The materials reviewed for Eureka Math² Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

The Assessment System includes lesson-embedded Exit Tickets, Topic Quizzes, and Module Assessments. According to the Implementation Guide, “Exit Tickets are not graded. They are paper based so that you can quickly review and sort them. Typical Topic Quizzes consist of 4-6 items that assess proficiency with the major concepts from the topic. You may find it useful to grade Topic Quizzes. Typical Module Assessments consist of 6-10 items that assess proficiency with the major concepts, skills, and applications taught in the module. Module Assessments represent the most important content taught in the module. These assessments use a variety of question types, such as constructed response, multiple select, multiple choice, single answer, and multi-part. There are two analogous versions of each Module Assessment available digitally. Analogous versions target the same material at the same level of cognitive complexity.” Examples of summative Module Assessments items that assess grade-level standards include:

• Module 1, Module Assessment 2, Item 9, “Consider the expression shown. $5,050\div75$. Write a word problem that can be solved by evaluating the given expression. Explain what the quotient and remainder represent.” (5.OA.2)

• Module 2. Module Module Assessment 2, Item 5, “Kayla needs to know whether she has more gold chain or more silver chain. She has 2 lengths of gold chain that measure $2\frac{1}{4}$in . and $\frac{3}{8}$in. She has 2 lengths of silver chain that measure $1\frac{5}{8}$in. and $1\frac{3}{4}$in. Without finding the actual sum, determine whether Kayla has more gold chain or more silver chain. Explain how you know.” (5.NF.2)

• Module 4, Module Assessment 2, Item 7, “Mr. Evans buys 3 new books and 3 used books. He spends $104.16 altogether. The used books cost $18.82, $11.32, and $16.51. Each of the 3 new books costs the same amount. How much does each of the new books cost?” (5.NBT.B)

• Module 5, Module Assessment 1, Item 3, students are shown an image of a rectangular prism, and told, “The volume of the right rectangular prism shown can be found by using the expression $(6\times10)\times3$. Part A. Enter a number in each box to show what the measurements of the prism could be.” (5.MD.C)

• Module 6, Module Assessment 1, Item 5, students are shown a coordinate plane and told, “A rectangle has a vertex at (6,10). The rectangle has an area of 30 square units. Plot the given vertex and three other possible vertices. Then draw the rectangle in the coordinate plate.” (5.G.2)

The materials reviewed for Eureka Math² Grade 5 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. 

According to the Grades 3-5 Implementation Guide, “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. Suggested questions, including key questions related to the objective, help students synthesize the day’s learning.” 

Instructional materials engage all students in extensive work with grade-level problems through the consistent lesson structure. Examples include:

• Module 1, Place Value Concept for With Whole Number, Lesson 17, and 18 engage students with extensive work with 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them). Lesson 17, Fluency, Whiteboard Exchange: Write and Evaluate Expressions, “Students express an addition, subtraction, multiplication, or division statement as an expression and evaluate the expression.” Teacher displays the statement: “11 more than 73. Write an expression to represent the statement. Write the value of the expression.” Lesson 18, Launch, “Students use parentheses to write expressions to match a word problem context. Teacher displays the expression $5+2$ to the class and pairs students to use the Co-construction routine to have students create a real-world situation that could apply to the expression. Teacher invites students to share their ideas and explain the relationship to the expression with the class. Teacher directs students to adjust the situations they created to match this expression: $3\times(5+2)$.” Land, Exit Ticket, “Write a word problem that can be solved by using the expression shown. $(6+7)\times11-34$.”

• Module 3, Multiplication and Division with Fractions, Lessons 12 and Lesson 21, engages students with extensive work of 5.NF.7 (Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions). Lesson 12, Learn, Use a Tape Diagram to Divide, Classwork, Problem 1, students divide a nonzero whole number by a unit fraction using tape diagrams. “Use the Read-Draw-Write process to solve each problem. A family makes 3 pans of brownies for a bake sale. They plan to sell gift bags that each hold $\frac{1}{2}$ of a pan of brownies. How many gift bags can the family make?” Lesson 21, Land, Exit Ticket, “Use the Read-Draw-Write process to solve the problem. Shen bought 20 pounds of ground beef. He used $\frac{1}{4}$ of the beef to make tacos. He used $\frac{2}{3}$ of the remaining beef to make $\frac{1}{4}$-pound burgers. How many burgers did he make?”

