The materials reviewed for Eureka Math² Grade 3 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 1, Topic C, Lesson 11: Demonstrate the commutative property of multiplication using a unit of 𝟒 and the array model, Learn, Tape Diagrams to Represent an Array, students develop conceptual understanding by drawing tape diagrams to represent the rows and columns in an array. “Let’s relate our fours array to tape diagrams. Show the 10 fours tape diagram. What multiplication equation can we use to represent this tape diagram, where the first factor is the number of groups? What multiplication equation can we use to represent this tape diagram, where the first factor is the number of groups? Show the 4 tens tape diagram. What multiplication equation can we use to represent this tape diagram, where the first factor is the number of groups? How are the tape diagrams similar to the array we made to show $10\times4=4\times10$? How are the tape diagrams different from the array? Teacher asks students to draw tape diagrams to represent $5\times4$ and $4\times5$. Place emphasis on the tape diagrams being the same length because they have the same total.” (3.OA.1)

• Module 2, Topic C, Lesson 17: Use place value understanding to subtract efficiently using take from a ten, Learn, Take from One Ten When Subtracting One-Digit Numbers, students develop conceptual understanding by subtracting “one-digit numbers from two-digit numbers by using a number bond to to take out ten to subtract. What do you notice about this work? From your observations, what do you wonder? What steps did this student take? How do you know? Advance the discussion to focus on the take from a ten strategy, and encourage student thinking that makes connections to using place value strategies to efficiently subtract, Let’s focus on the take from a ten strategy. Where do you see that in these problems? How does using the take from a ten strategy change how you subtract? How is a simplifying strategy, such as the take from a ten strategy, helpful?” (3.NBT.2)

• Module 5, Topic D, Lesson 18: Compare fractions with like units by using a number line, Learn, Fraction Position and Comparison on a Number Line, students develop conceptual understanding as they use fraction positions on the number line to compare fractions. “Draw a number line, labeling the starting mark as 1 and the ending mark as 4. Invite students to do the same. Write the following fractions: $\frac{2}{2}$, $\frac{7}{2}$. What should we do to our number line to help us locate these fractions? Partition and label the number line to show the whole numbers and then partition again to show halves.  Invite students to do the same. Where does 2 halves belong on the number line? How do you know? Label $\frac{2}{2}$ on the number line. Direct students to label $\frac{2}{2}$, and invite them to think-pair-share about where to label $\frac{7}{2}$. This number line doesn’t show 0. Where is 0? Which fraction is closer to 0, $\frac{2}{2}$ or $\frac{7}{2}$? Which fraction is greater, $\frac{2}{2}$ or $\frac{7}{2}$? How do you know? Write $\frac{5}{2}$ and $\frac{8}{2}$ next to $\frac{2}{2}$ and $\frac{7}{2}$.  Invite students to work with a partner to place them on a number line.  Let’s draw a box around the fractions that are equivalent to whole numbers. Which fractions will be in a box? Draw a box around $\frac{2}{2}$ and $\frac{8}{2}$. Direct students to do the same. 5 halves is not equivalent to a whole number. How did you decide where to place it? Which fraction is less, $\frac{5}{2}$ or $\frac{8}{2}$? How do you know?” (3.NF.2)   

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. The Problem Set, within Learn, consistently includes opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of conceptual understanding. Examples include:

• Module 1, Topic B, Lesson 7: Model measurement and partitive division by drawing equal groups, Launch, students independently demonstrate conceptual understanding as they determine whether 5 represents the number of equal groups or the number in each group. Teachers, “Display the equal-groups pictures one at a time. For each picture, ask the following: Is 5 the number of equal groups or the number in each group?” The teacher then “Invites students to turn and talk about how they know whether 5 is the number of equal groups or the number in each group.” Several pictures are provided, 2 plates each with 5 crackers, 5 plates each with 2 crackers, 5 circles each with 4 dots, and 4 circles each with 5 dots. (3.OA.1)

• Module 4, Topic B, Lesson 6: Tile rectangles with squares to make arrays and relate the side lengths to area, Land, Exit Ticket, students independently demonstrate conceptual understanding as they find the area of a rectangles by making arrays and relating the side lengths to areas. “Use the rectangle shown for parts (a)-(d). a. Use a ruler to find the unknown side length. b. Draw the missing tiles in the rectangle. c. Write an equation to show how to find the area of the rectangle. d. Area: ___ square centimeters.” A rectangle with one side label 7 cm and one side label ___ cm is shown. (3.MD.7)

• Module 5, Topic D, Lesson 18: Compare fractions with like units by using a number line, Exit Ticket, students independently demonstrate conceptual understanding of comparing fractions with like units by using a number line. “Use the number line for parts (a) and (b). a. Partition each whole number interval into thirds. b. Label $\frac{7}{3}$, $\frac{2}{3}$, and $\frac{4}{3}$ on the number line.” (3.NF.2) and (3.NF.3)

The materials reviewed for Eureka Math² Grade 3 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

The materials develop procedural skill and fluency throughout the grade level, within various portions of lessons, including Fluency, Launch, and Learn. Examples  include:

• Module 2, Topic C, Lesson 14: Use place value understanding to add and subtract like units, Fluency, Whiteboard Exchange: Make the Next Ten, students develop procedural skill and fluency as they identify the next ten and the number needed to make the next ten to prepare for using simplifying strategies to find sums and differences.  “Display the number 119. When I give the signal, read the number shown. Ready? What is the next ten? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond. Display the equation with the unknown addend. 119 plus what number equals 120? Write and complete the equation. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the completed equation. $119+$___$=120$.” (3.NBT.2)

• Module 2, Topic C, Lesson 15: Use the associative property to make the next ten to add, Fluency, Sprint: Multiply and Divide by 5, students develop procedural skill and fluency as they complete equations to build fluency with multiplying and dividing by 5. “Complete the equations. 1. $2\times5=$___. 2. $10\div5=$___.” (3.OA.7)

• Module 2, Topic C, Lesson 18: Use place value understanding to subtract efficiently using take from a hundred, Fluency, Ready, Set, Multiply, students develop procedural skill and fluency as they find the product and say a multiplication equation to build multiplication fluency within 100. “Let’s play Ready, Set, Multiply. Have students form pairs and stand facing each other. Model the action: Make a fist, and shake it on each word as you say, ‘Ready, set, multiply.’ At ‘multiply,’ open your fist, and hold up any number of fingers greater than 1. Tell students that they will make the same motion. At ‘multiply’ they will show their partner any number of fingers other than 1. Consider doing a practice round with students.Clarify the following directions: Only show 2, 3, 4, or 5 using one hand. Try to use different numbers each time to surprise your partner. Each time partners show fingers, have them both say the product. Then have each student say the multiplication equation, starting with the number of fingers on their own hand. See the sample dialogue under the photograph.” (3.OA.7)

Materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. The Problem Set, within Learn, consistently includes these opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of procedural skill and fluency. Examples include:

