Fractions As Division

Lesson LESSON 18 Overview Fractions as Division

Lesson Objectives Prerequisite Skills Lesson Vocabulary

Content Objectives • Understand division as equal groups or There is no new vocabulary. Review the sharing. following key terms. • Use visual fraction models to represent a • Understand that multiplication and • denominator the number below the fraction as division. division are inverse operations. line in a fraction that tells the number of • Solve word problems involving division • Divide whole numbers. equal parts in the whole. of whole numbers in which the quotient • fraction a number that names equal is a fraction or mixed number. • Multiply a fraction by a whole number. parts of a whole. A fraction names a point • Understand a fraction as a way to on the number line and can also represent division where the numerator Standards for Mathematical represent the division of two numbers. is divided by the denominator. Practice (SMP) • numerator the number above the line Language Objectives in a fraction that tells the number of SMPs 1, 2, 3, 4, 5, and 6 are integrated in equal parts that are being described. • Make a visual fraction model to represent every lesson through the Try-Discuss- • quotient the result of division. a fraction as a division of two whole Connect routine.* numbers and explain the relationship of • remainder the amount left over when In addition, this lesson particularly the model to the fraction. one number does not divide another emphasizes the following SMPs: • Draw a visual model and write an number a whole number of times. 2 Reason abstractly and quantitatively. equation to represent word problems involving a quotient of whole numbers 5 Use appropriate tools strategically. where the quotient is a fraction. 7 Look for and make use of structure.

*See page 305m to see how every lesson includes these SMPs.

Learning Progression

In previous grades students have In this lesson students extend their In later lessons students will learn to understood the meaning of division as understanding of both division and divide whole numbers by unit fractions equal groups or equal shares. They have and divide unit fractions by whole fractions to see that when a whole number interpreted fractions as equal parts of a numbers. In Grade 6 students will apply is divided by another whole number the whole or equal parts of a group. and extend their understanding of division quotient may be a fraction or a mixed with fractions to divide a whole number by 2 23 number, e.g., 2 4 5 5 and 23 4 7 5 . ··5 ··7 a fraction and to divide a fraction by a From exploring problems like this they fraction. recognize that a fraction can be interpreted as representing a division expression, one where the numerator is divided by the denominator. They further connect this idea to the inverse relationship of multiplication and division, 2 2 e.g., 2 4 5 5 and 5 3 5 2. ··5 ··5

373a Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. Lesson Pacing Guide Teacher Toolbox Whole Class Instruction Small Group Differentiation SESSION 1 Fractions as Division Additional Practice PREPARE Explore • Start 5 min Lesson pages 377–378 • Try It 10 min Ready Prerequisite Lesson 45–60 min • Discuss It 10 min Grade 4 • Connect It 15 min • Lesson 23 Understand Fraction Multiplication • Close: Exit Ticket 5 min RETEACH SESSION 2 Fractions as Division Additional Practice Develop • Start 5 min Lesson pages 383–384 Tools for Instruction • Try It 10 min Grade 4 45–60 min • Discuss It 10 min Fluency • Lesson 23 Understand Fraction Multiplication • Picture It & Model It 5 min Fractions as Division • Connect It 10 min Grade 5 • Close: Exit Ticket 5 min • Lesson 18 Interpreting Fractions as Division

SESSION 3 Fractions as Division Lesson Quiz REINFORCE • Start 5 min or Digital Math Center Activities Refine • Example & Problems 1–3 15 min Comprehension Check 45–60 min • Practice & Small Group Grade 5 Differentiation 20 min • Lesson 18 Fractions as Quotients • Close: Exit Ticket 5 min • Lesson 18 Relate Situations to Fractional Quotients

Lesson Materials EXTEND Lesson Activity Sheet: Number Lines Enrichment Activity (Required) Grade 5 Activities Per student: base-ten blocks (1 tens rods, 2 ones units), scissors • Lesson 18 Pizza Party Activity Sheets: Fraction Bars, Digit Cards, 1-Inch Grid Paper Math Toolkit fraction circles, fraction tiles, fraction bars, tenths grids, number lines, index cards Digital Math Fraction Models, Number Line Independent Learning Tools PERSONALIZE

i-Ready Lesson* Grade 5 • Fractions as Division

*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 18 Fractions as Division 373b LESSON 18 Connect to Family, Community, and Language Development

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

ACTIVITY Fra��Ns a� d��On Dear Family, 18 Do this activity with your child to explore fractions as division. This week your child is learning how fractions and Work with your child to fi nd opportunities to practice modeling a division division are related. situation as a fraction. • Together with your child, think of things that can be shared equally among He or she might see a problem like the one below. family members, such as boxes of crackers or bags of grapes. Three family members equally share 4 granola bars. How much does each family member receive? • Choose one idea. Work together with your child to show how to equally divide a number of the items among the people in your family. This word problem can be represented as a division problem. The family equally shares 4 granola bars among 3 people, so the division problem to solve is 4 4 3. Example: 4 family members equally share 7 bags of trail mix.

A model is a useful way to show the problem. • Have your child write the idea as a division problem. The model below shows 4 wholes. Each whole is divided into 3 parts. 7 Example: 7 4 4 5 ··4 • Have your child explain how much of the item each family member will get.

7 3 Example: Each person will get , or 1 , bags of trail mix. 1 ··4 ··4 Each family member receives of each of 4 whole bars. So, the answer to the 4 ··3 4 division problem 4 4 3 is . You can say that the fraction represents the ··3 ··3 division problem 4 4 3.

This shows how fractions and division are related. You can think of fractions as the division of two numbers.

4 Another way to write the fraction is to show it as a mixed number. So, each 4 1 ··3 family member receives , or 1 , granola bars. ··3 ··3 Invite your child to share what he or she knows about how fractions and division are related by doing the following activity together.

373 374 ©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 373 374 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted.

Goal Math Talk at Home The goal of the Family Letter is to help students and their families Encourage students to look for objects at home that can be divided understand how fractions and division are related. into equal parts to show relationships between fractions and division. Activity Conversation Starters Below are additional conversation starters Understanding how fractions and division are related will help students can write in their Family Letter or math journal to engage students build skills that are necessary to perform more complex family members: mathematical operations involving fractions. Look at the Fractions • What are some objects or items that can be shared equally among as Division activity and adjust it if necessary to connect with your our family members? students. • How can we represent that as a division? How can we represent it as a fraction?

373–374 Lesson 18 Fractions as Division ©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.

Sessions 1 and 3 Use anytime during these sessions. Session 2 Use with Try It. • Many problems in this lesson discuss dividing labor or items into • Decorating school hallways is a way to show school pride and equal parts. Ask students to make a personal connection to how encourage collaboration between different classrooms. The they divide chores or activities around their home. Ask if the chores collaboration may include developing a theme and organizing how are divided by the days of the week when they are done, the students will work together to complete the work. In the same way, number of people doing the chores, etc. cities and communities make murals and landscape public spaces • Sharing objects or items equally is an example of division that to show pride. Ask students to share any community work they students encounter in their daily life. An equal number of students might have been a part of and how the work was divided. assigned to each classroom is another example of dividing with whole numbers. • Division situations do not always involve dividing a given quantity by a number less than the given quantity. Dividing one whole (e.g., a snack) between two or more people can be represented by a fraction. As students think of how to represent this division situation as a fraction, encourage them to make personal connections to how they use fractions and division in their daily lives.

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 Try It.

Levels 1–3 Levels 2–4 Levels 3–5 Listening/Writing Read Try It aloud. Speaking/Writing Read Try It with Reading/Writing Have students read Choose five volunteers to act out the students. Organize students into pairs to Try It and use a known strategy to solve the problem. Use 4 index cards to represent the restate the problem in their own words. problem independently. Ask students to four ounces of paint. Demonstrate cutting Provide the following sentence frame to think about the steps they took to model and each index card into 5 equal shares. As you encourage students to think about using solve the problem. Have them write cut each index card into 5 pieces, share one fractions to solve the problem: directions for a partner to model and solve piece with each volunteer. As you hand each How many fifths are in 4 wholes; 4 ounces of the problem in the same way. Organize student a share say: One fifth of an ounce for paint ? students into pairs. If possible, partner you. When you have divided each index card, students who used different tools or methods Provide fraction tiles or fraction circles. Allow ask the volunteers: How many fifths do you to solve the problem. Ask students to read partners to choose their preferred tool to have? [four fifths] Organize students into and follow their partner’s directions. Ask: model and solve the problem. Have them pairs to draw fraction bars or a grid model to Were you able to solve the problem in a new work together to write a summary of their represent the problem. Encourage them to way? Why or why not? Give students the steps using the sequencing words first, next, visualize the role-playing activity as they opportunity to revise their directions as and then. model the problem. needed.

©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 374a LESSON 18 SESSION 1 Explore LESSON 18 SESSION 1

Purpose In this session, students draw on Explore Fractions as Division their understanding of division as equal sharing and fractions as equal parts of a whole. They Learning Target You know that division is used for equal sharing and that fractions share models to explore how to represent the • Interpret a fraction as division of the represent a number of equal parts of a whole. In this lesson, you will numerator by the denominator a learn how division and fractions are related. Use what you know to 5 a 4 b . Solve word problems division of a whole number by a greater whole 1 ··b 2 number. They will look ahead to think about try to solve the problem below. involving division of whole numbers leading to answers in the form of how division of whole numbers can lead to Mrs. Tatum needs to share 4 ﬂ uid ounces of red paint fractions or mixed numbers, e.g., by equally among 5 art students. How many ounces of red using visual fraction models or quotients that are fractions or mixed numbers. equations to represent the problem. paint will each student get? SMP 1, 2, 3, 4, 5, 6 Start TRY IT Math Toolkit Connect to Prior Knowledge Possible student work: • fraction circles or tiles • fraction bars Why Supports students in relating repeated Sample A • fraction models • tenths grids addition of a fraction to multiplying the fraction by a 1 ounce 1 ounce 1 ounce 1 ounce • number lines whole number. A B C D E A B C D E A B C D E A B C D E • index cards 1 1 1 1 1 4 How Have students complete the addition Each student gets of each ounce. Since 1 1 1 5 , each ··5 ··5 ··5 ··5 ··5 ··5 equation and the multiplication equation to 4 student gets of an ounce. represent the number of sixths shown in the fraction ··5 model. Sample B Start Solutions 4 ounces 5 40 tenths of an ounce Complete the equations to tell 1 1 1 3 1 1 5 how many sixths are shaded. ··6 ··6 ··6 ··6 1 3 3 3 5 ··6 ··6 1 1 1 5 ··6 1 40 tenths 4 5 5 8 tenths 3 5 ··6 Each student gets 0.8 ounce of red paint. DISCUSS IT Ask your partner: Why did you choose that strategy? Tell your partner: At ﬁ rst, Grade 5 Lesson 18 Session 1 | Explore Fractions as Division ©Curriculum Associates, LLC Copying is permitted. I thought . . . TRY IT 375 Make Sense of the Problem ©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 375 To support students in making sense of the Common Misconception Look for students who are not comfortable with the idea problem, have them identify that the number of of dividing a whole number by a greater whole number. As students present students sharing the paint, 5, is greater than the solutions, have them specify why they represented the 4 ounces of paint the way number of ounces of paint, 4. they did.

DISCUSS IT Select and Sequence Student Solutions Support Partner Discussion One possible order for whole class discussion: Encourage students to use the Discuss It question • concrete models showing 5 groups of 4 fifths and sentence starter on the Student Worktext page • drawings to represent the problem as part of their discussion. • number lines marked in fifths Look for, and prompt as necessary for, • equations showing 4 as an equivalent fraction with numerator divisible by 5 understanding of: • 4 ounces as the amount to be shared equally Support Whole Class Discussion • 5 students to get equal shares Prompt students to note the relationship between the numbers in each model and the numbers in the problem. • the amount of each share is the unknown Ask How do [student name]’s and [student name]’s models show the ounces of paint divided? Each student’s share of paint? Listen for Each ounce can be divided into 5 equal parts. Each student gets the same 1 4 number of equal parts for their fair share. They each get four s, or , or 8 tenths. ··5 ··5

375 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. LESSON 18 EXPLORE SESSION 1

CONNECT IT CONNECT IT 1 LOOK BACK 1 LOOK BACK Look for understanding that each whole ounce of Explain how to fi nd the amount of paint each student gets. 1 Possible answer: I drew a picture to show that each student gets of each paint can be divided into equal parts and each ··5 1 1 1 1 4 student’s equal share is the sum of the equal parts ounce. I added 1 1 1 to get ounce for each student. ··5 ··5 ··5 ··5 ··5 that is his share in each ounce. 2 LOOK AHEAD Suppose Mrs. Tatum wants to share 8 fl uid ounces of paint equally among the 5 students. You can think about this quotient in two ways. 1 1 Hands-On Activity a. Think of each student getting of each ounce. Shade of each ··5 ··5 Use fraction bars to divide. whole in the model below to show one student’s share. If . . . students are unsure about dividing 4 wholes into equal parts that are fractions, Then . . . use this activity to model Try It. 1 ounce 1 ounce 1 ounce 1 ounce 1 ounce 1 ounce 1 ounce 1 ounce 1 8

Materials For each student: scissors, Activity 8 ounces 4 5 5 8 3 ··5 5 ··5 ounces Sheet Fraction Bars (4 bars for fifths) b. Think of 8 ounces as 5 ounces + 3 ounces. Explain how the shaded part of the model below shows one student’s share. • Explain to students that they will model the 5 ounces 3 ounces Try It problem. • Explain that each fraction bar represents 1 Possible explanation: Each student gets 1 full ounce of paint and of each 1 ounce of paint. Have students write the ··5 of the remaining 3 ounces. letters A through E in the sections of each 8 3 5 1 fraction bar, each letter representing one of c. Write the quotient 8 4 5 as a fraction and as a mixed number. ··5 ··5 the 5 art students sharing the ounces of paint. REFLECT Prompt students to identify each art student’s 3 2 1 How would you write the fraction as a division expression? Write a word problem ··5 share of each ounce as of an ounce. 2 ··5 that can be represented by your expression and by the fraction . • Have students cut the fraction bars into fifths ··5 2 4 5; Possible problem: How much paint will each student get if 2 ounces of and collect each art student’s shares in a paint are shared equally among 5 students? separate pile. Have them count fifths to find 376 the size of one share. Discuss the results. 376 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. • Repeat activity, representing other situations such as 4 students sharing 3 bags of popcorn. Close: Exit Ticket 3 REFLECT 2 LOOK AHEAD Look for understanding that a fraction can represent division of the numerator by the Point out that sometimes when you divide whole denominator. The problem situation students describe should have 2 as the quantity numbers, the quotient may be in the form of a being divided and 5 as the number of equal shares it is divided into. fraction greater than 1 or a mixed number. Common Misconception If students confuse which number in a fraction represents Students should be able to show or interpret the the dividend and which represents the divisor in the equivalent division expression, equal parts comprising one share for both ways of then write the three division expressions from the Student Worktext page for this dividing the 8 ounces of paint into 5 equal shares. session: 4 4 5, 8 4 5, 2 4 5. Have students pair each expression with the fraction they Have students think about using multiplication to found that represents it. For each expression/fraction pair have students circle the explain why the quotient of two whole numbers can numerator in the fraction and identify where that number is in the expression. be represented by a fraction. Real-World Connection 8 Ask Your work in problem 2 shows that 8 4 5 5 . ··5 Encourage students to think about everyday places or situations in which What is the related multiplication equation that dividing a quantity into equal shares may result in an amount that includes a shows this same relationship? fractional part. Have volunteers share ideas. Examples: cooking 5 eggs for breakfast to 8 8 (5 3 8) 40 feed 3 people, making 10 place mats out of 6 feet of fabric, needing 3 cups of yogurt Listen for 5 3 5 8; 5 3 5 5 5 8 ··5 ··5 ······5 ··5 to make 24 servings of fruit smoothie.

©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 376 LESSON 18 SESSION 1 Additional Practice Name: LESSON 18 SESSION 1

Solutions Prepare for Fractions as Division

Support Vocabulary Development 1 Think about what you know about division. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can. Possible answers: 1 Have students say the terms: fraction, division expression, quotient and remainder. Guide a Word In My Own Words Example conversation with students in which they discuss a number that names equal parts of 4 fraction what they know about division. Then ask them to a whole ··5 discuss the definition for each term and write it in the In My Own Words column. division an expression that has the 4 4 5 Call on volunteers to share what they wrote for each expression operation of division term. Correct any misconceptions and ask students to revise their answers if necessary. As students share their graphic organizer, encourage quotient the result of division 16 4 8 5 2 them to use this sentence frame:

• My explanation for is . the amount left over when one remainder number does not divide another 20 4 3 5 6 R 2 2 Read the problem. Have students discuss with a number a whole number of times partner how you use multiplication to check an answer to a division problem. Then ask students to 3 2 Write the fraction as a division expression. 3 4 4 confirm their work with another set of partners. ··4 How could you use multiplication to check your answer? 3 3 Possible answer: If 3 4 4 5 , then 4 3 should be 3. Supplemental Math Vocabulary ··4 ··4 3 4 3 3 4 3 5 • dividend ··4 ·····4 12 5 • divisor ···4 5 3 • quotient

377 ©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 377

377 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. LESSON 18 SESSION 1

3 Assign problem 3 to provide another look at 3 Solve the problem. Show your work. fractions as division. Mrs. Tatum needs to share 3 grams of glitter equally among 8 art students. How many This problem is very similar to the problem about grams of glitter will each student get? 1 1 1 Mrs. Tatum sharing red paint among her art gram gram gram students. In both problems, students are given a word problem that requires them to divide one whole number by a greater whole number. The question asks how many grams of glitter each Possible student work using pictures: student will get. 1 gram 1 gram 1 gram A B C D E F G H A B C D E F G H A B C D E F G H Students may want to use fraction bars or 1 grid paper. Each student gets of each gram. ··8 1 1 1 3 Suggest that students read the problem three times, 1 1 5 ··8 ··8 ··8 ··8 asking themselves one of the following questions 3 each time: Each student gets gram of glitter. Solution ··8 • What is this problem about? 4 Check your answer. Show your work. • What is the question I am trying to answer? Possible student work: • What information is important? 3 3 3 3 3 3 3 3 3 8 3 5 1 1 1 1 1 1 1 ··8 ··8 ··8 ··8 ··8 ··8 ··8 ··8 ··8 24 Solution: 5 ··8 1 Each student gets of each gram. Since 5 3 ··8 1 1 1 3 3 3 1 1 5 , each student gets gram of glitter. So, each student gets gram. ··8 ··8 ··8 ··8 ··8 ··8 Medium

4 Have students solve the problem a different way, or use multiplication, to check their answer.

378 378 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. English Language Learners: Prepare for Session 2 ELL Differentiated Instruction Use with Apply It.

Levels 1–3 Levels 2–4 Levels 3–5 Listening/Speaking Read Apply It Listening/Speaking Read Apply It Reading/Writing Have students read problem 9 to students. Ask students to problem 9 with students. Ask them to state Apply It problem 9 individually. Ask students underline the phrase into each. Have students the problem in their own words. In partners, to write the steps that are needed to solve discuss why they think that a division have them decide on the first step they will the problem, and to also draw a visual model. expression is needed to solve the problem. use to solve the problem. Have students trade their steps with another Provide a sentence frame to guide students’ When complete, have students work with student and compare the steps and visual discussions: I need a division expression another set of partners to compare their first models they plan to use to solve the because objects are separated, shared, or steps. Then ask them to solve the problem problem. Ask them to solve the problem distributed into equal groups. individually. individually. Confirm that students understand the terms Ask students to explain their thinking as they After students solve the problem and write remainder form and mixed number. Organize explore writing the answer as a mixed their answers in remainder form and as a students into pairs and give them base-ten number. Ask guiding questions, such as: How mixed number, have them pair up again to blocks to model the problem. Encourage do you know what the denominator will be? debrief. Ask them to take turns explaining students to use gestures, pictures, numbers, How many wholes are there? How many tenths how they found the mixed number. and words to communicate ideas. are remaining? How is the mixed number related to the remainder form of the answer?

