3rd-5th Grade - Gateway 1

The materials reviewed for Eureka Math² Grade 3 through Grade 5 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

According to the Grades 3-5 Implementation Guide, “All Eureka Math² assessments are considered formative because they are intended to inform instruction. The assessments may also be considered summative when you choose to use the data to produce a grade or report that becomes part of a student, school, or district record.” The Assessment System includes lesson-embedded Exit Tickets, Topic Quizzes, and Module Assessments. According to the 3-5 Implementation Guide, Assessment, Components, “Exit Tickets are short, paper-based assessments that close lessons. These assessments use at least one problem, question, or writing prompt to assess whether a student has learned the basic skills and concepts needed for success in upcoming lessons. Items reflect the minimum that students must demonstrate to meet the lesson objective. You may look for evidence of the mathematical practice identified as the focus mathematical practice for the lesson in student work on the Exit Ticket. 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. Module Assessments, 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 Assessment items that assess grade-level standards include:

• Grade 3, Module 1, Module Assessment Version 2, Problem 3 states, “Pablo puts photos in his photo album. He puts the photos in 3 rows of 6. How many photos does Pablo put in his photo album altogether? Part A. Use the sliders to make an array to represent the problem. Part B. Which equations can be used to solve the problem? Select the two correct answers. 1. 6\div3= , 2. 6\times3= , 3. 3\times3= , 4. 6+6+6= , 5. 3+3+3+3+3= . Part C. How many photos does Pablo put in his photo album altogether? Pablo puts ____ photos in his photo album altogether.” (3.OA.1)

• Grade 4, Module 3, Module Assessment Version 1, Problem 7 states, “A pet store owner has 243 goldfish. She sells 67 of them. She places the remaining goldfish in equal groups among her goldfish bowls. Each goldfish bowl holds 8 goldfish. Part A, Which is the closest estimate for the number of bowls the pet store owner needs for the remaining goldfish? 20, 30, 40, 160 Part B, How many bowls does the pet store owner need for the remaining goldfish? The pet store owner needs ___ bowls. Part C, Is your answer reasonable? Explain.” (4.OA.3)

• Grade 5, Module 5, Module Assessment Version 1, Problem 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. Write a number on each blank to show what the measurements of the prism could be. Part B. Use the diagram of the prism to explain why the expression (6\times10)\times3. can also be used to find the volume.” (5.MD.C)

The materials reviewed for Eureka Math² Grade 3 through Grade 5 meet expectations for having assessment information included in the materials to indicate which standards are assessed. Materials consistently identify the standards and practices assessed for formal assessments. 

The publisher states, “Every formal Eureka Math² assessment item is tagged to at least one CCSS standard, at least one associated Achievement Descriptor, and at least one Mathematical Practice.” Under the overarching Assessments tab in the Great Minds Library, the Standards link for Topic Quizzes and Module Assessments identifies the standards addressed on each assessment. The digital version also identifies the aligned standard below each problem in the Topic Quizzes and Module Assessments; the PDF print versions do not include these standard alignments. The publisher also states, “Across each grade level, all eight SMPs were tagged to multiple items, allowing students to demonstrate the full intent of the grade-level practices.” For each Module Assessment, the digital platform provides a dot chart titled “Mathematical Practice Alignment to Module Assessment Items” that identifies the Mathematical Practices aligned to individual assessment items.