• Module 2, Topic B, Lesson 8: Read temperatures on a thermometer using number line concepts, Fluency, Flip: Relating Multiplication Models, students independently demonstrate procedural skill and fluency with 2, 3, 4, 5, and 10 multiplication facts while playing a card game. “Place all the cards in a pile facedown. Take turns flipping over a card and saying a complete multiplication sentence that matches. For example, a student may flip over $2+2+2$ and say ‘$2\times3=6$.’ Continue until all cards are used.” (3.NBT.2, 3.OA.7)

• Module 3, Topic C, Lesson 17: Identify and complete patterns with input-output tables, Fluency, Whiteboard Exchange: Relating Division and Multiplication, students independently demonstrate procedural skill and fluency as they complete division equations by using a related multiplication equation. “$48\div6=$___, $35\div=$___, $49\div7=$___, $40\div8=$___, $64\div8=$___, $42\div6=$___, …” (3.OA.7)

• Module 3, Topic D, Lesson 24, Organize, count, and represent a collection of objects Fluency, Whiteboard Exchange: Add or Subtract Within 1,000, students independently demonstrate procedural skill and fluency as they add or subtract within 1,000 to build fluency with the operations. “$784-199=$___, $709+194=$___, $902-356=$___, $564+239=$___, $600-277=$___.” (3.NBT.2)

The materials reviewed for Eureka Math² Grade 3 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 2, Topic E, Lesson 22: Represent and solve two-step word problems using the properties of multiplication, Practice Set, students independently solve routine application word problems. Problem 2, “7 students share 28 markers equally. a. How many markers does each student get? b. What is the total number of markers shared with 3 of the students?” (3.OA.8)

• Module 2, Topic A, Lesson 6: Use all four operations to solve one-step word problems involving liquid volume, Classwork, students solve routine application problems with teacher guidance. “Use the Read-Draw-Write process to solve the problem. 1. Oka mixes 167 milliliters of lemon juice with 754 milliliters of iced tea. How many milliliters of lemon juice and iced tea are there altogether?” (3.MD.2)

• Module 6, Topic A, Lesson 5: Solve time word problems where the change in time is unknown, Problem Set, Problems 1-4, students independently solve routine word problems as they find the elapsed time in word problems. Problem 3, “Science class starts at 1:05 p.m. and ends at 1:52 p.m. How many minutes long is science class?” (3.MD.1)  

Examples of non-routine applications of the math include:

• Module 1, Topic E, Lesson 23: Represent and solve two-step word problems using drawings and equations, Problem Set, Problem 4, students solve a non-routine word problem with teacher assistance. “Ivan has a bag of 18 fruit snacks. There is an equal number of peach, cherry, and grape fruit snacks. Ivan eats all the grape fruit snacks. How many fruit snacks does Ivan have left?” (3.OA.3)

• Module 4, Topic A, Lesson 3: Tile polygons to find their areas, Exit Ticket, Problems 1 and 2, students tile polygons to measure area and use tiles to make different polygons with the same area as they solve non-routine problems. Problem 1, “Use squares to cover the shape. Draw lines to show where the squares meet. Then find the area of the shape.” Problem 2, “Use squares to make a different shape with the same area in problem 1. Sketch your shape.” (3.MD.6)

• Module 4, Topic D, Lesson 19: Apply area concepts to complete a multi-part task, Land, Exit Ticket, students independently solve a non-routine real-world problem. “Ivan wants to put a tile path around his swimming pool. The shaded area shows the path. a. How many square feet of tiles does Ivan need? b. Ivan has 50 square feet of tiles. Each box of tiles covers 10 square feet. How many boxes of tiles does Ivan need to buy to cover the rest of the path? How do you know?” There are two rectangles shown, the smaller of which is inside the larger rectangle. The sides of the larger rectangle are labeled 20 feet by 10 feet and the sides of the smaller rectangle are labeled 16 feet by 8 feet. The space between the two rectangles is shaded gray. (3.MD.7)

The materials reviewed for Eureka Math² Grade 3 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. 

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 C, Lesson 18: Use place value understanding to subtract efficiently using take from a hundred, Learn, Take from One Hundred to Subtract, students attend to procedural skills and fluency as they subtract two-digit by three-digit numbers. “Write $230-96=$__. Subtracting 96 from 230 requires unbundling. Let’s use a simplifying strategy to make a problem we can do in our heads. What benchmark number is 96 close to? Where can I get the 100 from? 230 is 100 and what? Let’s draw a number bond to show breaking 230 into 130 and 100. Draw the number bond and write $100-96$. What is $100-96$? Write 4 to complete the equation. Cross out 96 and 100 and write 4 next to it. Write  $130+4=$. What is $130+4=$? Write 134 to complete the equation.” (3.NBT.2)

• Module 4, Topic B, Lesson 9: Multiply side lengths to find the area of a rectangle, Learn, Relate Side Lengths to Area, students attend to conceptual understanding as they solve for area. “Display the picture of the 4 by 7 rectangle with missing squares. How many rows are in this incomplete array? How many square units are in each row? How are the side lengths of the rectangle related to the area? Can you multiply any two side lengths to find the area? It gets confusing to call all the sides of the rectangle side lengths. We can use the terms length and width to tell one side length from the other. For this rectangle, let’s call this side length the length and the other side length the width. What is the width of this rectangle? What is the length of this rectangle? We can multiply the length and the width of the rectangle to find its area. What multiplication equation can we use to find the area of this rectangle?” (3.MD.7b) 

• Module 5, Topic A, Lesson 2: Partition different wholes into fractional units concretely, Learn, Fraction Stations, students attend to application as they break objects into fractional units. “Your group will partition each object at your station into the fractional unit on the sign. Each item at your station represents 1 whole. Partition the entire item into the assigned fractional unit. For example, use all the clay at your station when you partition it into smaller pieces. Partition the whole amount of water by pouring all of it into the other cups in equal amounts. The water in each cup represents an equal part of the whole. Fold the wax craft stick. Fold or draw on the paper. Do not cut the paper or wax craft stick. Assign students to groups of three and assign each group to a station. Provide time for groups to work. Direct students to draw representations in the table in their books.” (3.G.2)

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

• Module 1, Topic A, Lesson 2: Interpret equal groups as multiplication, Launch, students engage in conceptual understanding and procedural skills and fluency as they determine an efficient way to organize and count an unknown number of objects. “Gather the class and invite 10 students to stand in front of the room. How many students are standing? How many arms does each student have? How many groups of 2 arms are there? What addition expression can we write to represent our groups of 2? Write the addition expression. How many twos do we have? What is the value of 10 groups of 2? Instead of adding, what is another way we can find the total of 10 twos? Which way is more efficient: repeated addition of 2 until we have 10 twos or skip-counting by twos 10 times? Ask students to work with a partner and add to find the total.” (3.OA.1) 

• Module 3, Topic A, Lesson 2: Count by units of 6 to multiply and divide by using arrays, Fluency, Whiteboard Exchange: Interpreting Tape Diagrams, students engage in conceptual understanding and application as they model partitive division and write an equation to build an understanding of two interpretations of division. “Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the tape diagram with 2 equal parts and a total of 6. What is the total? Does this tape diagram show the number of groups or the size of each group? Write a division equation to represent this tape diagram where the quotient is the size of each group. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the equation.” (3.OA.5 and 3.OA.7) 