©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 378 LESSON 18 SESSION 2 Develop LESSON 18 SESSION 2

Purpose In this session students solve a Develop Fractions as Division problem that requires finding the quotient 5 4 3 and representing it as both a fraction and a Read and try to solve the problem below. mixed number. Students model the division either on paper or with manipulatives. The Jared, Monica, and Heather have 5 hallways to decorate for the purpose of this problem is to develop strategies student council. If they share the work equally, how much will each student decorate? for finding quotients that are fractions or mixed numbers. TRY IT Math Toolkit Possible student work: • fraction circles or tiles Start • fraction bars Sample A • fraction models • tenths grids Connect to Prior Knowledge • number lines Materials For each student: Activity Sheet • index cards Number Lines J M H J M H J M H J M H J M H Why Support students’ facility with writing 5 Each student decorates of a hallway. fractions greater than 1 as mixed numbers. ··3

How Have students label a number line to show Sample B 7 14 5 fourths, locate the fractions and on the number 5 4 3 5 ··4 ··4 ··3 2 line, and then write the equivalent mixed numbers. 5 1 ··3 Start 2 Each person decorates 1 hallways. ··3 Solutions Locate each fraction on a number line labeled in fourths. Check students’ Then write the fraction as a number lines. 7 3 mixed number. 5 1 4 4 DISCUSS IT ·· ·· Ask your partner: Do you 14 2 1 7 14 5 3 , or 3 agree with me? Why or 5 5 4 4 2 ··4 ··4 ·· ·· ·· why not? Tell your partner: I disagree with this part because . . . DevelopGrade 5 Lesson 18 Session 2 | Develop Fractions Language as Division ©Curriculum Associates, LLC Copying is permitted. Why Review the terms dividend, divisor, and 379 ©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 379 quotient to facilitate student discussions. How Write the terms on the board and explain that DISCUSS IT they are used to identify the parts of a division Support Partner Discussion equation. Write a division equation on the board Encourage students to name the model they used as they discuss their solutions. and have students work to label each of the numbers with the term dividend, divisor, or quotient. Support as needed with questions such as: Discuss and correct any misconceptions. Have • How would you describe your model? students revise their work as needed, and copy the • What was it about this problem that made you think of using that model? labeled equation onto a class chart for students to Common Misconception Look for students who accurately model the problem but use as a reference. have difficulty identifying what constitutes one equal share from all the equal parts represented. As students present solutions, ask them to identify Jared’s share in TRY IT the model. Make Sense of the Problem Select and Sequence Student Solutions To support students in making sense of the One possible order for whole class discussion: problem, help them recognize that 3 students are decorating 5 hallways. • concrete models showing 3 groups of 5 thirds Ask The problem asks you to find how much each • drawings to represent the problem student will decorate. What does how much mean in • number lines marked in thirds 5 this situation? What will be the unit for your answer? • equations showing 5 4 3 can be represented by ··3

379 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. LESSON 18 DEVELOP

Support Whole Class Discussion Explore diff erent ways to understand fractions as quotients. Compare and connect the different representations Jared, Monica, and Heather have 5 hallways to decorate for the and have students identify how they are related. student council. If they share the work equally, how much will each student decorate? Ask Where does your model show the number of hallways? The part of the hallways Jared decorates? Pi��e It Monica decorates? Heather decorates? You can use a fraction model to picture how the students divide up the work. Listen for Students should recognize that There are 5 hallways for 3 students to decorate, which is 5 4 3. 1 accurate representations show 5 wholes divided If they share the work equally, each student can decorate of each hallway. ··3 into thirds, with each student’s share identified. Responses may include each student’s share is 5 equivalent to . ··3

m��l It You can use a number line to model each student’s share of the work. PICTURE IT & MODEL IT If no student presented these models, connect The number line is numbered from 0 to 5 because there are 5 hallways. It is divided into thirds because each student can decorate one third of each hallway. them to the student models by pointing out the ways they each represent: J M H J M H J M H J M H J M H

• the 5 wholes 0 1 2 3 4 5 • the number of equal parts The thirds can be rearranged to show each student’s share of the work. • the number of parts decorated by each student Jared Monica Heather Ask What is the total number of equal parts shown in the fraction model? On the number line? Is it the 0 1 2 3 4 5 same or different? Why? Listen for 15 is the total number of equal parts shown in both the fraction model and on the number line. It is the same number because each model shows 5 wholes divided into thirds, so there are 5 3 3 thirds, or 15 thirds in all. 380 380 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. For the fraction model, prompt students to identify how color is used to help represent the problem. Deepen Understanding • Why are three different colors used to shade the Remainders thirds in each whole? SMP 2 Reason abstractly and quantitatively. For the number line model, prompt students to When discussing the number line model, have students use a mixed number to recognize that the same information is shown on 2 identify what is shown as each student’s share of the work. 1 hallways both number lines. 3 ··3 4 • How many thirds are part of Jared’s share in the top Ask If the 3 students were sharing 5 paintbrushes instead of 5 hallways, would 2 number line? The bottom number line? it make sense for them to each get 1 paintbrushes? How many paintbrushes ··3 • How is the first number line similar to the fraction would each student get? How many paintbrushes would be left over? model? How is it different? 2 Listen for of a paintbrush does not make sense. Each student would ··3 • Think of each whole on the number line as get 1 paintbrush with 2 brushes left over. representing 1 hallway. How does each way of shading the number line show a different way the Prompt students to consider how in the past when they have divided whole students can divide up the work? numbers that did not divide evenly, they wrote the part left over as a remainder. Ask them how 5 4 3 would be written using this method. [1 R 2] Discuss division situations where they would want to show a quotient as a mixed number, and situations where they would want to show it as a whole number and a remainder.

©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 380 LESSON 18 SESSION 2 Develop SESSION 2

Co�� It CONNECT IT Now you will use the problem from the previous page to help you understand • Remind students that one thing that is alike about fractions as quotients. all the representations is they show whole-number 1 How many thirds of a hallway are there to decorate in 5 hallways? 15 thirds division that results in a quotient that is a fraction. 2 How many thirds of a hallway will each student decorate? 5 thirds • Explain that on this page they will look at two 5

Write this as a fraction. ··3 of a hallway different ways to think about the division and two 5 5 4 3 5 different ways to show the quotient. 3 Write a division equation that shows the quotient as a fraction. ··3 5 3 3 5 5 Write a multiplication equation to check this equation. ··3 Monitor and Confirm 4 How many whole hallways can each student decorate? 1 1 – 3 Check for understanding that: How many hallways remain after those are done? 2 2 • there are 15 thirds in all How much of the 2 remaining hallways will each student decorate? ··3 • 15 thirds 4 3 5 5 thirds Write a mixed number to show how many hallways each student will decorate. 5 2 • the quotient can be written as the fraction 1 ··3 ··3 hallways • as with whole numbers, you can write related 5 Calculate using remainder notation: 5 4 3 5 1 R 2 multiplication and division equations Compare this answer to the mixed number. How are they alike? The whole number is the same as the quotient without the remainder. Support Whole Class Discussion The numerator is the same as the remainder. 4 – 5 Have students think about modeling the 6 How does the bar in a fraction represent division? way of dividing up the work described in problem 4. The bar means that the number in the numerator is divided by the number in Guide them to connect writing the quotient in the denominator. remainder form and as a mixed number. 7 REFLECT Ask How would you change the fraction model in Look back at your Try It, strategies by classmates, and Picture It and Model It. Picture It to show this way of dividing up the work? Which models or strategies do you like best for fi nding fraction quotients? Explain. What would a number line model of this way Students may respond that they like using fraction models or number lines to look like? visualize dividing an amount into equal shares, or that they like representing

Listen for In Picture It, each of the first three a problem as a division equation that shows the quotient as a fraction. rectangles would be fully shaded in a single color, 381 pink, blue, and green. On a number line, you could ©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 381 shade from 0 to 1 in pink, from 1 to 2 in blue, and from 2 to 3 in green. For the other two sections, Hands-On Activity 1 1 1 shade of each section pink, blue, and green. ··3 ··3 ··3 Connect fractions to equivalent division expressions. Ask Does the mixed number or the remainder form If . . . students are unsure about how to interpret a fraction as division, better represent the solution? Then . . . use this activity to rewrite fractions as equivalent division expressions. Listen for The mixed number gives an exact Materials For each student: base-ten blocks (1 tens rod, 2 ones units), Activity amount each person decorates. The remainder Sheet Digit Cards (3, 4, 5) form indicates that each decorate 1 full hallway • Distribute materials to students. Have students use the digit cards and base- 5 and then some of the remaining 2 hallways. ten blocks to “build” the fraction used to solve the Try It problem, , using the ··3 rod as the fraction bar and placing a digit card for 5 above the rod and a digit 6 Look for the idea that the bar in a fraction can card for 3 below it. be interpreted as meaning divided by—the • Ask students to alter the fraction they built to show the division expression numerator is divided by the denominator—just as used to represent the problem, 5 4 3, moving the digit cards and using the the division symbol in an expression does. ones units along with the rod to make a division symbol (4). Discuss where 7 REFLECT students placed the numerator and denominator to make the expression. • Repeat activity, using the situation from the Explore Try It: 4 ounces of paint Have all students focus on the strategies used to shared equally by 5 students. This time, have students first show the division solve this problem. If time allows, have students expression that models the problem and then turn it into the fraction quotient. share their preferences with a partner.

381 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. LESSON 18 DEVELOP SESSION 2

Ap�� i� APPLY IT Use what you just learned to solve these problems. For all problems, encourage students to draw some kind of model to support their thinking. Allow some 8 Five friends are equally sharing 3 packs of football cards. How many packs of cards will each friend get? Use a visual model to support your answer. leeway in precision; dividing wholes and number Possible student work: lines into equal parts accurately can be very difficult. 1 pack 1 pack 1 pack 3 8 of a pack; Students may use a model to see A B C D E A B C D E A B C D E ··5 that there are 15 fifths in 3 packs, so each friend gets 3 of a pack 3 of the fifths. See possible model on the Student Solution ··5 Worktext page. Students may also show a number 9 Elena made 10 ounces of apple chips. She puts the same amount of apple chips into each of 4 containers. How many ounces of apple chips are in 1 container? line divided into fifths marked to show 5 shares of Write a division expression to represent the problem and solve. 3 fifths each. Write the solution in remainder form and as a mixed number. Use a visual model to support your answer. 2 Possible student work: 9 10 4 4 5 2 R 2; 10 4 4 5 2 ; Each container has ··4 2 1 2 , or 2 , ounces of apple chips. See possible visual ··4 ··2 model on the Student Worktext page. Students 0 1 2 3 4 5 6 7 8 9 10 2 should see that the mixed number best answers the 10 4 4 5 2 R 2; 10 4 4 5 2 ··4 question because it gives an exact amount. 2 1 2 R 2 or 2 or 2 ounces of apple chips Solution ··4 ··2 Does the remainder form or the mixed number form best answer the question? Close: Exit Ticket Explain. The mixed number tells exactly how many ounces, so it best answers the question of how many ounces of apple chips. 10 C; The expression 12 4 7 shows the numerator, 12 10 Which expression is equivalent to ? 12, divided by the denominator, 7. ··7 Ꭽ 12 2 7 Ꭾ 7 2 12 Error Alert If students choose A, B, or D, then Ꭿ 12 4 7 ൳ 7 4 12 remind students that the bar in a fraction can mean the numerator is divided by the denominator. Have 12 them read the fraction , inserting the words 382 ··7 382 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. divided by for the fraction bar.

©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 382 LESSON 18 SESSION 2 Additional Practice Name: LESSON 18 SESSION 2

Solutions Practice Fractions as Division 1 Study the Example showing whole-number division with a fraction quotient. Number Line A; This number line shows Then solve problems 1–5. 4 wholes divided into sixths so it can be used to solve 4 4 6. Basic Example There are 4 packages of printer paper to be divided equally among 6 classrooms. How much paper will each classroom get? 2 The model would change to show 5 equal parts There are 4 packages for 6 classrooms to share, which is 4 4 6. in each rectangle; the answer would change 1 4 If you divide each package into sixths, each classroom would get one sixth of each to 3 4, or of a package. 1 4 ··5 ··5 package. So, of each package from 4 packages is the same as of a package. Medium ··6 ··6

4 5 4 4 6 5 ··6

3 5

4 Each classroom gets of a package. ··6

1 Circle the number line you would use to solve the problem in the Example.

Number Line A

0 1 2 3 4

Number Line B

0 1 2 3 4 5 6

2 Look at the Example. Suppose only 5 classrooms share 4 packages. How would the model in the Example change? How would the answer change? Each rectangle would be divided into 5 equal sections instead of 6; the 4 answer would change to of a package. ··5 383 ©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 383

Fluency & Skills Practice Teacher Toolbox

Assign Fractions as Division Fluency and Skills Practice

Name: In this activity students solve Fractions as Division Solve each problem.

real-world division problems in 1 Roger has 4 gallons of orange juice. He puts 2 Marta has 8 cubic feet of potting soil and the same amount of juice into each of 3 fl ower pots. She wants to put the same 5 pitchers. How many gallons of orange amount of soil in each pot. How many cubic which the quotient is a fraction or juice are in 1 pitcher? feet of soil will she put in each fl ower pot? mixed number. Students may encounter similar situations in 3 Greg made 27 ounces of potato salad to 4 Chandra spends 15 minutes doing 4 math serve to 10 guests at a picnic. If each serving problems. She spends the same amount of everyday life. For example, students is the same size, how much potato salad will time on each problem. How many minutes each guest receive? does she spend on each problem? may need to divide 5 cubic meters of peat moss evenly among

4 garden plots. Or, they may need 5 Taylor has 5 yards of gold ribbon to decorate 6 DeShawn is using 7 yards of wire fencing 8 costumes for the school play. She plans to to make a play area for his puppy. He wants use the same amount of ribbon for each to cut the fencing into 6 pieces of equal costume. How many yards of ribbon will she length. How long will each piece of to determine how many 2-cup use for each costume? fencing be? servings of milk are in 25 cups of milk. 4 7 __ What is a division word problem that can be represented by 3?

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

383 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. LESSON 18 SESSION 2

7 1 3 , or 2 cans; Students may use fraction models 3 Trish is taking care of the Han family’s dogs. The Hans leave 7 cans of dog food for ··3 ··3 the 3 days they will be away. How much food will the dogs get each day if Trish or number lines divided into thirds, or the feeds them an equal amount each day? Show your work. Write the answer in expression 7 4 3, to show how much food the remainder form and as a mixed number. dogs will get each day. Students should explain Students might use number lines, equations, or some other method to show the quotient of 7 4 3. that the mixed number better expresses the amount of food per day because it gives an exact amount. The remainder just indicates that

more than 2 cans will be used. 7 1 2 R 1 cans; , or 2 cans Medium Solution ··3 ··3 Which best answers the question, the remainder form or the mixed number? 4 No; See possible explanation on the student Explain. page. Possible answer: The mixed number, because it tells exactly how many cans Challenge to use each day. The remainder form shows to use more than 2, but not an exact amount.

5 Less than 1 cup; Students may use fraction 4 Raul plans to run 30 miles this week. He wants to run the same number of miles 7 models divided into sevenths, the expression each day of the week. He says he will run mile each day. Is he correct? Explain. ··30 48 4 7, or some other method to find each 48 6 No; Possible explanation: Raul divided the number of days by the number of person will get , or 6 ounces. That amount is ··7 ··7 miles, 7 4 30. He needed to divide the number of miles by the number of less than 8 ounces. 30 2 days, 30 4 7. Raul will run , or 4 , miles a day. Medium ···7 ··7

5 Gus makes 48 fl uid ounces of spiced cider. If he serves an equal amount to each of 7 people, will each person get more than 1 cup of cider or less than 1 cup? (1 cup 5 8 fl uid ounces) Show your work. Students might use fraction models, equations, or some other method to 48 6 6 show that 48 4 7 5 or 6 . 6 ounces < 8 ounces ···7 ··7 ··7

Solution less than 1 cup 384 384 Lesson 18 Fractions as Division ©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/Speaking Read Apply It Reading/Writing Read Apply It problem 1. Reading/Writing Have students read problem 1 to students. Ask them to point to Have students read the problem with a Apply It problem 1 with a partner and work the words equal amount. Ask students to partner and work together to decide on a together to decide on a strategy to use to discuss with a partner what operation they strategy to use to solve for the amount of solve for the amount of space Erica will give will need to solve the problem and why. space Erica will give each vegetable. each vegetable. Ask partners to solve the Provide the following sentence frame: When Have students work together to solve the problem. you break up a bigger number into problem using the strategy they chose. Have students work with a different group equal smaller amounts, you need and discuss the different approaches each to divide . group selected to solve the problem. Provide a sentence starter and encourage students to justify their thinking: We know our answer is correct because .

©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 384 LESSON 18 SESSION 3 Refine LESSON 18 SESSION 3

Purpose In this session students solve word Refine Fractions as Division problems involving division of whole numbers leading to answers in the form of fractions or Complete the Example below. Then solve problems 1–9. mixed numbers and then discuss and confirm

their answers with a partner. 2 quarts are shared equally Before students begin to work, use their EXAMPLE by 3 friends, so I know that Luke, Carter, and Ava have 2 quarts of juice. They want to each friend will have less than 1 quart of juice. That responses to the Check for Understanding to share it equally. How many quarts of juice will each of determine those who will benefit from means the them get? quotient is a additional support. fraction. Look at how you could show your work using a model As students complete the Example and and equations. problems 1–3, observe and monitor their reasoning to identify groupings for differentiated instruction.

1 2 4 3 5 2 3 PAIR/SHARE ··3 Model the problem for Start 5 2 ··3 3 quarts of juice divided equally among Luke, 2 quart Carter, Ava, and Ava’s little Check for Understanding Solution ··3 brother. Materials For remediation: Activity Sheet Fraction Bars (7 bars for thirds) 3 sheets of paper, scissors Ap�� i� Why Confirm understanding of fractions as division. 1 Erica has 7 square feet of space in her rectangular garden to Each vegetable will get at How Have students solve the problem using any plant carrots, beans, peppers, and lettuce. Suppose she gives least 1 square foot of strategy and write a division equation to show the each vegetable an equal amount of space. How much space garden space. How will will each vegetable get? the rest of the space be solution. Show your work. divided up? Start Possible student work using a model: Solution Solve the problem and write a 7 1 C B P L C B P L C B P L C B P L , or 2 , lawns each; division equation. ··3 ··3 7 1 1 ft2 1 ft2 1 ft2 1 ft2 1 ft2 1 ft2 1 ft2 7 4 3 = , or 2 Andy, Este, and May mow ··3 ··3 PAIR/SHARE What are some ways you 7 lawns and share the work 7 3 , or 1 , square feet can check your solution? equally. How many lawns Solution ··4 ··4 does each person mow? 385 ©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 385

Grade 5 Lesson 18 Session 3 | Refine Fractions as Division ©Curriculum Associates, LLC Copying is permitted. Error Alert If the error is . . . Students may . . . To support understanding . . .

First, have students solve a simpler problem: 3 students mow 6 lawns and share the work equally. How many lawns have reversed the numerator 3 4 7 or 3 does each person mow? ··7 and the denominator. [6 4 3 5 2] Then have students apply this same reasoning to the given problem.

Have students cut out fraction bars to model 7 lawns and label a sheet of paper with the name of each person in the 7 3 3 5 21, problem. Ask: How many lawns does each person get? Have not understand that they need 7 2 3 5 4, students share the lawns equally by placing fraction bars to divide to solve the problem. or 7 1 3 5 10 or equal portions of fraction bars on each sheet of paper. Point out that this is an equal sharing problem and guide students to recall that division is used for equal sharing.

385 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. LESSON 18 REFINE

Example 2 Deon needs to make 36 pizza crusts. He has 120 ounces of dough How many whole ounces of 2 and wants to use the same amount of dough for each crust. He dough will each crust get? quart; the fraction model and equations show one weighs a portion of dough for 1 crust on a scale. The weight, in ··3 What will way to solve the problem. Students could also solve ounces, should fall between what two whole numbers? happen with Show your work. the remaining ounces? the problem by drawing a number line from 0 to 2 Possible student work: 2 120 2 that is divided into 3 equal sections of . 120 ounces 4 36 pizza crusts 5 or 3 ounces per crust ··3 ····36 ··6 2 Look for You divide 2 wholes into 3 equal shares. 3 is between 3 and 4. ··6 When the divisor is greater than the dividend, the quotient is a fraction. PAIR/SHARE Create a diff erent division 120 story to represent . APPLY IT Solution between 3 and 4 ounces ···36 1 7 3 , or 1 square feet; Students could solve the 3 Jonas is doing a science experiment with his class. The teacher has ··4 ··4 About how much water will problem by drawing a model showing 21 fl uid ounces of pond water to share equally among 10 pairs of each pair of students 1 square foot of the garden given to each of the students. How much pond water will Jonas and his science receive? Will it be more or less than 2 uid ounces? 4 vegetables, and the remaining 3 square feet of partner receive? 10 Ꭽ fl uid ounce garden divided into fourths, with a total of ··21 1 3 fourths given to each vegetable. Ꭾ 1 fl uid ounces ··10 DOK 2 Ꭿ 2 fl uid ounces

Look for The solution is a mixed number since 21 ൳ fl uid ounces there is enough garden space for each ··10 vegetable to get at least 1 square foot of space Olivia chose Ꭽ as the correct answer. How did she get but not enough for each to get 2 square feet. that answer? Possible explanation: She wrote a fraction to show 10 divided 2 Between 3 and 4 ounces; See possible work on by 21 instead of a fraction to show 21 divided by 10. the Student Worktext page. Students could also solve the problem by using rounding. If 36 is PAIR/SHARE Does Olivia’s answer make rounded down to 30, each dough would weigh sense? 120 4 30 5 4 ounces. If 36 is rounded up to 40, 386 each dough would weigh 120 4 40 5 3 ounces. 386 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. So, 120 4 36 must be between 3 and 4 ounces. DOK 2 Look for The solution requires two steps: the first is to find the mixed number quotient of 120 4 36 and the second is to identify the two whole numbers that the quotient falls between.

3 D; Students could solve the problem using the 21 equation 21 4 10 5 . ··10 Explain why the other two answer choices are not correct: 21 1 1 B is not correct because 5 2 , not 1 . ··10 ··10 ··10 21 C is not correct because . 2. ··10 DOK 3

©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 386 LESSON 18 SESSION 3 Refine SESSION 3

32 2 4 D; 32 fluid ounces 4 5 5 , or 6 fluid ounces. 4 Teddy makes 32 fl uid ounces of hot cocoa. He pours equal ··5 ··5 2 amounts of cocoa into 5 cups. The amount of hot cocoa in each 6 is between 6 and 7. ··5 cup will fall between which two amounts? DOK 2 Ꭽ 3 and 4 fl uid ounces Ꭾ 4 and 5 fl uid ounces 5 B; Each lap takes almost 1 minute. Ꭿ 5 and 6 fl uid ounces 8 8 minutes 4 10 5 minute. ··10 ൳ 6 and 7 fl uid ounces DOK 2 5 Pierce swims 10 laps in a pool in 8 minutes. He spends the same amount of time on each lap. How much time does each lap take him?

6 A (Yes); 2 Ꭽ minute D (No); ··10 8 Ꭾ minute E (Yes); ··10 10 H (No); Ꭿ minutes ··8 I (Yes) 2 ൳ 1 minutes DOK 2 ··8 6 Dani needs 8 equal sections from a board that is 13 feet long. Does Error Alert Students may not recognize that the the expression represent the largest possible length of 1 section of the board length can be expressed as a multiplication board, in feet? expression. Yes No 5 1 Ꭽ Ꭾ ··8

8 Ꭿ ൳ ···13 13 ൴ ൵ ··8 8 4 13 ൶ ൷

1 13 3 ൸ ൹ ··8

387 ©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 387 Differentiated Instruction RETEACH EXTEND

Hands-On Activity Challenge Activity Model dividing two whole numbers with a fraction quotient. Model fractions as quotients. Students struggling with the concept of dividing a lesser number by a greater number Students who have achieved proficiency Will benefit from additional work with concrete models of fraction quotients Will benefit from deepening Materials For each student: 8 squares of paper, each one a 3 3 3 array of squares cut from understanding of fractions as division Activity Sheet 1-Inch Grid Paper, scissors • In groups of 3, students pass their work • Pose the following problem: 7 cups of juice are shared equally among 9 students. How much to the next person after each round. juice does each student get? • Round 1: Each student writes a division • Have students use 7 paper squares to model 7 cups. Set aside the remaining square. word problem in which the quotient is a 1 1 fraction or a mixed number. • Have students shade in each square to show that each student gets from each cup. ··9 ··9 1 • Round 2: Students draw a visual model • Have students cut out the shaded s and reassemble them on the remaining square to ··9 to represent the problem they receive. 7 show how much juice one student gets. cup 3 ··9 4 • Round 3: Students write an equation that • Reassemble the squares and repeat the activity for numbers of cups less than 7, such as represents the model they receive. 3 cups and 5 cups, and sharing them equally among 9 students. • The group checks each other’s work.