• Module 4, Place Value Concepts for Decimal Operations, Lesson 1 engage students with extensive work of 5.NBT.1 (Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and $\frac{1}{10}$ of what it represents in the place to its left). Fluency, Counting on the Number Line by Tenths, students count by tenths in fractional and decimal form to extend their place value understanding to the thousandths place. “Use the number line to count by tenths to 10 tenths and then back down to 0 tenths. The first number you say is 0 tenths. Ready. Now count forward and back by tenths again. This time use whole numbers and decimal numbers. The first number you say is 0. Ready?” Teachers display each number one at a time on the number line as students count. Launch, students use division to relate adjacent place units to tenths.Teacher uses a video (Decomposing 1 Liter) to activate students prior knowledge of place value unit decomposition. A picture of a 1 L bottle of water decomposed into parts is shown. “Label the liter bottle 1,000 mL. Point to the 1,000 mL bottle. When 1,000 mL is poured equally into 10 containers, how many milliliters are in each container? 1,000 mL divided equally into 10 containers is 100 mL. When 100 mL is poured equally into 10 containers, how many milliliters are in each container? When 10 mL is poured equally into 10 containers, how many milliliters are in each container? What do you notice about how each unit was decomposed, or divided?” Learn, Decompose 1 One into Thousandths, Classwork, Problem 1, “Complete the Equations.  1 one = 10 ____; 1 one = 100 ____; 1 one = 1000____.”  Learn, 10 Times As Much As and $\frac{1}{10}$ As Much As, “students relate adjacent place value units by using 10 times as much as and $\frac{1}{10}$ as much as.” Teachers display the picture of the 1 ones disk and 10 tenths disks, tells students that decimal numbers can be represented by place value disks, and invites students to describe the relationship between 1 one and tenths. The following questions are used to guide the discussion:. “How many tenths make 1 one? So 1 one is how many times as much as 1 tenth? What equation could we write to represent the relationship between 1 one and 1 tenth?” Land, Debrief, “students model and relate decimal place value units to thousandths.” Teachers facilitate class discussion about decimal place value, relating adjacent lace value units using prompts: “How is a place value unit related to the next larger unit? How is a place value unit related to the next smaller unit?” Exit Ticket, Problem 1, “Consider the tape diagram. a. Write the value that A represents in decimal form. b. The value of A is ____ as much as 0.01.”

The instructional materials provide opportunities for all students to engage with the full intent of standards. Examples include: 

• Module 3, Multiplication and Division with Fractions, Lessons 3 and 9 engage students with the full intent of 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). Lesson 3, Learn, Multiply a Whole Number by a Fraction Less Than 1, students use a tape diagram to multiply a whole number by a fraction, ”Write $\frac{2}{3}\times6=$___. Describe what $\frac{2}{3}\times6$ means. Is the product greater than or less than 6? How do you know?” Teachers are directed to Invite students to work with a partner to find the product. Lesson 9, Learn, Use Unit Language to Multiply, “students make a simpler problem by reasoning about factors before they multiply.” Classwork, Problem 4. “Fill in the blanks to find the product $\frac{1}{5}\times\frac{10}{11}$, $\frac{1}{5}$ of 10 is ___ , $\frac{1}{5}$ of 10 elevenths is ___ elevenths.“ 

• Module 4, Place Value Concepts for Decimal Operations, Lessons 10, 14, 21, and 22 engage students with the full intent of standard 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction…). Lesson 10, Launch, “students consider possible methods for adding decimal numbers.” Teachers direct students to display the picture and invite students to notice and wonder and share the scenario. “Sasha sells spices at a market. She weighs 5.64 grams of garlic and 2.7 grams of cumin for a customer.” Teachers are directed to “display $5.64+2.7=$___, prompt students to think–pair–share about possible methods they can use to figure out how many grams of spices Sasha sells to the customer, and  encourage students to consider how the different methods may or may not help them add these addends.” Lesson 14, Learn, Problem Set, Problem 13, “$4\times6.24=$___”. Lesson 21, Learn, Problem Set, Problem 2, “Draw on the place value chart to divide then record your work in vertical form. $5.4\div2=$___.” A place value chart with columns labeled ones, tenths, hundredths is shown. Lesson 22, Learn, Use a Different Method, students use different methods to solve an equal groups word problems. Classwork, Problem 3, “Tara pours 40.25 cups of juice equally into 23 glasses. How much juice is in each glass?”

• Module 5, Addition and Multiplication with Area and Volume, Lesson 17 and 19, engage students with the full intent of 5.MD.4 (Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units). Lesson 17, Learn, Pack Prisms, “students pack containers shaped like right rectangular prisms with centimeter cubes to find volumes of solids with the same dimensions.” Teacher directs students to form pairs, and provides each pair a $4 cm\times3 cm\times2 cm$ prism and 40 centimeter cubes. Classwork, Problem 1, “Sketch to show the number of unit cubes visible on the faces of the right rectangular prism. In the blank, write the total number of unit cubes it takes to pack the prism. Number of unit cubes: ___.” An image of a rectangular prism is provided. Lesson 19, Land, Exit Ticket, “The right rectangular prism shown is composed of centimeter cubes. a. Draw lines to decompose the prism into layers. b. Use the layers you created in part (a) to complete the following sentences.The prism has ___ layers.There are ___ centimeter cubes in each layer. The volume of the prism is ___ cubic centimeters. c. How does decomposing a prism into layers help you find the volume?” A picture of a rectangular prism is shown.

The materials reviewed for Eureka Math²  Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major work of each grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade. 

• The number of modules devoted to the major work of the grade (including assessments and supporting work connected to the major work) is 4.5 out of 6, approximately 75%.