• Module 6, Topic A, Lesson 3: Solve time word problems where the end time is unknown, Learn, Finish Time on a Number Line, students engage in developing conceptual understanding and application as they solve real-world problems involving time. “Pablo puts noodles into boiling water at 5:27 p.m. The directions say they need to cook for 16 minutes. What time will the noodles be done?” (3.MD.1)

The materials reviewed for Eureka Math² Grade 3 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 MP1 and MP2 across the year and they are identified for teachers within margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Grade 3-5 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 2, Topic A, Lesson 7: Solve one-step word problems using metric units, Learn, Solve a One-Step Difference Unknown Comparison Word Problem, Classwork, Problem 2, students make sense of problems and persevere in solving them as they use the Read-Draw-Write process to understand the problem and assess the reasonableness of solutions. “Amy swims 255 meters more than Mia. Mia swims 475 meters. How many meters does Amy swim? Read the entire problem aloud. Prompt students to reason about the situation by asking questions such as the following: Who is the problem about? What are Amy and Mia doing? Do we know who swims more meters? Who swims fewer meters? What can we draw?”

• Module 4, Topic D, Lesson 16: Solve historical math problems involving area, Learn, Area of a Square Inside a Square, Classwork, students make sense of problems and persevere in solving them as they encounter a new problem type - finding the area of a shaded part of a larger shape and look for entry point to its solution. “Ask the following questions to promote MP1: What can you figure out about the area of the shaded part by looking at the area of the whole square? What are some strategies you can try to start finding the area?”

• Module 6, Topic B, Lesson 12: Reason about composing polygons by using tangrams, Learn, Compose Quadrilaterals, Land, Exit Ticket, “Students make sense of problems and persevere in solving them (MP1) as they visualize what type of shape they are supposed to make and try different configurations of tangrams until they find one that works. Liz uses at least 4 tangram pieces to make a trapezoid. She does not use the square piece. Sketch how she might create her trapezoid.”

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 B, Lesson 9: Round two-digit numbers to the nearest ten on the vertical number line, Learn, Round on a Vertical Number Line, Classwork, students reason abstractly and quantitatively (MP2) “as they consider rounding in different contexts (such as water in a graduated cylinder) as well as on a number line. Show a graduated cylinder containing 73 milliliters of water. This graduated cylinder has 73 milliliters of water in it. Let’s use a vertical number line to help us round that measurement. How many tens are in 73? What is 1 more ten than 7 tens? Our number line needs to show the interval from 7 tens, or 70, to 8 tens, or 80. What number is halfway between 7 tens and 8 tens? Ask the following questions to promote MP2: How does the graduated cylinder help you see what numbers to mark on your number line? What does your number line tell you about the amount of water in the graduated cylinder?” 

• Module 4, Topic D, Lesson 14: Reason to find the area of composite shapes by using grids, Learn, Shade and Add to Find Area, Classwork, Problem 1, students reason abstractly and quantitatively (MP2) as they “find the area of composite shapes by using properties of operations and pictorial models. Decompose the area of the shape and write an equation to show your thinking. Ask the following questions to promote MP2: What does the shading in your picture tell you about how to find the area? How does your expression represent your picture?”

• Module 6, Topic C, Lesson 16: Solve problems to determine the perimeters of rectangles with the same area, Learn, Compare the Perimeters of Rectangles with the Same Area, Classwork, students reason abstractly and quantitatively (MP2)as they “use equations and their drawn rectangles to reason about whether and why two rectangles can have the same area but different perimeters.” Problem 3, “Complete the table for rectangles with an area of 36 square units. Ask the following questions to promote MP2: What do the square units in your rectangles tell you about their areas? About their perimeters? How do the units involved in finding area and perimeter help you think about why rectangles with the same area can have different perimeters?”

The materials reviewed for Eureka Math² Grade 3 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 3-5 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 C, Lesson 10: Demonstrate the commutative property of multiplication using a unit of 2 and the array model, Launch, students construct viable arguments and critique the reasoning of others as they “explain their categories and analyze categories of their peers.” Students compare representations of equal groups. “Introduce the Which One Doesn’t Belong? routine. Display the picture with four arrangements of objects and invite students to study them.” The teacher is prompted to ask the following questions to promote MP3: “Why are your categories correct? Convince the class. What parts of your classmates’ categories do you question? Why?”

• Module 3, Topic A, Lesson 1: Organize, count, and represent a collection of objects, Learn, Classwork, Share, Compare, and Connect, students construct viable arguments and critique the reasoning of others as they record and share “their work with their peers and consider other students’ work and compare it to their own.” Students analyze two students' sample work (Robin and Luke’s Way), (David and Oka’s Way), and (Eva and Shen’s Way). The teacher is prompted to ask the following questions to promote MP3: “How did your organization help you find the total? What challenges did you face when organizing your collection? How did you work through the challenges? How does your drawing show how you organized your pictures? How did you use tens to help you think about nines?”

• Module 4, Topic B, Lesson 6: Tile rectangles with squares to make arrays and relate the side lengths to area, Exit Ticket, Problem 7, students construct viable arguments and critique the reasoning of others as they justify and explain different strategies for finding the area of a rectangle, and compare sample students’ work. “Shen and Jayla each make a rectangle with an area of 24 square centimeters. a. Label the unknown side length of each rectangle. b. Explain how Shen’s and Jayla’s rectangles have the same areas but different side lengths.”

• Module 5, Topic D, Lesson 18: Compare fractions with like units by using a number line, Launch, students construct viable arguments and critique the reasoning of others as they “justify the placement of their index cards on the number line. The class critiques each student’s reasoning as they discuss whether they agree or disagree and why (MP3). Ask the following questions to promote MP3: Is the place you put your index card a guess, or do you know for sure that is where it goes? How do you know for sure? What questions can you ask your classmate to make sure you understand their reasoning.” Students are shown an interactive number line from 0 - 4. “Select one of the prepared fraction index cards and use a think-aloud to model reasoning about its location on the number line. My card has 7 fourths on it. I think I should put this card just before 2 because I know that 8 fourths is equivalent to 2 and that 7 fourths is less than 8 fourths. Mix up the order of the fraction index cards and invite students, one at a time, to place a fraction on the number line and justify its placement. Invite the class to agree or disagree with the placement. What is the largest number on this number line? How do you know? How does the number line show that 3 fourths is greater than 1 fourth?”