387 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. LESSON 18 REFINE SESSION 3

25 7 Which situations can be represented by ? 7 A; Divide 25 yards of paper by the number of ··9 banners, 9, to find how much paper is used for Ꭽ Melanie equally shares 25 yards of paper to make 9 banners. each banner. Ꭾ Quill gives away 9 baseball cards from a pack of 25 cards. E; Divide 25 ounces by the number of equal Ꭿ George invites 25 kids and 9 adults to his birthday party. servings, 9, to find how many ounces in each ൳ Becca makes 9 rows with 25 buttons each. serving. ൴ Joe makes 9 equal servings from a 25-ounce bag of peanuts. DOK 2 7 8 Paco is trying to explain to his friend that 7 4 2 = . ··2 7 8 Part A Part A Draw a model or number line showing 7 4 2 = . ··2 See possible number line on the Student Possible student work using a number line: Worktext page. Part B 0 1 2 3 4 5 6 7 7 7 4 2 5 See possible explanation on the Student ··2 1 5 3 Worktext page. Students may also explain that to ··2 7 model 7 4 2, you can show 7 divided into 2 equal Part B Explain the equivalence of 7 4 2 and using words. 1 ··2 parts. Each part is equal to 3 , which is equivalent Possible explanation: Seven halves is equivalent to seven ··2 7 divided into two equal sections because both equal three and to . ··2 one half. DOK 3 9 MATH JOURNAL Write a division word problem that can be represented by the expression 12 4 5. Then explain how to solve your problem. Possible problem: Five friends want to share 12 ounces of juice

equally. How many ounces of juice should each friend get?

Possible explanation: You can solve the equation 12 4 5 5 n: 12 12 2 n 5 . Each friend gets , or 2 , ounces of juice. ···5 ···5 ··5

SELF CHECK Go back to the Unit 3 Opener and see what you can check off . 388 388 Lesson 18 Fractions as Division ©Curriculum Associates, LLC Copying is not permitted. Close: Exit Ticket REINFORCE PERSONALIZE 9 MATH JOURNAL Problems 4–9 Student responses should include a word problem Interpret fractions as division. with 12 as the number of wholes to be shared and 5 Provide children with as the number of equal shares. Students should All students will benefit from additional work with opportunities to work explain that the quotient 12 4 5 can be represented using fractions to represent division of whole numbers 12 on their personalized by the fraction . 5 by solving problems in a variety of formats. instruction path with ·· • Have students work on their own or with a partner to i-Ready Online Error Alert If students reverse the numerator and solve the problems. Instruction to: denominator in the fraction quotient, then have • Encourage students to show their work. • fill prerequisite gaps them use reasoning to determine which two whole • build up grade-level numbers the quotient of 12 4 5 falls between and 12 5 assess which of the two possible fractions, or , skills ··5 ··12 is between those two numbers.

SELF CHECK Have students consider whether they feel they are ready to check off any new skills on the Unit 3 Opener page.

©Curriculum Associates, LLC Copying is not permitted. Lesson 18 Fractions as Division 388 Lesson LESSON 22 Overview Multiply Fractions in Word Problems

Lesson Objectives Prerequisite Skills Lesson Vocabulary

Content Objectives • Multiply fractions. There is no new vocabulary. Review the • Add fractions. following key terms. • Represent real-world problems involving • equation a mathematical statement multiplication of fractions and mixed • Write equivalent fractions. that uses an equal sign (5) to show that numbers using visual models and • Write a mixed number as a fraction two expressions have the same value. equations. greater than 1. • factor a number that is multiplied. • Solve real-world problems involving • Write a fraction greater than 1 as a mixed multiplication of fractions and mixed number. • mixed number a number with a whole number part and a fractional part. numbers using visual models and • Use visual models to represent problem equations. situations. • product the result of multiplication. Language Objectives • Draw pictures to represent word Standards for Mathematical problems involving multiplication of Practice (SMP) fractions and mixed numbers. • Write equations to represent word SMPs 1, 2, 3, 4, 5, and 6 are integrated in problems involving multiplication of every lesson through the Try-Discuss- fractions and mixed numbers. Connect routine.* • Compare a visual model and an equation In addition, this lesson particularly that represent the same problem emphasizes the following SMPs: situation. 2 Reason abstractly and quantitatively. 4 Model with mathematics. 5 Use appropriate tools strategically.

*See page 305m to see how every lesson includes these SMPs.

Learning Progression

In previous Grade 5 lessons students In this lesson students draw on their work In the remaining lessons in this unit acquired a conceptual understanding of in previous lessons to solve word problems students will use what they know about multiplying fractions, using visual models that involve multiplying fractions and fraction multiplication to divide with unit to make sense of the meaning of fraction mixed numbers, including finding a fractions and solve word problems about multiplication. Students also used area fraction of a quantity, finding the area of a division with unit fractions. In later grades, models to multiply fractions and found rectangle, and finding the dimensions of a students will apply their understanding of areas of rectangles with fractional side resized object. fraction multiplication to solve problems lengths. They wrote equations to represent involving proportional relationships. They fraction multiplication and recognized that will also extend their understanding of using a model or writing an equation both fraction multiplication to multiply rational result in the same product. Students also numbers. conceptualized multiplication as scaling when they understood a product as a relationship between a quantity and a resizing, or scaling, factor. They reasoned about the size of a product when one factor is greater than 1, equal to 1, or less than 1.

435a Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. Lesson Pacing Guide Teacher Toolbox Whole Class Instruction Small Group Differentiation SESSION 1 Interactive Tutorial* (Optional) Additional Practice PREPARE Explore Prerequisite Review: Lesson pages 439–440 Solve Multiplicative Comparison Ready Prerequisite Lesson 45–60 min Problems Grade 4 • Lesson 7 Multiplication and Division in Multiplying Fractions in Word Word Problems Problems • Start 5 min RETEACH • Try It 10 min • Discuss It 10 min Tools for Instruction • Connect It 15 min Grade 4 • Close: Exit Ticket 5 min • Lesson 7 Solve Comparison Problems SESSION 2 Multiplying Fractions in Word Additional Practice Grade 5 • Lesson 22 Multiplying Fractions to Solve Develop Problems Lesson pages 445–446 • Start 5 min Word Problems 45–60 min • Try It 10 min Fluency • Discuss It 10 min Multiplying Fractions in REINFORCE • Picture It & Model It 5 min Word Problems Math Center Activities • Connect It 10 min • Close: Exit Ticket 5 min Grade 5 • Lesson 22 Write a Word Problem SESSION 3 Multiplying with Mixed Numbers in Additional Practice • Lesson 22 Real-World Multiplication Situations Develop Word Problems Lesson pages 451–452 • Start 5 min 45–60 min • Try It 10 min Fluency EXTEND • Discuss It 10 min Multiplying with Mixed • Picture It & Model It 5 min Numbers in Word Enrichment Activity • Connect It 10 min Problems Grade 5 • Close: Exit Ticket 5 min • Lesson 22 Plant Growth

SESSION 4 Multiplying Fractions in Word Lesson Quiz Problems or Digital Refine • Start 5 min Comprehension Check 45–60 min • Example & Problems 1–3 15 min • Practice & Small Group Differentiation 20 min • Close: Exit Ticket 5 min

Lesson Materials Lesson Activity Sheet: Fraction Bars (Required) Activities Per student: drawing paper (1 sheet) Per pair: 1 set of fraction tiles, 1 set of fraction circles, tracing paper (2 sheets), ruler, colored pencils or crayons (2 different colors), tape Math Toolkit fraction tiles, fraction circles, fraction bars, grid paper, number lines, index cards Digital Math Fraction Models, Number Line, Multiplication Models Tools

*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 22 Multiply Fractions in Word Problems 435b LESSON 22 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 Multiply Fractions in Word Problems ACTIVITY m��Ly�� f��On� i� w�� p��leMs Dear Family, 22 Do this activity with your child to multiply fractions in word problems. This week your child is learning about multiplying Together with your child, make up and solve real-world problems about fractions in word problems. multiplying fractions or use the problems below. Below are examples of problems you could solve. He or she might see a problem like this: 5 1 5 3 2 1. Pete found of a party sandwich left in the refrigerator. He took of the of Michael found of a pizza in the refrigerator. He ate of it. ··6 ··2 ··6 ··8 ··3 the sandwich to his neighbor. How much of the original sandwich did Pete How much of the original whole pizza did Michael eat? take to his neighbor? • One way to understand this problem is to draw a picture. Your child could draw 3 of a pizza. ··8 To show the part of the pizza that Michael ate, 2 your child could shade 2 of the 3 pieces to show . ··3 The shaded parts show how much of the original whole pizza 2. Shawn had 3 of a gallon of paint left in the can. Michael ate. Michael ate 2 , or 1 , of the original whole pizza. ··5 ··8 ··4 He used 2 of it to paint a cabinet. How much of ··3 the gallon of paint did he use?

• Another way your child could solve the problem is to write a multiplication equation.

2 3 2 3 of means 3 . ··3 ··8 ··3 ··8 3. Renee made some money babysitting. She saved 3 of the money. She spent ··4 2 3 2 3 3 6 3 5 5 2 of the money she saved to buy a shirt. What fraction of the money did ··3 ··8 ·····3 3 8 ··24 ··5 6 2 1 Renee spend on the shirt? So, is equivalent to , or . ··24 ··8 ··4 The answer is the same using either way to solve the problem. Michael ate 1 of the original whole pizza. ··4 Invite your child to share what he or she knows about multiplying fractions and word problems by doing the following activity together.

Answers: 5 6 6 3 1. __ ; 2. __ or __ 2 ; 3. __ or __ 435 436 12 15 5 20 10 ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 435 436 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted.

Goal Math Talk at Home The goal of the Family Letter is to provide opportunities for Students and family members solve the problems in the activity or students and their families to practice multiplying fractions in word use them as a model to make up other real-world problems. problems. Conversation Starters Below are additional conversation starters Activity students can write in their Family Letter or math journal to engage family members: Word problems provide real-world contexts for students to apply • What are some items that you might need to divide into smaller mathematical skills such as multiplying fractions. Look at the parts and then divide those parts into smaller parts? Multiplying Fractions in Word Problems activity and adjust if necessary to connect with your students. • How do you calculate what is left when you have a part of a whole and then you use a part of that part?

435–436 Lesson 22 Multiply Fractions in Word Problems ©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.

Sessions 1 and 2 Use anytime during these sessions. Session 3 Use with Try It. • Many carpenters, architects, and engineers use fractions in their • Have students customize the problem by researching what types of measurements. The tools they use, such as measuring tapes, have flowers or plants are commonly grown in their neighborhoods or markings with fractions indicated. Understanding fractions is key to their countries of origin. If time allows, discuss what type of climate making accurate measurements and interpreting them and soil is needed to grow the flowers or plants that students appropriately. selected. • Share that in woodworking there are special precision marking and measuring tools that allow for measuring in fractions of a unit such 1 1 1 1 1 as , , , , and . Have students note that each fraction in the ··2 ··4 ··8 ··16 ··32 1 list is one-half of (or times) the previous number in the list. ··2 • Have students work in small groups to research how different trades use fractions.

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 Try It.

Levels 1–3 Levels 2–4 Levels 3–5 Reading/Speaking Read the Try It Reading/Speaking Read the Try It Reading/Speaking Have students read problem aloud. Ask students to underline the problem with students. Organize them into and solve the Try It problem independently. quantities in the problem. Draw a sketch with pairs to discuss solution strategies and create Organize them into pairs to defend their Greyson’s house and the park on it. Call on a visual model to represent the problem. Ask: thinking. Tell students to convince their volunteers to provide the distances in the How confident are you in your answer? Have partners that their answer makes sense. problem. Use the information to create a partners signal with two thumbs up for very Encourage them to use language for diagram. Organize students into pairs to confident, one thumb up and one thumb constructing arguments such as because, model and solve the problem using tools down for fairly confident and two thumbs shows, represents, proves, validates, explains, from the Math Toolkit. Ask questions to assess down for not confident. Display the following interpret, conclusion, and reasonable. After understanding, such as: How does your model word bank to support partners as they both partners defend their answers, ask: Was 4 show mile? How many equal parts does the construct arguments to defend their answers: your partner’s answer reasonable? Have them ··5 model show? How does the model show the because, shows, represents, explains, signal thumbs up or down. Challenge distance Greyson walked? conclusion, equation, whole, part, numerator, students to identify a strength and an area Encourage students to use gestures, pictures, denominator. Provide support to students for improvement in their partner’s argument. numbers, and words to defend their answers. who expressed low confidence in their Have them share ideas. answers.

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 436a LESSON 22 SESSION 1 Explore LESSON 22 SESSION 1

Purpose In this session, students draw on Explore Multiplying Fractions in Word Problems their knowledge of fraction multiplication. They share visual models and reasoning about a Learning Target Now that you have learned how to multiply fractions, you will use • Solve real world problems involving problem situation that involves finding a what you know in problem situations. Use what you know to try to multiplication of fractions and mixed numbers, e.g., by using visual fraction of a fraction. They will look ahead to solve the problem below. fraction models or equations to represent the problem. think about how to write equations, make 4 Grayson lives mile from the park. He has already SMP 1, 2, 3, 4, 5, 6 ··5 estimates, and draw models to solve word 3 walked of the way to the park. How far has Grayson problems involving multiplying fractions. ··4 walked? Use a visual fraction model to show your thinking. Start TRY IT Connect to Prior Knowledge Possible student work: Why Supports students’ facility in using visual Sample A models to show their thinking about fraction Park Math Toolkit multiplication. • fraction tiles or circles 0 1 2 3 4 1 • fraction bars 1 3 5 5 5 5 How Have students show 3 using any visual • fraction models 4 ··2 ··5 If the park is mile from Grayson’s house, the distance between his • grid paper model they choose. Have them write the product. ··5 • number lines 1 house and the park is 4 equal parts, each of which is mile. 3 of • index cards ··5 Start • multiplication models 3 3 Solution: those parts is of one mile. ···10 ··5 Draw a visual model to show Look for models that the expression. Write the 1 Sample B show the factors product. ··2 3 4 3 3 Find of mile. 1 3 and and the ··4 ··5 4 3 ··5 3 4 12 ··2 ··5 3 of is . product . 4 5 20 ··10 ·· ·· ···

4 5 DISCUSS IT Ask your partner: Can you explain that again? Grade 5 Lesson 22 Session 1 | Explore Multiplying Fractions in Word Problems ©Curriculum Associates, LLC Copying is permitted. Tell your partner: A model TRY IT I used was . . . It helped Make Sense of the Problem me . . . To support students in making sense of the 437 3 ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 437 problem, have them identify that of the way to the ··4 3 park refers to of the distance from Grayson’s house Common Misconception Look for students who are not comfortable with this ··4 to the park. problem as a multiplying situation for fractions and may try to subtract to find the answer. As students present their solutions be sure to have them specify why they chose the operation they did to solve problem. DISCUSS IT Support Partner Discussion Select and Sequence Student Solutions Encourage students to use the Discuss It question One possible order for whole class discussion: and sentence starter on the Student Worktext page • physical parts showing fifths and fourths as part of their discussion. • drawings to represent the problem Look for, and prompt as necessary for, • number lines marked in fifths and then fourths 12 3 understanding of: • equations to find the distance walked is , or , mile 4 ··20 ··5 • mile as the distance to the park ··5 3 Support Whole Class Discussion • as the fraction of the distance Grayson has ··4 Prompt students to note the relationship between the numbers in each model and walked the numbers in the problem. 4 3 • mile as the whole of which is the part ··5 ··4 Ask How do [student name]’s and [student name]’s models show the distance from • the fraction of a mile walked as the unknown the house to the park? How far Grayson has walked as a fraction of that distance? 4 1 Listen for The distance to the park is mile, or 4 equal parts each mile. The ··5 ··5 3 4 fraction Grayson walked is of mile, or 3 of the 4 equal parts. ··4 ··5

437 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 EXPLORE SESSION 1

CONNECT IT Co�� It 1 LOOK BACK 1 LOOK BACK 3 4 Explain how you can use a visual model to show how far Grayson has Look for understanding of how to model of mile ··4 ··5 already walked. and interpret the model to find that Grayson has 1 Possible explanation: Using a number-line model, you can see that each of 3 ··4 1 3 3 walked mile. the distance is mile, so of the distance is mile. ··5 ··5 ··4 ··5 2 LOOK AHEAD Hands-On Activity You can use what you know about multiplying fractions to think through and Find a fraction of a fraction. solve word problems involving fractions. Consider this word problem: 1 3 Ehrin spills of a -pound box of cereal. How many pounds did she spill? If . . . students are unsure about drawing models ··2 ··4 1 1 a. Finding of a quantity is the same as multiplying by . What equation to show fraction multiplication, ··2 ··2 Then . . . use this activity to have them model could you write for the cereal problem? Use p for the unknown amount in the problem. the Try It and similar problems. 1 3 3 5 p ··2 ··4 Materials For each pair: 1 set of fraction tiles b. Estimate the product. Is the amount of cereal Ehrin spills on the fl oor 3 3 more than pound or less than pound? Why? • Have students use the 5 one-fifth tiles. ··4 ··4 3 Possible answer: The amount of cereal Ehrin spills is less than pound Explain that, as a group, the 5 tiles represent ··4 1 3 a whole of 1 mile. Ask: What does each tile because you are finding of . 1 ··2 ··4 represent? mile c. Complete the area model to show the problem. 3 ··5 4 4 How many pounds of cereal did Ehrin spill on • Have students model the -mile distance 3 1 ··5 from Grayson’s house to the park. [4 tiles] the fl oor? ··8 pound Possible model is shown. 2 3 Point out that Grayson walked of the 3 REFLECT ··4 4 3 4 mile. Review that finding of is the How does writing an equation, making an estimate, and drawing a model help you ··5 ··4 ··5 3 4 think through the problem? same as finding 3 . ··4 ··5 Possible answer: The equation shows the operation involved in solving the 3 • Ask students to use the tiles to find of ··4 problem. The estimate helps me think about a reasonable solution. Drawing a 4 3 mile and to explain how they did it. mile; ··5 3 ··5 model shows a visual of the problem. 3 of the 4 tiles represent 3 of the distance to ··4 438 the park. Since each tile represents 1 mile, ··5 438 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. 3 tiles represent 3 mile. ··5 4 • Repeat the activity for additional fractions of Close: Exit Ticket 4 2 4 mile, such as and . ··5 ··4 ··4 3 REFLECT Look for understanding of how to apply the skills they have to solving word 2 LOOK AHEAD problems. Student responses should include references to equations showing the operations needed, estimates giving a frame of reference for the answer, and drawing Point out that students know how to both write models as a way to picture a problem situation. fraction multiplication expressions and model the multiplication. This along with reasoning and Common Misconception If students are unclear or incomplete in their descriptions evaluating whether an answer makes sense of how they can use equation writing, estimating, and visual models to represent and prepares them to solve word problems involving solve problems, then have students look at and describe how they used each skill to fraction multiplication. solve the cereal problem on the Student Worktext page. Ask Why can you replace the word of in the 1 3 Real-World Connection phrase of with a multiplication symbol? ··2 ··4 Encourage students to think about everyday places or situations where people 1 3 Listen for You can think of finding of as ··2 ··4 might need to multiply fractions. Have volunteers share their ideas. Examples: 1 3 1 2 finding of a group of size , which is a preparing of a recipe that includes fractional measurements, watching of a ··2 ··4 ··3 ··3 3 5 multiplication situation. half-hour television show, cutting of a -yard piece of fabric for a craft project. ··4 ··8

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 438 LESSON 22 SESSION 1 Additional Practice Name: LESSON 22 SESSION 1

Solutions Prepare for Multiplying Fractions in Word Problems

Support Vocabulary Development 1 Think about what you know about fractions. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can. Possible answers: 1 Ask students to brainstorm models or pictures Examples they have used in the past to represent a fraction of 1 3 a fraction. Remind them that the root frac- means of ··3 ··4 “to break.” Examples Examples 1 A fraction of a fraction is a portion of a portion of 0 1 2 3 1 2 4 4 4 equal parts. Have students record their examples To find a fraction of a 2 1 2 of and compare with other students. ··3 ··2 3 fraction, you can multiply the two fractions. 2 Read the problem. Have students write the fraction of a expression with a partner. Write the following on the fraction board, aligning the words and numbers as shown: Examples Examples 4 fraction of a fraction 4 2 4 2 5 of 5 3 3 1 3 1 ··5 ··3 ··5 ··3 1 3 of 5 3 2 ··4 ··2 ··4 ··2 8 of a Examples 5 3 ··5 ··8 3 ···15 3 3 5 Ask: Is any fraction of less than ? ··8 2 5 2 5 8 8 of 5 3 ·· ·· ··3 ··6 ··3 ··6 Then have partners write the multiplication 2 3 5 5 ·····3 3 6 expression and answer the question in the problem. 10 5 ···18 Supplemental Math Vocabulary

• equal 1 3 2 Write a multiplication expression that can be used to fi nd of . ··5 ··8 1 3 • part 3 ··5 ··8 3 Why is the product less than ? ··8 3 1 3 1 Possible answer: The product is less than because it is of and is less ··8 ··5 ··8 ··5 than 1. A fractional part less than 1 is a smaller part. Multiplying a quantity by

a fraction less than 1 reduces the size of the quantity. 439 ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 439

439 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 SESSION 1

3 Assign problem 3 to provide another look at 3 Solve the problem. Show your work. 3 multiplying fractions in word problems. Lola lives mile from the basketball court. She has ··4 2 This problem is very similar to the problem about already walked of the way to the basketball court. ··3 Grayson walking to the park. In both problems, How far has Lola walked? Use a visual fraction model students will use a visual model to show finding a to show your thinking. fraction of a fraction to solve a word problem. Possible student work using a picture:

Students may also represent the situation with a se product of fractions. The question asks how far Lola o

has walked. 2

Students may want to use fraction bars or number 3 The basketball court is mile from Lola’s house, so the distance lines or draw models with pencil and paper. ··4 between her house and the basketball court is 3 equal parts, each of which Suggest that students read the problem three times, 1 2 is mile. 2 of those parts is of one mile. asking themselves one of the following questions ··4 ··4 each time: • What is this problem about?