• The number of lessons developed to the major work of the grade (including supporting work connected to the major work) is 102 out of 133, approximately 77%. 

• The number of days devoted to the major work of the grade (including assessments and supporting work connected to the major work) is 116 out of 133, approximately 87%. 

A lesson-level analysis is most representative of the instructional materials as the lessons include major work and supporting work connected to major work. As a result, approximately 77% of the instructional materials focus on major work of the grade.

The materials reviewed for Eureka Math² Grade 5 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Materials are designed so supporting standards are connected to the major work standards and teachers can locate these connections on a tab called, “Achievement Descriptors and Standards” within lessons. Examples include:

• Module 1, Topic B, Lesson 9: Multiply two- and three-digit numbers by two-digit numbers by using the standard algorithm, Learn, Multiply Three-Digit Numbers by Two-Digit Numbers, Classwork, Problem 3, connects the supporting work of 5.OA.2 (Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them) to major work of 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm). Students multiply using the standard algorithm relating it to area models to determine the more efficient strategy. “Flatback turtles lay 52 eggs in a nest. How many turtle eggs would there be in 427 nests?” The teacher directs students to work with a partner to discuss what they know and do not know from the story and asks, “What do we know and what do we need to know?” Teacher displays a tape diagram and asks, “What expression can we use to determine the number of turtle eggs?” Teacher displays area models and asks, “What do you notice? What is the same and what is different? Think back to the connection between area models and the standard algorithm. How many partial products are there in each of the models? How do you know? We know we can designate either factor as the unit. If we are using the standard algorithm, which factor should we designate as the unit? Why? We know we can designate either factor as the unit. If we are using the standard algorithm, which factor should we designate as the unit? Why?” Teacher directs students to work with a partner to find the partial products $2\times427$ and $50\times427$ by multiplying using the standard algorithm and asks, “What is $2\times7$? $2\times20$? $2\times400$? What is $50\times7$? $50\times20$? $50\times400$? What product did you find? Is it reasonable based on your estimate?”

• Module 3, Topic A, Lesson 5: Convert larger customary measurement units to smaller measurement units, Learn, Multiply to Convert Units, Classwork, Problem 2, connects the supporting work of 5.MD.1 (Convert among different-sized standard measurement units within a given measurement system) to the major work of 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction). Students multiply whole numbers and fractions to convert larger measurement units to smaller measurement units. “$\frac{7}{4}c=$____ fl oz. How many fluid ounces equal $\frac{7}{4}$ cups? What multiplication expression did you use? What is different about this example compared to previous examples? How did that affect your product? Why?” Learn, Conversions in the Real World, Classwork, Problem 3, “students apply their understanding of converting units to real-world situations. Use the Read–Draw–Write process to solve each problem. Mr. Sharma spends $\frac{3}{8}$ of a day at work. He spends the rest of the day at home. How many hours does he spend at home? What do we know? Can we draw something? What can we draw? What labels can we add to our tape diagram based on what we know?” The teacher draws a tape diagram, directs students to do the same, reads the problem to the class, and asks: “What does the question ask us to find? Where can we put the question mark in our model? What do you notice about the measurement unit in our tape diagram and the measurement unit in the question? How can we show 1 day as hours in our model? What conclusions can you make from the tape diagram so far? How many hours does Mr. Sharma spend at home? Is your answer reasonable? How do you know?” Land, Debrief, Students convert larger units to smaller measurement units. Teacher facilitates a discussion about converting larger to smaller measurement units by encouraging students to restate or add on to their classmates’ responses using the following prompts: “What do all the measurement unit conversions today have in common? What operation did all our equations involve when we needed to convert from a larger measurement unit to a smaller measurement unit? How can we use multiplication to convert larger measurement units to smaller measurement units?” Teacher writes, “$\frac{3}{4}yd=\frac{3}{4}\times1yd=\frac{3}{4}ft=\frac{9}{4}ft$. Compare $\frac{9}{4}$ and $\frac{3}{4}$. Which is greater? Does it make sense for the number of feet in $\frac{3}{4}$ yards to be greater than $\frac{3}{4}$? Why? How does the product $\frac{3}{4}\times3$, or $\frac{9}{4}$, compare to 3? Does it make sense that the product $\frac{3}{4}\times3$ is less than 3? Why?”

• Module 6, Topic D, Lesson 16: Interpret graphs that represent real-world situations, Learn, Problem Solving with the Coordinate Plane, Classwork, Problem 2, connects the supporting work of 5.G.2 (Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation) to the major work of 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used). Students use point coordinates to solve real-world problems. “The graph shows the total number of miles Kelly drove after a given number of hours on a road trip. a. How many miles did Kelly drive in the first hour of her trip? b. How many hours did it take Kelly to drive a total distance of 150 miles? c. How many miles did Kelly drive between hours 3 and 4? d. Kelly drove 180 miles in 5 hours. Plot a point to represent this information on the graph.” Teacher directs students to share their answers to parts a and b, and asks, “For part (c), how did you determine how many miles Kelly drove between hours 3 and 4? What is the ordered pair for the point you plotted for part (d)? How would the coordinates of your point for part (d) be different if Kelly had driven more than 180 miles in 5 hours? How would the coordinates of your point for part (d) be different if Kelly had taken longer than 5 hours to drive 180 miles?” A coordinate plane with five points, (1,45), (2,75), (3,135), (4,150), (5, 180) with the x-axis labeled Hours and the y-axis labeled Total Distance (miles), is shown.

