it helps me to add in my head.)” (2.NBT.7) 

• Module 4, Topic B, Lesson 7: Use concrete models to add and relate them to written recordings, Launch, students develop conceptual understanding alongside procedural skill and fluency as they add and subtract within 1000 using concrete models. “Model $136+285$ with disks, without referring to the addends orally or in writing. ‘What do you notice?’ (I notice two numbers on the chart. I notice more than 10 tens and 10 ones. I notice 3 hundreds. I notice the chart is not labeled. I notice 5-groups.) ‘What do you wonder?’ (I wonder how I can show this with numbers. I wonder what the total is. I wonder if more disks will be added or if disks will be taken away.) Invite students to think–pair–share about an addition expression they can write to represent the disks. $136+285$; $285+136$. Transition to the next segment by framing the work. ‘Today, we will show addition by using a place value model and vertical form.’” (2.NBT.7)

The materials reviewed for Eureka Math² Grade 2 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with these practices across the year and they are identified for teachers within margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic D, Lesson 19: Solve compare with difference unknown word problems in various texts, Learn, Compare with Difference Unknown Problem, students build experience with MP1 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students solve comparison problems, they make sense of problems and persevere in solving them (MP1). In measurement problems, models such as the measuring tape and tape diagram help students represent the problem more concretely. Ask the following questions to promote MP1: What are some things you could try to start solving the problem? Does your drawing make sense with the problem?” Teacher directions state, “‘How can you represent the information in the problem?’ (I can draw a number line. I can use a measuring tape. I can draw a tape diagram. I can write an equation.) Prompt students to solve the problem independently by choosing from the measurement tools provided. Regardless of their choice, encourage students to record their strategy. Circulate and observe student work. Select a few students to share their strategies in the next segment. Look for work samples that help advance student understanding toward more abstract representations of finding the difference.” 

• Module 2, Topic D, Lesson 25: Use place value drawings to subtract with two decompositions, Learn, Subtract from a Three-Digit Total with Two Decompositions, students build experience with MP1 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students look for an entry point to subtract from three-digit totals when there are not enough ones and tens, they make sense of problems and persevere in solving them (MP1). Ask the following questions to promote MP1: Can you subtract the ones without renaming? Where can we get more ones? Can you subtract the tens without renaming? Where can we get more tens?” Teacher directions state, “Invite students to think–pair–share about whether we can subtract 87 from 154 without renaming. (No, we need to rename 154 because we don’t have enough in the ones place. We don’t have enough ones to subtract 7 ones from 4 ones. No, we need to rename 154 because there aren’t enough in the tens place. There aren’t enough tens to subtract 8 tens from 5 tens.) ‘Let’s use a place value drawing to help us rename 154 so we can subtract.’ Guide students in drawing and labeling a three-column place value chart, as you do the same. ‘What number should we draw on our place value chart? Why?’ (We should draw 154 because it is the total. When we subtract, we take away the part we know from the total, so we need to draw 154 and take away 87. 87 is one of the parts. We are going to take away 87 from 154 to find the other part, so we need to draw 154.) ‘How many hundreds, tens, and ones are in the total? Say it in unit form.’ (1 hundred 5 tens 4 ones) ‘What is the part being subtracted? Say it in unit form.’ (8 tens 7 ones) ‘Look at the ones place. Can we subtract 7 ones from 4 ones?’ (No.) Invite students to think–pair–share about where they can get more ones. (From the tens place. I know 1 ten is the same as 10 ones. You can decompose a ten and draw 10 ones in the ones place.) Direct students to decompose 1 ten into 10 ones, as you do the same. ‘How did we rename, or regroup 154 so we had enough ones?’ (1 hundred 4 tens 14 ones) ‘Look at the tens place. Can we subtract 8 tens from 4 tens?’ (No.) ‘Where can we get more tens?’ (We can get more tens from the hundred. I know 1 hundred is the same as 10 tens. You can decompose 1 hundred into 10 tens. 10 tens and 1 hundred are the same amount, just in different units.) Direct students to decompose 1 hundred into 10 tens, as you do the same. ‘How did we rename 1 hundred 4 tens 14 ones so that we had enough tens?’ (We renamed 1 hundred as 10 tens. Now we have 14 tens 14 ones.) Invite students to think–pair–share about how 14 tens 14 ones has the same value as 154. (I know that 14 tens has the same value as 140 and 140 + 14 = 154. I know that 10 tens is 100, so 14 tens is 140. I know 10 ones is 1 ten, and 10 more than 140 is 150. Then 150 and 4 ones is 154.) ‘Are we ready to subtract?’ (Yes. We can take 7 ones from 14 ones and 8 tens from 14 tens. Yes, now we have enough to subtract like units. Take the tens from tens and the ones from ones.) Direct students to find the difference by completing the subtraction on their place value drawing. ‘What is 14 ones − 7 ones?’ (7 ones) ‘What is 14 tens − 8 tens?’ (6 tens) Point to the place value drawing as students chorally count how many tens and ones are left over.”

• Module 4, Topic E, Lesson 22:  Solve compare with smaller unknown word problems, Launch, students build experience with MP1 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “Students make sense of problems and persevere in solving them (MP1) as they use the Read–Draw–Write process to decontextualize problems into mathematical models and equations. Ask the following questions to promote MP1: How could you explain this problem in your own words? What are some things you could try to start solving the problem?” Teacher directions state, “Display the problem and read it aloud. ‘Lan picks 18 more blueberries than Jill. Lan picks 64 blueberries. How many blueberries did Jill pick?’ Invite students to turn and talk about what information they know from the problem. ‘What can we draw?’ (We can draw Lan’s blueberries in 5-groups. We can draw a tape diagram to compare Lan’s and Jill’s blueberries.) ‘Before we draw, who picks more blueberries, Lan or Jill? How do you know?’ (Lan has more. It says he picks 18 more than Jill.) Guide students through the process of drawing a comparison tape diagram on their whiteboards. ‘We know how many blueberries Lan picks. Let’s draw a tape to represent Lan’s  blueberries. Does Lan pick 18 blueberries?’ (No, Lan picks 64 blueberries.) Draw the first tape and label it as students do the same. ‘Lan has more blueberries. So, will the tape for Jill’s blueberries be longer or shorter than Lan’s?’ (The tape for Jill’s blueberries will be shorter than Lan’s tape.) Direct students to complete the tape diagram. Circulate and check that students have drawn a second tape that is shorter than the first, labeled the difference as 18, and labeled the unknown with a question mark. Invite students to think–pair–share about how their tape diagram represents the problem. (I drew another tape that was shorter for Jill’s blueberries, because Lan picks 18 more blueberries than Jill. I labeled the extra part outside of Jill’s tape with 18, since that’s the difference. I put a question mark inside the second tape since we want to know how many blueberries Jill picks.)”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 2, Topic A, Lesson 3: Use compensation to add within 100, Learn, Apply the Compensation Strategy to a Measurement Context, students build experience with MP2 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “Students reason abstractly and quantitatively (MP2) as they model the strategy of compensation by using a close benchmark number on an open number line. By attending to the meaning of the quantities, students make problems simpler as they recognize and use the benchmark number. Ask the following question to promote MP2: How does the open number line help you use a benchmark number to add?” Teacher directions state, “Give students a minute to draw to represent the problem. Then guide students in modeling the problem on the open number line. ‘Put your finger on 0 and slide to 23 cm to show the length of Jill’s fish. How much did the fish grow?’ (19 cm) ‘Is one of these numbers close to a benchmark? Which one?’ (19 is close to 20.) ‘What is $23+20$?’ (43) ‘How did you know the answer without writing anything down?’ (I added the tens to the tens and the ones didn’t change. I added like units. When I add the tens, the ones stay the same.) Invite students to think-pair-share about how they can use an open number line and the benchmark number, 20, to solve the problem. (Jill’s fish is 23 cm long, so start at 23 and make a hop of 20. That gets you to 43. Then move back 1. 20 is 1 more than 19. So I add 20 and then take away 1. I know that $23+20=43$ and $43-1=42$.) Draw an open number line as you revoice student thinking. ‘The open number line is similar to your measuring tape. How are they similar or different?’ (They both have numbers, and the numbers get larger as you move farther to the right. A bigger hop is like hopping a longer distance on a measuring tape. The open number line doesn’t have tick marks for every number, but the measuring tape does. I see every number on the measuring tape, but I only see some numbers on the open number line.)”