Examples include:

• Grade 3, Module 2, Module Assessment Version 1, Problem 3 states, “A shark weighs 286 kilograms more than an octopus. The shark weighs 315 kilograms. How much does the octopus weigh? A. 29 kg; B. 31 kg; C. 171 kg; D. 601 kg.” (3.MD.2)

• Grade 4, Module 3, Module Assessment Version 2, Problem 2 states, “Multiply 4,396\times4=_____; 27\times85=_____.” (4.NBT.5, MP7)

• Grade 5, Module 2, Module Assessment Version 1, Problem 3 states, “Mr. Perez cooks 8 cups of pasta. He delivers the pasta equally between 5 bowls. Mr. Perez then pours 1\frac{1}{4} cups of sauce into each bowl. How many cups of pasta and sauce are now in each bowl? There are _____ cups of pasta and sauce in each bowl.” (5.NF.1, MP1, MP2, MP7)

The materials reviewed for Eureka Math² Grades 3 through Grade 5 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

The Grades 3-5 Implementation Guide explains that the assessment system uses multiple assessments and sources of data to develop a complete picture of student understanding. The materials state, “On its own, a single assessment does not show a complete picture of student progress” and direct teachers to “use a combination of assessments and data to get a definitive picture of or to produce a grade for the subject.” The materials further explain that some standards are developed and assessed across multiple modules and that students are not expected to demonstrate full proficiency until all related content has been taught.  Across the series, students engage with Exit Tickets for each lesson, Topic Quizzes, Module Assessments, Benchmark Assessments, and Performance Tasks that include a variety of item types and modalities, such as constructed response, multiple choice, multiple select, multi-part questions, writing, and modeling. Module and Topic Assessment items also identify associated grade-level content standards and Mathematical Practices. Collectively, these assessment opportunities allow students to demonstrate the full intent of grade-level standards and Mathematical Practices across the series.

Examples include:

• Grade 3, Module 2, Module Assessment Version 1, Problem 5 states, “Add. 328+239=___.” Problem 7 states, “Subtract. 428-169=___.” The materials assess the full intent of 3.NBT.2 as students fluently add and subtract within 1,000 using strategies or algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

• Grade 4, Module 5, Module Assessment Version 2, Problem 2 states, “Add. \frac{3}{10}+\frac{25}{100}=____, \frac{81}{100}+\frac{5}{10}=___ , 2\frac{17}{100}+1\frac{6}{10}=___” The materials assess the full intent of 4.NF.5 as students express fractions with denominator 10 as equivalent fractions with denominator 100 and use this equivalence to add fractions with respective denominators 10 and 100.

• Grade 5, Module 3, Module Assessment Version 1, Problem 10 states, “Consider the expression shown. 6\div\frac{1}{4}. Write a word problem that could be solved by evaluating the given expression. Find the answer to your word problem.” The materials assess the full intent of MP4 as students model with mathematics by creating a real-world situation that can be represented by the given mathematical expression and using the expression to solve the problem they created.

The materials reviewed for Eureka Math² Grade 3-5 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 consists of four sections: Fluency, Launch, Learn, and Land. Fluency provides distributed practice with previously learned material and prepares students for new learning by activating prior knowledge and bridging small learning gaps. Launch activities create an accessible entry point to the day’s learning by building context and often creating productive struggle. Each Launch ends with a transition statement that establishes the learning goal. Learn presents new learning related to the lesson objective through a series of instructional segments and includes opportunities for direct instruction, guided instruction, group work, partner activities, interactive video, and digital elements. The Problem Set, included in Learn, provides students with an opportunity for independent practice. During Land, teachers facilitate a brief discussion using suggested questions, including key questions related to the objective, to help students synthesize the day’s learning.

Examples include:

• Grade 3, Module 5, Topic A, Lesson 1, engages students with the full intent of 3.G.2 (Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole). Students partition shapes into equal-area parts, identify and name the resulting fractional units, and connect each equal part to a unit fraction of the whole. In Launch, students analyze partitioned shapes to determine whether the parts are equal and name the fractional units: “Is the shape partitioned into equal parts or unequal parts? If the shape is partitioned into equal parts, how can we name the equal parts?” In Learn, Draw to Equally Share More Than One Object, students use the Read-Draw-Write process to solve an equal-sharing problem, “4 friends have 13 granola bars to share equally. How many bars will each get?” Students determine how to partition the remaining granola bar into four equal parts as teachers ask, “What will happen with the extra granola bar? How could the friends equally share the extra granola bar? How much does this friend get? What about this other friend? Does everyone get an equal amount or number? Is there another way you could equally share the granola bars?” As students share their representations, they connect the equal partitions to fractional units and unit fractions of the whole as teachers ask, “Notice how the bars are partitioned. How many equal parts are there? How did you show how the leftover granola bar was partitioned? What did you name the parts?” Teachers establish the relationship between the equal-area parts and unit fractions, “Fractional units are the units we count when we partition a whole into equal parts. Fourths are fractional units. What are some other fractional units? How do you know everyone gets an equal amount, or an equal number of units? 1 fourth is a fraction.” Students also consider the importance of equal areas when representing unit fractions, “Do these parts look like fourths of a whole granola bar? Can I really say the drawing shows fourths? Size is important when we make drawings of parts. We need to be as precise as possible when drawing equal parts of a whole.”

• Grade 4, Module 4, Topic A, Lesson 3 and Topic D, Lesson 19, engage students with the full intent of 4.NF.3 (Understand a fraction \frac{a}{b} with a>1) as a sum of fractions \frac{1}{b}). Students decompose fractions greater than one unit fraction into sums of unit and non-unit fractions and use their understanding of fractions as sums of like units to add and subtract fractions with like denominators. In Lesson 3, Learn, Decompose a Fraction into Non-Unit Fractions, students decompose fractions and relate each decomposition to a sum of fractions. Students fold a paper strip into eighths and use the resulting fractional units to decompose \frac{7}{8} as teachers ask, “What equation can we write to show \frac{7}{8} as a sum of a unit fraction?” Students then examine different decompositions of \frac{7}{8} into unit and non-unit fractions and connect the parts of a fraction strip to equations as teachers ask, “How many eighths are represented by the first color? How many eighths are represented by the second color? How could we record this decomposition as a sum of non-unit fractions?” In Lesson 19, Land, Exit Ticket, students apply their understanding of fractions composed of like fractional units to subtract fractions with like denominators and may use a number line to support their reasoning, “Add or subtract. Write the sum or difference in fraction form. You may use a number line to help you. 1. \frac{7}{8}-\frac{6}{8}=___.”

• Grade 5, Module 4, Topic B, Lessons 10 and 11, Topic C, Lesson 14, and Topic D, Lesson 21 engage students with the full intent 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). Students add, subtract, multiply, and divide decimals to hundredths, consider and use strategies for operating with decimals, solve contextual problems involving decimal operations, and use place value representations to support their calculations. In Lesson 10, Launch, students consider possible methods for adding decimal numbers in the context, “Sasha sells spices at a market. She weighs 5.64 grams of garlic and 2.7 grams of cumin for a customer.” Teachers “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.” In Lesson 11, Learn, Problem Set, Problem 20, students use the Read-Draw-Write process to add and subtract decimals in a contextual problem, “Use the Read-Draw-Write process to solve each problem. Kayla earns $5.75 washing the dishes and $2.50 taking out the trash. She spends $4.89 on a notebook. How much money does Kayla have left?” In Lesson 14, Learn, Problem Set, Problem 13, students multiply a decimal to the hundredths, “4\times6.24=___” In Lesson 21, Learn, Problem Set, Problem 2, students use a place value representation to divide a decimal and record their calculation in vertical form, “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, and hundredths is provided to support students in using place value to divide.

The materials reviewed for Eureka Math² Grades 3 through Grade 5 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade as included in the following grade-level breakdowns. 

Grade 3: 

• The approximate number of modules devoted to major work of the grade (including assessments and related supporting work) is 4 out of 6, approximately 67%.

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 111 out of 140, approximately 79%.

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 123.5 out of 172, approximately 72%. 

Grade 4: 

• The approximate number of modules devoted to major work of the grade (including assessments and related supporting work) is 5.5 out of 6, approximately 92%.

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 114 out of 140, approximately 81%. 

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 144 out of 176, approximately 82%. 

