4th Grade - Gateway 1

The materials reviewed for Eureka Math² Grade 4 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 1, Item 3, “Round 453,182 to the given place. Nearest hundred thousand:____ Nearest thousand: _____ Nearest hundred: _____.” The options are 1(500,000), 2(453,000), 3(453,200). (4.NBT.3)

• Module 3, Module Assessment 1, Item 7, “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? 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)

• Module 5, Assessment 1, Item 6, “Ivan says that 0.9 is less than 0.41 because 9 is less than 41. Do you agree with Ivan? Explain.” (4.NF.7)

• Module 6, Module Assessment 1, Item 6, students are shown a fraction with 360 as the denominator and asked, “What fraction of a circle is a 58\degree angle?” (4.MD.5)

• Module 6, Module Assessment 2, Item 1, students are shown six figures (three oval and three heart figures with lines drawn through each) and asked, “Which figures appear to show a line of symmetry? Select the three correct answers.” (4.G.3)

The materials reviewed for Eureka Math² Grade 4 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 Concepts for Addition and Subtraction, Lessons 7, 8, and 9 engages students with extensive work with 4.NBT.2 (Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons). Lesson 7, Fluency, Whiteboard Exchange: Unit to Standard Form, “students write the standard form of a two- or three-digit number given in unit form to prepare for writing numbers within 1,000,000. Display 1 ten 7 ones = _____. When I give the signal, read the number shown in unit form. Ready? Write the number in standard form.” Lesson 8, Learn, Write Numbers in Standard and Word Forms, “students group thousands to write numbers in word form and standard form. Write the number 1894 on the Place Value Chart to Millions. Where did you place the comma in the number? How do you know it belongs there? How do we read the number?” A place value chart to millions is shown. Lesson 9, Land, Debrief, “students compare numbers within 1,000,000 by using < , = , and >.” Teacher uses prompts to guide students in a discussion about place value and comparing numbers: “Write 51,034 and 510,034 with the 5 and 1 aligned as shown. Casey says these two numbers are equal because they both start with 51 and have most of the same digits. Do you agree or disagree? Why? How could you use ten times as much to compare the value of the 5 in the two numbers? The value of the 1? How are units important when comparing numbers? How are digits important when comparing numbers?” 

• Module 2, Place Value Concepts for Multiplication and Division, Lesson 5 engages students in extensive work of grade level problems with 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models). Fluency, Whiteboard Exchange: Multiply in Unit and Standard Form and Add, “students multiply tens and ones in unit form, write equations, in standard form, and add two products to build fluency with place value strategies for multiplying.” Teacher displays the equation  2\times3 tens = ___ and asks, “What is 2\times3 tens in unit form? Raise your hand when you know. Repeat the process with the following sequence: 2\times4 tens =___; 2\times6 ones =___; 3\times3 tens =___; 3\times2 ones = ___; 4\times5 tens =___; 4\times4 ones = ___.” Learn, Use the Place Value Chart to Multiply, “students draw on a place value chart to help them multiply and identify partial products.” Teacher instructs students, “Write 4\times12 and direct students to do the same. Let’s draw a place value chart and represent 12 by using tens and one.” A labeled place value chart with tens and ones is shown. Land, Debrief, “students multiply by using place value strategies and the distributive property.” Teacher facilitates a discussion about decomposing a factor into place value units and using the distributive property to multiply using writing prompts: “How can we use what we know about multiplying by place value units to multiply a two-digit number by a one-digit number? Why do we use the distributive property to find partial products when multiplying a two-digit number by a one-digit number?” Exit Ticket, “Draw on the place value chart to represent the expression 3\times41. Complete the equations.” A two column table labeled tens and ones is shown.

