Understand Division with Unit Fractions

Lesson LESSON 23 Overview Understand Division with Unit Fractions

CCSS Focus Lesson Objectives Prerequisite Skills Domain Content Objectives • Multiply and divide whole numbers. • Multiply whole numbers by unit Number and Operations—Fractions • Identify situations that involve dividing a fractions. Cluster unit fraction by a whole number. • Multiply unit fractions by unit fractions. B. Apply and extend previous • Identify situations that involve dividing a a 1 understandings of multiplication and • Understand that 5 a 4 b 5 a 3 . whole number by a unit fraction. ··b ··b division to multiply and divide fractions. • Use a visual fraction model to find the quotient of a unit fraction divided by a Lesson Vocabulary Standards whole number or the quotient of a 5.NF.B.7 Apply and extend previous whole number divided by a unit fraction. There is no new vocabulary. Review the understandings of division to divide unit following key term. fractions by whole numbers and whole • For a given division equation with a unit • unit fraction a fraction with a numbers by unit fractions. fraction and a whole number, write a numerator of 1. Other fractions are built a. Interpret division of a unit fraction by a multiplication equation that also can be used to find the quotient. from unit fractions. non-zero whole number, and compute such quotients. For example, create a Language Objectives 1 story context for 4 4, and use a ··3 • Draw a visual model to illustrate dividing visual fraction model to show the a unit fraction by a whole number. quotient. Use the relationship • Draw a model to illustrate using between multiplication and division multiplication to find the quotient of a 1 1 to explain that 4 4 5 3 12 unit fraction and whole number. 1 ·· 1 ·· because 3 4 5 . ··12 ··3 • Draw a visual model to illustrate dividing b. Interpret division of a whole a whole number a by unit fraction. number by a unit fraction, and • Draw a model to illustrate using compute such quotients. For example, 1 multiplication to find the quotient of a create a story context for 4 4 , and ··5 whole number and a unit fraction. use a visual fraction model to show the quotient. Use the relationship between multiplication and division 1 to explain that 4 4 5 20 because 5 1 ·· 20 3 5 4. Learning Progression ··5 Since Grade 3 students have used visual fractions to gain a conceptual Standards for Mathematical models such as arrays and area models to understanding of dividing with unit Practice (SMP) build foundational understanding of fractions. They divide whole numbers by SMPs 1, 2, 3, 4, 5, and 6 are multiplication and division of whole unit fractions and divide unit fractions by integrated in every lesson through the numbers. Students have multiplied with whole numbers, using area models, bar Try-Discuss-Connect routine.* fractions in Grade 4 and in previous models, fraction bars, number lines, and In addition, this lesson particularly Grade 5 lessons. Students have developed pictures to represent the division. emphasizes the following SMPs: an understanding of fractions as quotients In the next lesson students will apply 2 Reason abstractly and and learned that dividing a whole number their understanding of the concepts in this quantitatively. by a whole number is equivalent to lesson when they divide with unit 7 Look for and make use of structure. multiplying a whole number by a unit fractions in order to solve word problems. 8 Look for and express regularity in fraction. In Grade 6 students solve mathematical repeated reasoning. In this lesson students use their and real-world problems involving understanding of division with whole division of fractions by fractions. *See page 305m to see how every numbers and multiplication with unit lesson includes these SMPs.

457a Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. Lesson Pacing Guide Teacher Toolbox Whole Class Instruction Small Group Differentiation SESSION 1 Division with Unit Fractions Additional Practice PREPARE Explore • Start 5 min Lesson pages 461–462 • Model It 10 min Ready Prerequisite Lesson 45–60 min • Discuss It 5 min Grade 4 • Model It 10 min • Lesson 23 Understand Fraction Multiplication • Discuss It 10 min • Close: Exit Ticket 5 min RETEACH

SESSION 2 Understanding of Division with Unit Additional Practice Tools for Instruction Fractions Lesson pages 465–466 Develop Grade 4 • Start 5 min • Lesson 23 Understand Fraction Multiplication 45–60 min • Model It: Area Models 5 min Fluency • Discuss It 5 min Understanding of Grade 5 Division with Unit • Model It: Number Lines 5 min • Lesson 23 Dividing with Unit Fractions Fractions • Discuss It 5 min • Connect It 15 min REINFORCE • Close: Exit Ticket 5 min Math Center Activity SESSION 3 Ideas About Division with Unit Lesson Quiz Grade 5 • Lesson 23 Quotients—Greater Than 1 or Refine Fractions or Digital • Start 5 min Comprehension Check Less Than 1? 45–60 min • Apply It 35 min • Close: Exit Ticket 5 min EXTEND

Enrichment Activity Lesson Materials Grade 5 Lesson none • Lesson 23 Mystery Equation (Required) Activities Per student: 1 set of fraction circles or tiles Per pair: 2 sets of fraction circles or tiles Digital Math Fraction Models Independent Learning Tool PERSONALIZE

i-Ready Lesson* Grade 5 • Understand Division with Unit Fractions

*We continually update the Interactive Tutorials. Check the Teacher Toolbox for the most up-to-date offerings for this lesson.

©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 457b LESSON 23 Connect to Family, Community, and Language Development

The following activities and instructional supports provide opportunities to foster school, family, and community involvement and partnerships.

