Divide Decimals - Full Lesson

©Curriculum Associates, LLC Copying is not permitted.345a Lesson 17 Divide Decimals

Lesson Overview

LESSON 17

Divide Decimals

Prerequisite Skills

• Understand place value.

• Know properties of operations.

• Recall division and multiplication basic facts.

• Divide three- and four-digit dividends by two-digit divisors.

• Understand the relationship between multiplication and division.

Lesson Vocabulary

There is no new vocabulary. Review the following key terms.

• dividend the number that is divided by another number.

• divisor the number by which another number is divided.

• estimate (verb) to give an approximate number or answer based on mathematical thinking.

• quotient the result of division.

Lesson Objectives

Content Objectives• Divide decimals to hundredths.

• Explain the reasoning used to divide decimals to hundredths.

Language Objectives• Describe orally how to use multiplication

to understand a division problem.

• Estimate the quotient of decimals and justify using place-value reasoning.

Learning Progression

Previously in Grade 5 students divided whole numbers with up to four-digit dividends by two-digit divisors. Students learned to recognize division as a missing factor problem in which the quotient is the unknown factor.

In this lesson students apply their understanding of division to dividing with decimals. They use place-value understanding to connect patterns in decimal division and whole-number division. They use visual models and the relationship between multiplication and

division to divide decimal numbers by whole numbers, tenths decimals, and hundredths decimals. Students use multiplication to check the results of decimal division and explain how to divide decimals using their own words.

Later in Grade 5 students will use their knowledge of decimal division when they convert from larger units of measurement to smaller units. In Grade 6, students will continue to multiply and divide with decimals.

CCSS FocusDomainNumber and Operations in Base Ten

ClusterB. Perform operations with multi-digit

whole numbers and with decimals to hundredths.

Standard5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Additional Standard5.NBT.A.1 (See Standards Correlations at the end of the book for full text.)

Standards for Mathematical Practice (SMP)SMPs 1, 2, 3, 4, 5, and 6 are integrated in every lesson through the Try-Discuss-Connect routine.*

In addition, this lesson particularly emphasizes the following SMPs:

1 Make sense of problems and persevere in solving them.

2 Reason abstractly and quantitatively.

5 Use appropriate tools strategically.

8 Look for and express regularity in repeated reasoning.

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

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Lesson Pacing Guide

PERSONALIZE

i-Ready Lessons*Grade 5• Divide Decimals• Practice: Divide Decimals

Independent Learning

PREPARE

Ready Prerequisite LessonGrade 4• Lesson 15 Divide Four-Digit Numbers

RETEACH

Tools for InstructionGrade 4• Lesson 15 Divide Four-Digit by One-Digit

Numbers

Grade 5• Lesson 17 Divide Decimals

REINFORCE

Math Center ActivityGrade 5• Lesson 17 Cover Up Division

EXTEND

Enrichment ActivityGrade 5• Lesson 17 Building Projects

Small Group DifferentiationTeacher Toolbox

Lesson MaterialsLesson (Required)

Per pair: base-ten blocks (2 tens rods, 20 ones units)Activity Sheets: Number Lines**, Hundredths Grids

Activities Per pair: play money (20 nickels, 50 pennies) Per group: 2.7 m of string, meter stick, scissors, marker Activity Sheet: Number Lines**

Math Toolkit base-ten blocks, decimal grids, number lines, fraction bars, thousandths decimal place-value charts, counters, base-ten grid paper, play money

Digital Math Tools

Fraction Models, Number Line

**Used for more than one activity.

SESSION 1

Explore45–60 min

Interactive Tutorial* (Optional) Prerequisite Review: Multiply Decimals

Additional PracticeLesson pages 349–350

Dividing Decimals • Start 5 min• Try It 10 min• Discuss It 10 min• Connect It 15 min• Close: Exit Ticket 5 min

SESSION 2

Develop45–60 min

Dividing a Decimal by a Whole Number • Start 5 min• Try It 10 min• Discuss It 10 min• Picture It & Model It 5 min• Connect It 10 min• Close: Exit Ticket 5 min


Fluency Dividing a Decimal by a Whole Number

SESSION 3

Develop45–60 min

Dividing by Tenths • Start 5 min• Try It 10 min• Discuss It 10 min• Picture It & Model It 5 min• Connect It 10 min• Close: Exit Ticket 5 min


Fluency Dividing by Tenths

SESSION 4

Develop45–60 min

Dividing by Hundredths• Start 5 min• Try It 10 min• Discuss It 10 min• Picture It & Model It 5 min• Connect It 10 min• Close: Exit Ticket 5 min


Fluency Dividing by Hundredths

SESSION 5

Refine45–60 min

Dividing Decimals• Start 5 min• Example & Problems 1–3 15 min• Practice & Small Group

Differentiation 20 min• Close: Exit Ticket 5 min

Lesson Quiz or Digital Comprehension Check

Whole Class Instruction

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

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LESSON 17

Connect to Family, Community, and Language DevelopmentThe following activities and instructional supports provide opportunities to foster school, family, and community involvement and partnerships.

Connect to FamilyUse 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.

©Curriculum Associates, LLC Copying is not permitted.Lesson 17 Divide Decimals346

Do this activity with your child to divide decimals.

Work with your child to solve a real-world problem involving dividing decimals.

• Think of something you spend money on for the whole family, such as the grocery bill, tickets to the movies, or a new board game.

• Divide the cost by the number of people in your family. This will describe the cost for each family member.

For example: A book of puzzles costs $11.76. There are 4 people in the family.To fi nd the cost for each person, divide 11.76 by 4.

• Check that the answer is reasonable. In the example above, is 29.4 a reasonable answer for 11.76 4 4?

Be on the lookout for other real-world examples of dividing decimals that you can share with your child.

ACTIVITY DividinG DecimalS

346©Curriculum Associates, LLC Copying is not permitted. Lesson 17 Divide Decimals 345

Divide Decimals

17Dear Family,This week your child is learning to divide with decimals.Your child might see a problem like this:

Marty is running in a 3.2-kilometer race. Water stations are set up at 8 equal sections of the race. How far apart are the water stations?

One way to understand the relationship of the quantities in the problem is to use a bar model.

3.2 kilometers

?

The whole bar represents the length of the race, 3.2 kilometers. The bar has 8 equal sections. Find the length of each section to fi nd how far apart the water stations are.

Divide 3.2 by 8 to fi nd the length of each shorter section.

One way your child is learning to divide decimals is to think about multiplying decimals. Division and multiplication are related operations.

To fi nd 3.2 4 8, think 8 3 ? 5 3.2.3.2 5 32 tenths8 3 ? 5 32 tenths8 3 4 tenths 5 32 tenths

The answer, 4 tenths, is the length represented by each section of the bar model. The water stations are 0.4 kilometer apart.

Invite your child to share what he or she knows about dividing decimals by doing the following activity together.

345

GoalThe goal of the Family Letter is to provide students and family members with opportunities to learn about and explore division with decimals.

• By using real-life expenses such as a grocery bill to divide with decimals, the usefulness of the skill is reinforced.

ActivityStudents and family members use real-life situations in which they have to divide with decimals. Look at the Dividing Decimals activity and modify if needed to connect with students.

Math Talk at HomePoint out that dividing with decimals is similar to dividing with whole numbers. Remind students that dividing and multiplying are related operations and that multiplication can be used to verify the answers to division problems that involve decimals.

Conversation Starters Below are additional conversation starters students can write in their Family Letter or math journal to engage family members:

• To split a bill between two people, how do you calculate what each person needs to pay?

• The cost of restaurant meals and movie tickets are costs that can be split between two or more people. What are other examples of costs that can be divided?

Available in Spanish

Teacher Toolbox

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Connect to Community and Cultural ResponsivenessUse these activities to connect with and leverage the diverse backgrounds and experiences of all students.

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

Session 1 Use anytime during the session.

• Arts and crafts involve making objects such as decorations, furniture, and pottery by hand. Arts and crafts provide entertainment, and arts and crafts products are used to celebrate special occasions and are given as gifts.

• Ask students if there are any arts and crafts activities that they do with their families. Examples might include papier-mâché, making piñatas, and knitting.

• Say: When working with arts and crafts, having the right amount of material is important. Mathematics ensures that we know both how much material to use and how much material is left.

Session 3 Use with Additional Practice problem 7.

• Say: Art shows are a way to display drawings or other forms of art and share them with other people. Museums often display art permanently, but art shows are temporary. Ask students to share if they have participated in art shows at school before.

• Ask students to bring a copy of their favorite drawing from their culture and create a class art show.

Levels 1–3 Levels 2–4 Levels 3–5

Prepare for Session 1Use with Try It.

English Language Learners:Differentiated InstructionELL

Reading/Speaking Read the Try It problem aloud. Imitate pulling out a long piece of string and cutting it into pieces. Organize students into pairs to complete the following sentence frames:

• The length of the string that Mr. Kovich needs to cut is 2 meters .

• It needs to be cut into 0.2 meter pieces.

Display the expression: 2 4 0.2 5 ? Say: 2 meters of string need to be divided into 0.2 meter pieces. Point to the dividend and divisor as you say the corresponding number. Provide two decimal grids to each pair and have them think aloud as they model the solution.

Reading/Speaking Read the Try It problem with students. Organize them into pairs to restate the problem and discuss possible solution strategies. Provide a word bank to support discussion: dividend, divisor, quotient. Have students work together to write a division equation to represent the problem. Ask them to label the equation with words from the bank. Allow time for partners to work together to model and solve the problems using tools from the Math Toolkit. Call on students to explain their models. Ask: How does your model show the dividend? Divisor? Quotient?

Reading/Speaking Have students read Try It with a partner. Ask them to restate the problem, identify important quantities and discuss possible solution strategies. Provide a word bank to support discussion: dividend, divisor, quotient, tenths, decimal point, divide, multiply, unknown. Have students work together to write a division equation and a multiplication equation that could be used to solve the problem. Allow time for partners to model and solve the problem. Remind them to use known strategies to reason where to put the decimal point.

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LESSON 17

SESSION 1 Explore

Start

Connect to Prior KnowledgeMaterials For each pair: base-ten blocks (2 tens rods, 20 ones units), Activity Sheet Number Lines

Why Activate prior knowledge of division of whole numbers and multiplication of decimals in preparation for dividing with decimals.

How Have students use either base-ten blocks or a number line to model multiplication and division facts related to the problem they will work on in Try It.

©Curriculum Associates, LLC Copying is permitted.

Start

Grade 5 Lesson 17 Session 1 | Explore Dividing Decimals

Model each problem.