• Module 3, Topic D, Lesson 17: Relate the clock to a number line to count by fives, Learn, Count Groups of 5 Minutes, students build experience with MP2 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “Students reason quantitatively and abstractly (MP2) 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 to promote MP2: 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?” Teacher directions state, “Gather the class and place the piece of yarn in a circle on the chart paper where students can easily see it. Ensure the circle is large enough to fit 12 sticks of five cubes. ‘Let’s create, or build, a clock. What are some units that are shown on a clock?’ (Hours, minutes, and seconds) ‘Let’s count how many minutes are between the 12 and the 1 on the clock.’ Move the minute hand on the demonstration clock to each tick mark from the 12 to the 1, as students count. Stop at the 5-minute mark. ‘I’m going to put down 1 cube for each minute we counted. How many minutes have gone by?’ (5 minutes) Connect the five cubes and place the 5-stick on the inside of the yarn circle between the 12 and the 1. At the end of the 5-stick, attach the sticky note labeled 5 to the yarn with a clothespin. Repeat the process with the remaining individual cubes for the next 5-minute interval. Once the unit of five is established, adjust the process to place a 5-stick at each 5-minute interval and attach the corresponding sticky note with a clothespin. Ensure the sticks of five touch but are not connected. ‘Look at the clock. (Gesture to the demonstration clock.) Is every minute labeled?’ (No.) ‘What is labeled?’ (The hours) ‘When the hour hand points to the large numbers on a clock, it tells us the hour. What do you think each number represents when the minute hand points to them?’ (The minutes) Show 5:05 on the demonstration clock. ‘What time does the clock show?’ (5:05) What number do you see? (Gesture to the 1 on the clock.) (1) ‘Why is there a 1 here when 5 minutes have gone by?’ Invite students to think–pair–share about what the 1 represents when the minute hand is pointing to it. (The 1 represents 5 minutes. The 1 stands for 1 group of 5 minutes.) ‘The 1 represents 1 group of 5 minutes. Let’s count how many fives there are in 1 hour. Point and count each five in unit form (1 five, 2 fives, 3 fives, … , 12 fives).’ Label each group of five with a marker on the inside of the circle. Have students return to their seats. Refer them to their paper clocks. ‘Let’s practice counting by minutes and by groups of 5 minutes.’ Direct students to move the minute hand on their paper clocks as they count the minutes by five (5, 10, 15, 20, … , 60). Then repeat the process to count the number of fives in unit form (1 five, 2 fives, 3 fives, … , 12 fives).”

• Module 6, Topic A, Lesson 1: Compose equal groups and write repeated addition equations. Launch, students build experience with MP2 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “Students reason abstractly and quantitatively (MP2) when they create abstract models and equations based on the context of a real-world situation. Ask the following questions to promote MP2: How did you know you had equal groups? Does your answer make sense?” Students are presented with a word problem and the teacher is directed to use the Math Chat routine to engage students in mathematical discourse. Teacher directions state, “Give students 2 minutes of silent think time to manipulate their tiles and draw a picture to represent the problem on their personal whiteboards. Have students give a silent signal to indicate they are finished. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify two or three students to share their thinking. Purposefully choose work that allows for rich discussion about connections between strategies.Then facilitate a class discussion. Invite students to share their thinking with the whole group, and then record their reasoning.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP3 across the year and it is explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include:

• Module 1, Topic H, Lesson 34: Problem solve in situations with more than 9 ones or 9 tens., Launch, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students reason about showing numbers by using different place value chart representations, they are constructing viable arguments and justifying their reasoning to others (MP3). Students are explicitly asked to construct an argument for their answer.” Teacher directions state, “Use the Numbered Heads routine. Organize students into groups of three and assign each student a number, 1 through 3. Display the three place value drawings and read the following statement: The three representations all show a total value of ___. Give groups two minutes to study the place value drawings and to complete the statement. Remind students that any one of them could be the spokesperson for the group, so they should be prepared to answer. Groups should be prepared to share the following information: What they noticed about the place value units in each drawing, How thinking about the place value units helped them figure out the value of the drawings, How unit form helped them compare similarities and differences among the representations, Call a number 1 through 3. Have students assigned to that number share their group’s findings. Invite students to turn and talk about how they used what they know about place value units to find the value of the drawings.”

• Module 2, Topic B, Lesson 12: Use place value drawings to compose a ten and a hundred with two- and three-digit addends. Relate to written recordings, Learn, Relate Place Value Drawings to Written Recordings, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “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 (MP3). 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 to promote MP3: When is your classmate correct? Is there a general statement you can make that the class can agree to?” Students work with the problem, $84+47$. Teacher directions state, “Display the totals below work sample and a student work sample that shows expanded form written vertically. ‘Here is another way to record how to find the sum of 84 and 47. This written recording is called totals below.’ Invite students to think–pair–share about how the recordings are similar and different. (Both show the totals for each place value unit. I can see the sum of the ones and the sum of the tens in both recordings. Totals below doesn’t break apart the number into expanded form in writing, but still adds like units.) ‘Totals below is a written recording that shows the sum, or total, of each place value.’ Display the following statement: When I add two-digit numbers, I can add the tens first. Use the Always Sometimes Never routine to engage students in constructing meaning and discussing their ideas. Give students 2 minutes of silent think time to evaluate whether the statement is always, sometimes, or never true. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. (I think it is always true because I know I can add in any order. When I add 22 and 25, I can add 2 tens and 2 tens and get 4 tens. Then I can add 2 ones and 5 ones and get 7 ones. The answer is 47. I think it is never true. I learned you should always add ones first. I think it is sometimes true. You can’t add tens first when you compose a new unit. It doesn’t work.) Then facilitate a class discussion. Invite students to share their thinking with the whole group. Encourage them to provide examples and nonexamples to support their claims. Conclude by coming to the consensus that the statement is always true because you can add in any order and rename units at the end.”

• Module 4, Topic D, Lesson 20: Subtract by using multiple strategies and defend an efficient strategy, Land, Debrief, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students select, share, and defend their choice of most efficient strategy, they construct viable arguments and critique the reasoning of others (MP3). Ask the following questions to promote MP3: Will your strategy be the most efficient in other subtraction problems? How can you advise a classmate to choose the most efficient strategy?” Teacher directions state, “Objective: Subtract by using multiple strategies and defend an efficient strategy. Gather the class and facilitate a discussion about choosing efficient solution strategies. ‘How do you decide which strategy is the most efficient?’ (I look at the numbers in the problem to decide which strategy to use. If I don’t have to rename, I can subtract like units in my head. If the number I’m subtracting ends in a 9, 8, or 5, it is efficient to take from 10 or 100 or use compensation. I know my partners to ten so it is easy to add 1, 2, or 5 to the number that is left. If the number has zeros, it is efficient to use compensation, when I subtract the same amount from each number. Then I don’t have to rename to subtract.) ‘Will every subtraction strategy work for all subtraction equations?’ (You can use every strategy on any equation, but not all the strategies are efficient for every problem. Take from 10 and compensation are most efficient when the number you are subtracting is close to a ten or a hundred. Compensation, where you subtract 1 from each number, works best for me when the total has zeros.)”

• Module 5, Topic A, Lesson 5: Use different strategies to make 1 dollar or to make change from 1 dollar, Learn, Share, Compare, and Connect, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students analyze their peers’ work samples and discuss a variety of place value strategies to make $1, they construct viable arguments and critique the reasoning of others (MP3). Ask the following questions to promote MP3: Why are there many different solution strategies for the same problem? Which strategy do you think is most efficient for this type of problem? Defend your choice.” Teacher directions state, “Gather the class and invite students you identified in the previous segment to share their solutions one at a time. Consider intentionally ordering shared student work from an intuitive drawing to a more abstract model such as a tape diagram. As each student shares, ask questions to elicit their thinking and clarify the model used to represent the problem. Ask the class questions to make connections between the different solutions and their own work. Encourage students to ask questions of their own.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP4 and MP5 across the year and they are explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic C, Lesson 12: Model and reason about the difference in length, Learn, Relate Subtraction to Addition to Find the Difference in Length, students build experience with MP4 as described in the Teacher Note, Promoting the Standards of Mathematical Practice. “Students use tape diagrams to model with mathematics (MP4). Prompting students to show their thinking by annotating their diagrams encourages strong modeling practices. The following questions promote MP4: Where do you see the difference in length on your tape diagram? How can you show the part you need to add or take away on your tape diagram?” Teacher directions state, “Direct students to use their tape diagram to find the difference in length. Invite students to think–pair–share about the following question, ‘How did you find the difference between Imani’s estimate and the measurement?’ (I know 4 more than 16 is 20. 20 take away 4 is 16.) Highlight thinking that emphasizes adding to the smaller number to make the totals the same. ‘I heard someone say 4 more than 16 is 20. Let’s show that thinking on our tape diagram.’ Extend the tape that represents 16 centimeters by drawing dashes to outline a unit of 4. Consider using a different-color marker. Write a 4 inside the new part. The modified M tape should be the same length as the E tape. Have students say and write the addition equation, $16+4=20$. Invite students to think–pair–share about the following questions. ‘Where do you see the difference in length in the tape diagram? Where do you see the difference in the number sentence?’ (In the tape diagram, it’s the little box we added that shows the measurement. In the number sentence, it’s the number 4.) Direct students to underline the unknown, or the number that answers the question. ‘What is the difference in length in centimeters between Imani’s estimate and the actual length?’ (The difference in length is 4 cm.) Write the answer statement under the equation and direct students to do the same: The difference in length is 4 cm.”

• Module 3, Topic C, Lesson 13: Recognize that equal parts of an identical rectangle can be different shapes, Launch, students build experience with MP4 as described in the Teacher Note, Promoting the Standards of Mathematical Practice. “When students choose and draw a model to solve a real-world problem, they are modeling with mathematics (MP4). Ask the following questions to promote MP4: What key pieces of information should be in your model? Do you agree or disagree with a different approach? Why?” Teacher directions state, “Students decompose a whole to solve an equal-sharing problem to recognize that fourths of the same-size whole can be different shapes. Present the following prompt: (Picture is of five granola bars.) ‘4 friends have 5 granola bars. They want to share them equally. How many granola bars will each friend get?’ Allow students 1 minute to work independently. Circulate and select two or three students to share their work. Select work that shows partitioning the bars into fourths in various ways. Gather the class and invite two or three students to share their solutions, one at a time. As each student shares, ask questions to elicit thinking and clarify the model used to represent the problem. Ask the class questions to make connections between the different solutions and their own work.”