Connect to Family Use the Family Letter—which provides background information, math vocabulary, and an activity— to keep families apprised of what their child is learning and to encourage family involvement. Available in Spanish Teacher Toolbox Understand Division with Unit Fractions ACTIVITY DIVIDING w Un FrNs Dear Family, 23 Do this activity with your child to understand division with unit fractions. Work together with your child to solve real-life problems involving division with This week your child is exploring division with unit fractions. unit fractions. • Together with your child, use the picture to solve the problem below. 1. Suppose we want to give each person in our family half of a sandwich. A unit fraction is a fraction that has 1 as the numerator. 1 and 1 are examples of ··6 ··4 The sandwiches are shown below. How many people can we feed with these unit fractions. To learn about division with unit fractions, your child might see a 4 sandwiches? Do we have enough for our family, too many, or too few? problem like the one below.

A butcher wants to divide 3 pounds of meat into packages that will each contain 1 pound of meat. How many packages can she make? ··2 1 This problem can be solved by fi nding 3 4 . It can be helpful to use a number line ··2 model to understand the problem.

2 2 2 2 2 2

1 The model shows that 3 4 5 6. The butcher can make 6 packages that each ··2 1 contain pound of meat. ··2 Another way to say this is that the butcher can make 2 packages of meat per pound. An equation that shows this is 3 3 2 5 6.

1 • Look for similar situations in everyday life that involve dividing with a So, 3 4 5 6, and 3 3 2 5 6. Your child is learning that division and multiplication ··2 unit fraction. Below are some examples of problems you could solve. with fractions are related, just like division and multiplication with whole numbers

are related. 2. If you divide 2 hours of piano practice into sessions of 1 hour each, how many ··2 sessions do you have to practice? Invite your child to share what he or she knows about division with unit fractions by doing the following activity together. 3. One lap around the track is 1 mile. How many laps do you need to do to run ··4 3 miles?

Answers: 1 1 1 1. 4 4 __ 5 8; You can feed 8 people with the four sandwiches; 2. 2 4 __ 5 4 sessions; 3. 3 4 __ 5 12 laps 457 458 2 2 4 ©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 457 458 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted.

Goal Math Talk at Home The goal of the Family Letter is to provide opportunities for families Use the examples in the activity as a model to come up with other to help students develop an understanding of division with unit real-world problems involving dividing with unit fractions. Using fractions. food items and number of friends or family members might be a scenario that most family members can relate to. Activity Conversation Starters Below are additional conversation starters Encourage students to solve real-life problems involving division students can write in their Family Letter or math journal to engage with unit fractions. Look at the Dividing with Unit Fractions activity family members: and modify it if necessary to connect with your students. • What are some examples of whole objects that you might want to divide equally? • How many parts will you divide or split the object into? Why?

457–458 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. Connect to Community and Cultural Responsiveness Use these activities to connect with and leverage the diverse backgrounds and experiences of all students.

Session 1 Use with Model It. Sessions 1–3 Use anytime during the sessions. • Encourage students to visit a local pet shop or animal shelter. Ask • Encourage a conversation about units of measurement. Have students to work with a partner to come up with word problems students brainstorm measurement units they have used or seen in involving unit fractions that are based on their visits. Alternatively, the United States. Encourage them to think about units they may have students write problems about their pet or a friend’s pet have seen in furniture ads (inches, feet), labels for food products involving unit fractions. and drinks (ounces, pounds, gallons, cups), or in signs at the store or supermarkets (pounds). Remind students that these units are Session 2 Use with Model It. part of the U.S. customary system. Invite volunteers to share other • Ask students what they know about pancakes. Explain that units they have seen. Elicit units from the metric system and ask pancakes are a popular breakfast food in the United States. volunteers to tell what they are used for. Encourage students to share popular breakfast foods from other cultures and ask whether any of those foods resemble pancakes. If time permits, help students research pancake recipes and note if and how unit fractions are used in the recipes.

Connect to Language Development For ELLs, use the Differentiated Instruction chart to plan and prepare for specific activities in every session.

English Language Learners: Prepare for Session 1 ELL Differentiated Instruction Use with Model It.

Levels 1–3 Levels 2–4 Levels 3–5 Listening/Writing Read Model It problem Listening/Speaking Read Model It Listening/Speaking Read Model It 1 and have students follow along. Have problem 1 with students. Ask them to point problem 1 with students. Have partners take 1 1 students point to 1 lb and pound in the to 1 lb and pound in the model. Say: Show turns describing what they see in the model. 4 4 1 ·· ·· 1 Then have them tell how they will represent model. Point to and say: Show how you can how you can divide in three equal parts. Now ··4 ··4 the three cats in the model. divide this in three equal parts. Now show the show the part each cat gets. Ask: What part of the model tells you what part each cat gets. Have students work with partners to talk each cat gets? How much will each cat get? Have students work with partners to label the about the model. Provide the following Have partners discuss how they can model and write the answer to part b. sentence frames and sentence starters: represent the model with an equation. After • The whole model represents . they complete the equation, say: Compare the • Each column of the model represents . quotient you wrote in the equation to the parts • There are cats. in the model. What do you see? • We can show each cat’s part with . Ask: What part of the model tells you what each cat gets? How much will each cat get?