20 4 10 5

20 4 2 5

2 4 2 5

0.2 3 10 5

0.02 3 100 5

Possible Solutions20 4 10 5 2 20 4 2 5 102 4 2 5 1 0.2 3 10 5 20.02 3 100 5 2

TRY ITMake Sense of the ProblemTo support students in making sense of the problem, have them identify the amount of string Mr. Kovich has and describe what Mr. Kovich wants to do with his string.

DISCUSS ITSupport Partner DiscussionTo reinforce prior knowledge of division, encourage students to use dividend, divisor, and quotient as they talk to each other.

Look for, and prompt as necessary for, understanding of:

• 2 meters as the total length to be divided

• 0.2 meter as the “group size”

• the number of pieces of string as the number of groups

Purpose In this session, students draw on what they know about division with whole numbers and multiplication with decimals. They share models for finding 2 ÷ 0.2 as they begin to explore dividing with decimals. They will look ahead to think about place-value patterns in decimals division.

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TRY IT Math Toolkit• base-ten blocks• decimal grids• number lines • fraction bars• fraction models • thousandths decimal

place-value charts

DISCUSS ITAsk your partner: Why did you choose that strategy?

Tell your partner: The strategy I used to fi nd the answer was . . .

Now that you know how to multiply with decimals, you will learn how to divide with decimals. Use what you know to try to solve the problem below.

Mr. Kovich is preparing materials for a craft project. He needs to cut 2 meters ofstring into pieces that are 0.2 meter long. How many 0.2-meter pieces can he cutfrom 2 meters of string?

Learning Target• Add, subtract, multiply, and divide

decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

SMP 1, 2, 3, 4, 5, 6, 8

SESSION 1

Explore Dividing DecimalsLESSON 17

347

Possible student work:

Sample A

0 0.2 0.4

5 pieces in 1 meter

10 pieces in 2 meters

5 pieces in 1 meter

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Sample B

How many 0.2s in 2?

2 4 0.2 5 ? ? 3 0.2 5 210 3 0.2 5 2, because multiplying by 10 changes the placement of the decimal point. The value of the digit 2 in the tenths place becomes 10 times greater, which is 2 ones.

He can cut 10 pieces.

Common Misconception Look for students who are not comfortable with the idea of dividing by a number less than 1, as in 2 4 0.2. As students present solutions, have them make connections to money concepts by thinking about how many people $2 can be divided among if each person gets 20 cents ($0.20).

Select and Sequence Student SolutionsOne possible order for whole class discussion:

• concrete models to show 10 groups of 2 tenths

• number lines showing a length of 2 divided into 10 segments of length 0.2

• equations showing 2 4 0.2 or 20 4 2.

Support Whole Class DiscussionStudents may construct arguments to explain the reasoning for this problem in a variety of ways. Prompt students to explain and compare their reasoning.

Ask How do [student name]’s and [student name]’s models show different explanations for why 2 4 0.2 5 10?

Listen for Encourage students in presenting clear, logical explanations and making connections between models. Strategies may include using repeated subtraction or place-value reasoning along with division with whole numbers.


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LESSON 17 EXPLORE SESSION 1

Lesson 17 Divide Decimals

CONNECT IT1 LOOK BACK

Explain how you found how many 0.2-meter pieces are in 2 meters of string.

2 LOOK AHEAD You can look for patterns in division to see how dividing with whole numbers and dividing with decimals is alike and diff erent.

a. Complete the quotients in the table below to see patterns in division.

Expression Expression in Words Quotient

2 4 2 How many groups of 2 are in 2?

2 4 0.2 How many groups of 0.2 are in 2?

2 4 0.02 How many groups of 0.02 are in 2?

b. Use the division patterns in the table to complete the equations.

Dividend Divisor Quotient

2 4 2 5

2 4 0.2 5

2 4 0.02 5

c. How does the quotient compare to the dividend when the divisor is greater than 1? When the divisor is less than 1?

3 REFLECTThink about 2 meters ÷ 0.2 meter. Why is the quotient greater than the dividend when dividing by a decimal less than 1?

348

Possible answer: I drew a number line from 0 to 2.0, divided it into tenths, and then counted how many two-tenths are in 2 meters, 10.

The quotient is less than the dividend. The quotient is greater than the dividend.

Possible answer: 2 meters is split into 10 two-tenths size pieces. So, the

quotient, 10, is greater than the dividend, 2.

100

100

10

10

1

1

CONNECT IT 1 LOOK BACK

Look for understanding of a strategy for dividing the whole number 2 into 10 equal-size parts of size 0.2.

Visual ModelUse a number line to model dividing by a decimal less than 1.

If . . . students are unsure about the concept of dividing by a decimal,

Then . . . use this activity to provide them with a visual model.

Materials For each student: Activity Sheet Number Lines

• Have students label a number line to show 0 to 2, divided into tenths. Tell students that they will use the number line model to show how Mr. Kovich can cut his 2-meter long string into equal-size pieces of length 0.2 meter.

• Have students locate the tick mark that is 0.2 to the left of 2, and draw a jump arrow from 2 to that tick mark. Ask: What does this jump represent? [the first piece of string that Mr. Kovich cuts]

• Have students continue marking jumps of 0.2, moving left until they reach 0. Ask: How many jumps of 0.2 did you make? What does this represent? [10; Mr. Kovich can cut 10 pieces of string that are 0.2 meter long]

2 LOOK AHEADPoint out that division does not always result in a number less than the number you started with.

As students look for patterns in the tables, remind them that whole numbers have a “hidden” decimal point: 2 is 2.0, 1 is 1.0, 10 is 10.0, and 100 is 100.0.

Ask What patterns do you see in the placement of the decimal point in the divisors and the quotients?

Listen for Students should be able to relate the patterns in the tables to their understanding of the relationship between adjacent base-ten place values.

• As the placement of the decimal point in the divisor changes one place to the left (divide by 10), the placement of the decimal point in the quotient changes one space to the right (multiply by 10).

• When the divisor is divided by a power of 10, the quotient is multiplied by that same power of 10.

Close: Exit Ticket

3 REFLECTLook for understanding that when you divide a quantity by a decimal less than 1, you are forming groups that are smaller than 1 whole, so the number of groups is greater than the quantity you started with (the dividend).

Common Misconception If students think that dividing should always result in a quotient that is less than the dividend, then have use money concepts to show that 1 4 0.10 5 10, where 1 is 1 dollar and 0.10 is 1 dime. Dividing $1 into groups of size $0.10 results in 10 groups (10 dimes). The quotient, 10, is greater than the divisor, 0.10.

Real-World ConnectionEncourage students to think about everyday places or situations where people

might need to divide by a decimal. Have volunteers share ideas. Example: Suppose you have $3 and you want to buy bananas that cost $0.60 per pound. To find out how many pounds of bananas you can you buy, divide 3 by 0.6.


LESSON 17

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Name: LESSON 17 SESSION 1


2 Write a related multiplication equation for the division equation 5 4 0.2 5 n.

1 Think about what you know about multiplication and division. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can.

Examples

Examples

Examples

Examples

Examples

Examples

inverse operations

Prepare for Dividing Decimals

349

Possible answers:

Possible answer: n 3 0.2 5 5

3 3 4 5 12 12 4 3 5 44 3 3 5 12 12 4 4 5 3

Multiplication combines equal groups to find a total.Division separates a total into equal groups.

5 3 0.2 5 1.01.0 4 0.2 5 5

5 3 0.8 5 44 4 0.8 5 5

24 4 4 5 ?? 3 4 5 246 3 4 5 24

4 3 0.5 5 22 4 0.5 5 4

0 1 2

0.8 0.8 0.8 0.8 0.8

3 4

0

0.2

1

0.2 0.2 0.2 0.2

2

0.5 0.5 0.5 0.5

Solutions

Support Vocabulary Development

1 Have students say inverse operations.

Ask students to turn to their neighbor and discuss the relationship between multiplication and division using the term inverse operations. Check for understanding and provide support as needed.

Have students work in small groups to complete the graphic organizer. Call on volunteers to share their examples and encourage them to use complete sentences. Ask students to discuss with their group why they have chosen their example. Provide the following sentence starters to guide the discussion:

• I selected this example because …

• I like his/her example because …

• My example is similar in …

Listen to the group discussions and guide them as needed. Ask students to revise their graphic organizers if necessary.

2 Read the problem. Have students work with a partner to write a related multiplication equation.

Supplemental Math Vocabulary• combine

• equal

• groups

SESSION 1 Additional Practice




Speaking/Writing Ask students to form pairs and read Connect It problem 5. Have them discuss how to divide a decimal by a whole number. Collaborate with students to create a bank of terms to support their discussions, such as whole number, decimal point, place value, quotient, tenths, and hundredths. Then have partners work together to write the steps for dividing a decimal by a whole number using complete sentences and precise mathematical language.

Ask students to share their sentences with another pair and discuss how their steps are similar or different.

Speaking/Writing Read Connect It problem 5 and have students form pairs. Ask students to discuss with their partners how to divide a decimal by a whole number. Then have them work together to write a process for dividing a decimal by a whole number. Provide the following sentence starters to guide students:

• When dividing decimals by whole numbers you first .

• Then you divide and .

• Last, you place the decimal point .

Ask students to share their sentences with another pair and discuss how they are similar or different.

Writing Read Connect It problem 5. Display the following sentence frames and read them aloud as students point to the words. Have students then write the sentence frames and complete them with the missing terms. Provide the terms decimal, whole number, decimal point, and place value as needed.

• To divide a decimal by a whole number, you first write the decimal as a whole number of tenths or hundredths.

• You divide and then place the decimal point to show the place value of the quotient.


3 Solve the problem. Show your work.

Mrs. Carter is preparing materials for a craft project. She needs to cut 5 meters of ribbon into pieces that are 0.5 meter long. How many 0.5-meter pieces can she cut from 5 meters of ribbon?

Solution

4 Check your answer. Show your work.

LESSON 17 SESSION 1

350

Possible student work using pictures:

0 1 2 3 4

2 piecesin 1 meter

2 piecesin 1 meter

2 piecesin 1 meter

10 pieces in 5 meters

2 piecesin 1 meter

2 piecesin 1 meter

5


How many 0.5s in 5? 5 4 0.5 5 ? ? 3 0.5 5 510 3 0.5 5 5, because multiplying by 10 changes the placement of the decimal point to show that 5 tenths becomes 5 ones.

She can cut 10 pieces.

She can cut 10 pieces of ribbon.