Grade 5: 

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

• The approximate number of lessons devoted to major work of the grade (including assessments and related supporting work) is 96 out of 134, approximately 72%. 

• The approximate number of instructional days devoted to major work of the grade (including assessments and related supporting work) is 119.5 out of 165, approximately 72%.

An instructional day analysis across Grades 3 through Grade 5 is most representative of the instructional materials, as the days include major work, supporting work connected to major work, and the assessments embedded within each chapter. Approximately 72% of the materials in Grade 3, 82% of the materials in Grade 4, and 72% of the materials in Grade 5 focus on major work of the grade.

The materials reviewed for Eureka Math² Grades 3 through 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 that supporting standards are connected to the major standards of the grade. Teachers can locate these connections within the “Achievement Descriptors and Standards” tab in lessons.

Examples include:

• Grade 3, Module 5, Topic A, Lesson 4, Teach Book, Launch, connects the supporting work of 3.G.2 (Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole) to the major work of 3.NF.1 (Understand a fraction \frac{1}{b} as the quantity formed by 1 part when a whole is partitioned into “b” equal parts; understand a fraction \frac{a}{b} as the quantity formed by 𝑎 parts of size \frac{1}{b}). Students identify and relate numbers of equal parts to fractional units. The teacher displays the picture of the yellow hexagon and the next picture (1 half) and asks, “‘How much of the hexagon is red? Half, How many halves?’” The teacher displays the last picture in the halves sequence and invites students to count as each half is pointed to, and asks, “‘How much of the hexagon is red? 2 halves. How many halves make 1?’”  Learn, Identify and Count Fractional Units, “Display the picture of the rectangle with 1 half shaded. Trace the outline of the rectangle and ask the following series of questions: Are there equal parts? How many equal parts are there? What is the fractional unit? What is each part called? What is the unit fraction? ‘Count the parts to make 1.’ Repeat the process with the pictures of the shapes that show 1 fifth, 1 third, and 1 eighth shaded. Remove the scaffolding questions as students are ready. Consider drawing additional examples as needed.”

• Grade 4, Module 4, Topic D, Lesson 20, Teach Book, Learn, connects the supporting work of 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit…) to the major work of 4.NF.3d (Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem). Students subtract fractions from whole numbers to solve word problems. “Luke buys 15 pounds of rice. He uses \frac{1}{4} pounds of rice. How many pounds of rice does Luke have left?” The teacher asks, “What is this problem about? What is known? What can we draw to represent the known information?” The teacher draws and labels a tape diagram to represent the known information in the problem, directs students to do the same, and asks, “‘What is unknown? What in the tape diagram represents the amount of rice he has left? How can we show the value of this part is unknown? How many pounds of rice does Luke have left?’” The teacher labels the unknown part with the letter p, invites students to talk about how the tape diagram helps them solve the problem. The teacher directs students to write an equation using p to represent the unknown, find the value of p, write a statement to answer the questions, and asks, “How many pounds of rice does Luke have left?” The teacher guides a discussion to help students connect the equation and the tape diagram, using the prompts: “Where is the total amount of rice represented in the tape diagram? In the equation? Where are the pounds of rice that he used represented in the tape diagram? In the equation? Where are the pounds of rice that are left represented in the tape diagram? In the equation?” 