The materials reviewed for Eureka Math² Grade 3 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 3-5 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 3, Topic D, Lesson 22: Solve two-step word problems involving multiplication of single-digit factors and multiples of 10, Learn, Solve a Two- Step Array Problem, Classwork, Problem 1, “Students model with mathematics (MP4) as they use tape diagrams, place value charts, and other pictorial models to represent and make sense of word problems. The RDW process scaffolds students’ movement between the concrete problem and their abstract representation of it. Use the Read–Draw–Write process to solve the problem. Use a letter to represent each unknown. 1. There are 8 rows of 10 carpet squares in each classroom. How many carpet squares are in 4 classrooms?” The teacher is prompted to ask the following questions to promote MP4: “What can you draw to help you understand problem 1? What key ideas in problem 1 do you need to make sure are in your model?” 

• Module 4, Topic C, Lesson 11: Decompose to find the total area of a rectangle, Learn, Break Apart Rows to Find Area, Problem Set, Problem 5, “Students model with mathematics (MP4) as they apply their understanding of area to create array models and write expressions for representing and solving problems in context. Luke covers his kitchen and dining room with square tiles. Each $\square$ represents one tile, which is 1 square foot. What is the total area that the new tiles cover?”

• Module 5, Topic E, Lesson 25: Express whole numbers as fractions with a denominator of 1, Learn, Build Wholes from Unit Fractions, Classwork, Problem 2, “Students model with mathematics (MP4) as they model the bread problem in different ways (tape diagrams, equations, and number lines), which helps them see how whole numbers can be written as fractions James bakes 3 kinds of bread: rye, wheat, and white. He bakes 2 loaves of each kind of bread. He cuts the loaves of rye bread into thirds. He cuts the loaves of wheat bread into halves. He leaves the loaves of white bread whole.” The teacher is prompted to ask the following questions to promote MP4: “What math can you write to represent the loaves of bread? How do you represent the key ideas from the bread problem in your equation?”

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 1, Topic B, Lesson 9: Represent and solve division word problems using drawings and equations, Learn, Number in Each Group Unknown, Classwork, Problem 1, “Students use appropriate tools strategically (MP5) when they select their own solution strategies and decide which type of model to draw.” Use the Read–Draw–Write process to solve the problem. “1. There are 24 desks in Miss Wong’s classroom. She arranges the desks into 6 equal groups. How many desks does Miss Wong put in each group?” The teacher is prompted to ask the following questions to promote MP5: “What kind of drawing would be helpful?” 

• Module 2, Topic D, Lesson 21: Add measurements using the standard algorithm to compose larger units twice, Learn, Choose an Addition Strategy When Renaming Twice, Classwork, Problem 2, “Students use appropriate tools strategically (MP5) as they choose an addition strategy that makes sense to them. Find each sum. a. $566+347$, b. $477+253$, c. $634+288$.” The teacher is prompted to ask the following questions to promote MP5: “What strategy would be most efficient for finding $477+253$? Why? Why did you choose to make a benchmark number? Did that work well?”

• Module 6, Topic B, Lesson 10: Draw polygons with specified attributes, Learn, Draw Polygons with Given Attributes, “Students use appropriate tools strategically (MP5) as they use their ruler or right-angle tool to draw polygons with specific attributes. In particular, students select appropriate tools as they recognize which tool helps them draw which attributes. Invite students to draw a polygon with 1 angle larger than a right angle…” The teacher is prompted to ask the following questions to promote MP5: “What tool can help you draw a pair of parallel sides? Can you use your right-angle tool to help you draw your polygon? Why did you choose to use your ruler? Did that work well?”

The materials reviewed for Eureka Math² Grade 3 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 3-5 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 mathematics, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Module 3, Topic B, Lesson 10: Use parentheses in expressions with different operations, Learn, Same Problem, Different Solutions, “Students attend to precision (MP6) as they explore how parentheses, grouping, and order matter when finding the value of an expression.” Teachers, “Display the problem: Amy has 2 cartons of eggs. Each carton contains 6 brown eggs and 4 white eggs. How many eggs does Amy have in total? Display the names and equations as you say the following. (Ivan $2\times6+4=16$, Mia $2\times6+4=20$). Here are the equations Ivan and Mia wrote and their solutions to the problem. What do you notice? Invite  students to work with a partner to determine how Ivan and Mia got different answers and which answer correctly represents the total number of eggs Amy has. Partners should be prepared to explain their thinking with words or drawings. Provide students time to work. Then invite partners to share and justify their thinking. It seems that Ivan and Mia grouped their numbers differently. What can we use to show how to group numbers? Where should we put parentheses to show how Ivan grouped the numbers? Where should we put parentheses to show how Mia grouped the numbers? Direct students to turn and talk about why Mia’s equation with parentheses represents the problem.” Teachers are prompted to ask the following questions to promote MP6: “How are you using parentheses in your expressions? When finding the value of an expression involving multiple operations, what steps do you need to be extra careful with? Why?” 

• Module 4, Topic A, Lesson 4: Compose rectangles to compare areas, Launch, “Students attend to precision (MP6) as they distinguish between different square units and recognize how the choice of unit is related to the area of the shape.” Teachers, “Direct students to build a rectangle with 10 inch tiles. Circulate and observe student strategies. Look for work samples to show 1 row of 10 and 2 rows of 5. Display the picture of the two possible rectangles. Ask students to think–pair–share about the following questions. Are both shapes rectangles? How do you know? Do these rectangles have the same area? How do you know? Repeat the process with centimeter tiles. Display the picture of all four rectangles. Invite students to think–pair–share about the following questions. Amy says that the rectangles made with centimeter tiles have a different area than the ones made with inch tiles. Ivan says that because all the rectangles have 10 tiles, they must have the same area. Are the areas the same or different? What could we do with our tiles to show they are different areas? Invite students to turn and talk about their ideas for how they could describe the areas of the shapes. The squares are different sizes, so we need to be more precise when naming the square units. Transition to the next segment by framing the work. Today, we will precisely name units to describe and compare the areas of rectangles.” Teachers are prompted to ask the following questions to promote MP6: “What details are important to think about when measuring the areas of these rectangles? Is it exactly correct to say the area of this rectangle is 10? What can we say to be more precise?” 

• Module 6 Topic D, Lesson 19: Measure the perimeter of various circles to the nearest quarter inch by using string, Learn, Perimeters of Various Circles, Classwork, “Students attend to precision (MP6) when they carefully place their string around the various circles, mark it, and then measure the distance to the mark to the nearest quarter inch using a ruler.” Students, “Measure and record the perimeter of each circle to the nearest quarter inch.” The teacher  circulates and supports “students in measuring precisely. Invite students to confirm their measurements with a partner. Gather students for a discussion about their measurements. Consider questions such as the following. What was challenging about measuring the perimeter of each circle? How did you handle it? What do you notice about the perimeters? Invite students to turn and talk about how correctly determining the value of the tick marks on a ruler affects the precision of their measurements.” A table with two columns labeled Circle and Perimeter (inches) is provided. Teachers are prompted to ask the following questions to promote MP6: “When using your string to find perimeter, what steps do you need to be extra careful with? Why? Where is it easy to make mistakes when using your string to find the perimeter of a shape? How precise do you need to be?”