Prepare for Session 2Use with Connect It.

3 Assign problem 3 to provide another look at solving a problem that involves dividing with decimals.

This problem is very similar to the problem about Mr. Kovich cutting a string into pieces. In both problems, students are given a word problem that requires them to divide a whole number by a decimal to solve. The question asks how many 0.5-meter pieces Mrs. Carter can cut from 5 meters of ribbon.

Students may want to use base-ten blocks or decimal grids.

Suggest that students read the problem three times, asking themselves one of the following questions each time:

• What is this problem about?

• What is the question I am trying to answer?

• What information is important?

Solution:

Students should recognize that they need to find the number of 0.5s in 5. Since there are two 0.5s in one whole, there are ten 0.5s in 5 wholes. Mrs. Carter can cut 10 pieces of ribbon. Medium

4 Have students solve the problem another way to check their answer.


LESSON 17


LESSON 17


TRY IT

SESSION 2

Read and try to solve the problem below.

Coach Ann is setting up a 2.7-kilometer race. She uses fl agsto mark off 9 equal sections of the race. How far apart should she space the fl ags to mark off the sections?

Develop Dividing a Decimal by a Whole Number

Math Toolkit• base-ten blocks• counters• base-ten grid paper• number lines • fraction bars• fraction models

DISCUSS ITAsk your partner: Do you agree with me? Why or why not?

Tell your partner: I agree with you about . . . because . . .

351


Sample A

2 wholes 7 tenths

Trade 2 wholes for 20 tenths. Divide 27 tenths into 9 equal groups.

The distance between flags should be 0.3 kilometer.

Sample B

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

I can count off the tenths into 9 sections with 3 tenths in each group. The flags should be 0.3 kilometer apart.

Start

Connect to Prior KnowledgeWhy Prepare students to connect dividing with decimals to dividing with whole numbers.

How Have students express 0.4 and 5.2 as a whole number of tenths and as a whole number of hundredths.


Start

Grade 5 Lesson 17 Session 2 | Develop Dividing a Decimal by a Whole Number

Complete each equation with a whole number.

0.4 5 tenths

0.4 5 hundredths

5.2 5 tenths

5.2 5 hundredths

Solutions0. 4 5 4 tenths 0. 4 5 40 hundredths5.2 5 52 tenths 5.2 5 52 hundredths

Develop LanguageHow Develop an understanding of the phrase mark off.

How Ask student to circle the phrase mark off in the Try It. Explain to student that when you mark off something, you set a signal that is easy to see. In this problem, the coach marks off 9 equal sections of a race course with flags. Ask students to think of real-life situations, such as in their favorite sport, when something might be marked off.

TRY ITMake Sense of the ProblemTo support students in making sense of the problem, clarify that the race will be run on a course that is 2.7 kilometers long.

Ask What is Coach Ann using flags for?

DISCUSS ITSupport Partner DiscussionEncourage students to use the term tenths as they discuss the problem.

Support as needed with questions such as:

• What strategy or strategies did you use?

• How did you record your ideas?

Common Misconception Look for students who divide 27 by 9 for a solution of 3 kilometers. Encourage students to use number sense to recognize that 3 kilometers is not a reasonable answer. Ask: Should the length of each section be greater than or less than the total length of the race course, 2.7 kilometers?


• concrete models to show equal groups

• drawings or bar models to show equal groups

• number lines marked in tenths

• equations showing 2.7 4 9 5 ? or 9 3 ? 5 2.7

Purpose In this session students solve a problem that requires dividing 2.7 by 9. Students model the quotient 2.7 4 9 either on paper or with manipulatives to develop strategies for dividing a decimal by a whole number.

SESSION 2 Develop



Explore diff erent ways to understand dividing a decimal by a whole number.

Coach Ann is setting up a 2.7-kilometer race. She uses fl ags to mark off 9 equal sections of the race. How far apart should she space the fl ags to mark off the sections?

Picture ItYou can draw a bar model to represent the problem.

You know the number of kilometers to be divided into equal groups.

You know the number of equal groups.

2.7 kilometers

9 groups

k

You do not know the number of kilometers in each group.

Model ItYou can use the relationship between multiplication and division to understand the problem.

To solve 2.7 4 9 5 k, think 9 3 k 5 2.7.

Use place-value understanding to fi nd the missing factor.

2.7 5 27 tenths

9 3 ? tenths 5 27 tenths

LESSON 17 DEVELOP

352

Support Whole Class DiscussionCompare and connect the different representations and have students identify how they are related.

Ask Where does your model show the length of the race? The 9 equal sections where flags should be placed? How far apart the flags should be?

Listen for Students should recognize that accurate representations include showing the division of the length of the race in kilometers (2.7, or 27 tenths) into 9 equal lengths of 0.3 kilometers. For example, 2.7 kilometers may be represented by 27 tenths rods, which are then divided into 9 equal groups of 3 tenths rods.

PICTURE IT & MODEL ITIf no student presented these models, connect them to the student models by pointing out the ways they each represent:

• the number of kilometers to be divided

• the number of equal groups

• the unknown group size

Ask How is the length of the race represented in different ways in the models?

Listen for The bar model shows 2.7 kilometers as the length of the whole bar. In the division and multiplication equations, you see 2.7 kilometers as a dividend and as a product. It is also written as 27 tenths.

For the bar model, prompt students to identify how the model represents the problem.

• What does each section of the bar represent?

• What does the letter k represent?

For the equation model, prompt students to describe how the model uses the relationship between multiplication and division.

• Does the bar model in Picture It represent both the division equation and the multiplication equation? Why?

• Why can you use a multiplication equation to help you solve a division equation?

Deepen UnderstandingEquation ModelsSMP 8 Use repeated reasoning.

When discussing the equations in Model It, prompt students to see how the same basic fact can be used to solve a variety of division equations.

Ask What basic multiplication and division facts are related to the equations 9 3 ? tenths 5 27 tenths and 2.7 4 9 5 k?

Listen for 9 3 3 5 27 and 27 4 9 5 3

Ask How are the numbers 27 and 3 in the basic division fact 27 4 9 5 3 related to the numbers 2.7 and 0.3 in the equation 2.7 4 9 5 0.3?

Listen for 27 is ten times 2.7, and 3 is ten times 0.3.

Generalize Could you use 27 4 9 5 3 to solve other equations, such as 0.27 4 9 5 m and 270 4 9 5 n? Have students explain their reasoning. Listen for understanding that in any situation where you have a group of 27 of one unit, you can divide the group of 27 into 9 equal groups of 3 of that unit. 0.27 5 27 hundredths and 270 5 27 tens, so 0.27 4 9 5 0.03 and 270 4 9 5 30.


LESSON 17


Connect ItNow you will use the problem from the previous page to help you understand how to divide a decimal by a whole number.

1 Look at Picture It. Why is the bar divided into 9 equal parts?

What do you need to fi nd?

2 Look at Model It. How does the division expression 2.7 4 9 relate to the bar model?

3 Explain how you know that 2.7 5 27 tenths.

4 How far apart should Coach Ann space the fl ags? tenths kilometer

Write the distance as a decimal. kilometer

5 Explain how you could divide a decimal by a whole number.

6 REFLECTLook 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 decimal by a whole number? Explain.

SESSION 2

353

Coach Ann marks off 9 equal sections.

3

0.3

Possible answer: To find the size of each group, you need to divide the number of kilometers (2.7) by the number of groups (9).

Possible answer: You can write a related multiplication equation and find the missing factor by thinking of the decimal as a whole number of tenths (or hundredths). Then you can divide as with whole numbers and place the decimal point to show tenths (or hundredths).

Some students may like using a concrete model, such as fraction bars or base-

ten blocks, because it helps them visualize dividing the decimal amount into

equal groups. Other students may like to use place-value understanding to

write the decimal as a whole number of tenths or hundredths before dividing.

2 ones 5 20 tenths; 20 tenths 1 7 tenths 5 27 tenths

the length of each section

CONNECT IT• Tell students that Picture It and Model It show

models useful for decimal division problems.

• Explain that on this page they will examine the models and describe strategies for dividing a decimal by a whole number.

Monitor and Confirm

1 – 2 Check for understanding that:

• The bar model shows the 9 equal sections of the 2.7-kilometer race course.

• The bar model helps you identify each number’s role in the division equation: dividend, divisor, or quotient.

Support Whole Class Discussion

3 – 4 Tell students that these problems relate to using place-value understanding to divide decimals, and will prepare them to provide the explanation required in problem 5.

Ask Why is it helpful to write 2.7 as 27 tenths? After you divide by 9, how do you write the quotient as a decimal?

Listen for Writing the decimal as a whole number of tenths, 27, makes it easy to divide by 9. This lets you divide as with whole numbers. Then you place the decimal point to show 3 tenths as 0.3.

Ask How would you use a similar strategy to divide 0.27 by 9?

Listen for Write 0.27 as 27 hundredths. 27 hundredths divided into 9 equal groups is 3 hundredths, or 0.03.

5 Look for the idea that you can write the decimal as a whole number of tenths (or hundredths) and then divide the tenths (or hundredths) into the number of equal groups given by the whole-number divisor. Then place the decimal point to show tenths (or hundredths).

6 REFLECTHave all students focus on the strategies used to solve this problem. If time allows, have students share their preferences with a partner.

SESSION 2 Develop

Hands-On ActivityAct out the problem.

If . . . students are unsure about writing the equation 2.7 4 9 5 0.3,

Then . . . use this activity to relate the equation to a concrete model.

Materials For each group: 2.7 meters of string, meter stick, scissors, marker

• Tell students they are going to model the problem with a length of string. Explain that because 2.7 kilometers is too great a measurement to use in the classroom, they are going to use 2.7 meters instead. Emphasize that the numbers used are the same as in the problem and that only the measurement units are changed. Have students confirm the measurement of the string.

• Give students the task of cutting the string into 9 equal pieces. Suggest that they not make any cuts until they plan a strategy for dividing the 2.7 meters into 9 equal parts. They may wish to use a marker to mark cut points.

• When they are finished cutting the string, have students align the ends of the pieces of string and measure to find out about how long the pieces are. Precision is difficult in this activity. [about 0.3 meter] Ask pairs of students to explain their process for determining how to cut the string.



Apply ItUse what you just learned to solve these problems.