• Grade 5, Module 4, Topic E, Lesson 28, Teach Book, Launch, connects the supporting work of 5.MD.1 (Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems) 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 convert customary measurements by representing equivalent measurements in a single unit, renaming fractional measurements as decimals, and using decimal multiplication to justify equivalent measurements. “Present the picture of the containers of milk and use the Math Chat routine to engage students in mathematical discourse.” Students are shown three 1-gallon containers and two 1-quart containers of milk. “Give students 2 minutes of silent think time to represent the amount of milk shown by using one measurement unit. Have students give a silent signal to indicate they are finished. Then, ask students to discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. If possible, select a student who converted gallons to quarts and a student who converted quarts to gallons. Purposefully choose work that allows for rich discussion about connections between strategies.” Students verify their conversions by writing the equation, “3.5 gal=3.5\times4 qt=14 qt,” using decimal multiplication to convert gallons to quarts. Write 3\frac{2}{4} gal = 14 qt. “Is this equation true? How do you know?” (Yes. I know because we renamed a mixed number, 3\frac{2}{4}, as a decimal, 3.5. We did not change the value of either side of the equation, so the equation is still true.) Invite students to turn and talk about how they think we could have used the relationship between quarts and gallons to find the number of gallons in decimal form instead of fraction form. Transition to the next segment by framing the work. “Today, we will convert customary measurements involving decimals.”

The instructional materials for Eureka Math² Grade 3 through Grade 5 meet expectations for including problems and activities that connect two or more clusters in a domain, or two or more domains in a grade. 

Connections among the major work of the grade appear throughout the materials where appropriate. Each lesson includes Achievement Descriptors that describe what students should be able to do by the end of the lesson and identify the associated standards. The Teach books include a Module Achievement Descriptors by Lesson resource at the end of each module that identifies the Achievement Descriptors addressed in each lesson, allowing teachers to trace the standards addressed across lessons and modules. 

Examples include:

• Grade 3, Module 1, Topic E, Lesson 19, Teach Book, Learn, Problem 2, connects the major work of 3.OA.B (Understand properties of multiplication and the relationship between multiplication and division) to the major work of 3.OA.C (Multiply and divide within 100). Students use the distributive property to break apart a multiplication expression into known facts and solve the multiplication problem. In Problem 2, “Miss Diaz’s class is going on a field trip. The bus has 8 rows of 4 seats. How many seats are on the bus? Guide students through the process of using the break apart and distribute strategy by breaking apart the rows to find out how many seats are on the bus. 8\times4=(5\times4)+(3\times4)=20+12=32. There are 32 seats on the bus. Consider the following possible sequence: What multiplication fact represents the entire array? Write 8\times4 below the array. Let’s use the break apart and distribute strategy to break apart the 8 rows of 4. Break the array into 5 rows of 4 and 3 rows of 4 by shading 5 rows of 4 with your pencil.”

• Grade 4, Module 3, Topic F, Lesson 22, Teach Book, Learn, Problem 2, connects the major work of 4.OA.A (Use the four operations with whole numbers to solve problems) to the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic). Students use place value understanding to estimate quotients while solving multi-digit division word problems. In Problem 2, “Use the Read–Draw–Write process to solve the problem. Use a letter to represent the unknown. A florist has 1,174 flowers. He makes bunches of 4 flowers. How many bunches can he make?” Students work to solve the problem and then, as a class, discuss estimating to find the answer. “Let’s estimate the quotient by finding a multiple of 4 that is close to the total. How many hundreds are in 1,174? Is 11 hundreds a multiple of 4? Which multiple of 4 is close to 11 hundreds? How can that help you estimate the quotient? What is the estimated quotient?” In the Exit Ticket, “Use the Read-Write-Draw process to solve the problem. Explain why your quotient is reasonable. A factory makes 1,912 toys in 4 days. They make the same number of toys each day. How many toys do they make in 1 day?”

• Grade 5, Module 4, Topic C, Lesson 18, Teach Book, Learn, 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 their understanding of fraction multiplication, area models, and place value to multiply decimal numbers. In Learn, “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, “Relate decimal-number multiplication to fraction multiplication. How can you use what you know about multiplying fractions to multiply decimal numbers?”

The instructional materials reviewed for Eureka Math² Grade 3 through 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. 

Each Teach Book begins with a Module Overview that includes a topic-by-topic summary of the development of learning throughout the module. The Module Overview also contains Before This Module and After This Module coherence sections that look backward and forward to show the module's place in the development of learning within and across grade levels. In addition, The Why Behind the Lesson Design explains the organization of the module's content and structure, including how the module connects to previous and future grade-level learning. Each Topic Overview summarizes the development of learning within the topic and typically describes how the content connects to previous or upcoming learning, including learning from prior or future grade levels. Some Teacher Notes within lessons also explain connections between current learning and prior and future mathematical concepts.