• Module 5, Topic A, Lesson 3: Solve one- and two-step word problems to find the total value of a group of coins, Problem Set, Problem 2, students build experience with MP4 as described in the Teacher Note, Promoting the Standards for Mathematical Practice. “Students model with mathematics (MP4) when they draw a tape diagram or number bond or use a pictorial representation of physical coins to model and to help them understand the relationship between the parts and the total in a real-world problem. Ask the following question to promote MP4: How does your tape diagram represent the relationship between the total and the given part? How does your tape diagram help you set up the subtraction problem correctly? Which model, the tape diagram or the number bond, do you prefer? 2. Ann wants to buy a toy. She has 1 quarter, 2 dimes, and 8 pennies. She needs 45 cents more. How much does the toy cost?” Students are encouraged to use the Read-Draw-Write process.

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students choose tools strategically as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 2, Topic A, Lesson 4: Use compensation to add within 200, Learn, Model Compensation on a Number Line Diagram, students build experience with MP5 as described in the Teacher Note, Promoting the Standards of Mathematical Practice. “Students use appropriate tools strategically (MP5) when they choose from a measuring tape, the open number line, the arrow way, and other models to visualize the parts of each addition problem. Ask the following questions to promote MP5: What kind of model would be helpful in solving this problem? How could an open number line or the arrow way help you show how you used a benchmark number to add?” Teacher directions state, “Pair students and designate each student as partner A or partner B. Direct partners to show how to use compensation on an open number line and with the arrow way. Have partner A record on an open number line as partner B records by using the arrow way to find the total for $24+39$. Invite students to take turns modeling a think-aloud. Direct students to switch roles and then repeat the process with the following sequence: $124+69$, $48+116$, $59+125$ Circulate as students work and encourage them to explain why compensation works. Consider using the following prompts. ‘Where do you see each addend represented in the model? How do you know which addend to count on from? How does using a benchmark number help make the addition problem easier?’ Facilitate a discussion on how students used compensation to solve the problems. Encourage them to use the strategy name compensation in their answers. ‘How does compensation make these problems easier to solve?’ (Compensation makes it easier to add larger numbers if I can use benchmark numbers. It’s easier for me to add 50 to 116. Then I can subtract 2 to get the answer. I can add more efficiently because I add a benchmark number and then take away the extra amount.)”

• Module 4, Topic E, Lesson 23: Solve two-step addition and subtraction word problems, Learn, Solve Two-Step Word Problems Involving Data, students build experience with MP5 as described in the Teacher Note, Promoting the Standards of Mathematical Practice. “Students use appropriate tools strategically (MP5) when they use a graph or tape diagram to set up a two-step word problem to improve their understanding. Ask the following question to promote MP5: How does each tool help you see the relationship between the given quantities and the unknown quantity in the artwork? Is there one tool that you prefer?” Teacher directions state, “Students solve a two-step word problem by using contexts from a piece of art. Direct students to their books and chorally read the next problem. (Shown is a picture with people at a reception, a word problem, and a graph that relates to the picture.) ‘Let’s find the scale on the graph. What do you notice about the scale?’ (The scale doesn’t show a count of 1. It shows a count of 5.) Point to the scale and chorally count by fives. ‘Can we still read the graph the same way, even when the scale isn’t counting by ones? Why?’ (Yes, we can look at where the shading ends to see how many. Yes, we can count each box by fives to see how many there are of each category.) ‘How can the graph help us with the problem?’ (The graph can help us see how many adults and children are at the reception. We can add 15 more children to the graph, to show the 15 more children that come to the reception. We can see that there are more adults than children. We can use the graph to see how many more adults there are than children. The graph can help be part of our drawing. The bars on the graph are like a tape diagram.) Use the Math Chat routine to engage students in mathematical discourse. Give students 5 minutes of independent time to use. Read–Draw–Write to find out how many children are at the party and how many more adults than children are at the reception.”

• Module 5, Topic B, Lesson 11: Measure to compare differences in lengths, Launch, students build experience with MP5 as described in the Teacher Note, Promoting the Standards for Mathematical Practice. “When students select among available measurement tools to create a space creature, they use appropriate tools strategically (MP5). Ask the following questions to promote MP5: 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?” In this activity students reason about measurements and how to use a tool effectively for accuracy to compare space creatures. Teacher directions state, “Display the picture of the three space creatures. Have students turn and talk about how they can compare the height of the three creatures. ‘Let’s compare the three creatures.’ Invite students to think–pair–share to make comparison statements about the space creatures’ height. (Creature 1 is shorter than creatures 2 and 3. It’s the shortest. Creature 2 is the tallest. Creature 3 is taller than creature 1 but shorter than creature 2.) ‘What comparisons can you make about their arms and legs?’ (Creature 2’s legs are a lot longer than creature 1’s legs. Creature 1’s arms are shorter than creature 3’s arms.) ‘How can we check the length of the arms and legs so we can compare more precisely?’ (We could use a ruler to find the length of each arm and leg. Then we could find out how much longer or shorter one creature’s legs are than another’s.) ‘How can we measure the arms or legs if they are bent?’ (We can use mark-and-move-forward with our rulers to measure each bent part. Then we can add the parts together.) ‘We could put a piece of string on the arm that is bent, mark the length on the piece of string, and then measure the piece of string.’ Draw a bent arm and model by using the piece of string to measure the arm. Place the string on the arm, cut the string, and place it on the ruler to find the length.”

The materials reviewed for Eureka Math² Grade 1 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP6 across the year and it is explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

Students attend to precision in the mathematics, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Module 1, Topic B, Lesson 5: Connect measurement to physical units by iterating a centimeter cube, Learn, Make a Numberless Ruler, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students attend to precision (MP6) when they concentrate on creating same-size spaces on a unit ruler. Help students recognize that the distance between two tick marks is the same as the length of a centimeter cube and that they are counting length units, not tick marks. In later lessons, rulers show a tick mark for 0 and counting tick marks will result in an incorrect measurement.” Teacher directions state, “Distribute one paper strip and 20 cubes to each student. ‘We’ll use centimeter cubes to make a ruler.’ Direct students to line up their cubes along the bottom edge of the paper strip, end to end. Guide them to draw a tick mark up from the end of each cube along the bottom of the paper. Students should draw the first mark at the end of the first cube. Ensure that students do not number the tick marks. Invite students to think–pair–share about what they notice about their numberless rulers. (The marks have even spaces between them. There are no numbers on our ruler. The spaces are all the same size. Each space is 1 cm long.) ‘All the spaces on the ruler are the same size. We call each same-size space a length unit. The mark you make at the end of each cube is called a tick mark. Each tick mark shows where one length unit ends and the next one begins. Each tick mark represents a unit.’ Direct students to count each length unit by placing their finger in the space between the tick marks. ‘How many length units did you count?’ (20) Give each student a new, unsharpened pencil. Have them measure to confirm the length of the pencil by aligning the pencil with the endpoint of the numberless ruler. ‘How long is the pencil? How do you know?’ (It’s 19 cubes long. I pretended there were cubes on the ruler and counted them. It’s 19 length units. I counted 19 spaces.) Emphasize that saying nineteen does not refer to just the nineteenth length unit. ‘The first 19 tick marks show the distance covered by 19 length units. When we say nineteen, we refer to the distance covered by 19 length units.’ Consider having students whisper-count and slide a finger 1 length unit at a time as they cover the distance of 19 length units. ‘Whether we measure with centimeter cubes or with this ruler, the size of each length unit is the same.’”

• Module 2, Topic B, Lesson 9: Use place value drawings to compose a ten and relate to written recordings, Learn, Use Place Value Drawings to Add, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students communicate precisely to others (MP6) when they can express the proper representation of the addends in the place value chart. Ask the following questions to promote precision of units: Does the place value drawing represent the correct place value units for the addends? How do you know you have correctly found the sum on the place value chart?” Teacher directions state, “Direct students to their books and the problem $115+25=$__. ‘Let’s add 115 and 25. You can draw a place value chart and represent each addend with dots. Draw two long, vertical lines to have a place for hundreds, tens, and ones. Because the dots do not show their values like the disks do, you need to label the chart with place value units.’ Label the three columns 100s, 10s, and 1s, as students do the same. ‘Let’s show the first addend, 115, on the place value chart. How many hundreds are in 115?’ (1) ‘How many tens?’ (1) ‘How many ones?’ (5) ‘Let’s count to be sure we showed the correct number.’ Point to each dot while counting. (100, 110, 111, 112, 113, 114, 115) ‘Now let’s show the other addend, 25.’ Use a similar sequence to show 25. ‘Now we are ready to add. Look at the ones place. What do you notice?’ (I can make a ten. $5+5=10$) ‘You can compose a new unit of ten. Circle the 10 ones and draw an arrow into the tens place. Then draw the new unit of ten.’ Circle 10 ones and then draw an arrow and 1 ten, as students do the same. Now you can add the tens. What is 1 ten + 2 tens + 1 ten?’ (4 tens) ‘There is still 1 hundred in the hundreds place. We did not add any hundreds.’ Invite students to think–pair–share about how the place value drawing helps to show and solve the addition problem. (I can see all the units, so it’s easy for me to add them. We are adding like units. First, we show all the units, and then we add the ones, the tens, and the hundreds. It makes it easy for me to see when there’s a new ten.)”