7 How much will each person receive if $20.35 is split equally among 5 people? Show your work.

Solution

8 What is 0.99 4 11? Show your work.

Solution

9 What is 51.2 4 4?

� 1.28

� 12.08

� 12.8

� 120.8

SESSION 2 LESSON 17 DEVELOP

354


20 4 5 5 435 hundredths 4 5 5 7 hundredthsso $20.35 4 5 5 $4.07


0.99 4 11 5 n means 11 3 n 5 0.99.0.99 5 99 hundredths11 3 ? hundredths 5 99 hundredths11 3 9 hundredths 5 99 hundredths9 hundredths 5 0.09

$4.07

0.09

APPLY ITFor all problems, encourage students to draw some kind of model and write equations to support their thinking.

7 $4.07; Students may break apart the dividend into 20 + 35 hundredths and divide in parts. Students may also draw a picture to show how they could divide the money.

8 0.09; Students may use related multiplication and division equations to solve the problem as a missing factor problem. Students may also draw a bar model to represent the problem.

Close: Exit Ticket

9 C; Write the decimal as a whole number of tenths, then divide: 512 tenths 4 4 5 128 tenths, or 12.8.

Error Alert If students choose A, B, or D, then write the following frame on the board to help them use their knowledge of place value to solve the problem: 51.2 5 tenths 4 4 5 tenthsHave student write the solution as a decimal in the tenths.


LESSON 17


Name:


Study the Example showing one way to divide a decimal by a whole number. Then solve problems 1–5.

ExampleThe temperature rises 4.8 degrees in 6 hours. If the temperature rises by an equal amount each hour, how many degrees did it rise each hour?

You can represent this with a bar model.

4.8 degrees

6 hours

?

To fi nd 4.8 4 6, think 6 3 ? 5 4.8.

4.8 5 48 tenths 6 3 8 tenths 5 48 tenths 6 3 0.8 5 4.8 4.8 4 6 5 0.8

The temperature rose 0.8 degree each hour.

1 Look at the Example. Suppose the temperature rises 5.4 degrees in 6 hours. Complete the steps to fi nd 5.4 4 6.

a. 5.4 4 6 Think: 3 ? 5

b. 5.4 5 tenths 3 ? 5 tenths

c. 6 3 tenths 5 tenths

d. 5.4 4 6 5

2 Use numbers from the box. Write the number of tenths and hundredths in each decimal.

3.5 5 tenths 3.5 5 hundredths

0.79 5 tenths 0.79 5 hundredths

Practice Dividing a Decimal by a Whole Number

LESSON 17 SESSION 2

3.5 0.79 0.35350 35

7.9 79 790

355

6

9

0.9

35

7.9

350

79

54

54 6 54

5.4


Solutions

1 0.9; See completed steps on the student page.

a. Write the problem as a multiplication equation.

b. Write the decimal as a whole number of tenths. Write the problem as a multiplication equation with whole numbers of tenths.

c. Solve the equation as whole numbers of tenths.

d. Write the quotient as a decimal. Basic

2 3.5 5 35 tenths; 3.5 5 350 hundredths; 0.79 5 7.9 tenths, 0.79 5 79 hundredths. Basic

Fluency & Skills Practice Teacher Toolbox

Assign Dividing a Decimal by a Whole Number

In this activity students check whether a given quotient in a division problem is correct by multiplying that quotient by the divisor. If the answer is incorrect, students then find the correct answer. This activity gives students practice in checking whether the answers to division problems involving decimals are correct. Dividing a decimal by a whole number is a useful skill in daily life, as when students find the unit prices of similar items to determine which is the better deal. ©Curriculum Associates, LLC Copying is permitted for classroom use.

Name:

Fluency and Skills Practice

Dividing a Decimal by a Whole Number

Division Problems Student Answers

0.88 4 11 0.80.08

Product: 11 3 0.8 5 8.8

5.6 4 8 0.07

7.2 4 9 0.8

25.35 4 5 5.7

21.7 4 7 3.1

14.4 4 12 0.12

96.16 4 8 12.2

60.18 4 2 30.9

1 Can an answer be incorrect even if it looks reasonable? Explain.

Multiply to check if the student’s answer is reasonable. If not, cross out the answer and write the correct quotient.




Speaking/Writing Have students read the Try It problem independently and identify important information. Organize them into small groups to generate a list of three to four strategies that they could use to solve the problem. Encourage them to support their ideas with clear, detailed explanations. For example, if the group chooses to list modeling the problem with base-ten blocks as a strategy, they should talk about converting ones into tenths and dividing them into groups of 3 tenths. Then ask students to choose one of the strategies to use to solve the problem independently and report back to the group.

Speaking Read Try It with students. Organize them into pairs to restate the problem and discuss possible solution strategies. Provide tools from the Math Toolkit and have them model the problem.

Give each pair 10 index cards with the following numbers and symbols: 3.6, 3.6, 0.3, 0.3, ?, ?, 3, 4, 5, 5. Have them build two related expressions to represent the problem and talk about using a related multiplication fact to solve a division problem.

Encourage students to make connections between the model they used to solve the problem and the equations they constructed from the index cards.

Speaking/Writing Read Try It aloud with small groups of students. Help them make sense of the problem and identify the quantities they will use to solve the problem. Use base-ten blocks to model 3.6. Trade 3 ones for 30 tenths. Display 36 tenths. Say: Three and six tenths is the same as 36 tenths. Encourage students to think aloud with you to solve the problem. Ask: How many groups of 3 tenths are in 36 tenths? Have the group decide what to do to model the problem. Allow students to manipulate the base-ten blocks and solve. After students solve the division problem, work with them to co-construct a complete sentence to frame their answer.


3 Complete the steps for using an area model to solve 1.56 4 12.

1.56 4 12 is the same as 3 ? 5 .

1.56 5 hundredths

12 156 12

12 3 10 5

156

2

12 3 3 5

2

156 hundredths 4 12 5 hundredths

1.56 4 12 5

4 Conor earns $9 an hour for yard work. He raked leaves one afternoon and earned $29.25. How many hours did he rake leaves? Show your work.

Solution

5 Look at problem 4. How much does Conor earn for each minute he does yard work? (1 hour 5 60 minutes) Show your work.

Solution

LESSON 17 SESSION 2

356

12

13

0.13

156

10 3

36

36

36

36

120

120

0

1.56

Students might use a bar model, a place-value strategy, or some other method to find $29.25 4 $9.

Students might use a bar model, a place-value strategy, or some other method to find $9.00 4 60.

3.25 hours

$0.15


3 0.13; See the completed steps and area model on the student page. The steps use the following process.

• Write the division problem as a multiplication equation.

• Then write the decimal as a whole number of hundredths.

• Use an area model to solve the whole number division problem.

• Write the quotient as a whole number of hundredths, and then rewrite as a decimal. Medium

4 3.25 hours; Students may use a concrete model or place-value understanding to write the decimal as a whole number of hundredths before dividing. Medium

5 $0.15; Students may use a concrete model or place-value understanding to write the decimal as a whole number of hundredths to find $9.00 4 60. Challenge


LESSON 17


LESSON 17


TRY IT

Read and try solve the problem below.

Grant has 3.6 pounds of pretzels. He puts the pretzels into bags that each hold 0.3 pound. How many bags does Grant use to hold the pretzels?

SESSION 3

Develop Dividing by Tenths

DISCUSS ITAsk your partner: How did you get started?

Tell your partner: I started by . . .

Math Toolkit• base-ten blocks• counters• base-ten grid paper• number lines • fraction bars• fraction models

357


Sample AHow many groups of 0.3 in 3.6?3.6 4 0.3 5 ? 3.6 5 36 tenths? 3 0.3 5 3.6 ? 3 3 tenths 5 36 tenths 12 3 3 tenths 5 36 tenthsGrant uses 12 bags.

Sample B

3.6

12 groups of 0.3

He has 12 bags of pretzels.

Start

Connect to Prior KnowledgeMaterials For each student: Activity Sheet Number Lines

Why Support students in using both multiplication and division to express the relationship between two tenths decimals, in preparation for dividing tenths by tenths.

How Have students use a number line to model the multiplication equation 0.4 3 6 5 24 and then write the related division equation.


Start

Grade 5 Lesson 17 Session 3 | Develop Dividing by Tenths

Draw a number line model for this multiplication equation.

0.4 3 6 5 2.4

Then write a related division equation.

Possible SolutionLook for a number line from 0 to 3, divided into tenths, with 6 groups of 4 tenths circled.

2.4 4 6 5 0.4

Develop LanguageWhy Review the meaning of the academic word relationship.

How Explain that the word relationship refers to the way two or more things are connected or how they are related. Have students find the word in Model It. Ask: How are multiplication and division related?

TRY ITMake Sense of the ProblemTo support students in making sense of the problem, have them describe what Grant is trying to do with his pretzels.

Ask How many pounds of pretzels does Grant have? How many pounds of pretzels does Grant want to put into each bag?

DISCUSS ITSupport Partner DiscussionEncourage students to use the Discuss It question and sentence starter on the Student Worktext page as part of their discussion.


• What strategy did you use?

• How is your strategy similar to and different from your partner’s strategy?

Common Misconception Look for students who reverse the dividend and divisor and attempt to divide 0.3 by 3.6. Have students draw pictures or use concrete models to represent the situation.


• concrete models to show equal groups

• drawings to show equal groups

• number lines marked in tenths

• equations showing 3.6 4 3 or 36 4 3

Purpose In this session students solve a problem that requires dividing 3.6 ÷ 0.3. Students model the quotient 3.6 ÷ 0.3 either on paper or with manipulatives to develop strategies for dividing decimals in the tenths.

SESSION 3 Develop



LESSON 17 DEVELOP


Explore diff erent ways to understand how to divide with tenths.

Grant has 3.6 pounds of pretzels. He puts the pretzels into bags that each hold 0.3 pound. How many bags does Grant use to hold the pretzels?

Picture ItYou can picture dividing by tenths with decimal grids.

Each large square represents 1 pound of pretzels.

You know the amount to be shared in equal groups and the size of each group.

Model ItYou can use the relationship between multiplication and division to understand the problem.

To solve 3.6 4 0.3 5 n, think n 3 0.3 5 3.6.

Use place-value understanding to fi nd the missing factor.

3.6 5 36 tenths0.3 5 3 tenths

n 3 3 tenths 5 36 tenths

358


Ask Where does your model show how many pounds of pretzels Grant has? Where does it show how much of a pound he will place in each bag?

Listen for Students should recognize that accurate representations include the division of the total pounds of pretzels (3.6, or 36 tenths) into equal groups of size 0.3 pound. For example, 3.6 pounds may be represented by 36 tenths rods, which are then divided into 12 equal groups of 3 tenths rods.