The instructional materials reviewed for Eureka Math² Grade 5 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

Grade 5 lessons are coherent and consistent with the Standards. Teachers can locate standard connections on a tab called, “Achievement Descriptors and Standards” within lessons. Examples include:

• Module 2, Topic B, Lesson 5: Add and subtract fractions with related units by using pictorial models, Launch, connects the major work of 5.NF.B (Apply and extend previous understanding of multiplication and division to multiply and divide fractions) to the major work of 5.NF.A (Use equivalent fractions as a strategy to add and subtract fractions). “Students analyze models that show like units, related units, and unlike units.” Teacher displays the Vertical Block Drop digital interactive and asks, “What addition expression can we write to represent what we see in the model? What do you expect to see when I drop the blocks? The model represents a way to add fractions that have like units. In this case, the like units are fifths and our sum is also in fifths. Let’s analyze another model. What addition expressions do you see represented in this model? What do you expect to see when I drop the blocks? What do you think is the total?” A model of 13 and 36 is provided.

• Module 4, Topic C, Lesson 18: Relate decimal-number multiplication to fraction multiplication, Learn, Multiply Decimal Numbers by One Tenth, connects the major work of  5.NBT.B (Perform operations with multi-digit whole numbers and with decimals to hundredths) to the major work of 5.NF.B (Apply and extend previous understanding of multiplication and division to multiply and divide fractions). “Students use fraction form and place value understanding to multiply decimal numbers by 0.1. We can use what we know about fraction multiplication to find $0.1\times0.1$. What is 0.1 renamed in fraction form? What is $\frac{1}{10}\times\frac{1}{10}$? Let’s draw an area model to check whether the answer $\frac{1}{100}$ makes sense. What is another way to describe $\frac{1}{10}\times\frac{1}{10}$? We want to find $\frac{1}{10}$ of $\frac{1}{10}$. We start with $\frac{1}{10}$. How can we represent $\frac{1}{10}$ on an area model? What do we need to do now to show $\frac{1}{10}$ of $\frac{1}{10}$? How many equal parts does our model show now? What does each part represent? What is $\frac{1}{10}$ of $\frac{1}{10}$? Now that we’ve used an area model to see how $\frac{1}{10}\times\frac{1}{10}=\frac{1}{100}$ makes sense, let’s show this multiplication on a place value chart.” Learn, Multiply Decimal Numbers by Multiples of Tenths and Hundredths, “Students use fraction form, unit form, and place value understanding to multiply decimal numbers. Write $7\times0.2$. How can we rewrite this expression by using a fraction? What is the product in fraction form? What is the product in standard form?” Land, Debrief, “Students relate decimal-number multiplication to fraction multiplication. How can you use what you know about multiplying fractions to multiply decimal numbers?”

• Module 6, Topic B, Lesson 7: Generate number patterns to form ordered pairs, Launch, connects the supporting work of 5.OA.B (Analyze patterns and relationships) to the supporting work of 5.G.A (Graph points on the coordinate plane to solve real-world and mathematical problems). “Students notice and wonder about patterns of points in a coordinate plane.” Teacher displays the graph with the three sets of points in different colors and invites students to think–pair–share about what they notice and what they wonder. A coordinate plane with points (2,3), (4,6), (6,9), (8,12), (10,15) plotted is shown. Learn, Work with Two Number Patterns, Classwork, Problem 2, “Students generate two number patterns by using two rules.” Students are directed to complete the problem with a partner. “Leo and Sasha create number patterns. Leo’s pattern: Start at 6 and multiply by 4. Sasha’s pattern: Start at 85 and subtract 6. Record the first five terms of Leo’s pattern and of Sasha’s pattern in the table.” A table divided in half and labeled Leo’s Pattern and Sasha’s pattern is shown. Learn, Graph Number Patterns, Classwork, Problem 3, “Students form ordered pairs from corresponding terms of two patterns and graph the ordered pairs in the coordinate plane. Use the table to complete parts (a)–(c). a. Use the rules to complete the patterns. b. Write the ordered pair for each pair of corresponding terms by writing the number from pattern A as the x-coordinate and the number from pattern B as the y-coordinate.” A table with headings “Pattern A Add 2, Pattern B Add 3, Ordered Pair” is shown. c. Plot the points in the coordinate plane.” A coordinate plane numbered 0-15 on the x-axis and y-axis is shown.