©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 458a LESSON 23 SESSION 1 Explore LESSON 23 SESSION 1

Purpose In this session students explore the Explore Division with Unit Fractions idea of dividing a unit fraction by a whole number and dividing a whole number by a unit How is dividing with fractions related to Learning Targets • Interpret division of a unit fraction fraction. They use real-world contexts to give multiplying with fractions? by a non-zero whole number, and compute such quotients. meaning to these types of division problems, • Interpret division of a whole number and they use area models and bar models to by a unit fraction, and compute such Mo It quotients. visualize and show the quotients. Complete the problems below. SMP 1, 2, 3, 4, 5, 6, 7, 8 1 1 Mrs. Cook wants to share pound of fi sh equally among 3 cats. ··4 Start How much fi sh will each cat get? a. Draw on the area model to solve the problem. Possible model: Connect to Prior Knowledge Why Review using an area model to multiply unit fractions to prepare students for using an area cat 1 model to divide with unit fractions. cat 2 cat 3 How Have students draw an area model to find the 1 1 product of and . 1 3 5 ·· ·· 12 Start b. Each cat will get ··· pound of fi sh. Solution c. Complete the division equation to match your area model. Draw an area model to find 1 1 1 ; Area models may 4 3 5 15 ··4 ···12 the product. ·· 1 show a product of 1 1 ··15 2 Look at the problem about Mrs. Cook again. Complete the statements 3 5 ? ··3 ··5 by showing a whole below to solve the same division problem using multiplication. 1 1 divided into fifths a. Each cat will get ··3 of pound of fi sh. 1 ··4 with shaded as well ··5 b. Write a multiplication equation to tell how much fi sh each cat as fifths divided into DISCUSS IT will get. • Compare your model and 1 1 1 1 thirds with shaded. Possible equation: 3 5 equations to your partner’s ··3 ··3 ··4 ···12 model and equations. How Grade 5 Lesson 23 Session 1 | Explore Division with Unit Fractions ©Curriculum Associates, LLC Copying is permitted. are they alike? How are MODEL IT they diﬀ erent? 1 Read the question at the top of the Student Worktext • I think 4 3 is the same as ·4 1 1 3 because . . . page. Remind students that they know how to think ·3 ·4 a 1 about a fraction as both a 4 b and a 3 . ··b ··b 459 ©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 459 1 – 2 Tell students that they are going to use what they know about dividing whole numbers and DISCUSS IT multiplying with unit fractions to think about Support Partner Discussion dividing a unit fraction by a whole number. Then Encourage students to refer to their area model and equations as they clarify the task and have students complete the discuss dividing 1 by 3. problems. ··4 Look for models and equations showing: Common Misconception If students are unsure • 12 equal parts in the whole about the idea of dividing a fraction by a whole 1 • a quotient or product of number, then relate the problem shown to a similar ··12 problem involving just whole numbers. For example: Support Whole Class Discussion What equation would you write if Mrs. Cook wanted Prompt students to discuss how the models and equations represent the problem. to share 6 pounds of fish equally among 3 cats? [6 4 3 5 2] Ask How does the area model show the division? How does it show multiplication? 1 1 Listen for The area model shows being divided into 3 equal parts, 4 3. ··4 ··4 1 1 1 1 One of the equal parts in the model represents of or 3 . ··3 ··4 ··3 ··4 1 Ask How is dividing a number by 3 like multiplying the same number by ? ··3 Listen for Dividing a number by 3 means to divide the number into 3 equal parts, which is the same as finding one third of the number.

459 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. LESSON 23 EXPLORE SESSION 1

Mo It MODEL IT Complete the problems below. 3 – 4 Tell students that they will now think 1 3 Mr. Putnam wants to cut a 3-foot rope into -foot pieces. ··4 about dividing a whole number by a unit fraction. 1 How many -foot pieces can he cut? Clarify the task and have students complete the ··4 a. Draw on the bar model to solve the problem. problems. Possible model: Common Misconception If students do not divide each whole in the bar model into 4 equal parts, then t encourage them to think about the problem as how b. Mr. Putnam can cut 12 pieces of rope that are 1 many fourths are in 3 wholes. each foot long. ··4 c. Complete the division equation to match your model. 1 3 4 5 DISCUSS IT ··4 12 Look at your model again. Complete the statements below to solve DISCUSS IT Support Partner Discussion 4 • Look at problems 3 and 4. the same division problem using multiplication. Encourage students to refer to their bar models and Why can both a division a. There are 4 fourths in each whole foot. equation and a equations as they discuss. multiplication equation be b. Write a multiplication equation to fi nd the number of fourths used? Look for representations showing: in 3 feet. • I think dividing by a unit • 4 fourths in each whole 3 3 4 5 12 fraction is like dividing by a whole number because . . . 1 c. Mr. Putnam can cut 12 -foot pieces of rope. • a division equation with a quotient of 12 ··4 I think it is diﬀ erent because . . . • a multiplication equation with a product of 12 5 REFLECT 1 Explain what it means to divide 3 by . Include a diff erent real-world ··4 Hands-On Activity example from the problem above in your explanation. 1 Possible answer: 3 4 means finding out how many Use fraction circles or tiles to divide ··4 by a unit fraction. fourths are in 3 wholes, such as finding how many friends 1 can share 3 pizzas if each friend gets of a pizza. If . . . students are unsure about dividing a ··4