• the number of pounds of pretzels Grant starts with

• the number of pounds pretzels to be put in each bag

• the unknown number of bags

Ask How is the total number of pounds of pretzels shown in different ways in the models?

Listen for The decimal grids show 3.6 pounds by shading 36 columns of 10 squares. In the division and multiplication equations, you see 3.6 as a dividend and as a product. It is also written as 36 tenths.

For the decimal grids model, prompt students to interpret the model.

• Why does the model use four grids?

• What do the red vertical lines show?

For the equation model, prompt students to think of other ways to model the related division and multiplication equations.

• How could you show the division equation with a number line model?

• Would the same number line model show the related multiplication equation?

Deepen UnderstandingDecimal Grids ModelSMP 2 Reason abstractly and quantitatively.

To support discussion of the decimal grid model, prompt students to consider how to use the associative property in interpreting the model.

Display the decimal grid model. Write an expression to model the total shaded part of the grids, by writing an expression under the first grid, discussing how it represents the grid, and then having volunteers write the other expressions.

(0.3 1 0.3 1 0.3 1 0.1) 1 (0.2 1 0.3 1 0.3 1 0.2) 1 (0.1 1 0.3 1 0.3 1 0.3) 1 (0.3 1 0.3)

Then erase the parentheses and use the associative property to write the expression in a way that shows the 12 equal groups of 0.3.

0.3 1 0.3 1 0.3 1 (0.1 1 0.2) 1 0.3 1 0.3 1 (0.2 1 0.1) 1 0.3 1 0.3 1 0.3 1 0.3 1 0.3

Ask How could you interpret this way of writing the expression as “closing the gaps” between the grids?

Listen for You could push the grids together to join a 0.1 group to a 0.2 group, forming a group of 0.3.


LESSON 17


CONNECT ITNow you will use the problem from the previous page to help you understand how to divide with tenths.

1 Look at Picture It. How do you decide what to shade?

2 Why are the grids separated into groups of 3 columns?

3 Look at Model It. How do you know this is a division problem?

4 Explain why 3.6 5 36 tenths.

5 Both Picture It and Model It tell you that Grant uses bags of pretzels.

6 Explain how to divide tenths by tenths.

7 REFLECTLook back at your Try It, strategies by classmates, and Picture It and Model It on the previous page. Which models or strategies do you like best for dividing tenths by tenths? Explain.

SESSION 3

359

Possible answer: Each large square is 1 pound of pretzels, and each column is 0.1 pound, so shade 3 whole large squares and 6 columns of the fourth large square to show 3.6 pounds.

Possible answer: Grant puts the pretzels into 0.3-pound bags. Each column is 0.1 pound, so 3 columns represents 0.3 pound, or one bag of pretzels.

Possible answer: I know the entire amount and the size of each group and need to find how many groups.

Possible answer: You can express each decimal as a whole number of tenths. Then you can divide as with whole numbers.

Some students may like using decimal grids because they can separate the

dividend into equal groups of tenths and count the number of groups. Other

students may like using place-value reasoning to write the decimals as whole

numbers of tenths because then they can use whole-number division.

3 ones equals 30 tenths; 30 tenths 1 6 tenths 5 36 tenths.

12

SESSION 3 Develop



• Explain that on this page they will examine the models and describe strategies for dividing tenths by tenths.

Monitor and Confirm


• each grid stands for one whole, or 1 pound of pretzels

• each column of the grid stands for one tenth, or 0.1 pound of pretzels

• each three-column group represents one 0.3-pound bag of pretzels

• the problem involves dividing a total amount, 3.6 pounds, into equal groups of size 0.3 pound


4 – 5 Tell students that these problems relate to using place-value understanding to divide decimals, and will prepare them to provide the explanation required in problem 6.

Ask Why is it helpful to write 3.6 as 36 tenths and 0.3 as 3 tenths?

Listen for After both numbers are written as whole numbers of tenths, it is easy to divide 36 by 3 to find that there are 12 groups of 3 tenths.

6 Look for the idea that you can write each decimal as a whole number of tenths, and then divide the whole numbers to find the number of groups of tenths in the total.


Visual ModelRepresent the problem with an area model.

If . . . students are unsure about using the relationship between division and multiplication when dividing decimals,

Then . . . use this activity to help them visualize the relationships and find the missing factor in an area model.

• Have students draw a rectangle and then draw a vertical line to separate it into a large section to represent an area of 3.0 and a smaller section to represent an area of 0.6. Have them label the height of the rectangle as 0.3.

• Inside the large section, have students write the equation 0.3 3 ? 5 3.0.

• Inside the smaller section, have students write the equation 0.3 3 ? 5 0.6.

• Ask students to identify the missing factor in each section. [10 and 2]

• Have students use the missing factors to label the widths of the two sections of the rectangle. Ask students to identify the width of the large rectangle. [12]

• Ask students to write a division equation to match the area model. [3.6 4 0.3 5 12]


LESSON 17 DEVELOP

Apply ItUse what you just learned to solve these problems.

8 How many dimes are there in a jar of dimes worth $2.70? Show your work.

Solution

9 A fence is 52.5 meters in length. Posts divide the fence into sections that are each 3.5 meters in length. How many sections are in the fence? Show your work.

Solution

10 What is 42 4 0.7?

� 0.06

� 0.6

� 6

� 60

SESSION 3

360 Lesson 17 Divide Decimals ©Curriculum Associates, LLC Copying is not permitted.

360


There are 10 dimes in each dollar.2 dollars 5 20 dimes70 cents 5 7 dimes20 1 7 5 27

27 dimes

15 sections


52.5 4 3.5 5 ? 3.5 5 35 tenths 52.5 5 525 tenths525 4 35 5 15

APPLY ITFor these problems, encourage students to try at least two different strategies across the three problems. Tell students that using different strategies will expand their understanding of operations with decimals.

8 27 dimes; See possible work on Student Worktext page. Students may also use the relationship between division and multiplication, picture dividing with decimal grids, draw a bar model, or make an area model with a missing factor to solve the problem.

9 15 sections; Students may solve the problem by writing each number as a whole number of tenths, and dividing as with whole numbers.

Close: Exit Ticket

10 D; Students may write the multiplication equation ? 3 7 tenths 5 420 tenths to find 60 as the missing factor, and thus, the quotient for 42 4 0.7.

Error Alert If students choose A, B, or C, then help them see that they need to think of both numbers in the same units. If they change 0.7 to 7 tenths, they need to change 42 to 420 tenths, multiplying both numbers by 10 to write them as a whole number of tenths.


LESSON 17


Name:


Study the Example showing one way to divide a decimal by a decimal. Then solve problems 1–7.

ExampleWhat is 2.1 4 0.7?

You can represent this problem with decimal grids.

Each large square represents 1 whole.

To fi nd 2.1 4 0.7, think 0.7 3 ? 5 2.1.

The lines separate groups of 0.7.

2.1 5 21 tenths0.7 5 7 tenths

7 tenths 3 ? 5 21 tenths7 tenths 3 3 5 21 tenths

2.1 4 0.7 5 3

1 Look at the Example. How is the quotient, 3, represented by the grids?

2 Which of these expressions are represented by the decimal grids in the Example?

� 0.7 3 3 � 3 3 0.7

� 0.7 4 3 � 2.1 4 3

� 21 4 0.7 � 0.7 4 2.1

3 How many grids would you need to represent the problem 4.5 4 0.5? Explain.

LESSON 17 SESSION 3

Practice Dividing by Tenths

361

Answers will vary. Possible answer: It is represented by the 3 groups you make when you divide 2.1, or 21 tenths, into groups of 7 tenths.

5 grids; Possible explanation: In order to represent 4.5, I would use 4 decimal grids to represent 4 and 1 more to represent 0.5, for a total of 5 grids.


Solutions

1 Answers will vary; See possible answer on the student page. Students explain that the 3 groups of 7 tenths each are represented in the groups of 7 columns in the grids. Basic

2 A; The lines separate groups of 0.7; there are 3 groups of 0.7.

B; The lines separate groups of 0.7; there are 3 groups of 0.7.

D; 2 whole and 1 tenth column of the grids are shaded, showing 2.1. The lines divided the columns are into 3 equal groups. Medium

3 5 grids; Explanations should describe the 4 grids needed to represent the 4 wholes and 1 grid for the 5 tenths, or 0.5, for a total of 5 grids. Basic


Assign Dividing by Tenths

In this activity students practice dividing both whole numbers and decimals by tenths. Students may use this skill in everyday life. For example, they might need to calculate their average speed if they travel 150 miles in 2.5 hours. Or, they might want to find how many 1.5-foot lengths of wood can be cut from a single 7.5-foot plank.

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

Name:


Dividing by Tenths

1 2.4 4 0.4

4 12.1 4 1.1

7 75 4 2.5

10 120 4 0.2

13 112.5 4 7.5

2 4.8 4 0.6

5 12 4 0.2

8 0.32 4 0.4

11 45 4 0.3

14 1.32 4 1.2

3 56 4 0.7

6 4.5 4 1.5

9 7.5 4 1.5

12 99 4 0.9

15 234 4 6.5

The answers are mixed up at the bottom of the page. Cross out the answers as you complete the problems.

Answers

1.1

6

3

11

60

0.8

110

600

8

150

36

80

15

30

5




Listening/Speaking Organize students into small groups to read and discuss the Try It problem. Ask them to choose three or four tools from the Math Toolkit and discuss how each could be used to model the problem. Then ask students to choose one of the strategies to use to solve the problem independently and report back to the group.

Have students think about why they prefer their chosen strategy and complete the sentence starter:

• I prefer to use because .

Listening/Speaking Read Try It with students. Organize them into pairs to restate the problem. Ask them to choose two tools from the Math Toolkit and discuss how each could be used to model the problem. Have partners select their preferred strategy and model the problem. Provide a word bank to support conversations: equal groups, hundredths, represent, feet, cents, and dollars.

Have partners meet with another set of partners who solved the problem using a different strategy. Ask: How is your way like the other group’s way? How is your way different?

Listening/Speaking Read Try It aloud to small groups of students. Ask them to restate the problem in their own words. Display the question:

• How many groups of 8 cents are there in $1.20 ?

Provide play money and think aloud with students to model the problem. Encourage them to provide as much information as possible and use precise language such as cents, dollars, foot, and feet.

Have group members take turns modeling the problem with play money and explaining the steps.