The materials reviewed for Eureka Math² Grade 5 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

Content from future grades is identified within materials and related to grade-level work. These references are consistently included within Topic and Module Overviews to reveal coherence across modules and grade levels. The Topic Overview includes information about how learning connects to previous or future content. Some Teacher Notes within lessons enhance mathematical reasoning by providing connections/explanations to prior and future concepts. Examples include: 

• Module 1: Place Value Concepts for Multiplication and Division with Whole Numbers Module Overview, After His Module connects 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used) to work in Grade 6. “In Grade 5 Module 4 Students use place value knowledge and times as much as language to learn about decimal numbers. Students see how the strategies they use for whole-number operations extend to operations with decimal numbers. They convert metric measurements from smaller units to larger units. In Grade 6 Modules 2 and 4 In module 2, students learn to divide whole numbers with any number of digits by using the standard algorithm. In module 4, students build upon grade 5 knowledge by writing and evaluating numerical expressions with terms that have whole-number bases and exponents.”

• Module 5: Topic D: Volume and the Operations of Multiplication and Addition, Topic Overview, connects 5.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume) to work in Grade 6. “In topic D, students extend their understanding of volume concepts from topic C. They learn the volume formulas and use them to solve mathematical and real-world problems. In grade 6, students find the volumes of right rectangular prisms with fraction edge lengths by packing with unit cubes with fraction edge lengths and by applying the formulas $V=B\times h$ and $V=l\times w\times h$.”

• Module 6: Topic D: Solve Real-World Problems with the Coordinate Plane, Topic Overview,  connects 5.OA.3 (Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane) and 5.G.2 (Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation) to work in Grade 6. “In topic D, students build on this understanding when they use the coordinate plane to tell stories about relationships and data. Students build on their work with the coordinate plane and number patterns in grade 6, when they use all four quadrants to solve mathematical and real-world problems, and in later grades, when they graph linear relationships and construct and interpret scatter plots.”

Materials relate grade-level concepts from Grade 5 explicitly to prior knowledge. These references can be found consistently within Topic and Module Overviews and less commonly within teacher notes at the lesson level. In Grade 5, prior connections are often made to content from previous modules within the grade. Examples include:

• Module 1: Topic A: Place Value Understanding for Whole Numbers, Topic Overview, connects to 5.NBT.1 (Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left), 5.NBT.2 (Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10), and 5.MD.1 (Convert among different-sized standard measurement units within a given measurement system) to work in Grade 4. “In topic A, students apply their understanding of place value to multiply and divide by powers of 10 and their multiples. Prior to grade 5, students use place value understanding to round multi-digit whole numbers to any place. They compare quantities through multiplicative comparison and recognize that in a whole number, a digit in one place represents 10 times as much as what it represents in the place to the right.”

• Module 4: Place Value Concepts for Decimal Operations, Module Overview, Before This Module, connects 5.NBT.1 (Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and $\frac{1}{10}$ of what it represents in the place to its left) to work in Grade 4. “In grade 4 module 5, students relate decimal fractions to decimal numbers. They write and compare decimal numbers to hundredths by using models and by renaming them as fractions. Students also add decimal fractions with denominators 10 and 100 by renaming them as fractions. In grade 5 module 1 students apply place value understanding to multiply and divide whole numbers by powers of 10. They build fluency with the standard algorithm to multiply multi-digit whole numbers. Students also divide whole numbers by using tape diagrams, area models, and vertical form to record quotients and remainders."

• Module 6: Foundations to Geometry in the Coordinate Plane, Module Overview, Before This Module, connects 5.G.2 (Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation) to work in Grade 4. "In grade 5 module 4, students use the number line as a tool for counting, comparing, and operating with whole numbers. Earlier in grade 5, students compare and round decimal numbers in a similar manner to how they compare and round whole numbers, by using the structure of number lines and plotting points to solve problems. In grade 4 module 2, students apply their understanding of factors and multiples to find an unknown term in shape or number patterns. They recognize that they can use what they know about the earlier terms in a sequence to find a later term without having to list all the terms."

The materials reviewed for Eureka Math² Grade 5 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, 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 2, Topic A, Lesson 4: Solve word problems involving division and fractions, Learn, Model and Solve Division Problems, UDL: Action & Expression, “Consider posting guiding questions such as the following that encourage partners to monitor and evaluate their progress as they complete problems 2–4. Monitor, Is our answer reasonable? Should we try something else? Evaluate, What worked well? What might we do differently next time?”

• Module 3, Topic A, Lesson 6: Convert smaller customary measurement units to larger measurement units, Learn, Conversions in the Real World, Differentiation: Challenge, “Consider revising problem 2 so the number of days is not equal to a whole number of weeks. For example, 38 days = $5\frac{3}{7}$  weeks. If the family has to rent by the week, this means they need to rent for 6 weeks. Consider revising problem 3 so it requires converting twice when using the reference sheet. For example, a recipe needs 4 cups of milk. How many gallons of milk does the recipe require? In this instance, students need to consider how many cups are in 1 pint and then how many cups are in 1 gallon.”