whole number by a unit fraction, Then . . . use this activity to reinforce understanding by providing a similar problem. 460 Materials For each pair: 2 sets of fraction 460 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. circles or tiles Support Whole Class Discussion • Distribute fraction circles or tiles to each pair. Prompt students to compare dividing with unit fractions to dividing with whole • Provide students with the following problem: numbers. Suppose you have 2 sandwiches. How many children can you feed if each child gets one Ask How does the bar model help you understand dividing a whole number by a third of a sandwich? unit fraction? • Have students cover their 2 wholes with Listen for The bar model shows the wholes. You can divide each whole into the one-third pieces to represent cutting each parts of the size that the unit fraction represents. Then you can see all the parts sandwich into thirds. that are in the given number of wholes. • Ask: How many children can you feed if each child gets one third of a sandwich? How does your model show this? [6; The model shows Close: Exit Ticket there are 6 thirds in 2 wholes, so you can give one third of a sandwich to each of 6 children.] 5 REFLECT 1 1 Look for understanding that to divide 3 by , you think of 3 as the total and as the • Have students write a division equation to ··4 ··4 1 1 match the problem. 2 4 5 6 group size. You are finding the number of groups of in 3. 3 ··3 4 ··4 • Repeat the activity with other whole Common Misconception If students have difficulty writing a real-world example to 1 numbers and unit fractions, such as 2 4 . ··4 support their explanation, then provide a sentence frame to help, such as: How many 1 people can share 3 if each person gets of a ? ··4

©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 460 LESSON 23 SESSION 1 Additional Practice Name: LESSON 23 SESSION 1

Solutions Prepare for Division with Unit Fractions

Support Vocabulary Development 1 Think about what you know about unit fractions. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can. Possible answers: 1 Have students say unit fraction. Assess prior knowledge by asking students to discuss what they In My Own Words My Illustrations know about unit fractions and then complete In My Own Words.

Review numerator and denominator. Then have 2 students work in small groups to complete the a fraction with a numerator of 1 graphic organizer. Call on volunteers to share what they wrote for In My Own Words. Have students talk about the numerator of a unit fraction. Then have unit fraction them share their examples and non-examples. Examples Non-Examples 2 Read the problem. Have students work with a partner. Ask partners to count the units in the model aloud. Ask: How many units do you need to shade? 1 1 1 1 1 2 5 1 , , , , , , 1 Guide partners to decide where they would write ··2 ··3 ··4 ··5 ··6 ··3 ··4 ··2 each number in the fraction.

Supplemental Math Vocabulary • denominator

• numerator 2 Shade the fraction model to show a unit fraction. Write the unit fraction. 1 • one ··5

461 ©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 461

461 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. LESSON 23 SESSION 1

3 a. Students draw an area model of the cake Solve. 1 3 Mai See wants to share of a cake equally among 3 people. showing three thirds. They then divide the ··3 thirds into three equal parts to share equally How much of the cake will each person get? between the 3 people. a. Draw on the area model to solve the problem. Possible model: 1 1 b. 4 3 5 ··3 ··9 1 c. Each person will get of the cake. ··9 Medium eson eson 2 1 1 4 a. Each person will get of of the cake. ··3 ··3 eson 3 1 1 1 b. 3 = 3 3 9 ·· ·· ·· cae Medium b. Complete the division equation to match your area model. 1 1 4 3 5 ··3 ··9 1

c. Each person will get ··9 of the cake.

4 Look at your area model again. Complete the statements below to solve the division problem using multiplication. 1 1 a. Each person will get ··3 of of the cake. ··3 b. Write a multiplication equation to tell how much of the cake each person will get. 1 1 1 Possible equation: 3 5 ··3 ··3 ··9

462 462 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. English Language Learners: Prepare for Session 2 ELL Differentiated Instruction Use with Connect It.

Levels 1–3 Levels 2–4 Levels 3–5 Listening/Speaking Read Connect It Listening/Speaking Read Connect It Writing/Speaking Read Connect It problem 4 and have students chorally read problem 4 and have students read the problem 4 and have students read the the problem. Encourage them to ask each problem with a partner. Ask them to compare problem with a partner. Provide the following other questions about the models. Provide and contrast the area models and the vocabulary for students to write a description sentence starters and questions to guide number lines. Provide the following sentence of how the area model and number line their discussion: starters: are alike: divide, number line, parts, quotient, • I think the area model shows______. What do • The area model shows______. whole number line. you think? • The number lines show______. Have partners add examples to their descriptions and discuss with other partners. • I think the number line shows______. Do you • The area model and the number lines agree? both______. Encourage students to use the words agree or • The area models and the number lines are disagree to answer the questions. Allow alike because they both______. students to use drawings or simple phrases • The area models and the number lines are to explain their answers. different because______.

©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 462 LESSON 23 SESSION 2 Develop LESSON 23 SESSION 2

Purpose In this session students use fraction Develop Understanding of Division with bar area models and number line models to Unit Fractions show dividing with unit fractions. Then they compare the models and represent division with unit fractions using a model of their choice. Mo It: AREA MODELS Try these two problems.

1 Jemma made 5 cups of pancake batter. She uses a scoop that holds 1 Start cup to pour batter onto the skillet to make each large pancake. ··3 Use the model to show how many large pancakes Jemma can make. Connect to Prior Knowledge Possible model: Why Support students’ facility with dividing a whole number by a unit fraction. How Have students find the number of thirds in 1 whole, 2 wholes, and 3 wholes, drawing models to

support their answers. c c c c c Start 15 Solution Jemma can make pancakes. 1 Explain why your model represents 5 4 . Draw a model to show your Look for Models ··3 answer to each question. should show 3 thirds Possible answer: The model shows 5 cups that are split into 1 1 in 1 whole, 6 thirds in parts that each have a size of cup. How many s are in 1 whole? 3 ··3 2 wholes, and 9 thirds ·· 1 1 Possible model: How many s are in 2 wholes? 2 Suppose Jemma uses cup of pancake batter to ··3 in 3 wholes. ··3 1 How many s are in 3 wholes? make 4 mini pancakes. Use the model to fi nd the ··3 fraction of a cup of batter used for each mini pancake. 1

Each mini pancake uses ···12 cup of batter.