2 • What is the question I am trying to answer? Lola has walked mile. Solution ··4 • What information is important? 4 Check your answer. Show your work. Solution: Possible student work: 2 1 2 3 2 2 2 3 , or , mile; Check that visual fraction models Find of mile. “ of” means times, so I need to find 3 mile. ··4 ··2 ··3 ··4 ··3 ··3 ··3 ··4 3 2 2 1 show that of is , or . Students may also write ··4 ··3 ··4 ··2 3 2 2 1 the multiplication equation 3 5 , or . ··4 ··3 ··4 ··2 2 Medium

4 Have students solve the problem a different 2 3 6 way to check their answer. of mile is mile. ··3 ··4 ···12 6 2 5 ···12 ··4 440 440 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. English Language Learners: Prepare for Session 2 ELL Differentiated Instruction Use with Apply It.

Levels 1–3 Levels 2–4 Levels 3–5 Listening/Speaking Read Apply It Listening/Speaking Have students read Listening/Speaking Have students read problem 8 to students. Explain that the Apply It problem 8 with a partner. Ask them Apply It problem 8 with a partner. Ask them phrase calls for is used to indicate what is to underline the phrase calls for. Explain that to underline the phrase calls for. Have needed for a recipe. Write these sentence the phrase indicates what is needed for a volunteers explain what it means. Then have frames and have students complete and read recipe. Restate the problem if needed. Then them restate the problem. them aloud. ask: Will Stan need more or less eggplant than Remind students that something is • Stan needs to make . the recipe calls for? How do you know? Have reasonable when it makes sense. Have students respond using the sentence frame: • The recipe needs pound of eggplant. students discuss the type of answer they Stan will need less eggplant expect to get. Then have them solve the • Stan only needs to make of the recipe. because . [Possible answer: he wants problem with a partner and talk about the Before students solve the problem, ask them to make a fraction of the recipe] answer. Ask: Is your answer reasonable? Why? if Stan will need more or less eggplant than Provide sentence starters: the recipe calls for: I think the answer is reasonable Stan will need less eggplant. because . I don’t think the answer is reasonable because .

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 440 LESSON 22 SESSION 2 Develop LESSON 22 SESSION 2

Purpose In this session students solve a Develop Multiplying Fractions in Word Problems 2 3 word problem that requires finding of . ··3 ··4 Students model the fractions in the word Read and try to solve the problem below.

problem either on paper or with manipulatives 3 Brandon’s mother left of a pizza on the counter. ··4 to develop strategies for solving word problems 2 If Brandon eats of the leftover pizza, how much of ··3 that involve multiplying fractions. the whole pizza did Brandon eat?

Start TRY IT Possible student work: Connect to Prior Knowledge Sample A Math Toolkit Why Support students’ facility with writing • fraction tiles or circles • fraction bars equivalent fractions, in preparation for recognizing • fraction models equivalent ways to write the product of two • grid paper • number lines fractions. • index cards How Have students identify a common factor of the • multiplication models numerator and denominator of a fraction and then 6 He eats of the whole pizza. use division to write an equivalent fraction. ···12 Start Sample B Solutions 2 3 2 3 I need to find of pizza, so I find 3 . Write an equivalent fraction 3 1 ··3 ··4 ··3 ··4 5 2 3 2 3 3 6 by dividing the numerator ··15 ··5 3 5 5 8 4 2 ··3 ··4 ·····3 3 4 ···12 and denominator by the 5 or 6 1 ··20 ··10 ··5 Brandon eats , or , of a pizza. same factor. ···12 ··2 6 3 3 8 6 5 5 5 5 ··16 ··8 ··15 ··20 ··16 DISCUSS IT Ask your partner: How did you get started? DevelopGrade 5 Lesson 22 Session 2 | Develop Multiplying Language Fractions in Word Problems ©Curriculum Associates, LLC Copying is permitted. Tell your partner: I am not sure how to ﬁ nd the answer Why Clarify the meaning of the term leftover. because . . . How Ask students: Did Brandon’s mother leave the 3 441 whole pizza on the counter? [No, she left of the ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 441 2 ··4 pizza.] Explain: Brandon then eats of the leftover ··3 pizza. The leftover part is the part that is still on the DISCUSS IT counter, the part that was left after Brandon’s mother Support Partner Discussion ate some. Encourage students to share what did not work for them as well as what did. Support as needed with questions such as: TRY IT • Did the first strategy you tried work? Make Sense of the Problem • Would you use the same strategy next time for this type of problem? To support students in making sense of the Common Misconception Look for students who find the fraction of the whole pizza problem, have them describe what the problem is 1 that remains after Brandon finishes eating , rather than the fraction of the whole 1 ··4 2 asking them to find. pizza that Brandon eats. As students present solutions, be sure to have them identify Ask How much of a whole pizza is left on the counter? which parts show what Brandon ate. 2 2 Does Brandon eat of a whole pizza or of the ··3 ··3 pizza left on the counter? Select and Sequence Student Solutions One possible order for whole class discussion: • physical parts showing fourths and thirds • drawings to represent the problem • number lines marked in fourths and then thirds 6 1 • equations to find the amount Brandon eats is , or , of a pizza ··12 ··2

441 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 DEVELOP

Support Whole Class Discussion Explore diff erent ways to understand strategies for solving word problems that involve Compare and connect the different representations fi nding a fraction of a fraction. 3 2 and have students identify how they relate. Brandon’s mother left of a pizza on the counter. If Brandon eats of the ··4 ··3 Ask Where does your model show how much pizza leftover pizza, how much of the whole pizza did Brandon eat? Brandon’s mother left on the counter? The fraction of that amount Brandon eats? How much of a whole Pi��e It You can draw a picture to help you understand the problem. pizza he eats? Show 3 of a pizza. Listen for Students should recognize that ··4 accurate representations include showing fourths 3 and thirds. Responses may include of a pizza as ··4 the amount Brandon’s mother left on the 2 counter; as the fraction of that amount Brandon ··3 6 1 eats; and , or , of a whole pizza as the amount ··12 ··2 Since Brandon eats 2 of what is left, outline 2 of the 3 pieces that are left. You can see he eats. ··3 from the outlined parts how much of the whole pizza Brandon ate.

PICTURE IT & MODEL IT If no student presented these models, connect them to the student models by pointing out the ways they each represent: • the fraction of a whole pizza there was to start • the fraction of the leftover pizza that Brandon ate • the fraction of the whole pizza Brandon ate Mo�� It You can write an equation to help you understand the problem.

Ask Do the picture and equation show the same You need to fi nd a fraction of a fraction: 2 of 3 of a pizza. ··3 ··4 number of equal parts for a whole pizza? Explain. 2 of 3 means 2 3 3 . ··3 ··4 ··3 ··4 Listen for Yes, both the picture and equation 2 3 3 5 2 3 3 ··3 ··4 ·····3 3 4 show 4 equal parts in a whole pizza. The picture 442 shows 3 pieces of pizza plus 1 missing piece. The 442 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. 3 equation shows , three out of four equal parts, to ··4 describe the pizza left on the counter. Deepen Understanding For a picture of the pizza, prompt students to Equation Model identify the fractions represented by the pictures. SMP 4 Model with mathematics. • What fraction is represented in the top pizza picture? When discussing the equation model, prompt students to consider how when people talk about modeling a problem they tend to think of methods such as • What fractions are represented in the bottom pizza acting it out or drawing a picture. Point out that writing an equation is also a picture? way to model a problem. For an equation, prompt students to relate the product shown by the equation to a model. Ask What are some ways of modeling problems other than acting it out, 2 3 3 drawing a picture, or writing an equation? • How would you show on a visual model like the ·····3 3 4 Listen for Students should be able to describe a variety of other ones in Picture It? [Divide a model into fourths and methods: for example, making a chart, list, or graph or using objects. then divide each fourth into thirds and then shade 6 of the equal parts.] Ask Think about the models shown on this page. How is writing an equation similar to drawing a picture? Listen for Both models represent the important quantities in a problem and show how they are related.

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 442 LESSON 22 SESSION 2 Develop SESSION 2

Co�� It CONNECT IT Now you will use the problem from the previous page to help you understand • Remind students that one thing that is alike about strategies for solving word problems that involve fi nding a fraction of a fraction. all these representations is the numbers. 1 Look at Picture It. Why do you outline 2 of the 3 parts of the pizza? 2 • Explain that on this page they will compare the Possible answer: The problem says that Brandon eats of what was left. 3 answers they got using the two strategies of ·· drawing a picture and writing an equation. 2 How much of the whole pizza did Brandon eat? Explain your reasoning. 2 1 , or ; Possible explanation: The whole pizza is divided into 4 equal parts. ··4 ··2 Monitor and Confirm Brandon ate 2 of the 4 parts of the whole pizza.

1 – 3 Check for understanding that: 3 Look at Model It. How do you know that you should multiply 2 3 3 ? ··3 ··4 2 3 Possible answer: You need to find 2 of 3 of a pizza. To find a fraction of a • of of the pizza is 2 of the 3 parts. ··3 ··4 ··3 ··4 2 1 number, you multiply the number by that fraction. • Brandon ate , or , of the whole pizza. ··4 ··2 6 2 3 2 3 4 What is 2 3 3 ? • of means 3 . ·····3 3 4 ···12 ··3 ··4 ··3 ··4 Is this product the same as your answer to problem 2? Explain. Yes; Possible explanation: 6 is equivalent to 2 , or 1 . Support Whole Class Discussion ···12 ··4 ··2 4 When discussing problem 4, review with 5 What strategies can you use to solve a word problem that involves fi nding a students how they can write equivalent fractions by fraction of a fraction? Possible answer: You can draw a picture to represent the problem and use dividing the numerator and denominator by the reasoning to find a solution. You also can write an equation that represents same factor. the problem as a product of fractions and then multiply the numerators and multiply the denominators. 6 2 Ask How do you use division to show that 5 or ··12 ··4 6 1 that 5 ? 6 REFLECT ··12 ··2 Look back at your Try It, strategies by classmates, and Picture It and Model It. Listen for You can divide both the numerator and Which models or strategies do you like best for solving word problems that involve 6 6 2 denominator of by 3 to show that 5 . You fi nding a fraction of a fraction? Explain. ··12 ··12 ··4 can divide both the numerator and denominator Possible answer: I like multiplying the numerators and the denominators of 6 6 1 of by 6 to show that 5 . the fractions because it does not require trying to draw equal-size pieces. ··12 ··12 ··2 It is a quick way to find a fraction of another fraction. 5 Look for descriptions of at least two strategies, 443 such as drawing a picture and writing an equation, ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 443 to solve word problems involving finding a fraction of a fraction. Hands-On Activity 6 REFLECT Act out the problem situation. Have all students focus on the strategies used to If . . . students are unsure about using visual model strategies, solve this problem. If time allows, have students Then . . . use this activity for a hands-on approach. share their preferences with a partner. Materials For each pair: 1 set of fraction circles 3 • Have students use fraction pieces to represent of a pizza. [3 one-fourth ··4 pieces] Tell students they can think of this as the amount that is left. 2 • Remind students Brandon ate of the amount that is left. Guide them in ··3 1 seeing there are 3 equal-sized pieces left, so each piece is of the amount that ··3 2 is left. Have students remove of the amount that is left. [2 pieces] ··3 • Have students place the 2 pieces they removed over a whole fraction circle. 1 Ask: How much of a whole pizza did Brandon eat? Be sure students 3 ··2 4 understand the answer is a fraction of a pizza, not a number of pieces of pizza. • Use one-fifth pieces to repeat the activity for this problem situation: A recipe for 4 1 2 loaves of bread uses pound of flour. Seth needs to use of that amount to ··5 ··2 2 make 1 loaf. How much flour does Seth need? pound 3 ··5 4

443 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 DEVELOP SESSION 2

Ap�� It APPLY IT Use what you just learned to solve these problems. For all problems, encourage students to draw some 8 3 7 Lewis walked of a mile. Todd walked of the way with Lewis. How many miles kind of model to support their thinking. Allow some ··10 ··4 leeway in precision; drawing equal parts in models did Todd walk with Lewis? Show your work. Possible student work: 1 2 3 can be difficult. 4 4 4

24 6 3 L L L L L L 7 , , or mile; See possible work on the and and and and and and L L ··40 ··10 ··5 T T T T T T Student Worktext page. Students may also draw a 0 8 1 visual model showing tenths and each tenth divided 10 into fourths. They may also write the equation 24 6 3 , , or mile 3 8 24 3 ···40 ···10 ··5 3 5 , or . Solution ··4 ··10 ··40 ··5 9 8 Stan has a recipe for vegetable lasagna that calls for pound of eggplant. ··16 18 3 2 8 , or , pound; See possible work on the He wants to make a batch of lasagna that is of the amount of the recipe. ··48 ··8 ··3 Student Worktext page. Students may also draw a How much eggplant will Stan need? Show your work. visual model showing sixteenths, and each sixteenth Possible student work: 2 9 2 3 9 18 3 5 5 divided into thirds. ··3 ···16 ······3 3 16 ···48 18 18 4 6 3 5 5 ···48 ······48 4 6 ··8 18 3 or pound Close: Exit Ticket Solution ···48 ··8 5 9 Jamie worked hour fi ling papers for her mother. She listened to music for ··6 20 4 2 4 9 , , or hour; See possible work on the of the time she spent fi ling. How much time did Jamie spend listening to music? ··30 ··6 ··3 ··5 Student Worktext page. Students may also draw a Show your work. Possible student work: picture of an analog clock. They may also write the F and M 4 5 20 2 equation 3 5 , or . ··5 ··6 ··30 ··3 F F and M Students’ solutions should indicate understanding of: F F and M • Accurate use of visual fraction models and/or and M 20 4 2 , , or hour equations to represent and solve a word problem Solution ···30 ··6 ··3 that involves finding a fraction of a fraction. 444 444 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. Error Alert If students’ solution is a fraction greater than 1, then remind them what they know about multiplying a given factor by a factor less than 1— that the product will be less than the given factor. Ask them what they know about the solution to this 5 problem. It is less than hour. 3 ··6 4

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 444 LESSON 22 SESSION 2 Additional Practice Name: LESSON 22 SESSION 2

Solutions Practice Multiplying Fractions in Word Problems

3 5 3 3 5 15 Study the Example showing one way to solve a word problem with fractions. 1 3 5 5 ··5 ··8 ·····5 3 8 ··40 Then solve problems 1–5. Basic Example 2 Yes; See possible explanation on the student 3 15 Vicky’s favorite beach towel is green and white and has a fi sh design. The green part page. Students should recognize and name 5 3 covers of the towel. A fi sh design is drawn on of that part. What part of the towel ··8 ··40 ··8 ··5 the same amount. has a fi sh design?

Medium You can draw a picture. 5 3 Show a towel with shaded green. Draw fi sh on of the green part. ··8 ··5

3 Because 3 of the 8 parts of the towel have fi sh drawn on them, of the towel has ··8 a fi sh design.

1 You can also write an equation to solve the Example. Write the numbers to complete the equation showing what part of the towel has the fi sh design. 3 5 3 5 of means 3 . ··5 ··8 ··5 ··8

5 3 3 15 3 3 5 5 5 3 ··5 ····8 ·······5 8 ····40

3 2 Is your answer to problem 1 the same as the answer of shown in the ··8 Example? Explain. 15 3 3 5 15 Yes; Possible explanation: is equivalent to because 3 5 . ···40 ··8 ··8 ··5 ···40

445 ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 445

Fluency & Skills Practice Teacher Toolbox

Assign Multiplying Fractions in Fluency and Skills Practice Word Problems Multiplying Fractions in Word Problems Name: Solve each problem.

__3 5 1 Blanca has pound of tuna salad. She uses 2 __ In this activity students solve word 4 Frank has a board that is 8 yard long. He __1 3 of the tuna salad to make sandwiches. __ 3 cuts off a piece that is 5 the length of the problems that involve multiplying How much of the tuna salad did Blanca use? board. How long is the piece of the board Frank cut off ? two fractions in which both factors are less than 1. Students may 2 1 4 3 __ __ 4 __ Sharon drinks 3 of 2 pint of lemonade. How James lives 5 mile from the library. He has 3 much lemonade did Sharon drink? already walked __ of the way. How far has encounter this situation in their 4 James walked? daily lives, as when determining distances traveled, pounds of meat 5 2 5 6 __ __ Ali worked on his math homework for Madison has 6 yard of fabric. She uses 5 of 2 3 __ hour. He spent __ of the time solving the fabric to make a pillow cover. How much bought, yards of fabric used, and so 3 4 multiplication problems. How much time fabric did Madison use for the pillow cover? did Ali spend solving multiplication on. Students may use various problems? strategies to solve these problems,

including drawing pictures, using 7 How could you draw a picture to solve problem 2? number lines, or producing area models.

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445 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 SESSION 2

3 4 3 Suppose that the green part of Vicky’s towel covers of the towel and the fi sh 3 of the towel has the fish design; See possible ··5 ··5 3 design is drawn on of that part. Draw a picture to fi nd the part of the towel that picture on the student page. ··4 Medium has the fi sh design. Then write the answer. Possible student work: 3 4 3 3 4 12 3 4 3 5 5 , or ··4 ··5 ·····4 3 5 ··20 ··5 Medium

1 5 Student problems should refer to taking of 3 ··6 of the towel has the fish design. 3 3 Solution ··5 the quantity or scaling a measurement of ··8 ··8 1 3 1 4 Write an equation to show the answer to problem 3. by ; , or , unit; See possible work on the ··6 ··48 ··16 student page. Students may also draw a visual fraction model divided into eighths and with 3 4 3 3 4 12 3 3 5 5 , or the eighths divided into sixths. Solution ··4 ··5 ·····4 3 5 ···20 ··5 1 3 Challenge 5 Write a word problem that can be solved by fi nding the product 3 . ··6 ··8 Then solve your problem.

1 3 Student problems should refer to taking of the quantity or Problem ··6 ··8 3 1 scaling a measurement of by . ··8 ··6 Show your work. Possible student work: 1 3 1 3 3 3 3 5 5 ··6 ··8 ·····6 3 8 ···48 3 3 43 1 5 5 ···48 ·····48 43 ···16

3 1 or unit Solution ···48 ···16 446 446 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. English Language Learners: Prepare for Session 3 ELL Differentiated Instruction Use with Connect It.

Levels 1–3 Levels 2–4 Levels 3–5 Speaking/Writing Read Connect It Reading/Writing Have students read Reading/Writing Have students read problem 5 to students. Ask them to work in Connect It problem 5 with a partner. Have Connect It problem 5. Ask students to talk small groups. Help groups make a multiple- students work together to make a multiple- about the strategies with a partner and write column chart to compare and contrast the column chart to compare and contrast the their reflections. strategies that were used to solve the strategies that were used to solve the Then have partners exchange their writing problem. Have groups use the information in problem. and ask and answer questions. Provide the chart to talk about the strategy they liked Have students use the information in the examples or sentence frames as needed: best. Provide a sentence starter: chart to talk about the strategy they liked Why do you prefer ? • The strategy that I like best is . . . best. Provide sentence starters: Why do you think is helpful? Ask simple questions about the strategies • The strategy that I like best is . . . students select. • I like this strategy because . . .

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 446 LESSON 22 SESSION 3 Develop LESSON 22 SESSION 3

Purpose In this session students solve a Develop Multiplying with Mixed Numbers 1 3 word problem that requires finding of 2 . ··2 ··4 in Word Problems Students model the fraction and mixed number

in the word problem either on paper or with Read and try to solve the problem below.

manipulatives to develop strategies for solving 3 Janie has a rectangular garden that is 2 yards in length ··4 1 word problems that involve multiplying and 1 yard in width. She grows roses in of her garden. ··2 fractions and mixed numbers. How many square yards in Janie’s garden has roses?

Start TRY IT Math Toolkit Possible student work: • fraction tiles or circles Connect to Prior Knowledge • fraction bars Sample A • fraction models Materials For each student: Activity Sheet Fraction • grid paper • number lines Bars (3 wholes, 3 bars for fifths) • index cards 1 3 • multiplication models Why Support students’ facility with writing mixed 1 of 2 yd2 of yd2 2 2 4 numbers as fractions. 1 1 3 3 4 of 2 5 1; of 5 How Have students use fraction bars to model 2 . ··2 ··2 ··4 ··8 ··5 3 1 square yards of Janie’s garden have roses. Then have them use the fraction bars to show how ··8 many fifths in 2 wholes and to find how many fifths in all. Sample B 3 Find half of 2 and half of . Start ··4 Solution 1 of 2 is 1. Use fraction bars to show 4 10 4 14 ··2 2 5 1 5 1 3 1 3 3 4 5 5 5 5 of is 3 5 . 2 . Then show how to write ·· ·· ·· ·· ··2 ··4 ··2 ··4 ··8 5 ·· Look for models that 3 3 the mixed number as a fraction. 1 1 5 1 ··8 ··8 reflect understanding 3 DISCUSS IT 1 square yards of Janie’s garden have roses. 4 4 4 4 ··8 Ask your partner: Do you 2 5 1 5 that 2 5 2 1 . ··5 ····5 ··5 ····5 ··5 ··5 agree with me? Why or why not? Tell your partner: At ﬁ rst, I thought . . .

DevelopGrade 5 Lesson 22 Session 3 | Develop Multiplying Language with Mixed Numbers in Word Problems ©Curriculum Associates, LLC Copying is permitted. 447 Why Review the meaning of the term model in ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 447 mathematical contexts. How Tell students that model is a multiple-meaning DISCUSS IT Support Partner Discussion word. Also explain that model can be a noun (thing) and a verb (action). Say: In math, we can use numbers, Encourage students to use the term square yards as they discuss their solutions. expressions, pictures, or graphs to represent different Support as needed with questions such as: situations. A model is a representation, a way of • How would you describe your model? helping us see or understand something. • How did you know when you had found the solution? 3 Common Misconception Look for students who add 2 yards and 1 yard and then ··4 1 3 1 3 TRY IT find of 3 instead of of 2 . As students present their solutions, be sure to have ··2 ··4 ··2 ··4 Make Sense of the Problem them specify how they determined the area of Janie’s whole garden. To support students in making sense of the problem, have them describe how a yard is different Select and Sequence Student Solutions from a square yard. One possible order for whole class discussion: • physical parts showing wholes, fourths, halves, and eighths Ask What is the length of Janie’s garden? the width? • drawings, including area models, to represent the problem 3 • equations that show 2 rewritten as a fraction ··4 1 3 • equations that use the distributive property to find 3 2 1 ··2 1 ··4 2

447 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 DEVELOP

Support Whole Class Discussion Explore diff erent ways to understand multiplying fractions and mixed numbers. Compare and connect the different representations 3 Janie has a rectangular garden that is 2 yards in length ··4 and have students identify how they are related. 1 and 1 yard in width. She grows roses in of her garden. ··2 Ask Where does your model show the area of the How many square yards in Janie’s garden has roses? whole garden? The fraction of the garden that has roses? Pi��e It You can use an area model to help you understand the problem. Listen for Students should recognize that accurate 3 3 The purple shaded region of the area model shows half of 2 . responses include 2 square yards as the area of ··4 ··4 3 1 2 the garden and of that area for growing roses. 4 ··2 3 Models may show 3 wholes with 2 of the model ··4 1 representing the garden and half of that area 2 representing the part that has roses.