• Module 4, Topic A, Lesson 8: Round decimal numbers to any place value unit, Fluency, Sprint: Multiply or Divide by Powers of 10, Teacher Note, “Count forward by 100,000 from 0 to 1,000,000 for the fast-paced counting activity. Count backward by 10,000 from 100,000 to 0 for the slow-paced counting activity.” Learn, Round Numbers in a Useful Way, Teacher Note, “Since Mr. Evans’s number has digits in the hundredths place but not in the thousandths place, it cannot be rounded to the hundredths. Otherwise all student responses are valid. Acknowledge all reasonable explanations for their choices.”

• Module 5, Topic B, Lesson 8: Find areas of square tiles with fraction side lengths by relating the tile to a unit square, Learn, Areas of Square Tiles with Fraction Side Lengths, Teacher Note, “For this lesson, the size of one piece of patty paper (4 inches by 4 inches) is one unit. Hamburger patty paper $(5\frac{1}{2}$ inches by $5\frac{1}{2}$ inches), available in boxes of 1,000, is also an option for this lesson. Any paper may be used if patty paper is not available. Divide students into groups of three. Give each group both sizes of patty paper. Hold up one of the larger squares of patty paper.”

The materials reviewed for Eureka Math² Grade 5 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 3: Multiplication and division with fractions, Module Overview, Why, “Why do students learn to multiply and divide with fractions before they do so with decimals? Students have conceptually worked with fractions for longer than they have worked with decimals. Starting as early as in kindergarten, students informally model fractions as parts of a whole (shape) and identify equal parts. In grade 3, the concept of a fraction is formalized, and in grade 4, operations work with fractions begins. It is not until grade 4 that the fractional units of tenths and hundredths are introduced as decimals. Because tenths and hundredths are both place value units and fractional units, students use what they know about fractions to support their conceptual development with decimals. For the same reason, students learn to multiply and divide with fractions before they do so with decimals.”

• Module 4: Place Value Concepts for Decimal Operations, Module Overview, Why, “Why does the term decimal number in topics A–D change to decimal in topic E? This module builds on learning from grade 4 module 5. Grade 4 uses decimal number exclusively to help avoid confusion with related terms, such as confusing decimal number and decimal point, and to highlight the idea that decimals are, in fact, numbers, simply written in a new form. In topics A–D, decimal number is used to both support a seamless transition from grade 4 and to notably contrast with whole number, particularly because a large emphasis is on place value understanding and on whole-number methods for the multiplication and division work. Grade 6 module 2 uses decimal exclusively. Mathematically, either decimal number or decimal is valid, so this module serves as an intentional transition point. Because the emphasis in topic E shifts to applying the operations, rather than focusing on the operations themselves, there is less need to notably distinguish between decimal numbers and whole numbers.”

• Module 5: Addition and Multiplication with Area and Volume, Module Overview, Why, “Why do students learn to multiply fractions again in module 5 when they already learned it in module 3? A subtle distinction must be made between using an area model to multiply fractions and finding the area of a rectangle. In module 3, students use an area model to find the product of fractions before learning that for whole numbers a, b, c, and d, $\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}$, given that b and d are nonzero, but students are not asked to find the area of a rectangle with fraction side lengths until module 5 topic B. Even then, they cannot immediately apply the familiar formula because they have only learned that the formula applies to whole numbers l and w. Thus, students must develop the conceptual understanding of what it means to find the area of a rectangle with fraction side lengths. To do this, topic B has students replicate their work with whole-number side lengths from grades 3 and 4: they tile a region with fraction unit squares and count the tiles to see that the area of the region is equal to the number of tiles it takes to cover the region. Students can then conclude that the area formula can be applied even when l and w are fractions.”

The materials reviewed for Eureka Math² Grade 5 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 3, Topic A, Multiplication of a Whole Number by a Fraction, Description, “Students extend their understanding of fractions from parts of a whole (e.g., 1 third of a shape) to parts of a set or a number (e.g., 1 third of a group of 12 items). They find fractions of a set and then transition to finding a fraction of a whole number. Students learn that finding a fraction of a whole number means they are finding the product of a fraction and a whole number. They apply this learning to converting customary measurement units.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards."

• Module 4, Topic A, Lesson 7: Round decimal numbers to the nearest one, tenth, or hundredth. Achievement Descriptors and Standards, “5.Mod4.AD11 Round decimals by using place value understanding (5.NBT.A.4).”

• Module 5, Topic D, Volume and the Operations of Multiplication and Addition, Description, “Students synthesize the work of topic C by determining that the volume of any right rectangular prism is calculated either by multiplying the area of the base by the height, $V=B\times h$, or by multiplying the three dimensions of the prism, $V=l\times w\times h$. They use these two formulas to find volumes and unknown dimensions of right rectangular prisms in both mathematical and real-world problems. Students find the volume of a figure composed of right rectangular prisms by decomposing the figure into right rectangular prisms, finding the volume of each prism, and adding the volumes together.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards."