Grade 5 Lesson 23 Session 2 | Develop Understanding of Division with Unit Fractions ©Curriculum Associates, LLC Copying is permitted. Explain why your model represents the quotient DISCUSS IT Develop Language • How did you and your 1 4 4. partner model the division Why Clarify the meaning of the word hold as it is ··3 c used in problem 1. problems? 1 1 Possible answer: The model shows lines going across for cup. • I think 5 4 is equal to ··3 ·3 How Explain that the word hold has different 1 1 5 3 3 because . . . I think Then each cup is split into 4 equal parts. Each part is cup. 1 1 1 ··3 ···12 4 4 is equal to 3 meanings. Say: To “hold” can mean to “carry” or to ·3 ·3 ·4 “support.” It can also mean to “contain.” Provide an because . . . example: My water bottle holds 16 ounces. Have 463 students find the word in the problem. Ask: How ©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 463 much batter does the scoop hold? DISCUSS IT Support Partner Discussion MODEL IT: AREA MODELS Encourage students to find even small differences in their representations or to think 1 – 2 Present the problems and have students about ways in which their representations could be different. complete. As students work, have them identify that Support as needed with questions such as: they are being asked to use area models to show dividing with unit fractions. Students should • How did you decide how many equal parts to divide each model into? recognize that the area model in problem 2 is one of • What does each equal part in your model represent? the fraction bars from problem 1. • Did you use shading in both problems 1 and 2, or only in one of the problems? Why? Common Misconception If students think that the Support Whole Class Discussion quotient in a division problem is always less than the dividend, then review the idea of division as For each problem, have several students share their representations. finding the number of groups of a certain size that Ask How does each model show the quotient? there are in the total. Point out how in problem 1 it Listen for For problem 1, the quotient is the number of equal parts. The parts makes sense that the number of pancakes Jemma can be shown by dividing each bar into thirds with two horizontal lines or with makes is greater than the dividend of 5. two vertical lines. For problem 2, the quotient is shown by one of the 12 equal parts of the whole or the part of the model where the shading overlaps.

463 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. LESSON 23 DEVELOP SESSION 2

Mo It: Nu Li MODEL IT: NUMBER LINES Use number lines to show dividing with unit fractions. 3 Present the problem and have students complete. As students work, have them identify that 3 Circle the quotient that matches the question. Then model the quotient on the number line. Write the division equation that represents your model. they are being asked to choose the division 1 1 1 a. What is divided into 2 equal parts? 2 4 OR 42 expression that matches each question. Then they ··4 ··4 ··4 model the division on a number line and write a Possible model: division equation. 1 1 4 2 5 Equation ··4 ··8 1 1 1 DISCUSS IT DISCUSS IT b. How many groups of are in 4? 4 4 OR 44 ··2 ··2 ··2 • How did you and your Support Partner Discussion Possible model: partner decide what to Encourage partners to connect the numbers in the label the tick marks given on the number lines? questions to the corresponding number line model. 2 3 • I think a word problem that 1 1 4 4 5 8 is described by 4 2 is . . . Support as needed with questions such as: Equation ··2 ·4 • How did you know how many equal parts to divide each number line into? Co It • How does you model show the dividend and divisor? Complete the problems below. 4 Look at the area models and the number lines. What do they show about how the Support Whole Class Discussion quotient compares to the dividend when you divide a whole number by a unit fraction and when you divide a unit fraction by a whole number? Have several students share their number lines. Possible answer: When you divide a whole number by a unit fraction, you Ask Where does each model show the quotient? split that whole number into parts smaller than 1. So, the result is more parts 1 and a quotient greater than the original whole number. When you divide a Listen for In problem 3a, the quotient of is the unit fraction by a whole number, you break the original fraction into smaller ··8 size of one equal part on the number line. In parts. So, the quotient is less than the original fraction. 1 5 Use any model to fi nd 5 4 . Possible model: problem 3b, the quotient of 8 is the number of ··2 equal parts on the number line. 2 3

1 5 4 5 2 10 Hands-On Activity 464 ·· Divide a unit fraction by a whole 464 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. number. If . . . students are unsure about dividing a unit CONNECT IT fraction by a whole number, 4 Student responses show understanding that the models for dividing a whole Then . . . use this activity to provide a model. number by a fraction show a quotient that is greater than the whole number, and the models for dividing a fraction by a whole number show a quotient that is less than Materials For each student: 1 set of fraction the original fraction. circles or tiles • Have students find a fraction piece to 1 represent . Close: Exit Ticket ··2 1 • Display 4 5 and ask students what ··2 5 Look for models that show 5 wholes, each divided into 2 equal parts, reflecting 1 1 1 this means. Divide into 5 equal groups. an understanding that 5 4 represents finding how many groups of size are in 5. 3 ··2 4 ··2 ··2 1 • Tell students that to model 4 5, they need 1 2 Common Misconception If students find 5 4 2 instead of 5 4 , then remind ·· ··2 1 1 to find 5 equal-sized pieces that exactly cover them that 5 4 represents the number of groups of in 5, not the number of groups 1 ··2 ··2 the piece. [5 one-tenth pieces] ··2 of 2 in 5. • Have students write a division equation to represent their model. 1 4 5 5 1 Discuss 3 ··2 ··10 4 that the quotient tells the size of each piece. • Repeat with other problems, such as 1 4 2. ··3