4 Complete the steps to solve 4.5 4 0.5.

a. 4.5 4 0.5 Think: 3 ? 5

b. 4.5 5 tenths and 0.5 5 tenths

c. 5 tenths 3 5 45 tenths

d. 4.5 4 0.5 5

5 Rewrite each division problem as a multiplication problem and solve.

a. 6.3 4 0.9 5 ? 3 ? 5 6.3 4 0.9 5

b. 3.2 4 0.4 5 ? 3 ? 5 3.2 4 0.4 5

c. 1.8 4 0.3 5 ? 3 ? 5 1.8 4 0.3 5

d. 2.4 4 1.2 5 ? 3 ? 5 2.4 4 1.2 5

6 The Razdan family drinks 0.5 gallon of milk a day. Will 2.5 gallons of milk last them more than 1 week? Explain. Show your work.

Solution

7 Mrs. Lang is hanging pictures for the school art show across a wall that is 2.8 meters wide. She determines that each picture, along with the space needed on the sides of the picture, will take up 0.4 meter across the wall. How many pictures can she hang in one row across the wall? Show your work.

Solution

LESSON 17 SESSION 3

362

0.5

45 5

9

9

0.9

0.4

0.3

1.2

6.3

3.2

1.8

2.4

7

8

6

2

4.5

Students might use decimal grids, write a multiplication equation, or use another method to find 2.5 4 0.5.

7 pictures

Students might use decimal grids, write a multiplication equation, or use some other method to find 2.8 4 0.4.

No. Possible explanation: 2.5 4 0.5 5 5 and 5 , 7, so 2.5 gallons

will only last 5 days, not more than 1 week.


4 See the completed steps on the student page. Students use the relationship between multiplication and division and their understanding of place value to solve the problem. Basic

5 See the completed equations on the student page. Students use the relationship between multiplication and division and their knowledge of basic facts to solve the problems. Medium

6 No; See possible explanation on the student page. Students may choose any strategy to divide decimals and show that 2.5 4 0.5 5 5 and that 5 is less than 7, which is the number of days in a week. Therefore, they can conclude that 2.5 gallons will not last a week. Medium

7 7 pictures; Students may use any strategy for dividing decimals to solve 2.8 4 0.4. Challenge


LESSON 17


LESSON 17


TRY IT

Read and try to solve the problem below.

Fiona has $1.20 with which to buy some ribbon. The ribbon is on sale for $0.08 per foot. How many feet of ribbon can Fiona buy?

SESSION 4

Develop Dividing by Hundredths

Math Toolkit• base-ten blocks• play money• base-ten grid paper• number lines • fraction bars• fraction models

DISCUSS ITAsk your partner: Can you explain that again?

Tell your partner: A model I used was . . . It helped me . . .

363


Sample A

$1.20 5 120 cents

$0.08 5 8 cents

120 4 8 5 15

Fiona can buy 15 feet of ribbon.

Sample B

5 feet of ribbon will cost 5 3 $0.08 5 $0.40.

5 feet 5 feet 5 feet

$0 $0.40 $0.80 $1.20

She can buy 3 3 5 feet 5 15 feet of ribbon.

155

108 q ······ 120 2 80

402 40

0

Start

Connect to Prior KnowledgeMaterials For each student: Activity Sheet Hundredths Grids

Why Reinforce students’ understanding of dividing a decimal in the hundredths by a whole number, in preparation for dividing by hundredths.

How Have students use decimal grid paper to model the problem $1.50 ÷ 10, write an equation, and solve the problem.


Start

Model on decimal grids.Write an equation and solve.

Misha divides $1.50 into 10 equal parts. How much is in each part?

Grade 5 Lesson 17 Session 4 | Refi ne Dividing by Hundredths

Possible SolutionsLook for 2 decimal grids, 1 with all columns shaded and the other with 5 columns shaded for a total of 15 shaded columns.

$1.50 4 10 5 $0.15

Develop LanguageWhy Review the term decimal grid.

How Remind students that they have used base-ten grid paper to model problems with decimals. Tell students that a grid is an arrangement of horizontal and vertical lines that intersect to form squares. A decimal grid has ten columns and ten rows of squares and represents a whole. Each square in the grid represents one hundredth. Ask: How does a decimal grid help you understand a problem better?

TRY ITMake Sense of the ProblemTo support students in making sense of the problem, have them identify the amount of money Fiona has to spend on ribbon.

Ask How is the ribbon being sold?

DISCUSS ITSupport Partner DiscussionEncourage students to share what did not work for them as well as what did as they talk to each other.


• How did your model help you try to solve the problem?

• What strategy did your partner use to solve the problem?

Common Misconception Look for students who try to divide $0.08 by $1.20 instead of $1.20 by $0.08. Remind students that the dividend is the amount that is going to be divided into equal groups.


• concrete models to show $1.20 ÷ $0.08

• drawings or bar models to show equal groups

• number lines showing equal groups or showing hundredths

• equations showing 1.20 4 0.08 or 120 4 8

Purpose In this session students solve a problem that requires dividing 1.20 by 0.08. Students model the quotient $1.20 4 $0.08 in the word problem either on paper or with manipulatives to develop strategies for dividing by a decimal in the hundredths.

SESSION 4 Develop



LESSON 17 DEVELOP


Explore diff erent ways to understand how to divide by hundredths.

Fiona has $1.20 with which to buy some ribbon. The ribbon is on sale for $0.08 per foot. How many feet of ribbon can Fiona buy?

Picture ItYou can picture the division problem using a bar model.

You know the total amount of money and the size of one group.

$1.20

?

$0.08

You do not know the number of groups.

Model ItYou can use place-value reasoning to relate dividing with decimals to dividing with whole numbers.

1.20 4 0.08 5 ? How many groups of size 0.08 are in 1.20?

120 hundredths 4 8 hundredths 5 ?

120 4 8 5 ? How many groups of size 8 are in 120?

Then you can divide the hundredths as you might divide with whole numbers, such as using partial quotients.

?5

108 q ······ 120

2 8040

2 400

364


Ask Where does your model show how much money Fiona has? How did you show the cost per foot?

Listen for Students should recognize that accurate representations include the division of the total amount of money ($1.20, or 120 cents) into equal groups of size ($0.08, or 8 cents).


• the amount of money Fiona has

• the cost per foot of ribbon

• the unknown number of feet of ribbon Fiona can buy

Ask How is the amount of money Fiona has represented in different ways in the models?

Listen for The bar model shows $1.20 as the length of the whole bar. In the division equations, you see 1.20 as a dividend and then re-written as 120 hundredths.

For the bar model, prompt students to identify how the model represents the problem.

• What does the section labeled $0.08 represent?

• Why is the middle part of the bar shown with dashed lines and labeled with a question mark?

For the equation model, prompt students to analyze relationships between the numbers in the equations.

• How is 120 related to 1.20?

• How is 8 related to 0.08?

Deepen UnderstandingPlace-Value ReasoningSMP 1 Make sense of problems.

When discussing the equations in Model It, prompt students to consider how place-value reasoning allows you to replace the original form of the problem with a simpler form of the problem.

Ask How does using place-value reasoning help you to simplify and make sense of the problem?

Listen for Place-value reasoning lets you look at the problem as whole-number division rather than as dividing decimals. You can see it as 120 cents divided by 8 cents. The solution is the number of groups of 8 cents that can be made out of 120 cents, or 120 4 8 5 15.

Students can recognize the relationship between the values of the digits in both equations. Ask: How can you tell that 120 is 100 3 1.2 and that 8 is 100 3 0.08? [All digits have moved two place values to the left.] Explain that the relationship between the numbers is the same in both equations, so the quotient of both equations is the same.


LESSON 17


Connect ItNow you will use the problem from the previous page to help you understand how to divide by hundredths.

1 Look at Picture It. What are you trying to fi nd in this problem?

2 What operation will solve this problem? How do you know?

3 Look at Model It. Explain why 1.20 5 120 hundredths.

4 How many feet of ribbon can Fiona buy? Explain how you know.

5 Check your answer using the decimals in a multiplication equation.

feet of ribbon 3 $ 5 $ .

6 Explain how to divide by a decimal in the hundredths.

7 REFLECTLook back at your Try It, strategies by classmates, and Picture It and Model It on the previous page. Which models or strategies do you like best for dividing by hundredths? Explain.

SESSION 4

365

the number of groups of 0.08

Possible answer: Division; if you know the total amount and the size of one group, you can divide to find the number of groups.

Possible answer: Write both numbers as hundredths. Then divide as whole numbers. Then place the decimal point in the quotient where it makes sense.

Student responses may refer to using place-value understanding to rewrite

the decimals in terms of hundredths and then dividing as with whole

numbers.

15 feet; Possible explanation: The sum of the partial quotients is 10 1 5, or 15. 120 4 8 5 15, so 1.20 4 0.08 is also 15.

1 one equals 100 hundredths; 100 hundredths 1 20 hundredths 5 120 hundredths.

15 0.08 1.20



• Explain that on this page they will examine the models and describe strategies for dividing by a hundredths decimal.

Monitor and Confirm


• the problem involves dividing a total amount, $1.20, into equal groups of size $0.08

• 1 whole equals 100 hundredths

• Fiona can buy 15 feet of ribbon


5 Have students articulate how to use the relationship between multiplication and division to check a quotient.

Ask How do you write a multiplication equation to check your answer to a division problem?

Listen for The factors for the multiplication equation are the quotient and the divisor of the division problem. The product should equal the dividend, or the total amount being divided into equal groups. Students might explain this using the problem: you multiply how many feet of ribbon Fiona can buy by the cost per foot to show the total amount of money Fiona had to spend on ribbon.

6 Look for the idea that you can write each decimal as a whole number of hundredths, and then divide the whole numbers to find the number of groups of hundredths in the total.


SESSION 4 Develop

Hands-On ActivityUse coins to model decimal division.

If . . . students are unsure about solving $1.20 ÷ 0.08 with representational drawings or abstract numbers,

Then . . . use this activity to let students model the problem concretely.

Materials For each pair: play money (20 nickels and 50 pennies)

• Have students work with partners.

• Ask them to work together to make enough groups of 0.08 cents from 1 nickel and 3 pennies to make $1.20.

• In so doing, have students keep track of how many groups of 0.08, or 8 cents, they need to make to get $1.20, or 120 cents.

• Have students write the equations 1.20 4 0.08 5 15 and 120 4 8 5 15 and discuss how they each show dividing a quantity into 15 equal groups.



LESSON 17 DEVELOP


Apply itUse what you just learned to solve these problems.