• Module 4, Topic B, Lesson 9: Use place value drawings to represent addition and relate them to written recordings, part 2, Learn, Add with Place Value Drawings and New Groups Below, students build experience with MP6 as described in the Teacher Note. Promoting the Standards for Mathematical Practice, “Students attend to precision (MP6) when they focus on making careful representations of addition in place value drawings and in vertical form. Ask the following questions to promote MP6: Why do we carefully align digits in columns in vertical form? Why is it important to be precise when you compose and record new units? Where is it easy to make mistakes in each recording?” Teacher directions state, “Direct students to problem 1 in their books. ‘Let’s add 585 and 269 by using place value drawings and new groups below.’ Direct students to make a place value drawing to represent $585-269$. Then have them write the problem vertically. Invite students to think–pair–share about how their drawing matches the vertical form. Encourage them to use place value language. (The drawing and vertical form  match because they both show 5 hundreds 8 tens 5 ones and 2 hundreds 6 tens 9 ones. Both recordings show both addends. The totals of each place value unit—ones, tens, and hundreds—are the same in the drawing and in vertical form. Let’s add the ones.) ‘What is 5 ones + 9 ones?’ (14 ones) Invite students to think–pair–share about what to do first by using the place value drawing and then vertical form. (You can compose a ten, so you circle it and draw an arrow to show the new ten in the tens place. You show the new unit on the line in the tens place and write 4 below the line in the ones place. That shows 14 ones as 1 ten 4 ones.) Have students show the work on both recordings. Then have them check a partner’s work to be sure the drawing and vertical form match. ‘We renamed 14 ones as 1 ten 4 ones. Now, what do we do?’ (We add the tens.) ‘What is 8 tens + 6 tens + 1 ten?’ (15 tens) Invite students to think–pair–share about what to do next on the drawing and then in vertical form. (You bundle 10 tens as 1 hundred. Draw an arrow and a dot to show the new hundred in the hundreds place. You rename 15 tens as 1 hundred 5 tens. Write 1 on the line in the hundreds place because we have to add the new hundred. Write 5 below the line in the tens place because there are 5 tens left over when you compose the hundred.) Direct students to show the work on both recordings. Then have them check a partner’s work to be sure their drawing and vertical form match. ‘Let’s complete the problem. What is 5 hundreds + 2 hundreds + 1 hundred? (8 hundreds) Read the equation. ‘$585+269=854$’ Invite students to turn and talk about how each step on the place value drawing matches the steps in vertical form. Have partners take turns pointing to each part of the drawing while the other points to the corresponding part in vertical form. Encourage students to use place value language as they share their thinking. Repeat the process to find the sums for problems 2 and 3 in their books.”

Students attend to the specialized language of mathematics, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Module 1, Topic G, Lesson 28: Use place value understanding to count and exchange $1, $10, and $100 bills, Learn, Count and Exchange Bills, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students communicate precisely to others (MP6) when they express a number and specify the units of that form. Look for students to be able to express the monetary unit values with the numbers as they make exchanges for larger units and communicate with others. If students do not state the appropriate units, prompt them with questions such as 10 tens make 1 what? 1 unit?” Teacher directions state, “Pair students and designate each student as partner A or partner B. Distribute a chart and a Money Tool Kit to each pair. Direct partner A to count by ones and place one-dollar bills in the first column of the chart until they reach 10 one-dollar bills. ‘What can we do with 10 one-dollar bills?’ (You can trade 10 one-dollar bills for 1 ten-dollar bill. We can put the ten-dollar bill in the next column.) ‘Another way to say that we switched 10 ones for 1 ten is exchange. When you trade one thing for another that has equal value, that’s called exchange. Let’s use that new word together. Repeat after me, I can exchange 10 one-dollar bills for 1 ten-dollar bill.’ (I can exchange 10 one-dollar bills for 1 ten-dollar bill.) Direct partner B to remove the 10 one-dollar bills from the chart and place the 10 bill in the second column of the chart. Have partners take turns counting 10 one-dollar bills and exchanging them for 1 ten-dollar bill on the chart until they reach $100. ‘Each time you counted 10 ones you exchanged them for 1 ten. How many 10 bills are on your chart?’ (10 ten-dollar bills) ‘What can we do with 10 ten-dollar bills?’ (We can exchange 10 ten-dollar bills for 1 hundred-dollar bill.) ‘Where do you think we should put the $100 bill?’ (In the column next to where the tens were) ‘What do you notice about how we organized the units on the chart? Use place value language and point to the chart as you share your thinking.’ (The place value units go from largest to smallest, hundreds, tens, and then ones, as we go this way. (Gesturing from left to right.) The place value units go from smallest to largest, ones, tens, and then hundreds as we go this way. (Gesturing from right to left.)) ‘What pattern keeps repeating as we move up the chart, from the smallest unit to the largest unit?’ (We keep exchanging 10 smaller units to make a new larger unit. I see that 10 ones make 1 ten and 10 tens make 1 hundred. 10 smaller units make 1 of the next larger unit.) Invite students to think–pair–share about how counting up to $124 with bills is different than counting up to $124 with craft sticks. (We bundled the sticks, but with bills, we exchanged 10 ones for 1 ten. We had to exchange $10 bills for a higher value bill. It’s like trading 10 small things for 1 big thing. With sticks, we kept the sticks and the size of the bundle got bigger. With money, we got a new bill with a different value on it.)”

• Module 3, Topic A, Lesson 3: Identify, build, and describe right angles and parallel lines, Learn, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students communicate with precision (MP6) when they observe the existence of a right angle as a distinguishing attribute in a polygon or shape. Ask the following questions to promote MP6: Does a shape have to have right angles to be a polygon?” Teacher directions state, “‘Look at your sticky note. What shape is it?’ (A square) ‘How many angles does it have?’ (4) Display the picture of the polygons from Launch. ‘Which polygons have angles that look like the angles on your sticky note?’ (The red, orange, and blue ones, the rectangle, the square, and the hexagon) ‘We call angles that have square corners right angles. We know a shape has a right angle when we can put our square sticky note on the inside of the angle and the sides line up.’ Demonstrate putting a sticky note inside the angle of the rectangle. ‘Let’s see how many right angles we can find in our classroom.’ Invite students to search for right angles in objects around the room, such as in the corners of a bulletin board or at the corner of a book. Direct students to put a new sticky note on the right angles they find to ensure the angle aligns. Show a sticky note in the top corner of the trapezoid. Invite students to think–pair–share about whether the angle is a right angle. (It’s not a right angle because the sticky note does not fit inside the angle perfectly. It’s not a right angle because only one side of the sticky note is lined up with the sides of the shape. The angle is too big to be a right angle.) Display a sticky note in the bottom corner of the trapezoid. Invite students to think–pair–share about whether the angle is a right angle. (It’s not a right angle because the sticky note is too big. The angle must be smaller than a right angle. It’s not a right angle because only one side of the sticky note is lined up with the sides of the shape.) Direct students to use their spaghetti pieces to build a polygon with at least one right angle on a piece of grid paper. When finished, invite them to draw it and use their sticky note to check their partner’s drawing to see that at least one angle is a right angle. ‘How did you know whether your partner’s polygon had a right angle?’ I put my sticky note inside the shape to see if any of the angles matched the sides of my sticky note. If the angle looked like an L, I knew it was a right angle. If the sticky note fit perfectly in the corner, I knew it was a square corner.)”

• Module 5, Topic B, Lesson 10: Measure an object twice by using different length units, and compare and relate measurement to unit size, Land, Debrief, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students communicate with precision (MP6) 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 to promote MP6: 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?” In this activity students discuss their prior work in the Problem Set, in which fictional students had measured an object using different units and they had checked the work. Teacher directions state, “Initiate a class discussion by using the following prompts. Encourage students to restate their classmates' responses in their own words. Direct students to problem 7 on the Problem Set. ‘Who measured correctly? How do you know?’ (Both students measured correctly. I know because I measured the fish tank sticker in inches and got 4 inches. Then I measured the fish tank sticker in centimeters and got 10 centimeters.) ‘Why are the two measurements so different?’ (They are measured by using different units. There are more centimeters because centimeters are a smaller unit than inches. There are fewer inches because an inch is longer than a centimeter.) ‘Does the relationship between the size of the unit and the number of units stay the same when we use different units, such as hundreds, tens, and ones, or pennies, nickels, dimes, and quarters?’ (Centimeters are a smaller unit than inches, so it takes more centimeters to measure an object than inches. Ones are a smaller unit than tens, so it takes more ones than tens to compose a number. For the number 50, there are 50 ones but only 5 tens. It takes more of a smaller unit than a larger unit to measure an object. It takes more ones than tens or hundreds to compose a number, just like it takes more inches to measure the length of an object than it does feet or yards. Yes, it takes more dimes than quarters to compose a dollar because dimes are worth less than quarters. Yes, there are more seconds in an hour than minutes because minutes are a larger unit than seconds.) ‘When we record measurements, why is it important to include the number and the unit?’ (It is important to include the number and the unit because the unit size affects how many units it takes to measure the length. 5 inches is much longer than 5 centimeters. If you need a bookshelf that is 3 feet tall and you don’t say the unit, someone might make it 3 inches tall, and it will be too tiny.)”