©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 464 LESSON 23 SESSION 2 Additional Practice Name: LESSON 23 SESSION 2

Solutions Practice Division with Unit Fractions 1 Study how the Example shows dividing a whole number by a unit fraction. The model shows 2 wholes with each whole Then solve problems 1–6. divided into 5 equal sections, so there are 10 fifths in 2 wholes. Medium Ex Teams of students in Mr. Reed’s classroom decorate rectangular bulletin boards to 1 show their team name, colors, and mascot. Each team uses of a bulletin board. 2 The model shows 2 groups of 5 fifths, so there ··5 are 10 fifths in all. The team decorations fi ll 2 bulletin boards. How many teams are there in all? Medium The 2 large rectangles represent the 2 bulletin boards. 1 1 Each team’s decorations use of a board, so 3 15 teams; 3 4 5 15; 3 3 5 5 15 ··5 ··5 each rectangle is divided into 5 equal sections. Medium 1 From the model, you can write the division equation: 2 4 5 10 ··5 You can also write the multiplication equation: 2 3 5 5 10

Both equations show 10 teams in all.

1 1 Explain how the model in the Example shows 2 4 5 10. ··5 Answers will vary. Possible answer: By dividing each rectangle into equal 1 sections that are each of the whole, it shows there are 10 fifths in 2 wholes. ··5

2 Explain how the model in the Example shows 2 3 5 5 10. Answers will vary. Possible answer: It shows 2 groups of 5 fifths. There are 10 fifths in all.

3 Suppose Mr. Reed’s class has 3 full rectangular bulletin boards showing team decorations. How many teams would there be Vocabulary in all? Show your solution by writing both a division unit fraction a fraction equation and a multiplication equation. with a numerator of 1. 1 15 teams; 3 4 5 15; 3 3 5 5 15 1 1 1 5 Examples: , , ·· ··2 ··3 ··4 465 ©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 465

Fluency & Skills Practice Teacher Toolbox

Assign Understanding of Division Fluency and Skills Practice Understanding of Division Name: with Unit Fractions with Unit Fractions

1 1 __ What is 3 divided into 2 equal parts? Draw on the area model to solve the problem. In this activity students use area models and number lines to divide

with unit fractions and whole

1 __ 4 2 5 numbers. These representations 3 1 2 How many groups of __ are in 5? Draw on the area model to solve the problem. help students visualize what 4

happens when dividing unit 1 5 4 __ 5 fractions and whole numbers. 4 1 3 What is __ divided into 2 equal parts? Draw on the number line to solve the problem. Students may encounter situations 4 0 1 2 3 1 4 4 4

1 __ 4 5 in everyday life requiring this skill. 4 2

1 4 How many groups of __ are in 3? Draw on the number line to solve the problem. For example, they may need to 5

1 determine how many people 2 4 __ 5 3 5 pizzas can feed if each person 5 Which method do you like best when dividing with unit fractions? 1 gets of each pizza. ··6

©Curriculum Associates, LLC Copying is permitted for classroom use.

465 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. LESSON 23 SESSION 2

1 4 Mr. Reed put 3 students on each team. The teams divides its -board space equally 4 a. See the completed model on the student ··5 1 page. of the rectangle is shaded vertically. so each student decorates the same amount of a board. Complete the steps below 5 ·· to see what fraction of a rectangular bulletin board each student decorates. b. See the completed model on the student 1 a. Use the square at the right. Shade of the square page. Students should draw 2 horizontal lines ··5 1 to show of the board for one team. Possible model shown. to divide the model into 3 equal parts. ··5 b. Divide the square into 3 equal parts to represent c. See the completed model on the student 3 students on each team. 1 1 page. of the model is shaded horizontally. c. Shade of the square so that the overlapping shading represents 1 student. ··3 1 ··3 1 d. What fraction of the board does each student decorate? ···15 d. ; The section of the model where the ··15 1 1 1 e. 4 3 5 shading overlaps represents of the board. ··5 ···15 ··15 1 e. 5 Look at the model in problem 4. Write the multiplication equation you can also ··15 1 1 use to fi nd of . ··3 ··5 Medium 1 1 1 3 5 ··3 ··5 ···15 1 1 1 5 3 5 1 6 a. Write a word problem that can be represented by the quotient 3 4 . ··3 ··5 ··15 ··3 Basic Possible answer: An artist has 3 gallons of art paint. She wants to use 1 -gallon containers to store the paint. How many containers does the 6 a. See possible word problem on the student ··3 page. artist need? 1 b. Use the number line below to solve your problem. Then write a multiplication b. 3 4 5 9; 3 3 3 5 9 ··3 equation that also solves the problem. Medium Possible student model shown.

1 3 4 5 9 ··3 3 3 3 5 9

466 466 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. English Language Learners: Prepare for Session 3 ELL Differentiated Instruction Use with Apply It.

Levels 1–3 Levels 2–4 Levels 3–5 Listening/Writing Have students chorally Speaking/Writing Have students chorally Speaking/Writing Have students read read Apply It problem 5. Help students read Apply It problem 5. In partners, ask Apply It problem 5 with a partner. Have brainstorm action words they can use to talk students to say words they can use to talk partners discuss words they could use to talk about division; for example, divide, separate, about division. about division. and share. Write the words on the board. Write the words on the board and have Have students write their problems and then Have students work in pairs. Ask partners to partners discuss how they can use some of swap with another set of partners. Have use some of the words on the board to write the words to write a division problem. students solve each other’s problems. a word problem. Encourage them to use their Monitor as students write their problems and Encourage students to explain their word own names in the problems. Provide provide support as needed. choices. sentence frames: • (Student’s name) has . • He/She divides the into equal parts.