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 2, Topic B, Lesson 12: Estimate sums and differences by rounding, Land, Exit Ticket, “Students attend to precision (MP6) when they correctly use the approximately equal sign, ≈, to represent rounded estimates, and the equal sign, =, to represent exact calculations. Shen practiced playing the trumpet for 157 minutes last week. He practiced for 245 minutes this week. a. Estimate the total amount of time Shen practiced by rounding each number of minutes. b. Shen says he practiced for a total of 402 minutes. Is Shen’s total reasonable? Explain your answer.”

• Module 4, Topic A, Lesson 1: Explore attributes of squares, rectangles, and trapezoids, Learn, Attributes of Polygons, Classwork, Problem 1, “Students attend to precision (MP6) as they group and discuss quadrilaterals based on their attributes.” Students cut out polygons and “observe the polygons and discuss what they notice about them with a partner.” Teachers, “Invite partners to think–pair–share about ways they can group the polygons. Direct partners to find all the polygons with four sides and lay them in a row. What do we call polygons with four sides? What do you notice about the quadrilaterals? The polygons look different, but they share the attributes of having four sides and four angles. 1. Use your quadrilaterals to complete the following table. Guide students to use the quadrilaterals to complete the table with the following possible sequence. Chorally read the next attribute in the table: at least 1 pair of parallel sides. Hold a ruler horizontally and run your finger along the top and bottom sides. Think of parallel sides like the sides of this ruler. Imagine these two lines go on forever. Do you think they will ever cross? Why? Turn the ruler so it is vertical and ask whether the sides are still parallel. Then turn the ruler so it is slanted and ask again. Direct students to find polygon A. Look at the long sides of the quadrilateral. The long sides are across from each other, so we call them opposite sides. Are the opposite sides parallel? How do you know? Model using a highlighter to trace one of the pairs of parallel sides of polygon A. Direct partners to examine the other pair of opposite sides to see whether they are parallel. Then have them highlight the other pair of parallel sides with a different color.” A three column table with the headings Attribute, Quadrilateral(s), Sketch of 1 Quadrilateral are shown. The column with attributes includes the attributes of 7 sides of seven different quadrilaterals.” Teachers are prompted to ask the following questions to promote MP6: “How can we describe this shape by using its attributes? Is it exactly correct to say that two sides that don’t touch are parallel? What can we add or change to be more precise?” 

• Module 5, Topic A, Lesson 3: Partition a whole into fractional units by folding fraction strips, Learn, Create Fraction Strips for Thirds and Sixths, “Students attend to precision(MP6) as they communicate carefully to name fractions and fractional units, and as they make sure the parts of their fraction strips are equally sized.” Teachers, “Distribute two paper strips to each student. Direct students to a paper strip. We need to fold this strip into 3 units. How could inches on a ruler help us be precise when folding so we know all the units are the same size? Invite students to measure the strip. How long is the paper strip? We know the strip is 6 inches long, and we want to have 3 units. Invite students to think–pair–share about how many inches long each part needs to be. How many inches does each part need to be? How do you know? Model skip-counting by 2 inches to place tick marks at 2 inches, 4 inches, and 6 inches on the paper strip, noting that 6 inches is the end of the strip. Direct students to fold at each mark, draw a line along each fold, and label each section. Circulate and provide support for precise measuring and folding of equal units, distributing extra strips as necessary. Read and complete the sentence frames chorally with students: There are 3 equal parts in all. The fractional unit is thirds. One unit is called 1 third. Direct students to flip the strip over and draw a line along each fold. Guide them in touching each part and counting by thirds to make 1 (i.e., 1 third, 2 thirds, 3 thirds). Invite students to think–pair–share about how they could make strips to show sixths. Invite students to choose a method and make a strip for sixths. Circulate and provide support for precise measuring and folding of equal units, distributing extra strips as necessary. Read and complete the sentence frames chorally with the class before flipping the strip over and chorally counting the units. Invite students to compare their thirds strip to their sixths strip and think–pair–share about the relationships between these units.” Teachers are prompted to ask the following questions to promote MP6: “How can you describe your fraction strip by using the term fractional unit? When folding your fraction strip, what steps do you need to be precise with? Why?”

The materials reviewed for Eureka Math² Grade 3 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 3-5 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 2, Topic A, Lesson 4: Connect the composition of 1 kilogram to the composition of 1 thousand, Learn, Compose 1,000 Grams Concretely, “Students look for and make use of structure (MP7) as they see 1 liter both as a single unit and as being composed of 1,000 smaller units (milliliters).” Teachers, “Arrange another set of 10 empty cups in 10-frame formation. Let’s decompose again and see if we can get down to 1 milliliter. This time we’ll pour the 100 milliliters in this container into 10 equal parts. Watch carefully and check to make sure each container has the same amount. Let’s figure out how many milliliters are in each container. Ten groups of what makes 100? Skip-count by tens to prove that 10 tens is the same as 100. We have 10 containers that each have 10 mLof liquid volume. Let’s show this equation with the units. 10$\times$10 ⁢mL = 100⁢ mL. Write 10$\times$10 ⁢mL = 100⁢ mL to show the equation with units. Write 100 mL$\div$10 = 10⁢ mL. Invite students to think–pair–share about how the division equation describes how the water was decomposed and how it relates to prior learning. Teachers are prompted to ask the following questions to promote MP7: How can what you know about place value help you decompose 1,000 into 10 equal parts? How are the liquid volumes in the different containers related to each other?”

• Module 4, Topic C, Lesson 10: Compose large rectangles and reason about their areas, Learn, Compose Areas to Solve a Word Problem, Classwork, “Students look for and make use of structure (MP7) as they find and combine the areas of smaller rectangles to find the area of a larger rectangle. Use the Read–Draw–Write process to solve the problem. Liz and Ray have a picnic. They put blankets together to sit on. Liz’s rectangular blanket is 5 feet in length and 7 feet in width. Ray’s blanket is a rectangle with a length of 8 feet and a width of 7 feet. a. Shade the rectangle to show Liz’s blanket. b. Label the length and width of the shaded and unshaded rectangles.c. What is the area of Liz’s blanket? d. What is the area of Ray’s blanket? e. What is the total area of Liz’s and Ray’s blankets?” Teachers are prompted to ask the following questions to promote MP7: “How are the smaller rectangles and the larger rectangle related? How can that help you find the area of the larger rectangle? How is finding the area of the larger rectangle similar to a multiplication problem you’ve solved before?” 

• Module 6, Topic C, Lesson 14: Measure side lengths in whole-number units to determine the perimeters of polygons, Land, Exit Ticket, Perimeter of a Rectangle, “Students look for and make use of structure (MP7) as they notice and discuss how repeated side lengths and multiplication can be used to find perimeter more efficiently. Measure and label each side length in centimeters. Then find the perimeter of the polygon. Equation to find perimeter: ____. Perimeter ___cm.” An image of a polygon is shown. 

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 2, Topic B, Lesson 10: Round two- and three-digit numbers to the nearest ten on the vertical number line, Land, Exit Ticket, Problems 1 and 2, “Students look for and express regularity in repeated reasoning (MP8) as they explore similarities between rounding two- and three-digit numbers to the nearest ten. 1. Round to the nearest ten. Use the number line to show your thinking. $26\approx$___. 2. $260\approx$___.” Each problem is accompanied with an open vertical number line. 