The materials reviewed for Eureka Math² Grade 2 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP7 and MP8 across the year and they are explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice." According to the Grade 1-2 Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic A, Lesson 4: Use information presented in a bar graph to solve compare problems, Learn, Use a Bar Graph to Solve Compare Problems, students build experience with MP7 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “As students answer comparison questions by using the bar graph, they look for and make use of structure (MP7). Ask the following questions to promote MP7: How can matching tiles help you figure out how many more pigs than cows there are? Can matching tiles help you compare other categories too? How can looking for missing tiles help you figure out how many fewer cows than pigs there are? Gather students and show different-colored square tiles in a pile. Display the Farm Animals table. ’The table shows that there are goats, cows, pigs, and hens on a farm. Let’s make a bar graph to compare how many of each type of animal are on the farm. Let’s use these colored tiles to represent each animal. How can we organize the tiles to make a bar graph?’ (We can put the tiles in four lines, like bars. We can use a different color for each kind of animal. We can make a group for each color. We can use tiles to count out the number for each animal in the table.) Call on volunteers to organize the tiles in rows by color. Confirm each category total. Direct students’ attention to problem 1 in their student book. Guide students as they label the categories and scale. Then have them complete the Farm Animals graph. Have the class pause before continuing to problem 2. Tell students that mathematicians use graphs to answer questions. Invite them to do the same. ‘Let’s compare the total number of pigs and cows. Are there more pigs or cows?’ (Gesture to the sentence frame. There are more pigs than cows.) ‘How many more pigs than cows are there? How do you know?’ (There are 3 more pigs than cows. I matched them up, 1 blue to 1 yellow. There are 3 more yellow tiles.) ‘Are there fewer pigs or cows?’ (Gesture to the sentence frame. There are fewer cows.) ‘How many fewer cows than pigs are there? How do you know?’ (There are 3 fewer cows than pigs. I can tell because it looks like 3 blue tiles are missing.)”

• Module 3, Topic A, Lesson 5: Relate the square to the cube and use attributes to describe a cube, Learn, Construct a Cube, students build experience with MP7 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “When students apply their understanding of the relationship between two-dimensional shapes and three- dimensional shapes, they are making use of structure (MP7). Ask the following questions to promote MP7: 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? (Hold up the cube.) ‘Which two-dimensional shape makes up the face, or the flat part, of this three-dimensional shape?’ (A square) ‘ This is called a cube. How many square faces are there on this cube? Let’s count each face together.’ (1, 2, 3, … , 6) Point to and count each face on the cube. To ensure each face is only counted once, mark each counted face with a numbered sticky note. ‘Now we know a cube has six faces, so it is composed of six squares. Let’s build a square by using our toothpicks and marshmallows. What attributes does our square need to have?’ (Four straight sides that are equal in length Four right angles Two pairs of parallel lines) Direct students to build the outline of a face by making a square with their toothpicks and marshmallows. (Hold up the cube.) ‘We’ve made one face of the cube by using four toothpicks. How many toothpicks will we need in all to make a cube? Let’s count the edges, or the places where two faces meet, to find out. Count with me.’ (1, 2, 3, … , 11, 12) Point to and count each edge on the cube. To ensure each edge is only counted once, systematically count all the bottom edges, then the middle edges, and finally, the top edges. If using a plastic cube, consider tracing each counted edge with a dry-erase marker. ‘We just found out that a cube has 12 edges, so we need to use 12 toothpicks in all. We have already used 4 toothpicks.’ Invite students to turn and talk about how many more toothpicks are needed to finish building the cube. ‘How many more toothpicks do we need to finish the cube?’ (8) Direct students to use 8 more toothpicks to finish building their cube. Circulate to support students as needed. Write vertices. ‘When there is more than one vertex, we say vertices. Let’s count how many vertices a cube has. Which material represents the vertices?’ (The marshmallows) Direct students to point to each vertex of their cube as they count the 8 vertices. Invite students to think–pair–share to describe a cube’s attributes. Encourage them to include the words faces, edges, and vertices in their description. (A cube has 12 edges.Each cube’s face is a square. The cube has 6 faces. I see 8 vertices in the cube.) ‘You found all the attributes of a cube. It is a three-dimensional shape composed of 6 square faces, 12 edges, and 8 vertices.’”

• Module 5, Topic B, Lesson 12: Identify unknown numbers on a number line by using the interval as a reference point, Launch, students build experience with MP7 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “As students relate the yard stick to a number line they look for and make use of structure (MP7). Ask the following questions to promote MP7: How can you represent a distance on a number line? Why would you use a number line instead of a yard stick to represent distance? In this activity students view a rocket launch video and represent the distance traveled. What do you notice?’ (The rockets go up high and travel across the field, then land on the field. The yellow rocket goes the highest. The blue rocket goes the farthest. The green rocket, which is the biggest, goes the shortest distance.) ‘What do you wonder?’ (I wonder how high each rocket travels. I wonder how far each rocket travels across the field. I wonder why some rockets travel farther than others. I wonder how long each rocket is.) Play part 2 that shows the distance each rocket travels from the launch pad to where it lands. Pause the video so students can refer to the screen. ‘How far did the blue rocket travel?’ (60 yards) ‘How can we use a yard stick to show how far the blue rocket traveled across the field?’ (We need 60 yard sticks to show 60 yards. We can use one yard stick and just mark-and-move-forward. Hold up a yard stick.) ‘Why might it be challenging to use one yard stick to show 60 yards?  Invite students to think–pair–share about how the yard stick and a number line are the same and different. (The yard stick has numbers on it and so does a number line. The yard stick shows the length unit between each tick mark and so does a number line. They both have equal spaces between the numbers.) ‘Let’s draw part of a number line to represent the distance the blue rocket traveled. What should be the starting and ending numbers on our number line?’ (0 and 60) Have students turn and talk about how they can represent the numbers between 0 and 60. Give students 2 minutes to draw a number line on their personal whiteboards to represent the distance from the launch pad to the blue rocket’s landing spot. Circulate and listen as students work. Identify a few students to share their number lines and their thinking. Choose work that shows different intervals, such as ones, fives, tens, or twelves. Invite students to think–pair–share about how the number lines are the same and different. (The number lines all start at 0 and end at 60. The number lines have different numbers between 0 and 60. One number line shows counting by fives, the other shows counting by tens, and the last one shows counting by twelves.)”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic A, Lesson 2: Draw and label a bar graph to represent data, Learn, Make a Bar Graph, students build experience with MP8 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students look for and express regularity in repeated reasoning (MP8) when they recognize that they can use the scale to find the last box they need and color the row over to that box. Ask the following questions to promote MP8: Do you color the same number of boxes when you count each box and when you use just the scale? Why? How do you know that using the scale to mark the last box you need to color will always give you the right number of boxes?” Teacher directions state, “Display the Our Birthdays table and the prepared blank graph on chart paper. Then conduct a class survey about students’ birthday months. Using the data from the table, guide students to complete the bar graph in their student book as you create the graph on chart paper. Begin by having them fill in the title and label the categories on the side. Have students label in the same order as the table, starting with Spring at the top. ‘Before we show our data on a bar graph, we fill in the scale along the bottom of the graph. This helps make it easier to count the totals for each category.’ Pause to fill in the scale as students do the same. Add the term scale to the terminology chart. ‘Look at the numbers on the scale. What does this look like?’ (The number path from first grade, The numbers on a ruler) ‘The numbers on the scale go in order, just as they do on a ruler or a number path. On this graph, the scale tells us that each box stands for 1 student’s birthday. How many boxes should we color in for the Spring category?’ (8 boxes) ‘Yes, we color in 8 boxes to match the data from our table.’ Have students follow along on their graphs as you model the following procedure: Confirm which row to color by putting a finger on the category label and moving it across the row. Put a finger at 8 on the scale. Slide it up to the appropriate row. Make a mark in that box to indicate where to stop coloring. Color in 8 boxes for the Spring category. Continue in this way to complete the graph.”

• Module 2, Topic C, Lesson 16: Use compensation to subtract within 200, Learn, Reason About Compensation, students build experience with MP8 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “As students reason about how compensation can be used to subtract with larger numbers and extend the strategy to numbers within 200, they are looking for and expressing regularity in repeated reasoning (MP8). Ask the following questions to promote MP8: When you look at the sequence of expressions, what is the same and what is different? What patterns do you notice about the two work samples? Do you think you can use the compensation strategy to create simpler problems when you work with larger numbers?” Teacher directions state, “Display the two open number lines. Invite students to analyze the two work solution strategies. ‘What do you notice? From your observations, what do you wonder?’ (I notice the strategy works the same way for both problems. I notice both problems use a benchmark number. For both, you take away 1 more than you’re supposed to, so you have to add 1 back. I wonder what would happen if we subtracted a different number from 145. I wonder how it would work if we used different numbers.) ‘What steps did this student take? How do you know?’ (First, she noticed she must take away 9, and that is close to the benchmark 10. Next she subtracted 10 from 145. Only the tens change, so that gets her to 135. Then since she took away 1 more than she was supposed to, she added 1 back.) Display the arrow way notation alongside the open number line. ‘Where do you see the compensation strategy in this work?’ (They both subtract a benchmark number, 10, but on the number line it’s a big hop back and a little hop forward. With the arrow way, they write −10 above an arrow to show it gets you to 135, and they write +1 above an arrow to show that now you’re at 136. They both show that you take away 10 and then add 1 back to compensate. On the number line, you can see a big backward hop of 10 and then a little forward hop of 1. With the arrow way, the arrows don’t show size; they just show what number you get to, and they write the −10 and +1 above the arrows.) ‘The open number line model helps us see why compensation works. The hops show movement along the number line. We move backward when we subtract and move forward when we add. The arrows on the number line show the sizes of the hops. The arrow way is a way to record your thinking. (Point to the steps in the arrow way model as you name them.) I subtract 10 and get to 135. Then I add back 1 and get to 136.’”