©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 466 LESSON 23 SESSION 3 Refine LESSON 23 SESSION 3

Purpose In this session students Refine Ideas About Division with Unit Fractions demonstrate their understanding of dividing with unit fractions as they talk through three problems. Then they choose a problem to model Ap It Complete these problems on your own. both visually and symbolically, using both division and multiplication. 1 COMPARE 1 1 1 Draw a model to represent 4 4 and a model to represent 3 . ··4 ··4 ··4 Start Explain the relationship between the two expressions. Possible student models: Connect to Prior Knowledge Why Reinforce student understanding of dividing a unit fraction into equal parts. 4 3 1 1 How Have students divide by 6 using any model Possible explanation: Both have a value of . ··2 ···16 they want. Discuss how their models show that 1 1 2 ANALYZE when is divided into 6 equal parts, each part is ··2 ··12 Mary has 12 boards. She cuts each board into pieces that are of the whole. 1 1 each of a board long. Helena tells Mary that 12 4 is 4. Draw a ··3 ··3 Start Solution model and use words to explain why Helena’s statement is not 1 reasonable. Use any model to find the ··12 quotient. Possible student model: 1 4 6 5 ? ··2 2 3 2

1 Possible explanation: 12 boards divided into s is 36 pieces, not 4 pieces. ··3 3 JUSTIFY 1 1 Show that 4 3 5 by using a model. Explain why the quotient ··2 ··6 1 Grade 5 Lesson 23 Session 3 | R e ﬁ n e Ideas About Division with Unit Fractions ©Curriculum Associates, LLC Copying is permitted. is less than the number you started with, . Possible student model: ··2 PAIR/SHARE APPLY IT Discuss your solutions to Have students work independently or with a 2 these three problems with partner. Possible explanation: When you divide a number into parts, a partner. each part will be smaller than the number you started with. 467 1 COMPARE ©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 467 Look for understanding that both expressions have 1 2 the same value. Both models should show divided ANALYZE ··4 into 4 equal parts. Look for understanding that dividing a whole number by a fraction less than 1 results in a number greater than the whole number. Have students draw their models on paper or small whiteboards and hold them up. Choose several pairs Prompt discussion with questions such as: to explain their models. Use the following to start a • How does your model show 12 boards? 1 discussion: • How does your model show -size pieces? ··3 • How did you know how many equal parts to show in • How does your model show Helen’s mistake? each model? Common Misconception If students fail to recognize why Helen’s statement is not • How does the model show division (or multiplication)? reasonable, then write 12 4 ? 5 4 on the board to help students recognize that 4 is 1 the quotient of 12 4 3, not of 12 4 . • How are the models the same? How are the models ··3 different? 3 JUSTIFY Look for understanding that when you divide a fraction less than 1 by a whole number, the result is less than the number you started with. Have students explain how they used their model to help write their explanation.

467 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. LESSON 23 REFINE SESSION 3

4 Before students begin, read through Use what you have learned to complete problem 4. problem 4 as a class. Make sure students understand 4 Choose one of the following problems to solve. their task is to choose one of the problems and then Circle the problem you choose.

model it with a drawing and two equations. Drew wants to run at least 6 miles this month. He plans to run 1 mile each day. How many days will it take Drew to run 6 miles? As students work on their own, walk around to ··4

assess their progress and understanding, to answer 1 Maya made quart of strawberry jam. She plans to share it their questions, and to give additional support, ··2 equally among 4 friends. How much jam will each friend get? if needed. Have students share their model with a partner Part A Draw a model to represent the problem. and then compare their models. Models will vary. A model for the first problem should show 1 6 4 5 24, and a model for the second problem should ··4 1 1 show 4 4 5 . Scoring Rubric ··2 ··8 Part A Part B Write a division equation and a multiplication equation Points Expectations that represent the problem. 1 Possible equations for first problem: 6 4 5 24; 6 3 4 5 24 The model correctly shows the dividend ··4 1 1 1 1 1 being divided into sections specified by Possible equations for second problem: 4 4 5 ; 3 5 ··2 ··8 ··4 ··2 ··8 2 the divisor (each whole into fourths, or the half into 4 equal parts). The quotient 5 MATH JOURNAL is correct and is clearly indicated. 1 Find 2 4 . Write a word problem and use a visual model to show ··3 the problem. The model correctly shows either the dividend or the divisor but not both. The 6; Possible word problem: Randall has 2 bananas. He cut each 1 1 quotient is correct but may not be clearly banana into smaller pieces that are each of a banana. ··3 indicated. How many smaller pieces of banana did Randall make? Of the dividend, divisor, and quotient, at Possible visual model: 0 least two quantities are represented incorrectly or not at all.