PICTURE IT & MODEL IT yd If no student presented these models, connect them to the student models by pointing out the Mo�� It ways they each represent: You can write equations to model the problem. 3 • the area of the rectangular garden You can write 2 as a fraction. ··4 3 3 • the part of the garden that has roses 2 5 2 1 ··4 ··4 8 3 5 1 Ask Why does the equation model include the ··4 ··4 11 11 fraction but this fraction does not appear in the 5 ··4 ··4 1 11 area model? You need to fi nd a fraction of a fraction: of square yards. ··2 ··4 Listen for The picture shows the area of the 1 11 1 11 of means 3 . 3 ··2 ··4 ··2 ··4 garden is 2 square yards. The equation shows the ··4 1 11 1 3 11 11 3 5 area in square yards as . They are the same ··2 ··4 ······2 3 4 ··4 amount written two different ways. 448 448 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. For the area model, prompt students to identify 3 how the model shows 2 square yards. ··4 Deepen Understanding • How is it helpful that the model shows 3 square yards Equation Model in all? SMP 2 Reason abstractly and quantitatively. • How is each square yard used in representing 3 When discussing the equation model, prompt students to consider how the 2 square yards? ··4 commutative property applies. For the equations, prompt students to describe the 1 11 Review that the problem asks you to find of yd2 and that you can replace of steps shown by the equations. ··2 ··4 with the multiplication symbol to write an expression that models the problem. 1 3 • Before of 2 is written as a multiplication ··2 ··4 Have a volunteer come to the board to write the multiplication expression, 1 11 11 1 expression, what is done first? 3 , and the expression with the factors reversed, 3 . ··2 ··4 ··4 ··2 • Why is that done first? Ask Why can you use either expression to solve the problem? Is one expression a “better” model that the other for this problem? Explain. Listen for Since the order of factors can be changed without changing the product, either expression can be used to solve the problem. Students may say that although either expression can be used, the expression that 1 11 1 11 is a more direct translation of the phrase of is the expression 3 ··2 ··4 ··2 ··4 and is the more accurate model of the problem.

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 448 LESSON 22 SESSION 3 Develop SESSION 3

Co�� It CONNECT IT Now you will use the problem from the previous page to understand how to • Remind students that one thing that is alike about multiply fractions and mixed numbers.

all the representations is that they show how to 1 Use the last equation in Model It to fi nd the area of Janie’s garden that has roses. 1 3 11 3 find of the mixed number 2 . or 1 ··2 ··4 Janie’s garden has ···8 ··8 square yards of roses. • Explain that on this page they will analyze strategies Explain how you can use the area model in Picture It to fi nd the area of Janie’s for multiplying a mixed number by a fraction. garden that has roses. Possible answer: The area model shows the area of Janie’s garden with roses 1 1 1 1 1 3 in parts: 1 1 1 1 , or 1 square yards. Monitor and Confirm ··2 ··2 ··8 ··8 ··8 ··8

1 Check for understanding that: 2 Look at the fi rst equation in Model It. Why is the mixed number rewritten as a fraction? • In the area model, the area of the garden that has Possible answer: Since you know how to find a fraction of a fraction, you can rewrite the mixed number as a fraction and use a method that you know. roses is represented by the shaded parts for one 1 3 3 3 How can you multiply 3 2 without changing 2 to a fraction? half of the first square yard, one half of the second ··2 ··4 ··4 1 1 3 3 Possible answer: Find of 2 and of and then add the products. square yard, and of the third square yard. ··2 ··2 ··4 8 ·· 3 3 1 1 3 • 1 square yards of the garden has roses. What is 3 2? 1 What is 3 ? ··8 ··8 ··2 ··2 ··4 3 3 1 1 Support Whole Class Discussion Add the two products. 1 ··8 5 ··8 Is this result the same as your answer to problem 1? Yes 2 – 3 Tell students that these problems will 4 How can you multiply a mixed number by a fraction? prepare them to provide the explanation required in Possible answer: You can write the mixed number as a fraction and then problem 4. multiply or you can break apart the mixed number, multiply each part by the Make sure students understand that problem 2 fraction, and then add the products. focuses on one strategy for multiplying a mixed 5 REFLECT number by a fraction and problem 3 focuses on a Look back at your Try It, strategies by classmates, and Picture It and Model It. different strategy. Both strategies result in the Which models or strategies do you like best for multiplying fractions and mixed answer that is shown by the area model. numbers? Explain. Possible answer: I like to use a visual model because I can see a fraction of a mixed

Ask How are the strategies alike? How are they number. I can think about how to move and combine the parts of my model to

different? write the final answer as either a mixed number or a fraction. 3 3 449 Listen for Both strategies break apart 2 as 2 1 . ··4 ··4 ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 449 The strategy in problem 2 (used in Model It) does this in order to write the mixed number as a Visual Model fraction greater than 1. The strategy in problem Use an area model to rewrite a mixed number as a fraction. 3 3 breaks apart 2 in order to use the distributive ··4 If . . . students are unsure about what to write as the numerator when rewriting a property. Both strategies involve multiplying mixed number as a fraction before multiplying, fractions: the strategy in Model It has only one Then . . . use this activity to connect an area model and the strategy of rewriting a multiplication, while the other strategy has two mixed number as a fraction to multiply. multiplications. Materials For each student: drawing paper (1 sheet) • Explain that you can use the area model shown in Picture It to show the 4 Look for the idea that to multiply a fraction and strategy of rewriting a mixed number as a fraction before multiplying. a mixed number you first rewrite the mixed number • Tell students to draw the 3 squares that represent the 3 square yards in the either as a fraction or as the sum of a whole number 3 area model. Have them shade the model to show 2 , dividing the last square and a fraction. If you do the latter, you multiply each ··4 part by the fraction in the original expression and into fourths by drawing vertical lines to do this. add the products. • Then tell students to draw vertical lines to show the number of fourths in each 3 11 whole, count the number of fourths in 2 , and write it as a fraction. ··4 3 ··4 4 5 REFLECT • Guide students in drawing a horizontal line to divide the model in half to Have all students focus on the strategies used to 1 11 represent the expression 3 . Have students identify what is shown as solve this problem. If time allows, have students ··2 ··4 the answer. 11 , or 1 3 , square yards share their preferences with a partner. 3··8 ··8 4

449 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 DEVELOP SESSION 3

Ap�� It APPLY IT Use what you just learned to solve these problems. For all problems, encourage students to draw some 1 3 6 Izzy has 3 yards of rope. She uses of the rope to attach a tire swing to kind of model to support their thinking. Allow some ··2 ··5 leeway in precision; drawing equal parts in models a tree in her yard. How many yards of rope does Izzy use for the tire swing? can be difficult. Show your work. Possible student work: 21 1 6 , or 2 , yards; See possible work on the ··10 ··10 3 3 3 9 3 Student Worktext page. Students may also use 1 1 5 ··5 ··5 ··5 ··5 ···10 3 7 3 1 9 18 either the expression 3 or 3 3 1 to write 5 ··5 ··2 ··5 1 ··2 2 ··5 ···10 18 3 21 equations to solve the problem. 1 5 ···10 ···10 ···10

45 9 3 21 1 7 feet, 3 feet, or 3 feet; See possible student or 2 yards ··12 ··12 ··4 Solution ···10 ···10 work on the Student Worktext page. Students may 5 7 Colin has a chain that is foot long. He adds links to his chain so that it is 5 1 ··6 also use the expression 3 4 1 to write 1 ··6 1 ··2 2 4 times as long as the original chain. How many feet long is his chain now? ··2 equations to solve the problem. They may also draw Show your work. a visual fraction model divided into thirds, with the Possible student work: 1 1 8 1 9 thirds divided into sixths. 4 5 4 1 5 1 5 ··2 ··2 ··2 ··2 ··2 5 9 5 3 9 45 3 5 _ 5 ··6 ··2 6 3 2 ···12 45 _3 3 15 15 3 5 5 5 3 Close: Exit Ticket ···12 3 3 4 ···4 ··4 45 9 3 _ , 3 12 or 3 feet 8 A; Students may use either the expression Solution ···12 ··4 3 14 3 5 5 3 3 or 3 1 1 to write equations to solve the 8 George has 1 yards of fabric. He plans to use of the fabric to make a pillow. ··4 ··9 ··4 1 ··9 2 ··9 ··4 problem. They may also draw a visual fraction model How many yards of fabric will George use for the pillow? 6 8 Ꭽ 1 Ꭾ 1 divided into ninths, with the ninths divided into ··36 ··13 17 5 Ꭿ 1 ൳ 1 fourths. ··36 ··12 Error Alert If students choose B, C, or D, then review the two strategies for multiplying a mixed 450 number by a fraction using equations, recording 450 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. steps for each. Have students choose one of the strategies and follow the steps, checking off each step as they complete it.

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 450 LESSON 22 SESSION 3 Additional Practice Name: LESSON 22 SESSION 3

Solutions Practice Multiplying with Mixed Numbers

1 2 2 1 2 Study the Example showing one way to solve a word problem with a mixed 1 3 3 5 3 3 1 3 ··4 ··3 1 ··3 2 1 ··4 ··3 2 number. Then solve problems 1–5. 6 2 5 1 ··3 ··12 Example 2 5 2 Mr. Urrego is painting his deck for the summer. He has painted a rectangular area that ··12 2 1 2 2 square yards. is 3 yards long and yard wide. How many square yards of deck are painted? ··12 ··4 ··3 You can use an area model. 3 1 yd Basic 4 The larger sections of the area 1 1 model are 3 1 5 square yard. 2 26 2 ··3 ··3 yd 2 , or 2 , square yards; See possible work on 3 ··12 ··12 The smaller sections of the area the student page. 1 1 1 model are 3 5 square yard. Medium ··3 ··4 ··12 1 yd The model shows the number of square yards painted is: 2 2 2 2 6 2 2 2 1 1 1 5 1 5 2 1 5 2 ··3 ··3 ··3 ··12 ··3 ··12 ··12 ··12

1 Write the missing numbers to complete the multiplication equation showing how much of the deck is painted. Multiply the length and width of the painted area:

1 2 2 1 2 6 2 2 3 5 3 __ 1 3 5 1 5 3 1 3 3 2 2 ··4 ····3 1 ····4 ··3 2 ····3 ····12 ··12 2 2 ···12 square yards

2 To multiply by a mixed number, you can also write the mixed number as a fraction 1 2 and then multiply. Use this method to fi nd the product 3 3 in order to fi nd how ··4 ··3 many square yards of the deck are painted. Show your work. 1 13 13 2 13 3 2 26 Possible student work: 3 5 ; 3 = 5 ··4 ···4 ···4 ··3 ······4 3 3 ···12

26 2 or 2 square yards Solution ···12 ···12 451 ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 451

Fluency & Skills Practice Teacher Toolbox

Assign Multiplying with Mixed Fluency and Skills Practice Multiplying with Mixed Numbers Name: Numbers in Word Problems in Word Problems In this activity students solve word Solve each problem. __1 __2 3 1 Neil has 2 pounds of apples. He uses of 2 __ 4 3 Kathy is riding her bike 1 5 miles to her the apples to make pies. How many pounds friend’s house. She has already traveled problems that involve multiplying a 7 of apples does Neil use to make pies? __ 8 of the distance. How far has Kathy mixed number by either a fraction already traveled? or another mixed number. Students 1 3 __ 4 Keisha spent 3 3 hours at the science Javier is planting a rectangular garden that 2 1 __ __ museum. She spent 5 of that time in the will be 7 2 yards long and 1 yard wide. He may use this skill in their daily lives, __1

planetarium. How much time did Keisha will plant 5 of the garden with tomatoes. spend in the planetarium? How many square yards of the garden will such as when multiplying mixed be planted with tomatoes? numbers involved in recipes,

5 Ed has two dogs. The smaller dog weighs 6 Shane designed a rectangular mural that is distances, areas, and weights. __1 __1 __3 __1 2 yards long and 1 yards high. What is 8 3 pounds. The larger dog weighs 1 2 times 4 3 as much as the smaller dog. How much does the area in square yards of the mural? Students may use different the larger dog weigh? strategies, such as drawing area models or making number lines, to 7 How could you use an area model to solve problem 4? solve these problems.

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451 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 SESSION 3

15 7 3 1 3 , or 1 , miles; Students may also use the 3 On Saturday, Kira ran mile. On Sunday, she ran 2 times as far as on Saturday. Use ··8 ··8 ··4 ··2 3 1 expression 3 2 1 to write equations to a multiplication equation to fi nd how far Kira ran on Sunday. Show your work. ··4 1 ··2 2 solve the problem. Possible student work: 1 1 4 1 5 2 = 2 1 = 1 5 Medium ··2 ··2 ··2 ··2 ··2 3 5 15 7 3 5 5 1 ··4 ··2 ···8 ··8 4 See possible visual model on the student page. Students may draw other visual models divided into fourths, with the fourths divided into

halves. 15 7 or 1 miles Medium Solution ···8 ··8 4 Use a visual model to show another way to fi nd the distance Kira ran on Sunday. 3 1 5 Less than 1 square yard; Students may also use Accept visual models that show 3 2 . Possible model: 5 3 3 1 ··4 ··2 either the expression 3 or 3 1 1 to ··4 ··4 ··4 1 ··4 2 write equations to solve the problem. They may also draw visual models divided into sixteenths. Challenge 5 The multipurpose room at the Cortez School is being set up for the annual book 1 sale. Graphic novels will be displayed in a rectangular area 1 yards long and ··4 3 yard wide. Will the graphic novels be displayed in an area greater than or less ··4 than 1 square yard? Show your work. Students may use area models, equations, or some other method to find that 1 3 15 1 3 5 square yard. ··4 ··4 ···16

Solution less than 1 square yard 452 452 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. English Language Learners: Prepare for Session 4 ELL Differentiated Instruction Use with Apply It.

Levels 1–3 Levels 2–4 Levels 3–5 Speaking/Writing Read Apply It Reading/Writing Have students read Reading/Writing Have students read problem 8 to students. Model writing your Apply It problem 8 with a partner. Apply It problem 8 with a partner. own problem using another Apply It problem Ask students to make a T-chart and label one Have partners write their word problem. as a guide. Change the names, ideas, or column “Names” and the other “Topics.” Encourage them to share their work with actions. For example, you may choose to Ask students to complete the T-chart and other partners and have each pair review and revise problem 3 to: swap with another set of partners. Have them comment on each other’s work. 3 3 Ms. Martinez has ··4 bags of popcorn. use the charts to write word problems for the 1 Her students ate ··2 of her popcorn. What expression provided. fraction of the popcorn bags did Ms. Martinez’s students eat? Have students to brainstorm a list of names and actions that can be used in their word problems. Ask students to use the bank of names and phrases and work with a partner to write a word problem for the expression provided.

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 452 LESSON 22 SESSION 4 Refine LESSON 22 SESSION 4

Purpose In this session students solve word Refine Multiplying Fractions in Word Problems problems involving multiplication of fractions and mixed numbers using visual fraction models Complete the Example below. Then solve problems 1–8. and equations and then discuss and confirm

their answers with a partner. Breaking apart a mixed Before students begin to work, use their Example number happens twice in this problem. Chris uses 4 1 tubes of paint. Nico uses 1 1 times as much responses to the Check for Understanding to ··4 ··2 determine those who will benefit from paint as Chris. How much paint did Nico use? additional support. Look at how you can solve this problem using equations.

As students complete the Example and 1 1 4 3 1 5 4 problems 1–3, observe and monitor their ··4 ··4 1 1 1 1 1 1 4 3 5 4 3 1 3 5 2 1 reasoning to identify groupings for ··4 ··2 1 ··2 2 1 ··4 ··2 2 ··8 1 1 1 1 2 1 3 differentiated instruction. 4 1 2 1 5 6 1 5 6 1 5 6 ··4 ··8 ··4 ··8 ··8 ··8 ··8 PAIR/SHARE 3 6 tubes of paint How does the product Solution ··8 1 compare to 4 ? Start ··4 Check for Understanding Ap�� i� 3 2 1 Josh exercises at the gym 3 hours a week. He spends of his time Why Confirm understanding of multiplying ··4 ··5 How do I know what fractions to solve word problems. at the gym lifting weights. How many hours a week does Josh operation to use to solve this problem? How Have students solve the word problem using spend lifting weights at the gym? Show your work. any strategy they want. Remind them to show their Possible student work using an equation: 2 3 2 6 6 3 3 1 3 5 1 work. 1 ··5 2 1 ··4 ··5 2 ··5 ···20 24 6 5 1 Start ···20 ···20 Solution 30 3 5 Don practices piano for hour ···20 3 1 ···20 3 1 hour every day. He spends 5 , or 1 PAIR/SHARE ··4 ··5 ··2 ··2 of that time practicing scales. What is a reasonable estimate for the number of How long does he practice hours Josh lifts weights 3 1 or 1 hours each week? scales? Solution ··2 ··2

453 ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 453

Grade 5 Lesson 22 Session 4 | R e ﬁ n e Multiplying Fractions in Word Problems ©Curriculum Associates, LLC Copying is permitted. Error Alert If the error is . . . Students may . . . To support understanding . . .

Discuss the meaning of the problem with students. 19 1 3 hour have added the fractions. Reinforce the idea that the word of in the phrase of is ··20 ··5 ··4 interpreted as multiplication.

1 3 Have students use a visual model to find of . Students ··5 ··4 may draw a number line showing 1 whole divided into have multiplied the 1 fourths and then divide each fourth into 5 equal parts. hour denominators and neglected ··20 Have students note that the whole is now divided the numerators. into twentieths. Have students shade 1 fifth of each of 1 3 3 3 fourths to see that of is . ··5 ··4 ··20

453 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 REFINE

5 3 2 A fi eld is in the shape of a rectangle mile long and mile wide. Example ··6 ··4 What model can I use to 3 What is the area of the fi eld? Show your work. help understand this 6 tubes of paint; The equations shown are one way problem? ··8 Possible student work using an area model: to solve the problem. Students could also solve the 1 17 1 3 5 problem by rewriting 4 as , rewriting 1 as , 6 ··4 ··4 ··2 ··2 and multiplying the two fractions. They could also 3 4 draw a visual model divided into fourths, with the fourths divided into halves. PAIR/SHARE Can you solve this problem Look for The solution requires multiplying a mixed 15 5 or square mile in another way? number by a mixed number. Explain that Solution ···24 ··8 3 1 parentheses can be used to show what operation 3 Ari had of a bag of popcorn. His friends ate of his popcorn. ··4 ··2 What equation can I write should be done first. In this case, the multiplication What fraction of the whole bag of popcorn did Ari’s friends eat? to solve this problem? must be done before the addition. Ꭽ 1 ··4 Ꭾ 3 ··8 APPLY IT Ꭿ 5 3 1 ··4 1 , or 1 , hours; Students could solve the 3 ··2 ··2 ൳ 2 3 2 ··2 problem using the expression 3 3 1 × . 1 ··5 2 1 ··4 ··5 2 DOK 2 Kayla chose Ꭽ as the correct answer. How did she get that answer? 2 3 1 3 PAIR/SHARE Look for The solution requires finding of 3 , Possible answer: Kayla subtracted from instead of ··5 ··4 ··2 ··4 Does Kayla’s answer make 2 3 1 3 so find the product 3 3 . multiplying by . sense? ··5 ··4 ··2 ··4 15 5 2 , or , square mile; Students could find the ··24 ··8 5 3 product 3 by drawing a unit square divided ··6 ··4 into sixths one way and fourths the other way, shading five of the sixths and three of the fourths and labeling the length and width of 5 the section of overlapping shading as mile ··6 3 454 and mile. ··4 454 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. DOK 2 Look for The solution requires finding the area of a rectangle.

3 B; Students could solve the problem by writing 1 3 3 the equation 3 5 . ··2 ··4 ··8 Explain why the other two answer choices are not correct: 3 1 C is not correct because it is the sum of and . ··4 ··2 D is not correct because it is the product of 3 1 and . ··2 DOK 3

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 454 LESSON 22 SESSION 4 Refine SESSION 4

1 24 4 C; First, find of 24, 5 4. Subtract 4 from 24 4 On Sunday, Kristen bought a carton of 24 bottles of water. ··6 ··6 to find number of bottles remaining after 1 20 Monday, 24 2 4 5 20. Then find of 20, 5 5. ··4 ··4 Subtract 5 from 20 to find number of bottles remaining after Tuesday, 20 2 5 5 15. Identify

the picture showing 15 bottles. 1 • On Monday, Kristen drank of the bottles in the carton. DOK 2 ··6 1 • On Tuesday, Kristen drank of the bottles that remained in the carton ··4 18 1 5 , or , cup; He’s making only 6 servings, so after Monday. ··36 ··2 3 6 multiply by . Which picture represents the number of bottles of water remaining in ··4 ··9 the carton after Kristen drank her water on Tuesday? DOK 2 2 1 Error Alert Students who wrote 4 , or 4 , cups ··4 ··2 may not have written the number of servings Milo Ꭽ Ꭾ 6 wants to make as a fraction of the whole recipe 1 ··9 2 3 but instead multiplied by 6. ··4

Ꭿ ൳

3 5 Milo’s pancake recipe makes 9 servings. It calls for cup milk. Milo wants to ··4 make 6 servings. How much milk will he need? 18 1 or ···36 ··2 cup

455 ©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 455 Differentiated Instruction RETEACH EXTEND

Visual Model Challenge Activity Model multiplication with mixed numbers. Solve a two-step problem. Students struggling with fraction multiplication involving a mixed number Students who have achieved proficiency Will benefit from additional work building visual models Will benefit from deepening Materials For each pair: tracing paper (2 sheets), ruler, colored pencils or crayons understanding of modeling problems (2 different colors), tape • Give students the following two-step 2 2 • Distribute materials to pairs. Write 3 1 on the board. problem: Tim made a snack mix with ··3 ··5 3 5 cup of raisins and cup of nuts. He • On one sheet of paper have students draw two squares that share a common side. Have ··4 1 ··8 them trace the same figure onto the other sheet of paper. ate of the snack mix. How many cups ··5 3 5 11 2 of the mix did he eat? 1 5 and • Have students draw vertical lines to divide one figure into fifths, shade it to show 1 , 3 ··4 ··8 ··8 5 ·· 1 3 11 5 11 ; Tim ate 1 1 cups. draw horizontal lines to divide the other figure into thirds, and shade it a different color to ··5 ··8 ··40 ··40 4 2 show . Have students tape one figure on top of the other, aligning the figures. • Have students solve the problem, using ··3 2 2 14 both visual models and equations to • Have pairs identify the intersection of the shadings to find 3 1 . ··3 ··5 3 ··15 4 1 1 represent the situation. Have students • Repeat with other fraction expressions involving mixed numbers, such as 2 3 . ··2 ··4 compare models and equations.