• Module 6, Topic A, Lesson 3: Use math drawings to represent equal groups and relate them to repeated addition, Learn, Compose Equal Groups Concretely, students build experience with MP8 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students look for and express regularity in repeated reasoning (MP8) 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 to promote MP8: 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?” Teacher directions state, “Direct students to show 1 group of 5 with their tiles on their whiteboard and draw a circle around them. ‘How many groups do you have?’ (1 group) ‘How many tiles are in the group?’ (5 tiles) ‘Repeat after me: There is 1 group of 5.’ (There is 1 group of 5.) Direct students to continue making groups of 5 with their tiles until they have 4 groups of 5.‘Let’s touch and count each group of 5.’ (1 group of 5, 2 groups of 5, 3 groups of 5, 4 groups of 5) ‘We counted 4 groups of 5. What does the 4 represent?’ (The 4 represents the number of groups.) Draw a line underneath each group of 5. ‘What is the number of tiles in each group?’ (5) ‘How many fives should we write? Why?’ (We need to write 4 fives because there are 4 groups of 5.) Write the repeated addition equation and direct students to do the same. (Gesture to the first 2 fives as you say the equation.) ‘What is $5+5$?’ (10) (Gesture to the third five.) ‘What is $10+5$?’ (15) (Gesture to the fourth five.)  ‘What is $15+5$?’ (20) ‘What is 4 groups of 5?’ (20) Invite students to think–pair–share about a more efficient way to add the 4 fives.(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.)”

The materials reviewed for Eureka Math² Grade 2 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

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

• Grade 1-2 Implementation Guide, Inside Teach, Module-Level Components, Overview, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.”

• Grade 1-2 Implementation Guide, Inside Teach, Module-Level Components, Why, “The Why section gives insight into the decisions made during the writing of the module. This insight helps you understand the underlying structure of the module, flow of the content, and coherence of the different parts of the curriculum.”

• Grade 1-2 Implementation Guide, Inside Teach, Module-Level Components, Achievement Descriptors, “The Achievement Descriptors: Overview section is a helpful guide that describes what Achievement Descriptors (ADs) are and briefly explains how to use them. It identifies specific ADs for the module, with more guidance provided in the Achievement Descriptors: Proficiency Indicators resource at the end of each Teach book.”

• Grade 1-2 Implementation Guide, Inside Teach, Module-Level Components, Lesson Structure, “Each lesson is structured in four sections: Fluency, Launch, Learn, and Land. Lessons are designed for one 60-minute instructional period. 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. Launch creates an accessible entry point to the day’s learning through activities that build context and often create productive struggle that leads to a need for the learning that follows. Every launch ends with a transition statement that sets the goal for the day’s learning. Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Suggested facilitation styles vary and may include direct instruction, guided instruction, group work, partner activities, interactive video, and digital elements. The Problem Set, an opportunity for independent practice, is included in Learn. Land helps you facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket. In the Debrief portion of Land, suggested questions, including key questions related to the objective, help students synthesize the day’s learning. The Exit Ticket provides a window into what students understand so that you can make instructional decisions.”

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

• Module 2, Topic C, Lesson 15: Use compensation to subtract within 100, Launch, provides a teacher note for Differentiation: Support. “While measuring tapes are provided in the next segment, consider making them available now for students who can benefit from the concrete support of a number line.”

• Module 3, Topic A, Lesson 1: Determine the defining attributes of a polygon, Launch, provides a teacher note with general guidance, Teacher Note. “Much of the terminology in this topic (e.g., closed, side, corner, and square corner) is familiar from previous grades. However, given the specificity of the word use and the time that has passed, it may be appropriate to introduce the terminology as if it were new. Consider creating an anchor chart with the terms and visuals for students to reference as needed throughout the topic.”

• Module 4, Topic C, Lesson 12: Take from a ten or a hundred to subtract, Learn, Reason about Efficiency, provides a teacher note for UDL: Representation. “Consider providing students with a copy of the student work samples to use as a reference as they participate in the Five Framing Questions routine.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

Materials consistently contain adult-level explanations, examples of the more complex grade/ course-level concepts, and concepts beyond the course within Topic Overviews and/or Module Overviews. According to page 7 of the Grade 1-2 Implementation Guide, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.” Page 9 outlines the purpose of the Topic Overview, “Each topic begins with a Topic Overview that is a summary of the development of learning in that topic. It typically includes information about how learning connects to previous or upcoming content.” Examples include:

• Module 2: Addition and Subtraction within 200, Topic D: Topics for Decomposing a Ten and a Hundred to Subtract, Topic Overview, connects the work being done in Grade 2 to the development of student understanding as they work towards using the standard algorithm in Grade 4. “In topic D, students build on their understanding of place value strategies and taking from a unit of ten or a hundred. Consistent use of place value disks and place value drawings helps students systematically model the steps they take when they decompose a larger value unit. When students relate models to the unit form, it primes them to use the vertical form in module 4. As with addition, students are not expected to master the standard subtraction algorithm until grade 4. Throughout the topic, students move through concrete and pictorial representations to develop a conceptual understanding of subtraction. First, students use place value disks to represent the total and subtract numbers concretely. Then they use place value drawings to represent subtraction problems. Students see they can decompose a unit of ten, and later a hundred, when they need more ones in the ones place or tens in the tens place to subtract. Problems gradually increase in complexity as students decompose once, and then twice, to subtract. As it was in topic C, the language is intentionally consistent and repetitive. This secures familiarity with the representations and anchors students’ understanding as they move toward work with more abstract numbers. Students relate place value drawings to more abstract recordings that show the minuend and subtrahend in unit form. While students are not expected to write unit form recordings independently, they make connections that deepen their place value understanding. For example, when they unbundle 1 ten into 10 ones in a drawing, students see how 1 hundred 2 tens 6 ones can be renamed as 1 hundred 1 ten 16 ones. The fact that the value of the minuend does not change when it is renamed is a key understanding in grade 2.”

• Module 4: Addition and Subtraction Within 1000, Module Overview, Why, “Why do you show new groups below when adding in vertical form? The decision to show newly composed units on the line below the addends, as opposed to above, has several advantages that support conceptual understanding with the standard algorithm: 1. The digits are written in close proximity to each other, so students do not see them as unrelated. The close proximity reduces the likelihood of students reversing the order of the numbers when recording the regrouping. 2. When composing a new unit, students write the teen number in order, for example as 1 new unit of ten on the line first and then the additional ones next to it below the line. It is natural for students to write numbers in their usual order (e.g., 1 then 6), rather than the reverse. 3. Since students typically add digits from the top down in a given column, the additional 1 can be easily counted on to a larger sum at the end. Why is there an entire topic devoted to simplifying strategies for subtraction but not for addition?”

• Module 6: Multiplication and Division Foundations, Topic C: Rectangular Arrays as a Foundation for Multiplication and Division, Topic Overview, relates the use of arrays to the more complex strategy of using the area module in Grade 3. “Topic C naturally follows topic B, where students compose and manipulate the rows and columns of an array. This topic is designed to deepen students’ understanding of spatial relationships and structure as they build and partition rectangles with rows and columns of same-size squares. To begin, students build a rectangle by making a tile array without gaps or overlaps. Then they build square arrays and recognize that only one repeated addition equation can be used to represent the array because the number of rows is equal to the number of columns. Next, students use square tiles to draw an array. They reason about how they can compose a larger rectangle by using smaller units. Much like students iterated a length unit in module 1 to create a centimeter ruler, now students iterate the square unit to construct a row or column and, ultimately, a rectangle. As they draw, they realize the structure of an array is a collection of same-size squares arranged in rows and columns. Students begin to see a row or column in two ways: as a composition of multiple units (e.g., 3 tiles) and as a single unit (1 row of 3). This supports the transition from repeated addition to multiplication and the area model in grade 3.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information and explanations of standards are present for the mathematics addressed throughout the grade level. The Overview section includes Achievement Descriptors and these serve to identify, describe, and explain how to use the standards. Each module, topic, and lesson overview includes content standards and achievement descriptors addressed. Examples include:

• Module 1, Topic A, Lesson 2: Draw and label a bar graph to represent data, Achievement Descriptors and Standards, “2.Mod1.AD8 Draw and label picture and bar graphs to represent a data set with up to four categories. (2.MD.D.10)”

• Module 3: Shapes and Time with Fraction Concepts, Achievement Descriptors and Standards, “2.Mod3.AD3 Label a given daily event as taking place in the a.m. or p.m. (2.MD.C.7)”

• Module 4, Topic C: Simplifying Strategies for Subtracting Within 1000, Description, “Students expand upon their toolbox of simplifying strategies for subtraction by using familiar models and recording methods from module 2. Students apply the take from a ten or a hundred strategy to subtract within 1,000. They also explore three types of compensation as shown in the table.” Achievement Descriptors and Standards are listed for the topic in the tab labeled, “Standards.”