8 How many quarters are in a jar of quarters worth $9.75? Show your work.

Solution

9 What is 16 4 0.16? Show your work.

Solution

10 Which division expression can be used to fi nd the quotient 17.6 4 0.04?

� 1.76 4 4

� 176 4 4

� 176 4 40

� 1760 4 4

SESSION 4

366

39 quarters

100


1 quarter 5 25 cents$9.75 5 975 cents975 4 25 5 39


16 5 1,600 hundredths0.16 5 16 hundredths1,600 4 16 5 100

APPLY ITFor all problems, encourage students to use one of the models from this session to support their thinking. Allow some leeway in precision for drawing bar models.

8 39 quarters; See possible work on the Student Worktext page. Students may also draw a bar model, write each decimal as a whole number of hundredths, or use partial quotients.

9 100; See possible work on the Student Worktext page. Students may also draw a bar model, write each decimal as a whole number of hundredths, or use partial quotients.

Close: Exit Ticket

10 D; Since 0.04 needs to be multiplied by 100 to become 4 hundredths, 17.6 also needs to be multiplied by 100 to become 1760 hundredths.

Error Alert If students choose A, B, or C, then have them multiply each number in the original problem by 100 and check this new expression against the answer choices shown.


LESSON 17


Name:


Study the Example showing one way to divide by hundredths. Then solve problems 1–6.

ExampleFind 1.8 4 0.04.

Find the least place. Write each decimal to the least place.

0.04 5 4 hundredths1.8 5 180 hundredths

Divide as you would with whole numbers, using partial quotients or another method.

180 hundredths 4 4 hundredths 5 45

1.8 4 0.04 5 45

455

404 q ···· 180 2 160

202 20

0

1 Complete the steps to solve 1.02 ÷ 0.06.

a. 1.02 5 hundredths

0.06 5 hundredths

b. 102 4 6 5

c. 1.02 4 0.06 5

2 Did you use partial quotients or another method to divide 102 by 6 in problem 1? Explain.

3 Check your answer to problem 1 by writing the decimals in a multiplication equation.

3 5

Practice Dividing by Hundredths

LESSON 17 SESSION 4

367

102

6

17

17

0.06 17 1.02

Answers will vary. Possible answer: I used mental math. I thought of 102 as 60 1 42. That’s (6 3 10) 1 (6 3 7), or 6 3 17. So, 102 4 6 5 17.


Solutions

1 a. 102 hundredths; 6 hundredths

b. 17

c. 17 Basic

2 Answers will vary; See a possible explanation on the student page. Medium

3 0.06 3 17 5 1.02 Basic


Assign Dividing by Hundredths

In this activity students divide both whole numbers and decimals by hundredths. This skill is useful when working with precise measurements, as when determining the average speed of a sprinter who runs a 100-meter dash in 9.85 seconds or finding how many 0.15-kilogram pieces of copper can be made from a single 2.25-kilogram copper bar. Noticing patterns in dividing by decimals helps students understand the place-value system and improves their mental math skills. ©Curriculum Associates, LLC Copying is permitted for classroom use.

Name:


Dividing by Hundredths

1 1 4 0.25

4 6.5 4 0.25

7 1.8 4 0.09

10 2.25 4 0.75

13 36.36 4 0.06

2 4 4 0.25

5 1.8 4 9

8 225 4 75

11 0.36 4 0.06

14 9 4 2.25

3 3.75 4 0.25

6 1.8 4 0.9

9 22.5 4 7.5

12 6.36 4 0.06

15 13.5 4 2.25

Divide.

16 Describe a pattern you noticed when you were completing the problem set.




Reading/Writing Read Apply It problem 5. Have students form pairs. Ask students to discuss with a partner how they can calculate the amount of peanuts Keith bought on all three days and how many pounds each friend will get.

Ask students to write complete sentences describing the process for solving the problem. After their processes are complete, have partners take turns reading them to each other.

Speaking/Listening Read Apply It problem 5. Have students form pairs. Ask students to discuss with a partner how they can calculate the amount of peanuts Keith bought on all three days and how many pounds each friend will get. Provide the following sentence frames and starters. Have students take turns saying the completed sentences to their partners.

• Keith bought peanuts on , , and for a total of pounds.

• Keith will share the peanuts between himself and friends.

• To solve this problem, I first add .

• Then I divide .

Speaking/Listening Read Apply It problem 5. Make a three-column chart with the headings Monday, Tuesday, and Wednesday. Ask students to say how many pounds of peanuts Keith purchased on each day. Record students’ responses. Ask: How many people do the peanuts need to be divided between? Display the following sentence frames:

• Keith bought plus plus pounds.

• Keith will divide the pounds by .

Call on students to say the missing information. Then, read the completed sentences and have students choral read.


4 Is each equation True or False?

True False

1.23 5 123 hundredths � �

0.5 5 50 hundredths � �

74 hundredths 5 7.4 � �

1,088 hundredths 5 10.88 � �

5 Jaden buys 1.15 pounds of cheese at the deli counter. If each slice is 0.05 pound, how many slices of cheese does she buy? Show your work.

Solution

6 Alejandro feeds his dog 0.12 kilogram of dry dog food each day. He wants to buy the smallest bag that has enough food to feed his dog for one month. Should he buy the bag that has 1.8 kilograms, 2.4 kilograms, or 4.2 kilograms of dog food? Show your work.

Solution

LESSON 17 SESSION 4

368

Students might represent the problem with a bar model and then use partial quotients, write a multiplication equation, or use another method to find 1.15 4 0.05.

Students might use partial quotients, write multiplication equations, or use another method to find 1.8 4 0.12 5 15, 2.4 4 0.12 5 20, and 4.2 4 0.12 5 35.

23 slices

the bag that has 4.2 kilograms

Prepare for Session 5Use with Apply It.

4 A (True);

C (True);

F (False);

G (True) Medium

5 23 slices; See the student page for suggestions of possible strategies student might use. Medium

6 The bag that has 4.2 kilograms; See the student page for the three equations that reveal the number of days each bag would feed Alejandro’s dog, indicating that only the largest bag would provide enough food for a month, 35 days. Challenge


LESSON 17


Refine Dividing Decimals LESSON 17


Complete the Example below. Then solve problems 1–8.

EXAMPLENancy ran a total of 35 miles to train for a race. She ran 2.5 miles each day. How many days did Nancy run to train for the race?

Look at how you could show your work using equations.

Let d 5 number of days.

2.5 3 d 5 35

2.5 5 25 tenths, 35 5 350 tenths

25 3 d 5 350

350 4 25 5 14

Solution

Apply it1 What number multiplied by 8 will give a product of 9.6?

Write an equation and solve. Show your work.

Solution

SESSION 5

The student wrote a related multiplication equation to solve the problem.

PAIR/SHARECan you solve the problem in another way?

What is a good estimate for your answer?

PAIR/SHAREHow could you model this problem with a number line?

369

Possible student work using equations:

n 3 8 5 9.6 9.6 4 8 5 n 96 tenths 4 8 5 n 12 tenths 5 n 1.2 5 n

1.2

Nancy ran 14 days to train for the race.

Start

Check for UnderstandingMaterials For remediation: Activity Sheet Base-Ten Grid Paper

Why Confirm understanding of dividing decimals.

How Have students find the quotient 8.4 ÷ 0.3 using any strategy they want.


Start

Use any strategy to find the quotient.

8.4 4 0.3 5 ?

Grade 5 Lesson 17 Session 5 | Refi ne Dividing Decimals

Solution28

SESSION 5 RefinePurpose In this session students solve word problems involving dividing with decimals and then discuss and confirm their answers with a partner.

Before students begin to work, use their responses to the Check for Understanding to determine those who will benefit from additional support.

As students complete the Example and problems 1–3, observe and monitor their reasoning to identify groupings for differentiated instruction.

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

2.8have placed the decimal point to match the dividend or divisor.

Have students rewrite 8.4 as 84 tenths and 0.3 as 3 tenths. Then have students divide the whole numbers of tenths.

0.28have divided the quotient by 100 (0.1 3 0.1).

Have students check their work by multiplying. Then have them rewrite 8.4 as 84 tenths and 0.3 as 3 tenths. Then have students divide the whole numbers of tenths.

280have multiplied the dividend by 10 before dividing.

Have students use base-ten grid paper to show 8.4 and then mark off groups of 0.3. Then review the place-value division process.

Error Alert



2 The length of a screw is 0.75 centimeter. How many screws can be placed end to end to make a row that is 18 centimeters long? Show your work.

Solution

3 What is 6.5 4 0.5?

� 3.25

� 6

� 7

� 13

Gwen chose � as the correct answer. How did she getthat answer?

LESSON 17 REFINE

Will the answer be greater than or less than 18?

PAIR/SHAREExplain how you decided what operation to use to solve the problem.

PAIR/SHAREDoes Gwen’s answer make sense?

I could draw a model to represent this problem.

370

Possible student work: 18 4 0.75 18 5 1,800 hundredths; 0.75 5 75 hundredths 1,800 4 75

22

2075 q ······ 1,800 2 1,500

3002 150

1502 150

0

20 1 2 1 2 5 24

24 screws

Possible answer: She multiplied 6.5 by 0.5 to get 3.25. She should have divided 6.5 by 0.5 to get 13.

EXAMPLENancy ran 14 days to train for the race; The related multiplication equation with a missing factor shown is one way to solve the problem. Students could also solve the problem by creating a division house or use another method to find 35 4 2.5.

Look for To solve 35 4 2.5 5 n you can think about the missing factor equation n 3 2.5 5 35, which you can write in the form n 3 25 tenths 5 350 tenths.

APPLY IT1 1.2; See possible equations shown on the

Student Worktext page. DOK 2

Look for The dividend, 9.6, and the divisor, 8, are both close to 10, so the quotient should be close to 10 4 10, or 1.

2 24 screws; Students could solve the problem by writing 18 as 1,800 hundredths and 0.75 as 75 hundredths and then solve the whole-number division problem 1,800 4 75. DOK 2

Look for The answer will be greater than 18. Each screw is less than a centimeter long (0.75 , 1), so more than 18 can be lined up end to end to make a row that is 18 centimeters long.

3 D; Write each number as number of tenths (65 tenths and 5 tenths); then divide. 65 4 5 5 13. Explain why the other two answer choices are not correct:

B is not correct because 6 is the difference of the 6.5 and 0.5, not the quotient.

C is not correct because 7 is the sum of 6.5 and 0.5, not the quotient. DOK 3


LESSON 17


4 Jordan has $3.80 to spend at the used book store. Each book costs $0.95. What is the greatest number of books Jordan can buy?