• Module 3, Topic C, Lesson 15: Reason about and explain patterns of multiplication and division with units of 1 and 0, Launch, Classwork, Problem 5, “Students look for and express regularity in repeated reasoning (MP8) as they generalize the patterns they see when looking at equations involving multiplication and division with units of 1 and 0. 5. ___ divided into ___ equal groups is ___ in each group. ___$\div$___ = ___. Provide partners time to work. Circulate and support their work by asking questions such as: What do you know? Do you know the number of groups or the size of each group? What is unknown? Is the unknown the number of groups or the size of each group? Invite partners to turn and talk about what patterns they notice in their equations. Partners will need their equations in the next segment. Transition to the next segment by framing the work. Today, we will look at patterns in multiplication and division equations to help us learn more facts.” Teachers are prompted to ask the following questions to promote MP8: “When you look at the answers to problem 1, is anything repeating? How could that help you multiply with 1 more efficiently? What is the same about the class’s drawings for problem 5?”

• Module 4, Topic A, Lesson 5: Relate side lengths to the number of tiles on a side, Land, Problem Set, Problem 2, “Students look for and express regularity in repeated reasoning (MP8)as they see, through a series of examples, that the number of square tiles along one side of a rectangle is related to the side length. 2. Use a ruler to measure the side lengths of the rectangle in inches. Mark each inch with a tick mark. Connect the tick marks to show the square inches. Then find the area. Include the units. Area: ___ .” Teachers are prompted to ask the following questions to promote MP8: “What pattern do you notice when you compare the side length of a rectangle to the number of tiles on that side? Will the number of tiles along one side always tell you the side length? Explain.”

The materials reviewed for Eureka Math² Grade 3 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 3-5 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 3-5 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 3-5 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 3-5 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. Launch creates an accessible entry point to the day’s learning through activities that build context and often create productive struggle that leads to a need for the learning that follows. Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Land helps you facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the 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 margin notes. These notes provide information about facilitation, differentiation, and coherence. 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 1, Topic A, Lesson 5: Represent and solve multiplication word problems by using drawings and equations, Fluency, Choral Response: Relating Multiplication Models, Teacher Note, “Arrays in the sequence of pictures with more than 5 rows are shaded to support students in quickly determining the number of rows in each array without having to count each one.” Learn, Equal Groups Word Problem, Teacher Note, “This is the first use of a context video. It is shown before a related word problem to build familiarity and engagement with the context. It also allows students to visualize and discuss the situation before being asked to interpret it mathematically.”

• Module 2, Topic A, Lesson 7: Solve one-step word problems using metric units, Learn, Solve a One-Step Bigger Unknown Comparison Word Problem UDL: Action & Expression, “Consider providing an opportunity for students to self-monitor their thinking. What helped you make sense of the problem? Which tape diagram helped you better understand the problem? How can you use what you learned in today’s lesson to solve other problems? Can you use what you learned to make your own comparison problem?”

• Module 4, Topic B, Lesson 6: Tile rectangles with squares to make arrays and relate the side lengths to area, Launch, Differentiation: Support, “Students reason about how to find an exact measurement for area. Open and display the Trying to Cover a Rectangle digital interactive. Does the rectangle take up space? Does it have area? We need to find the area of the rectangle. Invite students to think–pair–share about how they might use the circles to find the area of the rectangle. Cover the rectangle with circles, extending some over the edge of the rectangle. Overlap circles to cover all the white space. Does the total area of the circles represent the area of the rectangle? Will using the circles this way help us measure the area of the rectangle? Why or why not? Let’s try using the circles with no overlaps. Demonstrate making an array of circles inside the rectangle. Ask students how many circles are in the array. Invite students to think–pair–share about whether the area of the circles is an accurate representation of the area of the rectangle. How could we use the squares to find the area of the rectangle?”

The materials reviewed for Eureka Math² Grade 3 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 5 of the Grade 3-5 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 7 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: Place Value Concept Through Metric Measurement, Module Overview, Why, “Why relate metric units to the place value system? One of the advantages of the metric system of measurement is its base-ten structure. In grade 2, students connect the base-ten system and the metric system for measuring length by using centimeters and meters. Relating place value concepts to measurement provides a natural application to strengthen understanding and highlight connections. The composition and decomposition of 1 thousand as 10 hundreds, 100 tens, and 1,000 ones parallels the composition and decomposition of 1 kilogram as 10 hundred grams, 100 ten grams, and 1,000 grams and 1 liter as 10 hundred milliliters, 100 ten milliliters, and 1,000 milliliters. Tens and hundreds are used to show the progression from 1 to 1,000, but emphasis is placed on creating the new units of 1 liter and 1 kilogram from milliliters and grams.”

• Module 3: Multiplication and Division with Units of 0, 1, 6, 7, 8, and 9, Module Overview, Why, “How do students continue to develop fluency with multiplication facts after module 3? In module 4, students use their multiplication skills and strategies to find the areas of rectangles and rectangular arrays. In modules 4, 5, and 6, fluency activities reinforce multiplication and division concepts and skills through counting the math way, relating multiplication and division, and finding unknown factors. Sprints also provide practice with multiplication and division. A lesson at the end of module 6 includes activities that reinforce fluency. These activities can be implemented at any time after module 3 and can be repeated to aid in developing fact fluency.”

• Module 5: Fraction as Numbers, Module Overview, Why, “Why is distance from zero used to compare fractions, even though in later grades that strategy won’t work for negative numbers?Students conceptually understand a fraction on a number line in two ways: as a position or location on the number line, and as a distance from zero. So, it is natural for them to use the same two ways of thinking when comparing fractions: the fraction located to the right is greater, and the fraction further from zero is greater. The first way of thinking  will continue to work once negative numbers are introduced in grade 6. The second way of thinking will need to be amended because, for example, −5 is further from zero than −3, but −5 is less than −3. When comparing two negative numbers, the second way of thinking is the exact opposite: the number further from zero is less, not greater. This opposite relationship is consistent with the way students will learn about negative numbers themselves—as the (additive) opposites of positive numbers. The introduction of negative numbers in grade 6 will challenge students’ understanding of the number system in general, which is what makes the use of this comparison strategy acceptable for grade 3.” An Image of two number lines is displayed.

The materials reviewed for Eureka Math² Grade 3 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: Multiplication and Division with Units of 2,3, 4, 5, and 10, Description, “In module 1, students relate repeated addition, equal groups, and arrays to multiplication and division. With a focus on units of 2, 3, 4, 5, and 10, students use the commutative and distributive properties as strategies to multiply, and they write expressions with three factors as a foundation of the associative property. Students express division as both unknown factor problems and division equations and break apart and distribute the total to divide. They use their understanding of multiplication and division concepts to reason about and solve one- and two-step word problems.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards."