• Module 6, Topic A, Lesson 4: Describe the distance and direction between points in the coordinate plane. Achievement Descriptors and Standards, “5.Mod6.AD4 Solve real-world problems by using the first quadrant of the coordinate plane. (5.G.A.2)”

The materials reviewed for Eureka Math² Grade 5 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 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 5: Addition and Multiplication with Area and Volume, Module Overview, Why, “Topic A includes significantly more digital interactives than any other topic in grade 5. Why? The use of digital interactives provides pictorial representation to support the concrete- pictorial-abstract framework and deepen student understanding of quadrilaterals. Each study of a new quadrilateral in topic A introduces students to a concrete representation, such as constructing the quadrilateral by using paper, pencils, right-angle tools, and rulers. A digital interactive provides a pictorial representation that shows various figures that look different than, but are called the same name as, the quadrilateral students constructed. Digital interactives are an ideal way to make comparisons among two-dimensional figures because the measures of angles, sides, and diagonals visually shift as the figure is manipulated. Then students engage with abstract representation in the hierarchy by generalizing properties of the quadrilateral and recording those properties in the correct categories. The repeated sequence for each quadrilateral provides a structure for comparing and manipulating quadrilaterals, inviting students to generalize about the properties in each new digital interactive, which means that topic A has more digital interactives than do other grade 5 topics.” An image of two quadrilaterals is shown. 

• Module 6: Foundations to Geometry in the Coordinate Plane, Module Overview, Why, “Why does the coordinate plane module start with plotting coordinates on number lines, including number lines of different orientations? Beginning with plotting coordinates on number lines activates prior knowledge of plotting and determining locations on a number line as distances from 0. Students understand a number line as a coordinate system that can be used to describe the location of a point when they choose the location of 0 and an interval length. Students then build on this knowledge to construct a coordinate plane when they realize that a single number line is not sufficient to describe the location of points that are not collinear.” Images of three coordinate planes are shown.

The materials reviewed for Eureka Math² Grade 5 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 2, Topic C, Lesson 11: Add mixed numbers with unrelated units, Materials, “Teacher: Chart paper (15 sheets), Station Problems (in the teacher edition), Tape. Students: None. Lesson Preparation: Prepare three signs on chart paper. Label one sign Station 1, one sign Station 2, and one sign Station 3. Hang the signs in different locations in the classroom.Print or copy Station Problems and cut each page in half. Prepare enough so each pair of students has a copy of all the problems.”

• Module 4, Topic A, Lesson 8: Round decimal numbers to any place value unit, Materials, “Teacher: Paper (4 sheets). Students: Multiply or Divide by Powers of 10 Sprint (in the student book). Lesson Preparation: Consider tearing out the Sprint pages in advance of the lesson. Prepare four signs on paper. Label the signs Nearest Ten, Nearest One, Nearest Tenth, and Nearest Hundredth. Hang the signs in different locations in the classroom.”

• Module 6: Foundations to Geometry in the Coordinate Plane, Module Overview, Materials,  “96 Colored pencils, 1 Projection device, 25 Dry-erase markers, 24 Sticky notes, $1.5ʺ\times2ʺ$, 20 Index cards, 269 Sticky notes, $3ʺ\times3ʺ$, 24 Learn books, 25 Straightedges, 12 Markers, 1 Teach book, 11 Paper, sheets, 1 Teacher computer or device, 25 Pencils, 1 Timer, 25 Personal whiteboards, 24 4” Protractors, 25 Personal whiteboard erasers.”

The materials reviewed for Eureka Math² Grade 5 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 B, Lesson 10: Multiply three- and four-digit numbers by three-digit numbers by using the standard algorithm, Launch, “Language Support: Define the words halving and doubling for students. Halving: dividing by 2 Doubling: multiplying by 2.” Learn, Relate the Area Model to the Standard Algorithm, “UDL: Action & Expression: Support students in monitoring their progress by encouraging self-questioning when they use the standard algorithm. Emphasize the importance of thinking through decisions and changing course if a strategy is not working. Think aloud to model self-questioning by using problem 3 as an example. Discuss how asking questions such as these may have helped the student avoid the error and work more efficiently: Which number should I designate as the unit? Do I have the correct number of partial products? Did I fully distribute all the parts of the other factor? Should I do anything differently?”

• Module 3, Topic B, Lesson 8: Multiply fractions less than 1 pictorially, Learn, Use a Number Line, “UDL: Representation: The digital interactive Fraction of a Fraction on a Number Line supports students in composing the parts of each fractional interval to find the fraction of the fraction. Consider allowing students to experiment with the tool individually or demonstrating the activity for the whole class.” Learn, Use an Area Model, “Differentiation: Support: Students may say they can find $\frac{2}{5}$ of $\frac{4}{5}$ by simply shading 2 of the fifths. Address this misconception by covering the bracket and the label $\frac{4}{5}$. Then ask students what shading 2 columns would represent. They should realize that shading 2 columns would represent $\frac{2}{5}$ of 1, or $\frac{2}{5}$ of $\frac{5}{5}$. Reveal the bracket and the label $\frac{4}{5}$ and ask students what portion of the model they should shade to represent $\frac{2}{5}$ of $\frac{4}{5}$. They should realize that to show $\frac{2}{5}$ of $\frac{4}{5}$, they must shade only a fraction of each of the 4 fifths and not 2 entire fifths.” Learn, Choose a Method, “Differentiation: Support: Consider using the following questions to support students as they make decisions about how to make sense of the problem. What does the problem tell us? What does the problem ask us to find? What multiplication expression represents the problem? What can you draw to represent the problem?” Learn, Reason About the Size of the Product, “Language Support: Consider supporting students with the Always Sometimes Never routine with sentence frames for their reference. The product of two fractions less than 1 is _____ (always or sometimes or never) less than both the factors. For example, _____.”