• Module 3, Multiplication and Division of Multi-Digit Numbers, Lessons 21, 22, and 23 engages students with extensive work with 4.OA.3 (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding). Lesson 21, Learn, Remainders, students identify remainders using arrays. “17 chairs are placed in rows of 5. What is the greatest number of rows of 5 chairs you can make with 17 chairs? Is every chair in a row of 5? What is the quotient? What is the remainder?” Lesson 22, Launch, students relate division to word problems and tape diagrams. “Jayla collects 150 gemstones. She places all the gemstones into cases. Each case has 5 gemstones. How many cases have gemstones? What is the quotient? What does it represent? What division expression is represented by both tape diagrams?” Lesson 23, Learn, Error Analysis, students identify and correct the mistakes in multi-step word problems. The teacher reads the problem with the class, “There are 8 classes of fourth graders in the cafeteria. Each class has 24 students. The tables in the cafeteria each have 5 chairs. What is the fewest number of tables needed for all the students?, and asks, What is known, What is unknown?” An image of “Robin’s Way” is provided with the following solution, “The fewest number of tables needed for all the students is 38. Present the work that shows Robin’s way.” Teacher informs students that Robin’s solution is not correct and provides students with one minute to identify the error and share. 

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

• Module 2, Fractional Units, Lessons 18,19 and 20 engages students with the full intent of   4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems). Lesson 18, Launch, “students determine the distance around a soccer field.” Teacher displays a picture of the soccer field and presents the problem. “A soccer team warms up by running laps around the edge of the soccer field. What distance do they run in 1 lap around the field?” Teacher invites students to think–pair–share about how to find the distance around the soccer field. A picture of a soccer field with length of 100m and width of 50m is shown. Lesson 19, Learn, Find the Area and Perimeter, “students use the length and width of a rectangle to find the area and perimeter.” Classwork, Problem 1, Teacher directs students to Problem 1 in their books and read the problem chorally as a class. “Which picture did we just see that relates to this problem? A rectangular garden bed is 4 feet wide. It is 3 times as long as it is wide. a. Draw a rectangle to represent the garden bed. Label the side lengths, b. Is 40 feet of wood enough to build a frame for the garden bed? How do you know? c. What is the area of the garden bed?” Lesson 20, Learn, Multiple Comparisons in One Problem, “students represent and solve a variety of comparisons in a word problem.” Teacher directs students to form groups of three students and displays the problem, “The perimeter of rectangle A is 42 meters. The perimeter of rectangle A is 3 times as long as the perimeter of rectangle B. The perimeter of rectangle C is 4 meters less than the perimeter of rectangle A. The perimeter of rectangle D is 3 times as long as the perimeter of rectangle A. Find the perimeter of each rectangle. How was finding the perimeters of rectangle B and rectangle D similar and different? How was finding the perimeters of rectangles B and C similar and different? From least to greatest, how would you order the rectangles according to their perimeters? How does the tape diagram help you?”

• Module 4, Foundations for Fractions Operations, Lessons 3 and 19 engages students in the full intent of 4.NF.3 (Understand a fraction \frac{a}{b} with a>1 as a sum of fractions \frac{1}{b}). Lesson 3, Learn, Decompose a Fraction into Non-Unit Fractions, “students decompose a fraction and relate each decomposition to a sum of fractions.” Teacher distributes one paper strip to students and guides students into folding the strips into eighths. “What equation can we write to show \frac{7}{8} as a sum of a unit fraction?” Teacher invites students to share their work and guide a discussion relating to decomposing \frac{7}{8} into unit fractions and non-unit fractions. “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?” A fraction strip partitioned into eighths is shown. Lesson 19, Land, Exit Ticket, “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}=___.”