� 3

� 4

� 5

� 6

5 Keith bought 3.4 pounds of peanuts on Monday, 2.5 pounds on Tuesday, and 4 pounds on Wednesday. He is going to divide the peanuts equally between himself and two friends. How many pounds of peanuts will each friend get?

• • • • • •

2

3

8

9

4

0

6

1

7

5

2

3

8

9

4

0

6

1

7

5

2

3

8

9

4

0

6

1

7

5

2

3

8

9

4

0

6

1

7

5

2

3

8

9

4

0

6

1

7

5

2

3

8

9

4

0

6

1

7

5

6 If you put 0.7 in the box for each equation, is the equation true?

Yes No

3 5.2 5 36.4 � �

49 4 5 70 � �

4 3.5 5 0.02 � �

9.1 3 5 6.37 � �

SESSION 5

371

. 33

4 B; Divide 3.80 by 0.95 to find the number of books. DOK 2

5 3.3; First, add 3.4 1 2.5 1 4. Then, divide the sum, 9.9, by the number of people getting peanuts, 3. DOK 2

Error Alert Students who find an answer of 33 may have written 99 tenths 4 3, but then neglected to write the quotient 33 tenths as a decimal. Remind students to use estimation to confirm the placement of the decimal point or check their answers with multiplication.

6 B (No);

C (Yes);

F (No);

G (Yes)

DOK 2

SESSION 5 Refine

Differentiated Instruction

RETEACH EXTEND

Visual ModelUse number lines to estimate quotients with decimals.

Students struggling with division of decimals

Will benefit from using an estimate to check their answer

Materials For each pair: Activity Sheet Number Lines

• Tell students they can use a number line to model an estimate for the quotient 2.76 4 6.

• Ask: What is 2.76 rounded to the nearest whole number? [3] Have students make a number line from 0 to 3, divided into tenths. Ask: How can you divide 3 wholes into 6 equal groups of tenths? [3 5 30 tenths, so divide 30 tenths into 6 groups of 5 tenths each.]

• Have students write the division equation represented by their number line. [3 4 6 5 0.5]

• Now write the original problem 2.76 4 6 as 276 hundredths 4 6. Ask: What is 276 4 6? [46] What is 276 hundredths 4 6? [46 hundredths, or 0.46]

• Ask: How does your estimate help you know the decimal point is placed correctly in the quotient? [The quotient 0.46 is close to the estimate of 0.5.] Repeat activity for 4.16 4 8. [0.52]

Challenge ActivityDivide with decimals to solve a problem.

Students who have achieved proficiency

Will benefit from deepening understanding of dividing with decimals

• Challenge students with a problem that asks them to find the perimeter of a rectangle with a given area.

A rectangle has an area of 15.3 square meters. The length of one side of the rectangle is 0.9 meter. What is the perimeter of the rectangle? [35.8 meters]



LESSON 17 REFINE


7 Jamie has 5 jars to fi ll with beads for a carnival game. She has 7.5 bags of multi-colored beads. Jamie wants to put an equal amount of beads in each jar. How many bags of beads can she put into each jar?

Part A Use decimal grids to solve the problem. Explain your solution.

Part B Use multiplication to check your answer.

8 MATH JOURNALA sticker is 1.2 centimeters wide. How many stickers will fi t edge to edge on a strip of paper that is 108 centimeters long? Explain your thinking.

SESSION 5 LESSON 17 REFINE

SELF CHECK Go back to the Unit 3 Opener and see what you can check off .

372

Possible student work: Each grid represents 1 bag of beads.

Possible student work:I need to show that 5 3 1.5 5 7.5.

1.53 5

251 50

7.5

tenthstenths

1.5 bags; Possible explanation: Jamie can put 1 whole bag of beads into each jar. Then she can share the remaining 25 tenths of a bag among the 5 jars and put 5 tenths of a bag into each jar.

Possible answer: I need to find how many groups of 1.2 are in 108, so I solve 108 4 1.2 5 n. If I write 108 as 1,080 tenths and 1.2 as 12 tenths, then I can find how many groups of 12 are in 1,080. 1,080 4 12 5 90, so 1,080 tenths 4 12 tenths 5 90. The strip of paper can fit 90 stickers.

7 Part A

1.5 bags; See possible shading of the decimal grids, and possible explanation, on the Student Worktext page.

Part B

5 3 1.5 5 7.5; See possible work on the Student Worktext page. DOK 3

Close: Exit Ticket

8 MATH JOURNAL 90 stickers; Student responses should indicate understanding that they are asked to find how many groups of 1.2 are in 108, which can be answered by finding the quotient 108 4 1.2. Responses should include a description of how to find a quotient of 90.

Error Alert If students find an answer of 9 stickers, then have them use an estimate to explain why the answer should be closer to 100 than to 10.

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

REINFORCE PERSONALIZE

Problems 4–8Divide decimals.

All students will benefit from additional work with dividing decimals by solving problems in a variety of formats.

• Have students work on their own or with a partner to solve the problems.

• Encourage students to show their work.

Provide students with opportunities to work on their personalized instruction path with i-Ready Online Instruction to:

• fill prerequisite gaps

• build up grade-level skills

©Curriculum Associates, LLC Copying is not permitted.372a Lesson 17 Divide Decimals

LESSON 17

Lesson Quiz Teacher Toolbox

Choice Matrix Scoring Rubric

2 points 1 point 0 points

All answers are correct

1 incorrect answer 2 or more incorrect answers

Short Response Scoring Rubric

Points Expectations

2

• Correct computations, solutions, and/or calculations. (1 point)

• Well-organized, clear, and concise work that demonstrates thorough understanding of math concepts and/or procedures. (1 point)

1• Mostly correct solution(s).• Shows partial or good understanding of math

concepts and/or procedures.

0• Incorrect solution(s).• No attempt at finding a solution.

1©Curriculum Associates, LLC

Copying permitted for classroom use.Grade 5 Lesson 17 Divide Decimals

Name ___________________________________________________________ Date ____________________

Lesson 17 Quiz

Solve the problems.

1 Which expressions can be represented by the decimal grids?

Decide if each expression is correct.

Choose Yes or No for each expression.

Yes No

2.4 4 0.6 � �

0.6 3 4 � �

2.4 4 4 � �

0.4 3 6 � �

2.4 4 0.4 � �

2 Gabriel has a stack of nickels that is 0.72 inch tall. The thickness of a nickel is 0.08 inch. How many nickels does Gabriel have? Show your work.

Gabriel has nickels.

(2 points)


0.72 4 0.08 5 ?

0.72 5 72 hundredths

0.08 5 8 hundredths

72 4 8 5 9

9

(2 points)

Tested SkillsAssesses 5.NBT.B.7

Problems on this assessment form require students to be able to divide decimals to hundredths, identify related division and multiplication equations, and recognize expressions represented by a decimal grid. Students will also need to be familiar with hundredths grids and basic division facts.

Alternately, teachers may assign the Digital Comprehension Check online to assess student understanding of this material.

Error Alert Students may:

• place the decimal point incorrectly.

• not use estimation to verify their answers.

• reverse the dividend and divisor.

Solutions1 A (Yes);

C (Yes); E (Yes); H (No); J (No) 2 points 5.NBT.B.7, DOK 2

2 9; See student page for possible work. 2 points 5.NBT.B.7, DOK 2

Extended Response Scoring Rubric

Points Expectations

4

• Correct solution, and/or calculations. (2 points)• Well-organized, clear, and concise work that demonstrates

thorough understanding of decimal division and modeling division in tenths. (2 points)

3• Mostly correct solution(s). (1 point)• Shows strong understanding of decimal division and/or

modeling division in tenths. (2 points)

2• Shows partial to limited understanding of decimal division

and/or modeling division in tenths.

1

• Contains incorrect and poorly organized solution(s).• Incomplete work and explanations.• Demonstrates limited understanding of decimal division

and modeling division in tenths.

0 • No attempt to find a solution.

©Curriculum Associates, LLC Copying is not permitted. 372bLesson 17 Divide Decimals

Differentiated Instruction

RETEACH REINFORCE EXTEND

Teacher Toolbox

Enrichment ActivitiesStudents who have achieved proficiency with concepts and skills and are ready for additional challenges

Will benefit from group collaborative games and activities that extend understanding

Math Center ActivitiesStudents who require additional practice to reinforce concepts and skills and deepen understanding

Will benefit from small group collaborative games and activities (available in three versions—on-level, below-level, and above-level)

Tools for InstructionStudents who require additional support for prerequisite or on-level skills

Will benefit from activities that provide targeted skills instruction

2©Curriculum Associates, LLC

Copying permitted for classroom use.Grade 5 Lesson 17 Divide Decimals

Name ___________________________________________________________ Date ____________________

Lesson 17 Quiz continued

3 Heidi has $4.55. She needs to buy bolts at the hardware store. Bolts cost $0.13 each.

Part A

What division equation and related multiplication equation could Heidi use to fi nd how many bolts she can buy?

� 0.13 4 ? 5 4.55 � 0.13 4 4.55 5 ?0.13 3 ? 5 4.55 4.55 3 ? 5 0.13

� 4.55 4 ? 5 0.13 � 4.55 4 0.13 5 ?0.13 3 4.55 5 ? 0.13 3 ? 5 4.55

Part B

How many bolts can Heidi buy? Show your work.

Heidi can buy bolts.

4 Derek has 4.8 pounds of potting soil and some fl ower pots. He puts 0.8 pound of soil in each pot, and he uses all of the soil. Derek draws this model to show how many fl ower pots he can fi ll.

Is Derek’s model correct? Why or why not?

(2 points)

(1 point)


4.55 4 0.13 5 ?



455 4 13 5 35

35

(4 points)

Yes, Derek’s model is correct; Possible explanation: Derek’s model

shows 48 tenths divided into 6 equal groups of 8 tenths. Derek can fill

6 flower pots.

3 Part A D; Students could solve the problem by recognizing the total, $4.55, must be divided by the cost per bolt, $0.13, or 4.55 4 0.13. Also, students must use their knowledge of relating division and multiplication equations to determine that a division equation can be represented as an unknown factor multiplication equation. A is not correct because the division equation has the dividend and divisor reversed. B is not correct because the division equation has the divisor and dividend reversed, and in the multiplication problem the product is represented as one of the factors. C is not correct because the division equation has the divisor represented as the quotient. 1 point 5.NBT.B.7, DOK 2

Part B 35; See student page for possible student work. 2 points 5.NBT.B.7, DOK 1

4 Yes; See student page for possible explanation. 4 points 5.NBT.B.7, DOK 3

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Divide Decimals - Full Lesson