• Module 2, Topic D, Lesson 20: Add measurements using the standard algorithm to compose larger units once. Achievement Descriptors and Standards, “3.Mod2.AD2 Add and subtract within 1,000 fluently using strategies based on place value, properties of operations, or the relationship between addition and subtraction (3.NBT.A.2).”

• Module 5: Fraction as Numbers, Description, “In module 5, students develop an understanding of fractions as numbers. They partition a whole into equal parts and recognize 1 of a fractional unit as a unit fraction. Students compose non-unit fractions from unit fractions and use visual fraction models and written fractions to represent parts of a whole. Students use fractions to represent numbers equal to, less than, and greater than 1. They compare fractions by using visual fraction models and by reasoning about the size of fractions that have the same numerator or denominator. Students identify equivalent fractions, and they apply fraction concepts by using rulers to measure to the nearest quarter inch and by plotting fractional length data on line plots.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards."

• Module 6, Topic C, Lesson 14: Measure side lengths in whole-number units to determine the perimeters of polygons. Achievement Descriptors and Standards, ”3.Mod6.AD5 Solve real-world and mathematical problems involving perimeters of polygons, Students analyze two multiplication strategies for finding the perimeter of a quadrilateral (3.MD.D.8).”

The materials reviewed for Eureka Math² Grade 3 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The Grade 3-5 Implementation Guide includes a variety of references to both the instructional approaches and research-based strategies. Examples include:

• Grade 3-5 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 3-5 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 Implementation Guide for Grade 3-5, “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 include:

• Module 4: Multiplication and Area, Module Overview, Why, “Why is a trapezoid defined as a quadrilateral with at least 1 pair of parallel sides instead of a quadrilateral with exactly 1 pair of parallel sides? The term trapezoid can have two different meanings. Exclusive definition: A trapezoid is a quadrilateral with exactly 1 pair of parallel sides. Inclusive definition: A trapezoid is a quadrilateral with at least 1 pair of parallel sides. Both definitions are legitimate, and at grade 3 there is not a significant advantage to either. The inclusive definition is chosen primarily for the conveniences it allows at later grades and for consistency with most geometry textbooks for college-bound students. One nice consequence of using the inclusive definition is that it means trapezoids and parallelograms have a similar relationship to the relationship between rectangles and squares: In each case, the latter is always also the former.” Images of a trapezoid and a parallelogram are shown. 

• Module 6: Geometry, Measurement, and Data, Module Overview, Why, “Why do lessons 24 and 25 introduce place value units to 1 million?Lessons 24 and 25 intentionally extend student thinking around place value units in preparation for the major work of grade 4. By organizing, counting, and representing a collection with a total value greater than 1,000 in lesson 24, students study patterns to identify and broaden their understanding of the place value system. Students make sense of relationships and patterns in the place value system as they repeatedly bundle ten smaller units to compose one of the next larger unit, only to realize that much larger units are needed. Lesson 25 is an optional lesson where students repeatedly count and bundle bills to one million. This lesson builds the foundation for students seeing that the value of a digit is 10 times as much as the value of the same digit in the place to its right, a key concept for expanding whole-number relationships to decimals. Consider including lesson 25 to provide students with experience with numbers larger than 1,000 before grade 4.”

The materials reviewed for Eureka Math² Grade 3 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 C, Lesson 12: Demonstrate the distributive property using a unit of 4,  Materials, “Teacher: Interlocking cubes, 1 cm (40). Students: Interlocking cubes, 1 cm (40). Lesson Preparation: Prepare 40 interlocking cubes, 20 in one color and 20 in another color, per student and teacher.”

• Module 2: Place Value Concepts Through Metric Measurement, Module Overview, Materials, “9 2-liter containers, 24 Learn books, 1 Bottle of liquid food coloring, blue, 1 Pad of chart paper, 30 Clear plastic cups, about 150 mL (5 oz), 1 Pad of sticky notes, 1 Container, about 1 liter, 25 Pencils, 1 Container, greater than 1 liter, 9 Permanent markers, 1 Container, less than 1 liter, 25 Personal whiteboards, 1 Demonstration thermometer, 25 Personal whiteboard erasers, 6 Digital compact scale with 5-cup bowl, 9 Plastic pitchers, 1.5 L or larger, 25 Dry-erase markers, 6 Platform scales, 14 Envelopes, 1 Projection device, 25 Eureka Math^® place value disks set, ones to thousands, 1 Resealable bag, gallon size, 1 Eureka Math^® tape measure, 150 cm, 24 Resealable bags, sandwich size, 1 Eureka Math^® whole number place value cards,1 Roll of painter's tape, 1 Graduated cylinder, 100 mL, 1 Syringe, 10 ml, 1 Graduated cylinder, 1,000 mL, 1 Teach book, 11 Index cards, 1 Teacher computer or device, 2,443 Interlocking cubes, 1 cm.”

• Module 4, Topic C, Lesson 12: Find all possible side lengths of rectangles with a given area, Materials, “Teacher: Interlocking cubes, 1 cm (24). Students: Multiply and Divide by 6 Sprint (in the student book), Interlocking cubes, 1 cm (24), Centimeter Grid (in the student book), Chart paper, sheet (1 per student group), Marker set (1 per student group), Scissors (1 per student group), Sticky note (1 per student group). Lesson Preparation: Consider tearing out the Sprint pages in advance of the lesson. Consider whether to remove Centimeter Grid from the student books and place inside personal whiteboards in advance or to have students prepare them during the lesson. Prepare one sheet of chart paper, pair of scissors, marker set, and sticky note per group of three students.”

The materials reviewed for Eureka Math² Grade 3 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 3-5 Implementation Guide, Page 46, “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 D, Lesson 15: Model division as an unknown factor problem, Learn, Represent Measurement Division with a Tape Diagram, “Language Support: Make an anchor chart for the terms sum, difference, product, and quotient to help students associate the terms with the correct operations. Discussion of these terms should include how the parts and the whole are related in the operations. When we add and multiply, the sum and product represent the whole. But when we subtract, the difference is a part of the whole. The quotient represents either the number of groups or the size of the group. UDL: Action & Expression: Consider modeling a think aloud for drawing the tape diagram from problem 2 to guide students through drawing the tape diagram for additional problems. Students may need continued support drawing a tape diagram when the number of groups is unknown. In addition, encourage students to share their thought process by asking them to think aloud and address misconceptions as needed. UDL: Action & Expression: Consider including opportunities for students to self-reflect on their process by displaying the following sentence frames for students to refer to either independently or during partner work: After reading the word problem I ask myself ____. I look for ____. If I get stuck I can ____. It is important to ____.”

• Module 3, Topic C, Lesson 15: Reason about and explain patterns of multiplication and division with units of 1 and 0, Launch, “Differentiation: Support: Assign each set of partners a number with which the students have some proficiency. Partners need to focus on making sense of the situation and noticing the patterns, not struggling to find the products or quotients. Ensure that each number, 2 through 9, is assigned to at least one pair of students. Language Support: The statements throughout this lesson are very similar. Students must correctly interpret each statement and make distinctions between the statements to generalize the patterns. Consider highlighting the parts of the statement in different colors to support students in interpreting the statements and describing the patterns.” Learn, Multiply and Divide with 0, “UDL: Representation: Consider presenting the information in another format. Provide students with 10 cubes to act out the following situations concretely before moving to the problems in the book: Show me 1 group of 0. How many? Show me 2 groups of 0. How many? Show me 0 groups of 1. How many? Show me 0 groups of 9. How many?”