Part B 468 468 Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. Points Expectations

Both equations are correct. The Close: Exit Ticket multiplication equation can be 2 considered correct if the factors and the 5 MATH JOURNAL product are correct, regardless of the Students use their understanding of dividing a whole number by a unit fraction to order in which the factors are written. 1 write a word problem and draw a visual model to show 2 4 . Students’ models ··3 One equation is correct but the other is should show two wholes divided into thirds. not. The multiplication equation can be 1 considered correct if the factors and the Error Alert If students confuse the dividend and divisor when writing the word product are correct, regardless of the problem and/or drawing the model, then help students recognize that 2 is the number 1 order in which the factors are written. of wholes and that indicates to divide each whole into thirds or 3 equal parts. ··3 Neither equation is correct. (A multiplication equation can be 0 considered correct if the factors and the product are correct, regardless of the order in which the factors are written.)

©Curriculum Associates, LLC Copying is not permitted. Lesson 23 Understand Division with Unit Fractions 468 LESSON 23 Name ______Date ______Teacher Toolbox Lesson Quiz Lesson 23 Quiz

Tested Skills Solve the problems. 1 1 Which models represent 4 6? Choose all the correct answers. (2 points) Assesses 5.NF.B.7a, 5.NF.B.7b ··4 Problems on this assessment form require students to be able to divide a whole number by a unit Ꭾ fraction or a unit fraction by a whole number, to use Ꭽ visual fraction models, and to write related multiplication equations and division equations for a given situation.

Alternately, teachers may assign the Digital ൳ Comprehension Check online to assess student Ꭿ understanding of this material. Error Alert Students may:

• confuse the divisor and the dividend. • multiply instead of dividing. ൴ • count tick marks, instead of parts, on a number line. 1 • replace a unit fraction with its related whole 2 This model represents a fraction divided by a whole number. ··a number a or vice versa. Solutions 1 A, C, E; Students could solve the problem by Complete the division equation that the model represents. 1 identifying the models that show divided into Write your answers in the blanks. (1 point) ··4 1 1 6 parts, with one of these 6 parts shaded. ··3 4 3 5 ··9 B is not correct because the model shows a 4 1 result of instead of . ··24 ··24 D is not correct because the model shows a 6 1 result of instead of . ··24 ··24 2 points ©Curriculum Associates, LLC Grade 5 Lesson 23 Understand Division with Unit Fractions 1 Copying permitted for classroom use. 5.NF.B.7a, DOK 2 Short Response Scoring Rubric Equation Fill-in-the-Blank Scoring Rubric Points Expectations 2 points 1 point 0 points Response contains the following: • Correct computations, solutions, and/or calculations. • The dividend and One of the following More than one of (1 point) 2 divisor are correct. is incorrect: the following are • Well-organized, clear, and concise work that • The quotient is • The dividend or incorrect: demonstrates thorough understanding of math correct. divisor. • The dividend or concepts and/or procedures. (1 point) • The quotient. divisor. • The quotient. Response contains the following: • Mostly correct solution(s). 1 • Shows partial or good understanding of math concepts and/or procedures. Multiple Select Scoring Rubric Response contains the following: 2 points 1 point 0 points • Incorrect solution(s). 0 • No attempt at finding a solution. All answers 1 incorrect answer 2 or more • No effort to demonstrate an understanding of are correct incorrect answers mathematical concepts and/or procedures.

468a Lesson 23 Understand Division with Unit Fractions ©Curriculum Associates, LLC Copying is not permitted. Name ______Date ______

Lesson 23 Quiz continued

1 3 Sandro has 6 quarts of strawberries. He divides this into -quart servings. 1 1 ··3 2 4 3 5 ··3 ··9 How many servings does he make? (2 points) 2 points 5.NF.B.7a, DOK 2 0 654321

3 18; Students may divide the number line into Show your work. thirds and count the total number of thirds, or Possible student work:

they may write and solve an equation. See 6 4 1 5 18 student page for possible student work. ··3 2 points 5.NF.B.7b, DOK 2 18 servings

4 Part A 1 4 Wendy wants to run 3 miles. Each lap around the track at her school is mile. B; Students could solve the problem by using ··8 Wendy wants to fi gure out how many laps she needs to run. their knowledge of number lines to select the number line that represents 3 miles with each Part A mile divided into 8 equal parts. Which model could Wendy use to fi nd how many laps she should run? (1 point) A is not correct because the number line Ꭽ represents 8 miles with each mile divided into 0 54321 76 8 3 equal parts. Ꭾ C is not correct because the number line 0 1 2 3 represents 3 miles with each mile divided into Ꭿ 7 equal parts instead of 8. 0 1 2 3

D is not correct because the number line ൳ represents one mile divided into eighths, with 0 1 2 3 4 5 6 7 8 each eighth divided into thirds. 8 8 8 8 8 8 8 8 Part B 1 point 5.NF.B.7b, DOK 2 How many laps will Wendy need to run? Write a division equation and a multiplication equation to represent the problem. (2 points)

Possible student work: 3 4 1 5 24 and 3 3 8 5 24 Part B ··8 1 3 4 5 24 ··8 24 laps 3 3 8 5 24 24 laps ©Curriculum Associates, LLC Grade 5 Lesson 23 Understand Division with Unit Fractions 2 Copying permitted for classroom use. 2 points 5.NF.B.7b, DOK 2

Differentiated Instruction Teacher Toolbox RETEACH REINFORCE EXTEND

Tools for Instruction Math Center Activities Enrichment Activities Students who require additional support Students who require additional Students who have achieved proficiency for prerequisite or on-level skills practice to reinforce concepts and skills with concepts and skills and are ready for Will benefit from activities that provide and deepen understanding additional challenges targeted skills instruction Will benefit from small group Will benefit from group collaborative collaborative games and activities games and activities that extend (available in three versions—on-level, understanding below-level, and above-level)