5th Grade - Gateway 1

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."