• Module 5, Place Value Concepts, Lessons 4 and 8 engages students in the full intent of 4.NF.6 (Use decimal notation for fractions with denominators 10 or 100.). Lesson 4, Learn, Mixed Numbers and Decimal Form on a Number Line, ”students use a number line to represent and make connections between mixed numbers and numbers written in decimal form.” The teacher directs students (in pairs) to represent 1 one and 7 tenths by drawing a number line. “How did you draw a number line to represent 1 one 7 tenths? How did you label the point you plotted to represent 1 one 7 tenths? Why can the point on the number line be labeled in decimal form and as a mixed number in fraction form?” The teacher selects a pair to share their equation and write 1\frac{7}{10}=1.7 as they share. “How do we read the mixed number? How does saying the mixed number help us write the number in decimal form? What does the wordand represent in the mixed number?” Lesson 8, Land, Debrief, “students express decimal numbers in expanded form.” Teacher facilitates a discussion about expanded form and the value of digits in decimal numbers. “In what different ways can we represent decimal numbers in expanded form? How can place value cards and the place value chart help us write a number in expanded form? What number is shown on the place value cards? What is the value of the 5? 4? 9?” Teacher shows students  5.49 by using place value cards and invites students to turn and talk about how writing or thinking about a number such as 5.49 in expanded form might be useful as they learn more about decimal numbers.

The materials reviewed for Eureka Math² Grade 4 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 devoted to the major work of the grade (including supporting work connected to the major work) is 100 out of 140, approximately, 71%. 

• The number of days devoted to the major work of the grade (including assessments and supporting work connected to the major work) is 120 out of 140, approximately 86%. 

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 71% of the instructional materials focus on major work of the grade.

The materials reviewed for Eureka Math² Grade 4 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 E, Lesson 23, Express metric measurements of length in terms of smaller units, Learn, Relative Size of Units, connects the supporting work of 4.OA.5 (Generate a number or shape pattern that follows a given rule…) to the major work of 4.NBT.1 (Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right). “Students relate the relative sizes of metric length units to the place value system.” The teacher invites students to turn and talk about how many centimeters long their meter sticks are and directs students to slide their fingers from 0 to 10 and asks, “What distance is 10 times as long as 1 centimeter? What distance is 100 times as long as 1 centimeter? How many times is a length of 1 centimeter used to measure a length of 1 meter? We can say that 1 meter is 100 times as long as 1 centimeter. What equation can we write to represent the statement? The meter stick is 1 meter long. How many centimeters are the same length as 1 meter?” The teacher directs students to the chart relating meters and kilometers and asks, “What type of chart have we used before that these charts might remind you of? In the place value system, how do we rename 100 ones? In the metric measurement system, how do we rename 100 centimeters?” Teacher invites students to talk about how meters and centimeters are related and asks, “How many meter sticks could we line up end to end to show a distance that is 10 times as long as 1 meter? What is the length of 10 meter sticks? How many meter sticks could we use to show a distance that is 100 times as long as 1 meter? What is the length of 100 meter sticks? How many meter sticks could we use to show a distance 1,000 times as long as 1 meter? If we lined up 1,000 meter sticks end to end, the length would be 1,000 meters, or 1 kilometer. Kilometers are abbreviated as km. We can say that 1 kilometer is 1,000 times as long as 1 meter. What equation can we write to represent the statement?” Teacher says and writes 1 Kilometer is 1,000 times as long as 1 meter and asks, “What equation can we write to represent the statement? How many meters are the same length as 1 kilometer?” Teacher directs students to chart the relation between meters and kilometers and asks, “How do we rename 1,000 ones? How do we rename 1,000 meters?” Teacher invites students to turn and talk about how the metric system is similar to the place value system and how kilometers and meters are related. 