• Module 6, Topic B, Lesson 8: Compare and classify quadrilaterals, Learn, Classify Quadrilaterals, “UDL: Representation: Consider providing a hands-on experience: Have students cut the shapes apart and manipulate them as they check attributes. Copy the shapes onto another sheet of paper and enlarge the images to make them easier to cut and manipulate. Language Support: Terms such as parallelogram and rhombus are familiar from grade 2 and are used extensively throughout this topic. Consider creating an anchor chart with definitions and examples of the terms for students to refer to. Language Support: The sample dialogue uses rhombuses to describe more than one rhombus. Rhombi is also an acceptable term for describing more than one rhombus. Consider whether to introduce the term to students. Differentiation: Support Students may need support in understanding that using more specific attributes results in fewer polygons fitting the description. Consider demonstrating this concept by using student characteristics to identify a specific student and then facilitating a discussion. Use sensitivity, and avoid personal specifics such as gender, race, and religion. Instead, draw attention to more neutral descriptions such as location in the room, letters in their name, items on their desk, and the like. Consider the following example: I’m thinking of a student who is sitting in the group by the door, is wearing a red collared shirt, and has the letter J in their name. Who am I thinking of? Would you know who I was thinking of if I only said they were in the group by the door? How did the specific details help you correctly identify them?”

The materials reviewed for Eureka Math² Grade 5 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 7: Multiply by using familiar methods, Learn, Share, Compare, and Connect, Differentiation: Challenge, “Direct students to a work sample that finds the product by using the standard algorithm. Ask students whether they can see the partial products in the standard algorithm. Then ask them to explain.” 

• Module 3, Topic B, Lesson 9: Multiply fractions by unit fractions by making simpler problems, Learn, Use Known Products to Multiply, Differentiation: Challenge, “For students who recognize the repeated reasoning that is used while finding these products, consider challenging them to explore whether a product of three or more fractions can be found by using similar reasoning.”

• Module 6, Topic B, Lesson 8: Identify addition and subtraction relationships between corresponding terms in number patterns, Learn, Generate Coordinates, Differentiation: Challenge, “Challenge students to determine the 50th x- and y-coordinates in problem 2. Have them use the corresponding coordinates to form the 50th ordered pair.”

The materials reviewed for Eureka Math² Grade 5 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 1, Topic, Lesson 16: Divide four-digit numbers by two-digit numbers, Learn, Division Word Problem Without a Remainder, MLL students are provided the support to participate in grade-level mathematics as described in the Teacher Notes box, “Context videos for problems 1 and 2 are available. The videos may be used to remove language or cultural barriers and to provide student engagement. Before beginning each problem, consider showing the video and facilitating a discussion about what students notice and wonder. This supports students in visualizing the situation before they are asked to interpret it mathematically.”

• Module 5, Topic A, Lesson 1: Analyze hierarchies and identify properties of quadrilaterals, 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.”

• Module 6, Topic A, Lesson 1: Construct a coordinate system on a line, 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 5 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 2, Topic B, Lesson 6: Add and subtract fractions with related units by using area models to rename fractions, Fluency, Whiteboard Exchange: Equivalent Fractions, Materials, Student: Equivalent Fractions. “Students use an area model to generate an equivalent fraction for a unit fraction to prepare for adding and subtracting fractions with related units. After each prompt for a written response, 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 square. Display the model partitioned, shaded, and labeled.”

• Module 3, Topic A, Lesson 1: Find fractions of a set with arrays, Launch, Materials, Student: Centimeter cubes. “Students use centimeter cubes to find fractional units of a set. Distribute 12 cubes to each student and ask them to take out their whiteboards. Display the problem. Mr. Perez has 12 eggs. He uses $\frac{1}{3}$ of the eggs to make a cake. How many eggs does Mr. Perez use to make the cake? Invite students to turn and talk about what they notice about the problem and what the problem asks them to find. Direct students to use the cubes to solve the problem. Circulate and observe student work. Allow students the opportunity to struggle productively. They might not find the answer.”

• Module 5, Topic A, Lesson 2: Classify trapezoids based on their properties, Launch, Materials, Student: Quadrilateral cutouts. “Invite students to turn and talk to define trapezoid and identify a figure that is a trapezoid. Let’s sort the quadrilaterals into figures that are trapezoids and figures that aren’t. What makes a trapezoid different from other quadrilaterals? What does it mean for a quadrilateral to have at least 1 pair of parallel sides? Prompt students to sort the quadrilaterals as either trapezoids or non-trapezoids.”