455 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 22 REFINE SESSION 4

396 99 9 6 , , or 9 square units; Multiply the 6 Jillian draws a rectangle with the dimensions shown below. What ···40 ··10 ··10 12 33 396 is the area of Jillian’s rectangle? length by the width, 3 5 . ··5 ··8 ···40 DOK 1 2 2 units 5 15 7 7 The left tree is , or 1 , feet tall. The middle 1 ··8 ··8 4 units 1 35 3 8 tree is 2 feet tall. The right tree is , or 4 , feet 396 99 9 ··2 ··8 ··8 , , or 9 square units tall; For the left tree height, multiply the height Solution ····40 ···10 ···10 3 3 5 15 1 3 5 7 Lily paints 3 trees for a wall mural. The middle tree is 2 ft tall. of the middle tree by , . The ··2 ··4 ··4 ··2 ··8 3 The tree on the left is as tall as the middle tree. The tree height of the middle tree is given in the ··4 3 on the right is 1 times as tall as the middle tree. How tall is each problem. For the right tree height, multiply the ··4 7 7 5 35 tree? Show your work. height of the middle tree by , 3 5 . ··4 ··4 ··2 ··8 Possible answer: DOK 2 1 5 3 7 2 5 ; 1 5 ··2 ··2 ··4 ··4 3 5 15 7 Left tree: 3 5 5 1 ··4 ··2 ···8 ··8 7 5 35 3 Right tree: 3 5 5 4 ··4 ··2 ···8 ··8 15 7 The left tree is , or 1 , ft tall. The middle tree Solution ···8 ··8 1 3 is 2 ft tall. The right tree is 4 ft tall. ··2 ··8

8 MATH JOURNAL 1 1 Write a word problem for the expression 3 3 . Use a visual ··2 ··2 model or an equation to show how to solve your problem. 1 Possible answer: Mary wants to build a fence that is 3 yards ··2 1 long. She builds of her fence on Monday. How many yards of ··2 fence did she build on Monday? 1 7 1 7 7 3 5 ; 3 5 ··2 ··2 ··2 ··2 ··4 7 Mary builds yards of fence on Monday. ··4 SELF CHECK Go back to the Unit 3 Opener and see what you can check off . 456 456 Lesson 22 Multiply Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. Close: Exit Ticket REINFORCE PERSONALIZE 8 MATH JOURNAL Problems 4–8 Student responses should indicate understanding of Multiply fractions. problem situations that require multiplying a fraction Provide children with and a mixed number, such as finding the area of a All students will benefit from additional work with opportunities to work rectangle or scaling a quantity, and understanding of multiplying fractions by solving problems in a variety on their personalized how to use a visual model or equations to represent of formats. instruction path and solve a problem of this type. • Have students work on their own or with a partner with i-Ready Online Error Alert If students multiply only the two to solve the problems. Instruction to: 1 fractions in the expression, writing the answer as 3 , ··4 • Encourage students to show their work. • fill prerequisite gaps then review multiplying with a mixed number to • build up grade-level clarify that the 3 wholes must be included in the skills multiplication. Have them rewrite the product 1 1 as 3 1 3 and use the distributive property. 1 ··2 2 ··2

SELF CHECK Have students consider whether they feel they are ready to check off any new skills on the Unit 3 Opener page.

©Curriculum Associates, LLC Copying is not permitted. Lesson 22 Multiply Fractions in Word Problems 456 Lesson LESSON 24 Overview Divide Unit Fractions in Word Problems

Lesson Objectives Prerequisite Skills Lesson Vocabulary

Content Objectives • Divide whole numbers. There is no new vocabulary. Review the • Multiply whole numbers and fractions by following key term. • Represent and solve real-world problems unit fractions. • unit fraction a fraction with a involving division of unit fractions by numerator of 1. Other fractions are built whole numbers using visual fraction • Understand that multiplication and from unit fractions. models and equations. division are inverse operations. • Represent and solve real-world problems • Use a visual fraction model to find the involving division of whole numbers by quotient of a unit fraction divided by a unit fractions using visual fraction whole number. models and equations. • Use a visual fraction model to find the • For a given division equation with a unit quotient of a whole number divided by a fraction and a whole number, use the unit fraction. inverse relationship between multiplication and division to write a Standards for Mathematical related multiplication equation. Practice (SMP) Language Objectives SMPs 1, 2, 3, 4, 5, and 6 are integrated in • Draw visual models to represent word every lesson through the Try-Discuss- problems involving division with unit Connect routine.* fractions. In addition, this lesson particularly • Write equations to represent word emphasizes the following SMPs: problems involving division with unit fractions. 2 Reason abstractly and quantitatively. • Describe the relationship between a 4 Model with mathematics. visual model and an equation that both 5 Use appropriate tools strategically. represent the same problem situation. 8 Look for and express regularity in • Write a multiplication equation to check repeated reasoning. a quotient of a division equation. *See page 305m to see how every lesson includes these SMPs.

Learning Progression

In the previous lesson students acquired a In this lesson students apply their In Grade 6 students will solve conceptual understanding of dividing with understanding of division with unit mathematical and real-world problems unit fractions (fractions with a numerator fractions to solve word problems. Students involving division of fractions by fractions. of 1). They used visual models, such as area use models to help them make sense of models, bar models, number lines, and the problems and to write equations to represent the division with unit fractions in pictures to understand what it means to the problems. Students find quotients to divide with unit fractions. Students divided solve the problems, and they use the whole numbers by unit fractions and divided inverse relationship between unit fractions by whole numbers. They used a 1 multiplication and division to check their their understanding that 5 a 4 b 5 a 3 ··b ··b answers. to write multiplication equations to solve division problems with unit fractions.

469a Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. Lesson Pacing Guide Teacher Toolbox Whole Class Instruction Small Group Differentiation SESSION 1 Interactive Tutorial* (Optional) Additional Practice PREPARE Explore Prerequisite Review: Lesson pages 473–474 Understand Division with Unit Fractions Ready Prerequisite Lesson 45–60 min Grade 4 Dividing Unit Fractions in Word • Lesson 24 Multiply Fractions by Whole Problems Numbers • Start 5 min • Try It 10 min RETEACH • Discuss It 10 min • Connect It 15 min Tools for Instruction • Close: Exit Ticket 5 min Grade 4 • Lesson 24 Multiply a Whole Number and a SESSION 2 Dividing a Unit Fraction by a Whole Additional Practice Fraction Develop Number Lesson pages 479–480 • Start 5 min Grade 5 45–60 min • Try It 10 min Fluency • Lesson 24 Dividing Unit Fractions to Solve • Discuss It 10 min Dividing a Unit Fraction Word Problems • Picture It & Model It 5 min by a Whole Number • Connect It 10 min REINFORCE • Close: Exit Ticket 5 min Math Center Activity Grade 5 SESSION 3 Dividing a Whole Number by a Unit Additional Practice • Lesson 24 Find the Division Expression Develop Fraction Lesson pages 485–486 • Start 5 min Fluency 45–60 min • Try It 10 min EXTEND Dividing a Whole Number • Discuss It 10 min by a Unit Fraction Enrichment Activity • Model Its 5 min • Connect It 10 min Grade 5 • Close: Exit Ticket 5 min • Lesson 24 Switching Places

SESSION 4 Dividing Unit Fractions in Word Lesson Quiz Refine Problems or Digital • Start 5 min Comprehension Check Independent Learning 45–60 min • Example & Problems 1–3 15 min • Practice & Small Group PERSONALIZE Differentiation 20 min • Close: Exit Ticket 5 min i-Ready Lesson* Grade 5 Lesson Materials • Divide Unit Fractions in Word Problems Lesson Activity Sheet: Fraction Bars (Required) 1 Activities Per student: 1 set of fraction circles or tiles, yarn foot , scissors, glue, ruler, 1 ··2 2 sheet of paper, modeling clay (2 equal-sized portions) Activity Sheet: Fraction Bars Math Toolkit fraction tiles, fraction bars, number lines, grid paper, index cards, sticky notes, ribbon or yarn Digital Math Fraction Models Tool

*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 24 Divide Unit Fractions in Word Problems 469b LESSON 24 Connect to Family, Community, and Language Development

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

This week your child is learning about dividing with • Together with your child, solve the problem below about dividing by unit fractions in word problems. a unit fraction.

He or she might see a word problem like the one below. How many square tiles are needed to make a border along a wall? Each tile 1 measures foot on each side, and the wall is 6 feet long. 1 ··3 Molly used square yard of fabric to decorate 4 fl ags. She used an equal amount of ··4 fabric for each fl ag. How much fabric did she use for each fl ag?

1 1 This problem can be solved by fi nding 4 4. ··4 4 One way to understand this problem is to use a model.

The square shown at the right represents 1 whole square yard of fabric. The shaded rectangle represents the 1 square yard that Molly used to decorate the 4 fl ags. ··4

You can divide the shaded rectangle into 4 equal parts 1 4 to represent the 4 fl ags Molly decorated. • Now suppose you are going to use the tiles to make a border along a Flag 1 The part shaded dark blue shows the amount used wall in your own house. First, measure to fi nd the length of the wall in Flag 2 for one fl ag. 1 out of 16 parts of the whole square yard feet. Then round your measurement to the nearest foot. Last, divide that 1 1 Flag 3 number by to fi nd the number of tiles you would need. is used for 1 fl ag. Molly used square yard of fabric for 3 ··16 ·· Flag 4 each fl ag.

Your child can also write a division equation to solve the problem.

1 1 4 4 5 ··4 ··16 Invite your child to share what he or she knows about dividing with unit fractions in Answer: 6 4 __ 1 5 18 tiles word problems by doing the following activity together. 3

469 470 ©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 469 470 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted.

Goal Math Talk at Home The goal of the Family Letter is to provide opportunities for families Have students and family members talk about unit fractions at to help students reinforce their understanding of dividing with unit home. They can find examples as they talk about cooking, building, fractions in the context of word problems. or sharing.

Activity Conversation Starters Below are additional conversation starters students can write in their Family Letter or math journal to engage Encourage students to work with family members to solve family members: real-world problems that involve dividing with a unit fraction. • What unit fractions are easy to show by cutting a pizza or pie into Look at the Dividing by Unit Fractions activity and modify it if equal shares? necessary to connect with your students. • How do you divide granola bars in halves? How do you divide them in sixths?

469–470 Lesson 24 Divide Unit Fractions in Word Problems ©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 Try It. Session 2 Use with Apply It. • Ask students to share if they have participated in races in schools or • Tell students that some problems involving fractions use pizza. know anyone who runs. If time permits, ask students to research Pizza is a circular food that can be separated into parts. Ask any local or national races for a given time period, including the students to share what foods in their culture have a circular shape. distance of each race. Have students select one of the races and If time permits, have students describe their food and how they make a diagram that shows water stops. could divide it.

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 Try It. Levels 1–3 Levels 2–4 Levels 3–5 Speaking/Writing Read the Try It Speaking/Writing Read the Try It Speaking/Writing Have students read problem aloud. Help students make sense of problem with students. Organize them into and discuss the Try It with a partner. the problem by acting it out. Use 6 equal pairs to model and solve the problem using Encourage partners to explore strategies pieces of string to represent each mile of the fraction bars. Provide a bank of terms to together, but to work separately to create race. Place them end to end with no gaps or support them as they think aloud to model their own models. Then have them debrief overlaps. Ask volunteers to stand on the the problem: fraction, numerator, and compare their work. Ask: How are your string to represent each half mile mark and at denominator, whole, part, divide, multiply, models alike? How are they different? the end to represent the water stop at the half, and represent. Have students complete Have students reflect on their solutions. finish line. After the role-play activity, the sentence frames below and point out Ask: Is your answer reasonable, considering the organize students into pairs to model the how their model supports each statement: context of the problem? problem with fraction bars. Encourage them • The model shows 6 wholes. Instruct students to explain, in writing, why to use gestures, pictures, numbers, and words • The wholes are divided into halves . they think their answer is reasonable. to communicate ideas. Have them write an answer using the frame: • There are 12 water stops.

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 470a LESSON 24 SESSION 1 Explore LESSON 24 SESSION 1

Purpose In this session students draw on Explore Dividing Unit Fractions in Word Problems their understanding of what it means to divide with unit fractions. They share models to explore Learning Target Previously, you learned about what it means to divide • Solve real world problems involving how to represent problem situations with visual with unit fractions. Use what you know to try to solve the division of unit fractions by non-zero whole numbers and division models. They will look ahead to think about problem below. of whole numbers by unit fractions, e.g., by using visual fraction models equations they can use to represent and solve and equations to represent the Micah is running a 6-mile race. problem. these types of problems. 1 SMP 1, 2, 3, 4, 5, 6, 8 There are water stops every mile ··2 and at the 6-mile fi nish line. Start How many water stops are there in all? Use a visual model to show Connect to Prior Knowledge your solution. Why Support students’ facility with using a visual model to represent division of a whole number by a unit fraction. TRY IT Math Toolkit Possible student work: • fraction bars How Have students copy and then complete a • fraction models Sample A • number lines number line model to represent the expression • grid paper Use 6 rectangles to represent the 6 miles. 1 • index cards 2 4 . Discuss that the quotient is 10 and that this ··5 start finish • sticky notes 1 means there are 10 groups of in 2. ··5 Start first last Look for Number stop stop Copy and complete the lines should show 1 There are 12 water stops. model to show 2 4 . each whole divided 5 ·· into fifths, so there are Sample B 0 1 2 10 fifths in all. 6-mile race with 2 water stops each mile W W W W W W W W W W W W DISCUSS IT Ask your partner: Why did 1 1 1 1 1 1 0 1 1 2 2 3 3 4 4 5 5 6 you choose that strategy? start 2 2 2 2 2 2 nish Tell your partner: A model Grade 5 Lesson 24 Session 1 | Explore Dividing Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is permitted. There are 12 water stops in all. I used was . . . It helped TRY IT me . . . Make Sense of the Problem 471 To support students in making sense of the ©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 471 problem, have them identify that there are water Common Misconception Look for students who are not comfortable interpreting stops at half-mile intervals along the 6-mile race the location of water stops at every half mile as including water stops at whole- 1 1 1 route. The first water stop is at the half-mile mark number miles as well as at mile, 1 miles, 2 miles, and so on. As students present ··2 ··2 ··2 and the last one is at the 6-mile mark. solutions, have them specify how far beyond the first water station the second water station is and where along the race route it is located. DISCUSS IT Select and Sequence Student Solutions Support Partner Discussion One possible order for whole class discussion: Encourage students to use the Discuss It question • physical parts showing wholes and halves and sentence starter on the Student Worktext page as part of their discussion. • drawings to represent the problem Look for, and prompt as necessary for, • number lines marked in wholes and halves understanding of: • division or multiplication equations showing 12 water stops in all • 6 as the number of wholes 1 Support Whole Class Discussion • as the equal-size parts each whole is divided into ··2 Prompt students to note the relationship between the numbers in each model and the numbers in the problem. Ask How do [student name]’s and [student name]’s models show the number of wholes? The number of parts in each whole? 1 Listen for 6 is the number of wholes. Each part is , so there are 2 in each whole. ··2 471 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 EXPLORE SESSION 1

CONNECT IT CONNECT IT 1 LOOK BACK 1 LOOK BACK Look for understanding that you can show the How many water stops are there in all? Explain how you can use a number line to support your answer. 6 miles of the race on the number line, mark each 12 water stops; Possible explanation: You can use tick marks to show the 1 mile along it to represent the location of a water 1 locations of water stops at each -mile point of the 6-mile race. Then count ··2 ··2 stop, showing there are 12 water stops in all. how many water stops there are in all. 2 LOOK AHEAD Hands-On Activity On the previous page, you used a visual model to solve a word problem involving dividing a whole number by a unit fraction. You can also use equations to Use fraction bars for fraction division. represent and solve these types of problems. Consider this word problem. 1 If . . . students are unsure about using visual Micah now runs in a 5-mile race. There are water stops every mile and at the ··3 models to show division with fractions, 5-mile fi nish line in this new race. How many water stops are there in all? Then . . . use this activity to make a model. a. Complete the division equation below. 1

Materials For each student: Activity Sheet miles in race 5 4 ··3 5 w number of water stops Fraction Bars (6 wholes) fraction of a mile for • Distribute fraction bars. Explain to students each water stop that each bar represents 1 mile of the race. b. Complete the multiplication equation below. • Prompt students to identify the water stops 1 miles in race 5 3 3 5 w number of water stops for the first mile are at mile and 1 mile. ··2 Have them fold one fraction bar in half, open number of water it, and label one water stop on the fold and stops in each mile one at the end of the fraction bar. c. How many water stops are in this race? Explain how you know. • Guide students to see the stops for the first 15; Possible explanation: There are 5 miles and 3 water stops in each mile. mile represent the location of stops for each 3 REFLECT 1 1 Explain what it means to divide 5 by , or 5 4 . mile. Have them fold and mark the other ··3 ··3 fraction bars accordingly. Possible answer: It means finding out how many thirds are in 5 wholes. • Have students arrange the fraction bars 472 end-to-end and count the water stops along 472 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. the route, confirming there are 12 water stops. • Repeat activity for additional division Close: Exit Ticket 1 1 expressions, such as 3 4 and 4 4 . ··2 ··4 3 REFLECT Look for understanding of what it means to divide a whole number by a unit fraction. Student responses should include references to finding how many parts equal in size 2 LOOK AHEAD to the unit fraction there are in a given number of wholes. Point out that you can also model and solve Common Misconception If students think that the quotient cannot be greater than problems involving dividing whole numbers by unit the dividend, then have them write the quotients for the following expressions: 6 4 fractions using division or multiplication equations. 3, 6 4 2, 6 4 1 and discuss the patterns they see. Have them extrapolate from the Ask Why in one equation is the number of miles, 5, pattern how the quotient compares to the dividend when the divisor is less than 1. 1 divided by the fraction , and in the other equation 5 ··3 Real-World Connection is multiplied by 3? Encourage students to think about everyday places or situations where people Listen for The division equation represents might need to divide with unit fractions. Have volunteers share ideas. Examples: food 1 dividing the 5-mile race into -mile segments to ··3 preparation, construction, physical training. 1 find how many s there are in all; the ··3 multiplication equation represents multiplying the number of water stops in one mile, 3, by the number of miles in the race, 5.

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 472 LESSON 24 SESSION 1 Additional Practice Name: LESSON 24 SESSION 1

Solutions Prepare for Dividing Unit Fractions in Word Problems

Support Vocabulary Development 1 Think about what you know about fraction models. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can. Possible answers: 1 Have students discuss models they know. Then ask them to share what they know about fractions. What Is It? What I Know About It Ask: How can you represent a fraction? Have students think of shapes they can divide into equal parts. I can use a rectangle, a circle, or a number Then ask them to draw the shape and show how to a visual model that represents a fraction divide it into equal parts. line to represent a fraction. As students write examples in the graphic organizer, ask them to identify how the whole and the fraction are represented. fraction model 2 Read the problem. Ask: How many wholes do you need to divide? Have students draw a model to show Examples Examples Examples the 4 wholes. Then ask: How do you divide each whole 1 into parts of size ? 2 2 2 ··2 4 ··3 ··3

··5 2 Supplemental Math Vocabulary 3 • bar diagram 0 1 2 3 • number line

1 2 Draw a fraction model to show the expression 4 4 . ··2 Possible answer:

473 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 SESSION 1

3 Assign problem 3 to provide another look at 3 Solve the problem. Show your work. dividing with unit fractions in word problems. Adela has a ribbon that is 2 yards long. She cuts the ribbon into 1 This problem is very similar to the problem about pieces that are yard long. How many pieces of ribbon are there ··4 Micah running a 6-mile race. In both problems, in all? Use a visual model to show your solution. students are asked to draw a visual model to help them to divide a whole number by a unit fraction. The question asks how many pieces of ribbon are Possible student work using a picture: there in all. 2 yd Students may want to use fraction bars, number lines, string, or ribbon. 1 yd Suggest that students read the problem three times, 4 1 asking themselves one of the following questions 2 4 5 2 3 4 5 8 ··4 each time: • What is this problem about? • What is the question I am trying to answer? Solution There are 8 pieces of ribbon in all. • What information is important? 4 Check your answer. Show your work. Solution: Possible student work: 1 2 4 5 2 3 4 5 8. Check that models show 8 pieces. 2-yard ribbon with 4 pieces for each yard ··4 Medium 0 1 2 3 1 5 6 7 2 4 4 4 4 4 4 4 Have students solve the problem a different 2 3 4 5 8 way to check their answer. There are 8 pieces of ribbon in all.

474 474 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. English Language Learners: Prepare for Session 2 ELL Differentiated Instruction Use with Apply It.

Levels 1–3 Levels 2–4 Levels 3–5 Listening/Speaking Have students Listening/Speaking Have students Speaking/Writing Ask students to read chorally read Apply It problem 6. In partners, chorally read Apply It problem 6. In partners, Apply It problem 6 and to write the steps ask them to talk about the problem. Use ask them to talk about the problem. Use they will take to solve the problem. questions and sentence frames to guide questions to guide them: Have students trade the steps with a partner. them: • What food does Felipe want to share? Partners review the steps and revise as • What food does Felipe want to share? • How much of that does Felipe have? needed. Once partners agree on a set of steps, have them solve the problem and talk • How much of that does Felipe have? • How will Felipe share it? about the answer. • How will Felipe share it? • How much will each person get? Ask: Did you and your partner agree on all the • How much will each person get? steps? How much pizza does each friend get?

• Felipe wants to share . • He has of the . • Felipe will share it with . • Each person will get .