• Module 6, Topic C, Lesson 15: Recognize perimeter as an attribute of shapes and solve problems with unknown measurements, Launch, “Differentiation: Support:Students may need support in understanding why two of the shorter side lengths are not included in the perimeter of the larger rectangle. Consider displaying two index cards, not touching, and the larger rectangle side by side. Ask students to trace the border of each of the index cards and then the border of the larger rectangle. Then ask them to reason why the two smaller sides are not included in the perimeter of the larger rectangle.” Learn, Partner Practice, “Language Support: As partners compare solutions, consider asking students to use the Agree or Disagree section of the Talking Tool to support respectful and productive conversation. UDL: EngagementConsider facilitating personal coping skills and strategies as students participate in the Partner Practice activity. Ask students to recall that if they are struggling to find the perimeter, they can choose a different approach or ask their partner a clarifying question.”

The materials reviewed for Eureka Math² Grade 3 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 2, Topic D, Lesson 21: Add measurements using the standard algorithm to compose larger units twice, Learn, Renaming Twice to Add, Differentiation: Challenge, “As time allows, challenge students to find the sums by using a different strategy. Experience with multiple strategies will allow for richer discussion.”

• Module 3, Topic B, Lesson 10: Use parentheses in expressions with different operations, Learn, Same Problem, Different Solutions, Differentiation: Challenge, “Ask students to extend their thinking to write equivalent expressions by using parentheses. A student may write $(3\times5)−2=13$ as $(3\times5)−2=(7\times2)−1$.”

• Module 6, Topic B, Lesson 8: Compare and classify quadrilaterals, Learn, Decompose Quadrilaterals into Two Triangles, Differentiation: Challenge, “Challenge students to determine what happens to other polygons when a diagonal line is drawn inside of them. Is there a pattern?”

The materials reviewed for Eureka Math² Grade 3 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 3-5 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 C, Lesson 16: Use compensation to add, Learn, Error Analysis, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box, “Consider strategically pairing partners with different levels of English proficiency to complete the error analysis. Students may be able to identify the error, but need support with how to explain it or why it might have happened. Designate a student in each pair to write down their explanation of the error to prepare for sharing their thinking with the class.”

• Module 4, Topic, Lesson 10: Compose large rectangles and reason about their areas. Launch, MLL students are provided the support to participate in grade-level mathematics as described in the Teacher Note box, “A context video for this word problem is available. It may be used to remove language or cultural barriers and encourage student engagement. Before providing the problem to students, consider showing the video and facilitating a discussion about what students notice and wonder. This supports students in visualizing the situation before being asked to interpret it mathematically.”

• Module 5, Topic A, Lesson 1: Partition a whole into equal parts and name the fractional unit, Launch, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box, “Consider using strategic, flexible grouping throughout the module. Pair students who have different levels of mathematical proficiency. Pair students who have different levels of English language proficiency. Join pairs to form small groups of four. As applicable, complement any of these groupings by pairing students who speak the same native language.”

The materials reviewed for Eureka Math² Grade 3 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 3-5 Implementation Guide, “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 include:

• Module 1, Topic A, Lesson 1: Organize, count, and represent a collection of objects, Fluency, Counting on the Rekenrek by Tens, Materials, Teacher: Rekenrek. “Students count by tens in unit and standard form to develop an understanding of multiplication. Show students the rekenrek. Start with all the beads to the right side. Say how many beads there are as I slide them over. The unit is 10. In unit form, we say 1 ten. Say 10 in unit form. Slide the second row of beads all at once to the left side. How many beads are there now? Say it in unit form. Continue sliding over each row of beads all at once as students count. Slide all the beads back to the right side. Now let’s practice counting by tens in standard form. Say how many beads there are as I slide them over. Let’s start at 0. Ready? Slide over each row of beads all at once as students count.”

• Module 2, Topic B, Lesson 9: Round two-digit numbers to the nearest ten on the vertical number line, Whiteboard Exchange: Halfway on the Number Line, “Students identify the number halfway between consecutive units of ten on a number line to prepare for rounding to the nearest 10. Display the vertical number line. Draw the vertical number line. What number is halfway between 0 and 10? Label it on your number line. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.” Learn, Round on a Vertical Number Line, “Students represent measurements on a vertical number line and use the vertical number line to help them round the measurements to the nearest ten. Show a graduated cylinder containing 73 milliliters of water. This graduated cylinder has 73 milliliters of water in it. Let’s use a vertical number line to help us round that measurement. Display a vertical number line. Direct students to the first vertical number line in problem 1 in their books. How many tens are in 73? What is 1 more ten than 7 tens? Our number line needs to show the interval from 7 tens, or 70, to 8 tens, or 80. Invite students to label the number line as you model labeling the lowest tick mark as  $70 = 7$ tens and the highest tick mark as  $80 = 8$ tens. What number is halfway between 7 tens and 8 tens? Let’s label the halfway mark. Label the halfway tick mark as  tens and 5 ones. Is 73 more than or less than halfway between the two tens? Watch as I plot and label 73 on the number line. Say “Stop!” when my finger points to where 73 should be. Move your finger up the number line from 70 toward 75. Stop when the students say to stop. Put your pencil where 73 should be on your number line. Label the spot as $73 = 7$ tens and 3 ones . Now that we know where 73 is, we can see how to round 73 milliliters to the nearest 10 milliliters. Which ten is 73 closer to? How do you know? So what ten does 73 round to? Display the sentence frame: ___ rounded to the nearest ten is ___. Point to the sentence frame as you say: Complete the sentence. 73 milliliters rounded to the nearest ten milliliters is about how many milliliters?” Image of a vertical number line is shown.

• Module 5, Topic E, Lesson 26: Create a ruler with 𝟏-inch, half-inch, and quarter-inch intervals, Launch, Materials, Teacher: Paper strip. “Students summarize the characteristics of various fraction models. Display the picture of the models students have used to represent fractions. Invite students to think–pair–share about what is important about each model and how the models are alike and different. As students share, consider placing special emphasis on the ruler in preparation for Learn and highlight the following concepts: The ruler’s partitions must be equally spaced. For the model to be understandable, there needs to be enough information, such as the location of whole numbers, labeled. However, it is not necessary to label every mark on the number line or ruler. The wholes must be the same size to be used to compare fractions or find equivalent fractions. Consider inviting students to share what makes some models easier for them to use than others. Show the paper strip. I want to create a ruler with this strip of paper. My ruler needs to precisely measure to the nearest quarter inch. Invite students to think about strategies they have used to partition and suggest ways to create the ruler.” Images of various fraction models are shown.