• Module 2, Topic D, Lesson 17: Express measurements of length in terms of smaller units, Launch, 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. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale) to the major work of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations…). Students convert from larger customary length measurements to smaller customary length measurements. The teacher displays a picture of a ribbon, “What do you notice about the units on the spool of ribbon? Yards and feet are customary units of length. What do you notice about the numbers next to the units?” The teacher invites students to think-pair-share and discuss why the spool of ribbon is labeled with different measurements, then holds up a meter stick and asks, “What metric units of length are represented on the meter stick? Which unit is larger, meters or centimeters? How does the meter stick represent the relationship between the larger unit, meters, and the smaller unit, centimeters? Which customary unit is larger, yards or feet?” A picture of a spool of ribbon is shown. Learn Conversion Strategies, students represent length conversions using tape diagrams, number lines and conversion tables. The teacher shows three paper strips lined up end to end and asks, “How many feet are represented by my paper strips? Let’s draw a tape diagram to represent my paper strips.” Teacher guides students through the process of drawing a tape diagram with the total labeled as 3 feet and 3 equal units to represent the paper strips. “Write 3 feet =  _____and direct students to do the same. How can we use a number line and our tape diagram to figure out how many inches are in 2 feet? Draw a number line below the tape diagram. Draw tick marks on the number line aligned with 0 feet, 1 foot, 2 feet, and 3 feet on the tape diagram. Direct students to do the same and invite students to work with a partner to label the inches on the number line. Start at the first tick mark and slide your finger along the number line, pausing at each tick mark as you ask the following question. How many inches are represented by the tick mark? Label the tick marks 0 inches, 12 inches, 24 inches, and 36 inches as students share their answers. How many inches are equal to 3 feet? How do the number line and conversion table also show that 3 feet = 36 inches? What strategy did you use to label the tick marks with inches? Can you use similar strategies of repeated addition, skip-counting by twelve, or multiplying the number of feet by 12 when using one number line or a conversion table to convert feet to inches? How many feet are in 6 yards? What multiplication equation represents how to convert 6 yards to feet? What does 6 represent? What does 3 represent? What does 18 represent?” Land, Debrief, Students express customary measurements of length in terms of smaller units. Teacher facilitates a discussion about expressing customary measurements of length in terms of smaller units using the following prompts: “How are yards, feet, and inches related? How can we convert measurements with larger length units to measurements with smaller length units?”

• Module 4, Topic D, Lesson 20: Subtract a fraction from a whole number, Learn, Solve a Word Problem, 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 pto 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?” 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?”

The instructional materials reviewed for Eureka Math² Grade 4 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 4 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 E, Lesson 21: Find factor pairs for numbers up to 𝟏𝟎𝟎 and use factors to identify numbers as prime or composite, Launch, connects the supporting work of 4.OA.B (Gain familiarity with factors and multiples) to the supporting work of 4.MD.A (Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit). Students reasons about factors for a given number using arrays. “Write the equation l\times w=24.” Teacher, “Invite students to turn and talk about numbers that make the equation true and directs students to work with a partner to draw as many arrays with an area of 24 square units as possible. As they create each array, have them sketch the array and record the equation that represents the area. Provide students with 2 minutes to find all possible arrays, sketch the arrays, and record the equations. After students record the equations, direct them to work with another group to compare their equations and to discuss the factors of 24 based on the equations they recorded. Invite students to think–pair–share about how they know that they have found all the factors of 24. We started with 1 and thought about whether we could make an array with 1 row. Then we tried 2 rows and 3 rows. We kept going like that until we got up to 6 rows. When we got to 6 rows, we realized it was the same factors we had already listed. We thought about all the multiplication facts that have a product of 24. Invite students to turn and talk about how they might find all the factors of a number such as 96. Transition to the next segment by framing the work. Today, we will identify all factor pairs for a given number, and we will describe numbers based on how many factors they have.”

• Module 3, Topic F, Lesson 22: Represent, estimate, and solve division word problems, Launch, connects the major work of 4.OA.A (Use the four operations with whole numbers to solve problems) to major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic). “Students relate two types of division to word problems and tape diagrams.” Teacher displays a picture of two tape diagrams and invites students to think-pair-share about what is similar and different about the tape diagrams. Teacher displays and presents the problem, “Jayla collects 150 gemstones. She puts equal amounts of gemstones into 5 cases. How many gemstones are in each case? What is the quotient? What does it represent? Jayla collects 150 gemstones. She places all the gemstones into cases. Each case has 5 gemstones. How many cases have gemstones? What is the quotient? What does it represent? What division expression is represented by both tape diagrams? How is what is known and unknown in the second problem different from what is known and unknown in the first problem?” Learn, Estimate Quotients, Classwork, Problem 2. Students solve word problems by estimating the quotient and rounding to a multiple of the divisor. “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?” Land, 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?”