• Module 6: Multiplication and Division Foundations, “Students count and solve problems with equal groups of objects. Students organize equal groups into rows and columns to create rectangular arrays. As they compose and decompose arrays, students gain foundations for multiplication.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The Grade 1-2 Implementation Guide includes a variety of references to both the instructional approaches and research-based strategies. Examples include:

• Grade 1-2 Implementation Guide, What’s Included, “Eureka Math² is a comprehensive math program built on the foundational idea that math is best understood as an unfolding story where students learn by connecting new learning to prior knowledge. Consistent math models, content that engages students in productive struggle, and coherence across lessons, modules, and grades provide entry points for all learners to access grade-level mathematics.”

• Grade 1-2 Implementation Guide, Lesson Facilitation, “Eureka Math² lessons are designed to let students drive the learning through sharing their thinking and work. Varied activities and suggested styles of facilitation blend guided discovery with direct instruction. The result allows teachers to systematically develop concepts, skills, models, and discipline-specific language while maximizing student engagement.”

• Implement, Suggested Resources, Instructional Routines, “Eureka Math² features a set of instructional routines that optimize equity by increasing access, engagement, confidence, and students’ sense of belonging. The following is true about Eureka Math² instructional routines: Each routine presents a set of teachable steps so students can develop as much ownership over the routine as the teacher. The routines are flexible and may be used in additional math lessons or in other subject areas. Each routine aligns to the Stanford Language Design Principles (see Works Cited): support sense-making, optimize output, cultivate conversation, maximize linguistic and cognitive meta awareness.” Works Cited, “Smith, Margaret S., Victoria Bill, and Miriam Gamoran Sherin. The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussions in Your Elementary Classroom. 2nd ed. Thousand Oaks, CA: Corwin Mathematics; Reston, VA: National Council of Teachers of Mathematics, 2018. Zwiers, Jeff, Jack Dieckmann, Sara Rutherford-Quach, Vinci Daro, Renae Skarin, Steven Weiss, and James Malamut. Principles for the Design of Mathematics Curricula: Promoting Language and Content Development. Retrieved from Stanford University, UL/SCALE website: http://ell.stanford.edu/content/mathematics-resources additional-resources, 2017.”

Each Module Overview includes an explanation of instructional approaches and reference to the research. For example, the Why section explains module writing decisions. According to the Grade 1-2 Implementation Guide, “The Why section gives insight into the decisions made during the writing of the module. This insight helps you understand the underlying structure of the module, flow of the content, and coherence of the different parts of the curriculum.” The Implementation Guide also states, “Works Cited, A robust knowledge base underpins the structure and content framework of Eureka Math². A listing of the key research appears in the Works Cited for each module.” Examples from Module Overviews Include:

• Module 1: Place Value Concepts Through Metric Measurement and Data: Place Value, Counting, and Comparing Within 1,000, Module Overview, Why?, “Part 1: Place Value Concepts Through Metric Measurement and Data, Why does the year start with categorical data? During the first week of school, teachers and students spend time establishing a classroom community. By launching with categorical data, teachers can leverage getting-to-know-you activities to generate student data, create graphs, and answer questions. Bar graphs provide students with a concrete and visual experience of comparison. Comparing categories on a bar graph sets up students for solving compare word problems by using a more abstract model, the tape diagram. Labeling the categories on a bar graph supports the practice of labeling tape diagrams, where students must visualize the amount or length.” Works Cited include, “Common Core Standards Writing Team, Progressions for the Common Core (draft), Grades K–5, Counting and Cardinality & Operations and Algebraic Thinking, 9. These word problem types come from Progressions for the Common Core State Standards in Mathematics, Operations and Algebraic Thinking Progression, and an explanation and example of some types are included here. See the table for examples. Darker shading indicates the four Kindergarten problem subtypes. Grade 1 and 2 students work with all subtypes and variants. Unshaded (white) problems are the four difficult subtypes or variants that students should work with in Grade 1 but need not master until Grade 2.”

• Module 2: Addition and Subtraction Within 200, Module Overview, Why?, “Why are two topics devoted to simplifying strategies for addition and subtraction?By the end of grade 2, students are expected to add and subtract fluently within 100 by using strategies based on place value, properties of operations, and the relationship between addition and subtraction. Fluency means being able to operate with numbers flexibly, efficiently, and accurately. Because students are not expected to work fluently with the standard addition and subtraction algorithms until grade 4, topics A and C are intentionally devoted to Level 3 addition and subtraction methods, in which students use simplifying strategies to make simpler problems. This gives students time to work through and to make connections between various strategies. As students apply place value understanding from module 1 and leverage familiar tools, they develop confidence and flexibility. While students are not expected to master all of the Level 3 strategies, they are expected to reason about the numbers in a problem and to consider efficient solution paths by using tools and written recordings. This builds their capacity toward mental math.” Works Cited include, “Common Core Standards Writing Team, Progressions for the Common Core (draft), Grades K–5, Counting and Cardinality & Operations and Algebraic Thinking, 9.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Each module includes a tab, “Materials” where directions state, “The following materials are needed to implement this module. The suggested quantities are based on a class of 24 students and one teacher.” Additionally, each lesson includes a section, “Lesson at a Glance” where supplies are listed for the teacher and students. Examples include:

• Module 1, Topic D, Lesson 16: Use a measuring tape as a number line to subtract efficiently, Overview, Materials, “Teacher: Double-sided meter sticks(2), Chart paper. Students: Measuring tape (1 per student pair).”

• Module 4: Addition and Subtraction Within 1000, Module Overview, Materials, “Chart paper, tablet(1), Eureka Math2™ place value disks set, ones to thousands(25), Computer with internet access(1), Pencils(25), Dot dice, set of 12(1), Personal whiteboards(25), Dry-erase markers(25), Personal whiteboard erasers(25), Index cards(12), Projection device(1), Learn books(24), Sticky notes, pad(5), Markers(25), Teach book(1), Eureka Math2™ measuring tape(24), Unifix® Cubes, set of 300(1), Please see lessons 1 and 24 for a list of organizational tools (cups, bowls, plates, trays, or rubber bands) suggested for the counting collection.”

• Module 6, Topic C, Lesson 12: Reason about how equal arrays can be composed differently, Overview, Materials, “Teacher: Arrays (digital download), Scissors. Students: Eureka Math² Numeral Cards (1 set per student pair), Hidden Addends Mat (1 per student pair), Arrays (in the student book), Color tiles, plastic, 1”(25), Scissors. Lesson Preparation: Gather the Hidden Addends Mats used in the previous lesson. Tear out the Arrays page from the student books. Consider whether to prepare this material in advance or have students remove it during the lesson. Print one copy of the Arrays page to use for demonstration.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Suggestions are outlined within Teacher Notes for each lesson. Specific recommendations are routinely provided for implementing Universal Design for Learning (UDL), Differentiation: Support, and Differentiation: Challenge, as well as supports for multilingual learners. According to the Grade 1-2 Implementation Guide, Page 47, “Universal Design for Learning (UDL) is a framework based on current research from cognitive neuroscience that recognizes learner variance as the norm rather than the exception. The guiding principles of the UDL framework are based on the three primary networks of the brain. Although the concept of UDL has roots in special education, UDL is for all students. When instruction is designed to meet the needs of the widest range of learners, all students benefit. Eureka Math² lessons are designed with these principles in mind. Lessons throughout the curriculum provide additional suggestions for Engagement, Representation, and Action & Expression.” Examples of supports for special populations include:

• Module 1, Topic A, Lesson 4: Use information presented in a bar graph to solve compare problems, Learn, Use a Bar Graph to Solve Compare Problems, students use different strategies to compare data on a bar graph. “Differentiation: Support: A common student error is to color the wrong row. Have students put a finger between rows before coloring. Or, suggest that they put their finger on the label and move it across the row as they color. Another common student error is to color more than the total number of boxes for each category. Suggest that students put their finger on the total on the scale and slide it up to the appropriate row. They can put a mark in the box to signal where to stop coloring.” 

• Module 4, Topic B, Lesson 5: Use the associative property to make a benchmark number to add within 1,000, Learn, Add to Make a Ten or Make a Hundred, students decompose an addend to make the next ten or next hundred to add. “UDL: Action & Expression: Consider posting questions for students to think about as they strategize. This scaffold can be gradually released as individuals are ready to apply the strategy independently. Are the addends close to a benchmark number? What is the next ten or next hundred? Is it easier to make a ten or a hundred? What new addition problem can you write?”

• Module 6, Topic B, Lesson 6: Decompose arrays into rows and columns and relate them to repeated addition, Learn, Decompose Arrays into Rows or Columns, students decompose an array into rows or columns and write repeated addition equations. “Language Support: Consider making a chart with key terms from the module with an example for each term. Include the following terms: Group, Row, Column, Array.” The Teacher Note includes a sample chart with illustrations and words.

The materials reviewed for Eureka Math² Grade 2 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity. 