An example of a connection to future grades in Grade 3 includes:

• Module 5, Topic C, Lesson 15, Teach Book, connects 3.NF.2 (Understand a fraction as a number on the number line; represent fractions on a number line diagram) to future work in Grade 4, where students describe fractions greater than 1 as mixed numbers (4.NF.B). The Teacher Note states, “Students formally describe fractions greater than 1 as mixed numbers in grade 4. The emphasis here is on representing measurements with whole numbers and some fractional units (e.g., 5 and \frac{1}{2} inches). In the next lesson, students drop the word and as they represent fractional measurements greater than 1 on the number line to make line plots.”

An example of a connection to prior knowledge in Grade 3 includes:

• Module 1, Module Resources, Module 1 Overview, connects 3.OA.1 (Interpret products of whole numbers) to previous work in Grade 2, where students form equal groups, represent repeated addition, and use arrays to build a foundation for multiplication and division (2.OA.4). The Coherence: Before This Module states, “In grade 2, students complete foundational work for multiplication and division. They form equal groups, write repeated addition sentences, arrange objects in rows and columns to form arrays up to 5 by 5, and discover how adding the number of objects in the rows or columns relates to repeated addition. At first, arrays are made with gaps between rows and columns and, later, are made with no gaps. Students build and manipulate arrays and use part–total language to express the composition and decomposition. Grade 3 module 1 elevates the work of grade 2 through formal introduction of multiplication and division.”

An example of a connection to future grades in Grade 4 includes:

• Module 1, Topic A, Topic Resources, Topic A Overview, connects 4.OA.1 (Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations) and 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison) to future work in Grade 5, where students interpret multiplication as scaling (5.NF.5). The Topic Overview states, “Students use models and multiplicative comparison language to represent multiplicative relationships. Multiplicative comparison gives students another way to interpret multiplication. For example, they see 15=3\times5 as 15 is 3 times as many as 5. This interpretation of multiplication is foundational throughout grade 4 as students describe place value relationships, identify multiples of whole numbers and fractions, and convert measurement units. It also prepares students for multiplication as scaling in grade 5.”

An example of a connection to prior knowledge in Grade 4 includes:

• Module 4, Topic A, Topic Resources, Topic A Overview, connects 4.NF.3a (Understand addition and subtraction of fractions as joining and separating parts referring to the same whole) and 4.NF.3b (Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation) to previous work in Grade 3, where students develop fractions as numbers and represent fractions on the number line (3.NF.A). The Topic Overview states, “In topic A, students decompose whole numbers and fractions into sums of fractions. They also develop an understanding that a mixed number is the sum of a whole number and a fraction less than 1, which helps them as they rename fractions greater than 1 and rename mixed numbers.”

An example of a connection to future grades in Grade 5 includes:

• Module 1, Module Resources, Module 1 Overview, 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 future work in Grade 6, where students divide multi-digit whole numbers using the standard algorithm and write and evaluate numerical expressions with whole-number bases and exponents (6.NS.2, 6.EE.1-2). The Coherence: After This Module states, “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. 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, and they use prime factorization as one strategy to find the least common multiple and greatest common factor. In module 4, students build upon grade 5 knowledge by writing and evaluating numerical expressions with terms that have whole-number bases and exponents.”

An example of a connection to prior knowledge in Grade 5 includes:

• Module 1, Topic A, Topic Resources, Topic A Overview, 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), 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 previous work in Grade 4, where students use place value to round multi-digit whole numbers, compare quantities multiplicatively, and recognize that a digit in one place represents 10 times the value of the place to its right (4.NBT.1, 4.NBT.3, 4.OA.1-2). The Topic Overview states, “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.”