• Module 6, Topic A, Lesson 3: Draw right, acute, obtuse, and straight angles, Learn, Draw and Name an Angle, Classwork Problem 1, connects the supporting work of 4.G.A (Draw and identify lines and angles, and classify shapes by properties of their lines and angles) to the supporting work of 4.MD.C (Geometric measurement: understand concepts of angle and measure angles), as students draw right, acute, obtuse, and straight angles. “Draw an angle with the given ray. Label the angle. Use \overline{LM} to draw \angle KLM.” An image of a ray is shown.

The materials reviewed for Eureka Math² Grade 4 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: Topic A: Multiplication as Multiplicative Comparison, Topic 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 work in Grade 5. “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.”

• Module 3: Multiplication and Division of Multi-Digit Numbers, Module Overview, After This Module, connects 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) and 4.NBT.6 (Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to work in Grade 5. "In grade 5 module 1, students multiply multi-digit whole numbers and develop fluency with the standard algorithm for multiplication. Students also divide with two-digit divisors and continue building conceptual understanding of multi-digit whole number division. They find whole-number quotients and remainders. In module 2, students transition from finding whole-number quotients and remainders to fractional quotients. In modules 1, 3, and 4 of grade 5, students use multiplicative relationships to convert metric and customary units involving whole numbers, fractions, and decimals. In addition to expressing larger measurement units in terms of smaller units, they express smaller measurement units in terms of larger units."

• Module 5: Topic D, Lesson 14: Solve word problems with tenths and hundredths, Learn, Solve a Comparison Word Problem, Teacher Note, connects 4.NF.5 (Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100) and 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. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale) to work in Grade 5. “If students use vertical form to show addition of decimal numbers, consider acknowledging their method and asking them to show their work in fraction form. Students learn to add decimal numbers by using vertical form in grade 5.”

Materials relate grade-level concepts from Grade 4 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 4, prior connections are often made to content from previous modules within the grade. Examples include:

• Module 1: Place Value Concepts for Addition and Subtraction, Module Overview, Before This Module, connects 4.OA.1 (Interpret a multiplication equation as a comparison) to work from Grade 3. “In grade 3 module 1, students build a conceptual understanding of multiplication as a number of equal groups (e.g., 4\times3=12 can be interpreted as 4 groups of 3 is 12). In grade 3 module 2, students compose and decompose metric measurement units and relate them to place value units up to 1 thousand. They use place value understanding and the vertical number line to round two- and three-digit numbers. Grade 3 students also add and subtract two- and three-digit numbers by using a variety of strategies, including the standard algorithm.”

• Module 2: Place Value Concepts for Multiplication and Division, Module Overview, Before This Module, connects 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) and 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems) to work from Grade 3. "In grade 3, students develop conceptual understanding of multiplication and division and become fluent with multiplication and division facts within 100. They multiply multiples of 10 by one-digit numbers, recognizing that they can use familiar facts and only the unit changes. Models and methods used in grade 3 include drawing equal groups and arrays, using the break apart and distribute strategy, writing equations in unit form and standard form, and applying the commutative and associative properties. In grade 3, students find the areas and perimeters of rectangles. They tile rectangles and count the number of square units to determine the areas, eventually recognizing that the area can be found by multiplying the number of square units in each row by the number of square units in each column. Students find the perimeter by adding and by using multiplication, focusing on the relationships between the side lengths in rectangles and other polygons."

• Module 4: Topic A: Fraction Decomposition and Equivalence, Topic 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 work in Grade 3. “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.”