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

• Module 1, Topic B, Lesson 10: Reason about the relationship between the size of the unit and the number of units needed to measure, Learn, Compare Length Units, students analyze various length units to understand the relationship among them. “Differentiation: Challenge: Invite students to apply knowledge of mental benchmarks. Consider asking the following questions. Make a thoughtful guess: How many digits are in a cubit? What did you use as your mental benchmark?” 

• Module 4, Topic B, Lesson 10: Choose and defend efficient solution strategies for addition, Learn, Share and Defend Strategy Choices, students use place value understanding to defend the efficiency of their solution strategies. “Differentiation: Challenge: Encourage students to show multiple solution strategies, beginning with the one they feel is most efficient. As time allows, consider inviting students to explain their alternate strategies, or ask the class to reason about what steps they think the student took to solve. For example, for $399+499$, a student might solve the problem by thinking $400+500-2=900-2=898$ When students share their flexible thinking, it provides multiple access points for all members of the class.”

• Module 6, Topic B, Lesson 8: Use square tiles to create arrays with gaps, Learn, Develop Contexts to Match Arrays, students construct scenarios to match a given array. “Differentiation: Challenge: Challenge students to develop a context in which the total and the number of rows are known, but the number of objects in each row is unknown.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Support for active participation in grade-level mathematics is consistently included within a Language Support Box embedded within parts of lessons. According to the Grade 1-2 Implementation Guide, “Multilingual Learner Support, Multilingual learners, or learners who speak a language other than English at home, require specific learning supports for gaining proficiency with the English needed to access the mathematics. Research suggests that best practices for these learners include opportunities and supports for student discourse and for using precise terminology. In addition to precise domain-specific terminology, high-impact academic terminology that supports learners across learning domains is explicitly introduced and used repeatedly in various contexts to build familiarity and fluency across the grade levels. Eureka Math² is designed to promote student discourse through classroom discussions, partner or group talk, and rich questions in every lesson.” According to Eureka Math² How To Support Multilingual Learners In Engaging In Math Conversations In The Classroom, “Eureka Math² supports MLLs through the instructional design, or how the plan for each lesson was created from the ground up. With the goal of supporting the clear, concise, and precise use of reading, writing, speaking, and listening in English, Eureka Math² lessons include the following embedded supports for students. 1. Activate prior knowledge  (mathematics content, terminology, contexts). 2. Provide multiple entry points to the mathematics. 3. Use clear, concise student-facing language. 4. Provide strategic active processing time. 5. Illustrate multiple modes and formats. 6. Provide opportunities for strategic review. In addition to the strong, built-in supports for all learners including MLLs outlined above, the teacher–writers of Eureka Math² also intentionally planned to support MLLs with mathematical discourse and the three tiers of terminology in every lesson. Language Support margin boxes provide these just-in-time, targeted instructional recommendations to support MLLs.” Examples include:

• Module 2, Topic A, Lesson 7: Solve word problems by using simplifying strategies for addition, Learn, Share and Defend Solution Strategies, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “As the small group discussion unfolds, consider providing students with sentence frames and the key terminology as a scaffold for peer conversations. For example, provide a smaller version of the addition strategies chart for students to reference as they defend their strategy choice. Also include the following sentence frames: I chose the ___strategy because ___. My strategy ___ works because ___. When I used ___, it made an easier problem because ___. Students explain their reasoning for selecting a particular solution strategy. Introduce the Take a Stand routine to the class. Draw students’ attention to the signs hanging in the classroom. Take a moment to read aloud the strategy on each of the signs. (Signs show the different strategies students may have used to solve the problem.)  Invite students to stand beside the sign that shows the strategy they used for this problem. When all students are standing near a sign, allow 2 minutes for groups to discuss the reasons why they chose that strategy. Then call on each group to defend their strategy choice by sharing reasons for their selection. Invite students who change their minds during the discussion to join a different group. Show student work as students share their reasoning.”

• Module 2, Topic D, Lesson 20: Reason about when to unbundle a ten to subtract, Land, Debrief, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “Consider providing students with sentence frames that include key terminology to help students share their thinking with a partner. For example, post the following sentence frames: I think you ___(do, do not) need to unbundle because ___. I think I can rename 52 as ___. Objective: Reason about when to unbundle a ten to subtract. Gather students with their Problem Sets and invite them to think–pair–share about the following questions. ‘Tell your partner which problems you can answer without unbundling a ten. How do you know when you need to unbundle a ten to subtract?’ (In problem 1, I didn’t need to unbundle because I had enough ones in the ones place to subtract ones from ones. In problem 4, I needed to unbundle a ten because I couldn’t subtract 7 ones from 2 ones, so I had to unbundle a ten to get more ones. In problem 5, I didn’t need to unbundle a ten because I had enough ones. I took 4 ones from 8 ones. In problem 6, I needed to unbundle a ten because I couldn’t take 4 ones away from 3 ones.) Direct students to problem 3. ‘How did you rename 52 to subtract 6?’ (I unbundled one of the tens and renamed it as 10 ones, so then I had 4 tens and 12 ones. I didn’t have enough ones to subtract 6 ones, so I crossed off a ten and drew 10 ones. Then I had 40 and 12.)”

• Module 4, Topic B, Lesson 8: Use place value drawings to represent addition and relate them to written recordings, part 1, Launch, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “The place value recordings are intentionally labeled with letters rather than the name of each recording. This reduces the cognitive demand of students struggling with language and allows them to focus on making mathematical connections. While students are not expected to master the names of each recording, they have been introduced to those names. Consider supporting students by creating a chart that shows the various recordings alongside a sample of each. Students reason about similarities and differences between different representations of addition. Display the picture of the place value drawing, the totals below recordings, and the new groups below recording. ‘What do you notice? What do you wonder?’ (I notice they all show the same addition problem and have the same total. I notice they all show how we add like units. I notice the new ten in A and D. In B & C, they found the totals for each place value unit. I wonder whether D will work for all numbers. It looks like it has the fewest steps! I wonder why you can add in any order in some of the recordings, but in others you always start in the ones place.) Invite students to turn and talk about how the recordings are alike and different. Then facilitate a class discussion. Invite students to share their thinking with the whole group. As students discuss, highlight thinking that shows the role of place value understanding in each recording.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Each lesson includes a list of materials for the Teacher and the Students. As explained in the Grade 1-2 Implementation Guide, page 11, “Materials lists the items that you and your students need for the lesson. If not otherwise indicated, each student needs one of each listed material.” Examples of manipulatives include: 

• Module 1, Topic C, Lesson 13: Estimate and measure height to model metric relationships, Materials, Students: measuring tape. For Fluency, Add on the Measuring Tape, students add 10 cm to a measurement to build an understanding of length units. “Invite students to roll out the measuring tape to 2 m and lay it in front of them. Consider displaying your own measuring tape as a model. After asking each question, provide think time and then signal for students to respond. ‘Wait for my signal to say the answer to each question. Put your finger on 30 cm. What is 10 more centimeters?’ (40 cm) ‘Slide your finger up to 40 cm while you say the addition equation, starting with 30 cm. Ready?’ (30cm +10cm =40cm Slides finger from 30 cm to 40 cm.)”

• Module 4, Topic C, Lesson 14: Use compensation to keep a constant difference by adding the same amount to both numbers, Materials, Teacher: Unifix^® Cubes. For Learn, Show Compensation with a Tape Diagram, the teacher uses Unifix^® Cubes to support instruction about Tape Diagram models. “Let’s look at another way to show why this strategy works.’ Show two rows of 5 Unifix Cubes in one color. Add 3 cubes of another color to the right end of the top row ‘There are 5 cubes in the bottom row. How many cubes are in the top row?’ (8) ‘What is the difference between 8 and 5?’ (3) Write $8-5=3$. Then add 1 cube of a third color to the left end of each row. ‘Did the difference change?’ (No.) ‘What new number sentence can we write to represent the difference now?’ $(9-6=3.)$  Write $9-6=3$. Then draw a tape diagram to represent the two rows of cubes. ‘I started with 8 and 5. When I added 1 more to each row, I changed the amounts to 9 and 6, but the difference stayed the same. We can say $8-5=9-6$ because they both equal 3.’ Write $8-5=9-6=3$.”

• Module 6, Topic C, Lesson 9: Determine the attributes of a square array, Materials, Students: “Color tiles, plastic, 1″ (25).” For Learn, Compose Rectangular Arrays students use tiles to compose arrays. “Direct students to place 12 tiles into 2 equal groups. ‘How many groups are there?’ (2) ‘How many tiles are in each group?’ (6) Direct students to arrange the 2 equal groups into 2 rows with no gaps or overlaps. ‘How many rows are there? (2) ‘How many tiles are in each row?’ (6) Direct students to outline the array to show its shape. ‘What shape is the array?’ (It’s a rectangle.) Invite students to think–pair–share about how they know the array is a rectangle. (I know it’s a rectangle because it has 4 sides. It has 4 right angles. It has 2 opposite pairs of parallel sides.) ‘What repeated addition equation matches the rows?’ $(6+6=12)$ What repeated addition equation matches the columns?’ $(2+2+2+2+2+2=12)$ The array can be represented by two different repeated addition equations depending on whether we think of the rows or columns as groups.) Direct students to rearrange their tiles to show 4 equal groups.”