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 474 LESSON 24 SESSION 2 Develop LESSON 24 SESSION 2

Purpose In this session students solve a Develop Dividing a Unit Fraction by a Whole Number 1 problem that requires dividing by 3. Students ··6 model the numbers in the word problem either Read and try to solve the problem below.

on paper or with manipulatives to develop 1 Piper uses yard of ribbon to make strategies for solving word problems that ··6 a border around an equilateral triangle. involve dividing a unit fraction by a How long is the piece of ribbon that Piper uses whole number. for each side?

Start TRY IT Math Toolkit Possible student work: • fraction bars Connect to Prior Knowledge • fraction models Sample A • number lines Why Support students’ facility with using a visual 1 1 • grid paper Each piece of ribbon is of . model to represent division of a unit fraction by a ··3 ··6 • ribbon or yarn 1 1 1 So, 4 3 5 3 . • index cards whole number. ··6 ··3 ··6 1 1 1 3 5 How Have students copy and then complete a ··3 ··6 ···18 1 1 For each side, Piper uses a piece of ribbon that is yard long. fraction bar model to represent the quotient 4 2. ···18 ··5 1 Discuss that the quotient is . ··10 Sample B Start Look for Models Use a rectangle to represent 1 whole yard of ribbon. Copy and complete the should show each fifth side 1 1 model to show 4 2. side 2 ··5 divided into halves, so side 3 1 there are 10 parts in 6 1 all, to show as the ··10 The whole is divided into 18 equal parts. quotient. 1 Piper uses a -yard-long piece of ribbon for each side. ···18 DISCUSS IT Ask your partner: How did you get started?

Grade 5 Lesson 24 Session 2 | Explore Dividing a Unit Fraction by a Whole Number ©Curriculum Associates, LLC Copying is permitted. Develop Language Tell your partner: I Why Clarify understanding of the word border as it knew . . . so I . . . relates to the perimeter or length around an object. 475 How Explain that to make a border around ©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 475 something means to put something around its edge. Model with a ribbon, yarn, or a measuring DISCUSS IT tape and a shape or classroom object. Have students Support Partner Discussion find the word in the Try It problem. Ask: What is Piper Encourage students to name the model or strategy they used as they discuss. using to make a border around the triangle? Support as needed with questions such as: • How would you describe your model? TRY IT • How is your model similar to your partner’s? Make Sense of the Problem Common Misconception Look for students who confuse the dividend and divisor 1 1 To support students in making sense of the and find 3 4 instead of 4 3. As students present solutions, have them specify the ··6 ··6 problem, have them identify an equilateral triangle quantity being divided and the number of parts it is divided into. as one that has equal-length sides. Ask What is the whole length of ribbon Piper has? Select and Sequence Student Solutions How many pieces does she need? One possible order for whole class discussion: • physical parts showing fifths, thirds, and fifteenths • drawings to represent the problem • number lines marked to show the fractions 1 • division or multiplication equations showing a ribbon length of yard ··18

475 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 DEVELOP

Support Whole Class Discussion Explore diff erent ways to understand dividing a unit fraction by a whole number to Compare and connect the different representations solve word problems. 1 and have students identify how they are related. Piper uses yard of ribbon to make a border around an equilateral ··6 Ask Where does your model show the total length triangle. How long is the piece of ribbon that Piper uses for each side? of ribbon Piper has? Where does it represent the number of pieces she needs? How does it show the Pi��e It You can draw a picture to help understand the problem. length of each piece? 1 Draw a 1-yard length of ribbon and then draw and label a -yard length. Listen for Students should recognize that ··6 1 yd 1 6 representations should show of a whole yard ··6 divided into three equal parts. Responses may 1 include yard as the length, 3 as the number of 1 yd ··6 1 1 pieces, and yard as the length of each piece. Divide the -yard length into 3 equal parts, one for each side of the ··18 ··6 equilateral triangle.

1 6 yd

side side side PICTURE IT & MODEL IT 1 2 3 If no student presented these models, connect 1 yd them to the student models by pointing out the ways they each represent: Mo�� It • the ribbon length Piper has to start You can use equations to model the problem. • the number of equal parts needed Write a division equation. 1 4 3 5 s • the length of ribbon for each side ··6 length of each part Ask How are the ribbon pieces represented in the length of ribbon number of equal parts picture? In the equations? Is it the same or different? Write a multiplication equation. Listen for Each representation shows the pieces 1 1 1 3 5 s length of each part as 1 of 3 equal parts of . ··3 ··6 ··6 fraction of the ribbon length of ribbon For a picture of the ribbon, prompt students to 476 identify how the ribbon is labeled to represent the 476 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. problem. • How is it helpful that a whole of 1 yard is represented? Deepen Understanding • Why is the entire yard of ribbon divided into thirds, Equation Model and not just one sixth of the yard? SMP 2 Reason abstractly and quantitatively. For an equation model, prompt students to Prompt students to consider how the equations in Model It reflect different ways identify how both equations represent the situation. to represent the relationship between the quantities in the problem. 1 1 • Why is the length of ribbon, yard divided by 3 in Ask How does the equation 4 3 5 s represent the ribbon in the problem? ··6 ··6 1 1 one equation and multiplied by in the other? Explain the meaning of each number, and 3, and why the operation is division. ··3 ··6 1 1 Listen for represents the yard of ribbon that Piper uses. The triangle is ··6 ··6 an equilateral triangle, so each side is the same length. That means Piper 1 1 needs to divide the yard of ribbon into 3 equal parts, or 4 3. ··6 ··6 1 1 Ask How does the equation 3 5 s represent the ribbon in the problem? ··3 ··6 1 Explain the meaning of the number and why the operation is ··3 multiplication. 1 Listen for The triangle has 3 equal sides, so each piece of ribbon will be ··3 1 1 1 of Piper’s -yard ribbon. To find of , you multiply. ··6 ··3 ··6

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 476 LESSON 24 SESSION 2 Develop SESSION 2

CONNECT IT CONNECT IT Now you will use the problem from the previous page to help you understand • Remind students that one thing that is alike about how to divide a unit fraction by a whole number. all the representations is the numbers. 1 Look at Picture It. What does the fi rst diagram show? What whole is being divided? • Explain that on this page students will use what Possible answer: It shows 1 yard divided into 6 sections. Each section is they know about multiplying fractions to help 1 yard, which is the length of ribbon Piper uses. them find quotients when they divide a unit ··6 1 Why does the second diagram show each -yard section divided into fraction by a whole number. ··6 3 equal parts? 1 Possible answer: The -yard section of ribbon has to go around 3 sides of a Monitor and Confirm ··6 triangle. The sides have equal lengths. 1 and 3 Check for understanding that: 2 Look at Model It. How does each equation relate to the second diagram in 1 1 • a ribbon that is yard long is of 1 whole yard Picture It? ··6 ··6 1 1 Possible answer: The division equation shows the -yard ribbon divided into • the -yard length of ribbon needs to be divided ··6 ··6 into 3 equal pieces 3 equal parts just like in Picture It. The multiplication equation shows that 1 1 1 1 1 you are finding of yard. Each part in Picture It also is of yard. • yard is the length of each piece of ribbon ··3 ··6 ··3 ··6 ··18 3 How long is the piece of ribbon Piper uses for each side of the triangle? 1 yard Support Whole Class Discussion ···18 1 4 What is 4 3? How can you use a multiplication equation diff erent from the one 2 In discussing problem 2, check that students ··6 1 shown in Model It to check that your answer is correct? understand how the division expression 4 3 and 1 1 3 1 ··6 ; Possible answer: You can use the equation 3 3 5 , or . Because 1 1 ···18 ···18 ···18 ··6 the multiplication expression 3 each describe 1 1 1 1 ··3 ··6 3 3 5 , 4 3 5 . one way to interpret the second diagram in Picture It. ···18 ··6 ··6 ···18 Then prepare them for problem 4, in which they will 5 REFLECT think about a different multiplication equation Look back at your Try It, strategies by classmates, and Picture It and Model It. Which models or strategies do you like best for dividing a unit fraction by a whole related to the quotient 1 4 3. ··6 number to solve word problems? Explain. Ask Consider the whole number division problem Some students may like using a drawing because it helps them reason 48 4 12 5 n. How do you use the inverse relationship through the problem. Others may say that using equations and what they between multiplication and division to help you know about multiplying fractions is a quick way to solve a division problem. divide, or to check your answer? 477 Listen for You can think about the related ©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 477 multiplication equation 12 3 n 5 48 and find the missing factor, which is 4. To use multiplication to Hands-On Activity check that 48 4 4 5 12, you find the product Act out the problem. 12 3 4, which is 48. If . . . students are unsure about representing the division symbolically, Then . . . use this activity to connect a concrete model to an abstract equation. 4 Look for the idea that the inverse relationship 1 Materials For each student: yarn foot , scissors, glue, ruler, sheet of paper between multiplication and division shows that 1 ··2 2 1 1 1 1 3 1 • Tell students they will use the foot of yarn to make an equilateral triangle. ÷ 3 5 because 3 3 5 , or . 2 ··6 ··18 ··18 ··18 ··6 ·· • Have students use whatever strategy they choose to make 3 equal-length 1 5 REFLECT pieces from the -foot piece of yarn. Prompt students to write a division ··2 1 Have all students focus on the strategies used to expression to describe what they have done so far. 3 4 3 4 solve this problem. If time allows, have students ··2 share their preferences with a partner. • Tell students to glue the yarn pieces onto the paper to make an equilateral triangle, have them measure the length of each side as 2 inches, and guide 1 them in writing this length as a fraction of a foot: 2 inches 5 foot. ··6 • Ask: What equation can you write to find the length of each piece when you divide 1 1 1 a -foot piece of yarn into 3 equal parts? 4 3 5 Discuss how to use the ··2 3 ··2 ··6 4 1 3 1 related multiplication equation to check the answer. 3 3 5 , or 3 ··6 ··6 ··2 4

477 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 DEVELOP SESSION 2

APPLY IT APPLY IT Use what you just learned to solve these problems. For all problems, encourage students to draw some 1 6 Felipe has of a pizza. He wants to share it equally with a friend. How much kind of model to support their thinking. Allow some ··4 leeway in precision; drawing equal parts in fraction of the original whole pizza will each of them get? Show your work. models can be difficult. Possible student work:

1 6 Each friend gets of the pizza; See possible ··8 work on the Student Worktext page. Students 1 1 4 2 5 1 1 ··4 ··8 may also interpret the problem as finding of ··2 ··4 1 1 1 and use the multiplication equation 3 5 . ··2 ··4 ··8 1 Each friend gets of the pizza. Solution ··8 1 7 of the whole garden has red roses; See 1 ··12 7 Angela uses of her rectangular fl ower garden for roses. She plants equal ··3 possible model and division equation on the rectangular areas of red, white, pink, and orange roses in this part of the garden. Student Worktext page. What fraction of the whole garden has red roses? Draw a model and write a division equation to represent and solve the problem. Possible student work: Close: Exit Ticket Red roses White roses Pink roses 8 A; You can think of the problem situation as Orange roses 1 1 finding of , which can be represented by ··4 ··3 1 1 1 1 4 4 5 the expression 3 . ··3 ···12 ··4 ··3 1 of the whole garden has red roses. C; The related multiplication equation for the Solution ···12 1 1 1 1 division equation 4 4 5 is 3 4 5 . ··3 ··12 ··12 ··3 8 Look at problem 7. Which multiplication expressions can be used to represent the Error Alert If students choose B, D, or E, then situation or check the division equation? 1 1 1 Ꭽ 3 Ꭾ 4 3 review using the inverse relationship between ··4 ··3 ··3 1 Ꭿ 3 4 ൳ 3 3 4 multiplication and division to write related ··12 1 multiplication and division equations, starting with ൴ 3 3 ··4 whole numbers for both factors. Also have them 478 analyze the models they drew and guide them to 478 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. see how the models can be interpreted as 1 1 1 1 representing of , or 3 . ··4 ··3 ··4 ··3

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 478 LESSON 24 SESSION 2 Additional Practice Name: LESSON 24 SESSION 2

Solutions Practice Dividing a Unit Fraction by a Whole Number

1 1 1 Study the Example showing one way to solve a word problem involving dividing a 1 3 5 ··8 ··2 ··16 fraction by a whole number. Then solve problems 1–5. Basic

2 Less than; Students may reason that a lesser Example 1 Felicia makes gallon of fruit punch. She pours an equal amount of punch is poured into the same ··2 amount into 8 glasses. What fraction of a gallon of fruit number of glasses, so the amount in each glass lass will be less. See possible description for how punch is in each glass? lass 1 Find 4 8. lass the model would change on the student page. ··2 lass Medium The model shows a rectangle divided into halves and then lass divided into 8 equal parts. There are a total of 16 parts, and one part is the amount of fruit punch in 1 glass. lass lass 1 1 4 8 5 ··2 ··16 lass 1 The amount in 1 glass is gallon. ··16

1 What multiplication equation could you write to solve the Example? 1 1 1 3 5 ··8 ··2 ···16

1 2 Suppose Felicia had made gallon of punch and poured an ··4 equal amount into 8 glasses. Would the amount in each glass 1 be more or less than gallon? Explain how the model in the ··16 Example would change to show this. Less than; Possible explanation: The model would be divided into fourths instead of halves, so all the sections would be smaller.

Fluency & Skills Practice Teacher Toolbox

Assign Dividing a Unit Fraction by Fluency and Skills Practice Dividing a Unit Fraction by Name: a Whole Number a Whole Number

1 1 __ Diane has 2 gallon of frozen yogurt and some bowls. She puts an equal amount of frozen In this activity students divide unit yogurt into each bowl. For each given number of bowls, how much frozen yogurt will she put in each bowl?

fractions by whole numbers. This a. 2 bowls gallon skill is useful in everyday situations b. 3 bowls gallon c. 4 bowls gallon involving fractional measurements. d. 5 bowls gallon e. 6 bowls gallon One such situation might involve 1 2 Eli uses __ pound of apples to make 4 servings of fruit salad. He uses the same amount of apples 3 4 distributing gallon of juice equally for each serving. What amount of apples does he use for each serving of fruit salad? ··4 among 5 children. Another would pound

1 4 3 Feng has a piece of wire that is __ yard long. He cuts the wire into 2 pieces so that each piece is be dividing cup of cooking oil into 6 ··5 the same length. How long is each piece of wire? 4 equal amounts. yard

1 4 __ Tia walked 2 mile in 5 minutes. She walked at the same rate for the entire distance. How far did Tia walk in 1 minute?

mile

5 What is a pattern that you notice in problem 1?

479 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 SESSION 2

1 1 3 pound; See possible model and equations on 3 Donal buys a -pound package of cheese. There are 8 slices of cheese in the ··32 ··4 the student page. package. Each slice has the same weight. What fraction of a pound is each slice? Medium Draw a model and write a division equation to represent and solve the problem. Possible student work: 1 1 1 1 4 of the original stack; Students may show an 4 8 5 3 ··6 ··4 ··8 ··4 1 1 1 3 5 area model divided in half one way and thirds ··8 ··4 ···32 1 the other way, with overlapping shading in ··6 of the model. They may also use the expression 1 1 1 4 3 or 3 to solve the problem. ··2 ··3 ··2 Medium 1 pound 1 ···32 5 of the original stack; See possible division Solution ··8 equation on the student page. 4 Student volunteers are getting ready to hand out programs at a talent show. 1 Leah and Tomas are each given of a stack of programs to hand out. Leah Medium ··2 1 divides her equally among herself and 2 friends. What fraction of the original ··2 stack of programs do Leah and her 2 friends each have? Show your work. Students may use area models, equations, or some other method to find 1 4 3. ··2

1 of the original stack Solution ··6

5 Look at problem 4. If Tomas divides his stack of programs between himself and his 3 friends, what fraction of the original stack will each of his friends have? Write a division equation to represent and solve the problem. Possible work: 1 1 1 4 4 5 3 ··2 ··4 ··2 1 5 ··8 1 of the original stack Solution ··8 480 480 Lesson 24 Divide Unit Fractions in Word Problems ©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 Speaking/Writing Read Apply It problem Listening/Speaking Have students Listening/Speaking Have students read 11 aloud. Ask students to identify context chorally read Apply It problem 11. Say: Dylan Apply It problem 11 with a partner. clues for the word submarine (sandwich). cut the sandwich into sixths. Ask: What fraction Encourage them to discuss how they can 1 Have them label the picture in the Student shows the size of each piece? represent the information they know and use 3 ··6 4 Worktext “submarine sandwich.” Then say: Ask: How can you represent the action with that model to find the solution to the Dylan cut the sandwich into sixths. Ask: What numbers? Provide support as needed to help problem. fraction shows the size of each students state the division. Then ask: How Have students use complete sentences to 1 piece? 3 ··6 4 many pieces does Dylan get when he divides 3 explain the model. Provide the following 1 Ask: How many pieces does Dylan get when he sandwiches into pieces of size ? sentence starters: 1 ··6 divides 3 sandwiches into pieces of size ? ··6 Have students complete the sentence frame: • My partner and I know that . Have students complete the sentence frame: • Dylan gets pieces. • First, we show . • Dylan gets pieces. • Then we show . • Dylan gets .

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 480 LESSON 24 SESSION 3 Develop LESSON 24 SESSION 3

Purpose In this session students solve a Develop Dividing a Whole Number by a Unit Fraction 1 problem that requires dividing 2 by . Students ··4 model the numbers in the word problem either Read and try to solve the problem below. on paper or with manipulatives to develop Alex makes 2 pounds of bread dough. He splits the strategies for solving word problems that 1 dough into -pound loaves before baking them in ··4 involve dividing a whole number by a unit the oven. How many loaves does he make? fraction.

TRY IT Start Possible student work: Math Toolkit Sample A • fraction tiles Connect to Prior Knowledge • fraction bars 1 There are four -pound loaves in each whole pound. Materials For each student: Activity Sheet Fraction ··4 • fraction models • number lines 2 3 4 5 8 Bars (6 fifths bars) • grid paper Why Support students’ facility with writing whole Alex made 8 loaves. • index cards numbers as fractions. Sample B How Have students use fraction bars to model the 2 pounds are divided into fourths. whole numbers 1–6 as numbers of fifths and then write each whole number as a fraction.

Start 1 2 3 4 5 6 7 8 Solutions 15 25 Use fraction bars to show each 3 5 5 5 whole number as fifths. ··5 ··5 5 30 1 1 5 6 5 2 4 5 8 ··5 ··5 ··4 3 5 5 5 ····5 ····5 Look for Models Alex made 8 loaves. should show the DISCUSS IT 1 5 6 5 appropriate number ····5 ····5 Ask your partner: Can you of fifths fraction bars explain that again? for that number of Tell your partner: The wholes. strategy I used to ﬁ nd the Grade 5 Lesson 24 Session 3 | Explore Dividing a Whole Number by a Unit Fraction ©Curriculum Associates, LLC Copying is permitted. answer was . . . Develop Language 481 Why Reinforce understanding of the word split in ©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 481 the context of fractions. How Write the word split on the board. Explain that DISCUSS IT to split means to divide, separate, or break into Support Partner Discussion parts. Have students circle the word in the Try It Encourage students to use the terms fourths and pounds as they discuss their work. problem. Ask: What does Alex do when he splits the Support as needed with questions such as: dough? • How did you get started? • Why did you choose that strategy? TRY IT Common Misconception Look for students who are not comfortable counting the Make Sense of the Problem number of equal parts and instead count the vertical lines dividing their model into To support students in making sense of the equal parts, arriving at a solution of more than 8 loaves. As students present problem, have them identify the amount of dough solutions, have them point out and count the loaves. Alex makes and the amount of dough each loaf needs. Select and Sequence Student Solutions Ask What size parts does Alex split the dough into? One possible order for whole class discussion: • physical parts showing wholes and fourths • drawings to represent the problem • number lines marked to show wholes and fourths • division or multiplication equations showing that Alex makes 8 loaves

481 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 DEVELOP

Support Whole Class Discussion Explore diff erent ways to understand how to divide a whole number by a unit fraction Compare and connect the different representations in order to solve word problems. 1 and have students identify how they are related. Alex makes 2 pounds of bread dough. He splits the dough into -pound ··4 Ask How does your model show the whole amount loaves before baking them in the oven. How many loaves does he make? of dough? the equal parts each whole is divided into? Listen for Students should recognize that Mo�� It You can use a number line to help understand the problem. accurate representations show 2 wholes, each representing 1 pound of dough, divided into Draw a number line and label it to show the 2 pounds of bread dough. 1 4 equal parts, each representing one -pound loaf. Mark the number line to divide each whole into fourths. ··4

Mo�� It MODEL ITs You can use what you know about equations, equivalent fractions, and common If no student presented these models, connect denominators to solve the problem. them to the student models by pointing out the 1 The equation 2 4 5 n models the problem with n being the number of loaves ··4 ways they each represent: Alex makes. • the 2 pounds of dough Write the numbers in the equation with a common denominator. • the equal parts each pound is divided into 8 1 4 = n ··4 ··4 Ask In the second Model It, the number 2 is 8 1 8 Now you can divide into equal groups of . rewritten as the equivalent fraction . How does the ··4 ··4 ··4 8 number line in the first Model It show 2 is equal to ? ··4 Listen for The number line is divided into 8 fourths between 0 and 2.

For a number line model, prompt students to identify the greatest number on the number line and the number of divisions. • How is the number line divided? • Why is it done that way? 482 For the equation model, prompt students to 482 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. compare the two equations and tell how they are alike and how they are different. Deepen Understanding • Why was 4 chosen as the common denominator? Equivalent Fractions 8 • How is it helpful to think of 2 as ? SMP 8 Use repeated reasoning. ··4 When discussing the equation method shown in the second Model It, prompt students to consider where else they have used renaming as a strategy. Ask In your own words, how do you describe the strategy shown for dividing a whole number by a unit fraction? 8 Listen for You rename the whole number, 2, as the fraction so that the ··4 dividend and the divisor describe the same-size parts of a whole. You can then think of dividing 8 fourths into equal groups of 1 fourth.

Ask Where else have you used a strategy of renaming? What are examples? Listen for Students should recognize that they have used a renaming strategy in many situations. Examples may include regrouping ones as tens, or tens as hundreds, to add whole numbers; writing decimals in tenths as decimals in hundredths to subtract decimals; and writing fractions with common denominators to add fractions.

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 482 LESSON 24 SESSION 3 Develop SESSION 3

Co�� It CONNECT IT Now you will use the problem from the previous page to help you understand • Remind students that one thing that is alike about how to divide a whole number by a unit fraction. all the representations is that they show dividing 1 In the fi rst Model It number line, how are the 2 pounds of bread dough represented? 2 wholes into same-size fractional parts. The pounds of dough are the numbered points 1 and 2. • Explain that on this page students will use what 1 2 How are the -pound loaves represented on the number line? they know about equivalent fractions and the ··4 relationship between division and multiplication Each whole number, or pound, is divided into fourths using tick marks. to explain and check the quotient. 3 How many fourths are in one whole? 4 In two wholes? 8 1 4 Look at the second Model It. How was the equation 2 4 5 n changed to an Monitor and Confirm ··4 equation involving fractions with common denominators? 1 – 3 Check for understanding that: Possible answer: The number 2 was changed to a fraction in fourths, so it has 1 8 • 2 is the number of whole pounds of dough the same denominator as . 2 5 . ··4 ··4 • 4 fourths are in one whole; 8 fourths are in 1 8 8 1 5 How many groups of are in ? What is 4 ? Explain. ··4 ··4 ··4 ··4 two wholes 1 8 1 8 8 1 8, 8; Possible explanation: There are eight s in because 8 3 5 . So, 4 5 8. ··4 ··4 ··4 ··4 ··4 ··4 Support Whole Class Discussion 6 How many loaves does Alex make? How are the fi rst Model It and second 4 – 6 Tell students that these problems show Model It alike in showing how to fi nd the solution? 1 8 loaves; Possible explanation: The first Model It shows eight tick marks in 2 how to use equivalent fractions to find the number ··4 1 8 1 on the number line. The second Model It shows eight s in using an equation. of s in 2. In problem 7, they will think about how to ··4 ··4 ··4 1 use multiplication to check the answer. 7 What multiplication equation can you write to check your answer to 2 4 ? Explain. ··4 1 8 1 1 8 3 5 or 2. Because 8 3 5 2, 2 4 5 8. Be sure students understand that the reason for ··4 ··4 ··4 ··4 writing 2 as a fraction with a denominator of 4 is so 8 REFLECT both the dividend and the divisor are expressed as a Look back at your Try It, strategies by classmates, and Model Its. Which models or number of equal-size parts of 1 whole. The strategies do you like best for dividing a whole number by a unit fraction? Explain. numerator shows the number of those parts in Some students may say they like using a visual model to reason through the 2 wholes. problem. Others may choose using equations because they know how to use

Ask How do the strategies discussed in problems 4 the relationship between multiplication and division. 1 and 6 help you identify the quotient 2 4 ? 483 ··4 ©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 483 Listen for They help you think about how many 1 parts are in 2 wholes. They both show that 2 has ··4 8 one-fourth parts. Hands-On Activity Act out the problem. If . . . students are unsure about using multiplication to check division, 7 Look for the idea that to check the quotient of a Then . . . use this activity to connect a concrete model to both equations. division equation you can use the inverse relationship between multiplication and division to Materials For each student: modeling clay (2 equal-sized portions) write the related multiplication equation. • Tell students they will act out the problem of splitting up the bread dough. Explain that each portion of modeling clay represents 1 pound of dough. 8 REFLECT 1 • Guide students to divide each of the 2 pounds of bread dough into -pound ··4 Have all students focus on the strategies used to 1 parts for the individual loaves. Discuss the division represented, 2 4 . Have solve this problem. If time allows, have students ··4 1 students identify the quotient and write the division equation. 2 4 5 8 share their preferences with a partner. 3 ··4 4 • Remind students division and multiplication are inverse operations—that 1 multiplication undoes division. Have students put the -pound portions back ··4 together to make whole pounds, reforming the original 2 portions. Discuss the 1 multiplication represented, 8 3 . Have students write the multiplication ··4 1 equation that is related to the division equation above. 8 3 52 3 ··4 4

483 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 DEVELOP SESSION 3

Ap�� It APPLY IT Use what you just learned to solve these problems. For all problems, encourage students to draw a 1 9 Stacy has 4 sheets of paper to make cards. Each card requires sheet of paper. model to support their thinking. Allow some leeway ··2 in precision; dividing fraction models into equal How many cards can Stacy make? Draw a model and write a division equation to parts can be difficult. represent and solve the problem. Possible student work using a number line: 1 1 2 3 4 5 6 7 8 9 4 4 5 8; 8 cards; See possible model on the ··2 Student Worktext page. Students may also draw 0 1 2 3 4

4 rectangles, each divided in half, with the halves 1 4 4 5 8; 8 cards ··2 numbered consecutively from 1 to 8. They may Solution 8 1 10 Look at problem 9 above. Which multiplication expressions can be used to also write the division equation ÷ 5 8, ··2 ··2 represent the situation or check the division equation? 1 8 reasoning there are 8 groups of in . ··2 ··2 Ꭽ 8 3 2 1 1 Ꭾ 4 3 10 D; The multiplication expression 8 3 can be ··2 ··2 1 1 used to check the equation 4 4 5 8. Ꭿ 16 3 ··2 ··2 1 ൳ 8 3 E; 2 cards can be made from each sheet of ··2 3 paper. There are 4 sheets, so 4 2 represents ൴ 4 3 2 the situation. 11 Dylan makes 3 submarine sandwiches. He cuts each sandwich into sixths to share. He stacks all the sandwich pieces on a plate. How many Close: Exit Ticket sandwich pieces does Dylan stack on the plate? Show your work. Possible student work: 1 3 4 5 p 11 Dylan stacks 18 sandwich pieces on the plate; ··6 Students may write the division equation Write an equation with a common denominator. 18 1 18 1 1 4 5 18 4 5 18, reasoning there are 18 groups of ···6 ··6 ··6 ··6 ··6 18 in . They may also draw a fraction model, Solution Dylan stacks 18 sandwich pieces on the plate. ··6 such as 3 rectangles, each divided in sixths. 484 Students’ solutions should indicate understanding of: 484 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. 1 • 3 wholes divided into equal parts of size ··6 • There are 6 one sixths in each whole, so there are 3 3 6, or 18 one sixths in 3 wholes. Error Alert If students find a quotient other than 18, then have them write their division equation and then use multiplication to check their answer. 1 For example, if they find the quotient to be , have ··2 1 1 them write 3 4 5 (the incorrect division equation) ··6 ··2 1 1 and then use 3 to check. Point out that because ··2 ··6 1 1 1 1 1 3 5 , not 3, the quotient for 3 4 is not . ··2 ··6 ··12 ··6 ··2 1 Review that 3 4 means to divide 3 wholes into ··6 1 parts, each of which is of 1 whole. ··6

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 484 LESSON 24 SESSION 3 Additional Practice Name: LESSON 24 SESSION 3

Solutions Practice Dividing a Whole Number by a Unit Fraction 1 Study the Example showing one way to solve a word problem involving dividing a 2 3 5 5 10; Each whole is divided into 5 fifths, whole number by a fraction. Then solve problems 1–6. so 2 wholes show 2 3 5, or 10 fifths. Basic Example 1 Darius walks dogs at an animal shelter. He walks each dog for hour. He walks the 2 45 times; Students may write and solve the ··5 1 division equation 9 4 5 n or the multiplication dogs one at a time. How many dogs can Darius walk in 2 hours? ··5 1 Find 2 4 . expression 9 3 5 5 n. They may also extend the ··5 number line in the example to show the The number line shows two hours. Each hour is divided into fi fths. 1 number of s in 9. 1 2 3 4 5 6 7 8 9 10 ··5 Medium

3 See possible explanation on the student page. There are 10 fi fths in 2. Students may also explain how they counted 1 1 2 4 5 10 the number of s in a model they drew that ··5 ··5 showed 9 wholes. Darius can walk 10 dogs in 2 hours. Medium 1 What multiplication equation could you write to solve the Example? 2 3 5 5 10

2 Use the information from the Example. In one month, Darius spends 9 hours walking dogs. How many times does he walk a dog in one month? 45 times

3 Explain how you got your answer to problem 2. Answers will vary. Possible explanation: I know Darius can do 5 dog walks in 1 hour, so in 9 hours he can do 9 3 5, or 45, dog walks. 485 ©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 485

Fluency & Skills Practice Teacher Toolbox

Assign Dividing a Whole Number Fluency and Skills Practice Dividing a Whole Number Name: by a Unit Fraction by a Unit Fraction

1 Eric has 4 pounds of blueberries to make into pies. How many pies can Eric make if each pie In this activity students divide needs the given amount of blueberries? 1 __ a. 2 pound pies whole numbers by unit fractions. 1 __ b. 3 pound pies 1 c. __ pound pies This skill can be used in everyday 4 1 __ d. 5 pound pies 1 e. __ pound pies activities such as cooking. For 6

1 example, suppose a person is 2 Sunita has 5 quarts of apple cider to fi ll some empty glasses. She fi lls each glass with__ quart of 4 cider. How many glasses does she fi ll?

making cheese-stuffed potatoes. glasses 1 Each potato requires cup of milk, 1 3 Lana has 6 yards of fringe to decorate banners identically . She uses __ yard of fringe for 1 banner. ··4 3 and the cook has 2 cups of milk. By How many banners can she decorate if she uses all the fringe? 1 banners dividing 2 by , the cook finds that 1 4 Terrance has 2 empty pages in his stamp collection album. Each stamp uses __ of an album ··4 8 he or she has enough milk to make page. How many stamps can Terrance put on the empty pages? 8 cheese-stuffed potatoes. stamps 5 Write a rule to help you divide a whole number by a unit fraction without drawing a model.

485 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 SESSION 3

4 20 pieces of tape; Students may use the 4 Mrs. Wing will tape up posters made by her students on the wall. 1 1 1 20 1 She cuts tape into -foot pieces. How many -foot pieces can she expression 5 4 or 4 to solve the 4 4 ··4 ··4 ··4 ·· ·· problem. They may also draw a fraction model cut from 5 feet of tape? Show your work. Students may use area models, equations, or some other showing 5 wholes divided into fourths. 1 method to find 5 4 . Medium ··4

5 14 vases; See possible model and division equation on the student page. Students may 14 1 20 pieces also write the division equation 4 5 14. Solution ··2 ··2 Medium 5 Taylor is helping decorate tables with fl owers for a graduation 1 celebration. She has 7 bunches of tulips. She will put of each ··2 6 14 vases; Students should show a different way bunch in a vase. How many vases does she need? Draw a model of solving the problem than in problem 5. They and write a division equation to represent and solve the problem. 1 may use the equation 14 3 5 7 to check ··2 their answer. 1 7 4 5 14 Challenge ··2

Solution 14 vases

6 Look at how you solved problem 5. Use a diff erent way to solve the problem and show how a multiplication equation can be used to check the answer. Students may use area models, equations, or some other method to find 1 7 4 . It should be different than the way shown in problem 5. Students may ··2 1 use the equation 14 3 5 7 to check their answer. ··2

Solution 14 vases 486 486 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. English Language Learners: Prepare for Session 4 ELL Differentiated Instruction Use with Apply It.

Levels 1–3 Levels 2–4 Levels 3–5 Speaking/Listening Have students Speaking/Listening Have students Reading/Writing Have students chorally chorally read Apply It problem 3. Ask them to chorally read Apply It problem 3. Ask them to read Apply It problem 3. In partners, have state what is known and unknown about the state what is known and unknown about the students discuss what is known and problem: problem: unknown about the problem. Ask partners to I know: Devonte uses for each event. I know: Devonte uses and . make a model to show the number of events Devonte makes notes for. Devonte fills of one sheet of paper. I need to know: . Have students use sequence terms (first, then, I need to know: how many Devonte Have partners make a model and write a last) to write the steps they took to write a makes notes for. division expression. division expression. Have partners make a model and write a When complete, ask: Did Barry get the correct Then have them compare their answer with division expression. answer? Why? another group. When complete, have When complete, ask: Did Barry get the correct students work together to explain how Barry answer? got D as an answer.

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 486 LESSON 24 SESSION 4 Refine LESSON 24 SESSION 4

Purpose In this session students solve word Refine Dividing Unit Fractions in Word Problems problems involving dividing a whole number by a unit fraction or a unit fraction by a whole Complete the Example below. Then solve problems 1–9. number. They then discuss and confirm their

answers with a partner. The student used a model Before students begin to work, use their EXAMPLE to visualize the problem. responses to the Check for Understanding to Sierra has a photo album with 3 empty pages. Each 1 photo uses of an album page. How many photos can determine those who will benefit from ··6 additional support. Sierra put on the empty pages? As students complete the Example and Look at how you could show your work using rectangles. problems 1–3, observe and monitor their reasoning to identify groupings for differentiated instruction. PAIR/SHARE 6 photos will fit on each of the 3 pages. What related equations can Start you write to represent the Solution 18 photos problem? Check for Understanding Why Confirm understanding of solving a word Ap�� i� problem that involves dividing a unit fraction by a 1 1 Corrine picked gallon of blackberries. She poured equal whole number. ··4 Can you draw a model to amounts of berries into 4 containers. What fraction of a gallon help understand the How Have students draw a visual model and write a is in each container? Show your work. problem? division equation to solve the problem. Possible student work using a model:

Start 1 Solution 4 1 Draw a visual model and write gallon; a division equation to solve. ··25 1 1 4 4 1 4 5 5 A pitcher contains gallon of ··5 ··25 ··5 juice. If 5 friends share the juice Look for Models PAIR/SHARE 1 How will the answer equally, how much does each show divided into 1 1 ··5 gallon compare to gallon? person get? 5 equal parts of a Solution ···16 ··4

whole divided into 487 ©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 487 25 equal parts. Grade 5 Lesson 24 Session 4 | R e ﬁ n e Dividing Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is permitted.

Error Alert If the error is . . . Students may . . . To support understanding . . .

Have students draw an area model or a number line have thought that the answer is and identify 1 . Then, have students divide the fifth into 25 the number of parts in the whole ··5 5 equal-sized pieces. Point out that each friend gets one of after dividing. these pieces, which is 1 of 1 , or 1 3 1 . ··5 ··5 ··5 ··5

Write “ 1 3 5” and “ 1 4 5” and discuss what each expression ··5 ··5 means in the context of the problem. 1 3 5 could mean “ 1 ··5 ··5 1 have multiplied instead of of 5 friends” or “each of 5 friends has gallon.” These do 1 ··5 dividing. not describe the problem situation. 1 4 5 means “ 1 gallon ··5 ··5 divided (or shared) among 5 friends,” which describes the problem situation exactly.

487 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 REFINE

1 2 Cooper’s USB drive is full with 5 video fi les. Each video fi le is the Example ··2 How could I represent this 18 photos; the fraction model shown is one way to same size. What fraction of the USB drive does 1 video fi le use? problem using an equation? solve the problem. Students could also solve the Show your work. 1 Possible student work using an equation: problem using the expression 3 4 or 6 3 3. They ··6 1 4 5 5 f may also use a different kind of fraction model, such ··2 1 1 f is of . as a number line. ··5 ··2 1 1 1 Look for You find the number of s in 3. 3 5 f ··6 ··5 ··2 1 1 1 3 5 ··5 ··2 ···10 APPLY IT 1 1 gallon; Students could solve the problem by ··16 PAIR/SHARE How can you check your drawing a model divided to show fourths one 1 of the drive 10 answer? way and divided into 4 equal parts the other Solution ··· 1 1 way, with the overlapping shading showing . 3 Devonte is studying for a history test. He uses of a side of one ··16 ··8 Is this problem like one you DOK 2 sheet of paper to write notes for each historical event. He fi lls have seen before? 1 2 full sides of one sheet of paper. Which expression could be used Look for Find one of 4 equal parts of gallon. ··4 to fi nd how many events Devonte makes notes for? 1 1 2 of the drive; Students could solve the problem Ꭽ 2 3 ··10 ··8 1 by representing it with the equation 4 5 5 f, Ꭾ 2 4 1 ··2 ··8 1 1 using reasoning to think of it as finding of , 1 ··5 ··2 Ꭿ 3 2 1 1 ··8 or 3 5 f. 1 ··5 ··2 ൳ 4 2 ··8 DOK 2 1 Barry chose ൳ as the correct answer. How did he get that answer? Look for Find one of 5 equal parts of . ··2 Possible answer: Barry knew that he should write a division 1 expression with 2 and , but he thought that the order of the 3 B; Students could solve the problem by ··8 reasoning that the starting amount, 2 sides, is numbers didn’t matter in division. PAIR/SHARE 1 Does Barry’s answer make divided into groups that are page in size. sense? ··8 Explain why the other two answer choices are 488 not correct: 488 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. A is not correct because it represents 2 groups 1 that are each of a page. ··8 1 C is not correct because it represents of ··8 2 pages. DOK 3

©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 488 LESSON 24 SESSION 4 Refine SESSION 4

1 4 Elise picks 6 pounds of apples. She uses pound of apples to make 1 container APPLY IT ··2 of applesauce. How many containers of applesauce can Elise make with all 4 A; Each pound has two halves, so each pound the apples? can make 2 containers of applesauce. Elise has Ꭽ 12 containers 6 pounds. 6 3 2 5 12. 1 Ꭾ 6 containers DOK 2 ··2 1 Ꭿ 5 containers ··2 5 9; Divide each mile shown on the number line ൳ 3 containers in thirds, to show the distance each team 5 Students are running in a relay race. Each team will run a total of 3 miles. 1 1 member will run. Count the number of s. Each member of a team will run mile. How many students will a team need ··3 ··3 DOK 2 to complete the race? Circle the correct number below. 1 3 9 12 36 1 ··9 6 Each piece is of the cake; Represent cutting ··18 1 the cake into 6 equal pieces with the You may use the number line to help fi nd your answer. ··3 1 1 1 expression ÷ 6. Think of that as finding of ··3 ··6 ··3 1 1 and write it as the expression 3 . ··6 ··3 1 DOK 2 6 Tanya has of a cake left over from a party. She cuts the leftover ··3 cake into 6 equal pieces to store in the freezer. What fraction of Error Alert Students may write the whole number 1 the original cake is each piece? Show your work. 6 instead of the fraction when writing the related ··6 Possible student work: 1 1 expression for ÷ 6; writing instead 6 3 and 1 1 1 ··3 ··3 4 6 5 3 6 ··3 ··6 ··3 showing the answer . 1 1 1 ··3 3 5 ··6 ··3 ···18

1 Each piece is of the cake. Solution ···18

489 ©Curriculum Associates, LLC Copying is not permitted. Lesson 24 Divide Unit Fractions in Word Problems 489 Differentiated Instruction RETEACH EXTEND

Hands-On Activity Challenge Activity Use fraction circles to divide a whole number by a unit fraction. Write and solve equations. Students struggling with understanding dividing by a unit fraction Students who have achieved proficiency Will benefit from additional work with concrete representations Will benefit from deepening Materials For each student: 1 set of fraction circles or tiles understanding of division with unit 1 fractions • Distribute materials. Pose the following problem: A painter can paint of a room in an hour. ··3 How many painters are needed to paint 6 rooms in an hour? • Challenge students to write a division equation showing a whole • Have students trace the whole fraction circle six times to represent 6 rooms. Ask: How number divided by a unit fraction that much of a room can 1 painter paint in 1 hour? 1 of a room Ask: How many painters are 3 ··3 4 has a quotient of 12. needed to paint an entire room in 1 hour? [3 painters] Have students place one-third Possible answer: 3 4 1 5 12 3 ··4 4 fraction pieces in one circle to show this. • Challenge students to solve a division • Have students divide the remaining whole circles into thirds to show the painters needed equation involving a mixed number. Ask for each room. Ask students to write a division equation for the problem and a solution. 1 3 1 them to solve 4 4 3 5 n. 6 4 5 18; 18 painters Repeat the activity for other numbers of rooms, such as 3 or 5. ··2 3 ··2 4 3 ··3 4

489 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. LESSON 24 REFINE SESSION 4

7 Marina has a pattern to make bows 7 4, 8, 12, 16; See completed table on the Student Ribbon Length Number of Bows 1 1 that requires yard of ribbon for Worktext page. Each bow requires yard of 4 (yards) ··4 ·· ribbon. Each yard makes 4 bows. each bow. Fill in the table to show 1 4 how many bows she can make from DOK 2 2 8 a given length of ribbon. 3 12

8 Part A 4 16 1 Ted puts gallon of ice cream in each bowl. 1 ··24 8 Part A Ted serves gallon of ice cream. He puts an equal amount of ice cream ··6 See possible model on the Student Worktext in each of 4 bowls. How many gallons of ice cream does Ted put in each bowl? 1 1 1 page. of is . ··4 ··6 ··24 Use a visual model to support your answer. Part B Possible student model: 1 1 1 4 1 4 4 5 ; 3 4 5 , or . ··6 ··24 ··24 ··24 ··6 DOK 2

1 1 1 of is . ··4 ··6 ···24 1 Ted puts gallon of ice cream in each bowl. Solution ···24 Part B Write a division equation to represent this situation. Then write a multiplication equation you can use to check your answer.

1 1 1 1 4 4 5 and 3 4 5 Solution ··6 ···24 ···24 ··6

9 MATH JOURNAL 1 Write a word problem represented by 4 4. Explain or show how to fi nd the answer. ··5 1 Possible problem: Willa uses lb of clay to make 4 equal-sized beads for a ··5 necklace. How heavy is each bead? 1 1 1 1 Possible student work: 4 4 5 3 ; Each bead is lb. ··5 ··4 ··5 ···20

SELF CHECK Go back to the Unit 3 Opener and see what you can check off . 490 490 Lesson 24 Divide Unit Fractions in Word Problems ©Curriculum Associates, LLC Copying is not permitted. Close: Exit Ticket REINFORCE PERSONALIZE 9 MATH JOURNAL Problems 4–9 Student responses should indicate understanding of Divide unit fractions. problem situations that require dividing a unit Provide students with fraction by a whole number, such as dividing a All students will benefit from additional work with opportunities to work fractional measure of a quantity into equal portions, dividing unit fractions by solving problems in a variety on their personalized as well as an understanding of how to use a visual of formats. instruction path model or equations to solve a problem of this type. • Have students work on their own or with a partner to with i-Ready Online Error Alert If students confuse the dividend and solve the problems. Instruction to: the divisor when representing the expression with a • Encourage students to show their work. • fill prerequisite gaps word problem, then have them draw a visual model • build up grade-level for the expression and use it to make up a word skills problem. Have them identify the numbers that are the dividend and divisor in each representation.

SELF CHECK Have students consider whether they feel they are ready to check off any new skills on the Unit 3 Opener page.