Technology Assessment Management System

Journal page 270, Problems 5–7See the iTLG.

768 Unit 9 Fractions, Decimals, and Percents

Teaching the Lesson materials

Key ActivitiesStudents use an estimation strategy for dividing decimals. They solve decimal division problems that offer review and practice of the partial-quotients division algorithm.

Key Concepts and Skills• Identify place value in decimals through hundredths.

[Number and Numeration Goal 1]• Divide decimals by whole numbers.

[Operations and Computation Goal 4]• Round decimals and estimate quotients.

[Operations and Computation Goal 6]• Use conventional notation to write number sentences.

[Patterns, Functions, and Algebra Goal 2]

Ongoing Assessment: Recognizing Student Achievement Use journal page 270. [Operations and Computation Goal 6]

Ongoing Learning & Practice materialsStudents play Polygon Pair-Up to practice identifying properties of polygons.

Students practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students divide whole numbers by whole numbersand estimate quotients.

Students use bills and coinsto model division numberstories involving decimalsand whole numbers.

Students write and solvenumber stories involvingdivision of decimals.

� Teaching Masters (Math Masters,pp. 114 and 299)

� Teaching Aid Master (Math Masters,p. 428)

� coins

ENRICHMENTREADINESSREADINESS

3

� Math Journal 2, p. 272 � Student Reference Book, p. 258� Study Link Master (Math Masters,

p. 298)� Polygon Pair-Up Polygon Deck and

Property Deck (Math Masters,pp. 496 and 497)

2

� Math Journal 2, pp. 270 and 271� Study Link 9�8 � Teaching Aid Master (Math Masters,

p. 404; optional)� slate

1

Objectives To introduce division of decimals by whole numbers;

and to reinforce the partial-quotients division algorithm.

Lesson 9�9 769

Getting Started

Mental Math and ReflexesPose division problems. Have students estimate the quotient and write a number model to show how they estimated. Discussthe strategies used. Suggestions:Sample answers:

61 / 3 60 / 3 � 20 135 / 7 140 / 7 � 20 731 / 99 700 / 100 � 737 / 2 40 / 2 � 20 344 / 5 350 / 5 � 70 419 / 51 400 / 50 � 859 / 4 60 / 5 � 12 621 / 8 640 / 8 � 80 555 / 62 540 / 60 � 986 / 3 90 / 3 � 30 459 / 9 450 / 9 � 50 1,946 / 204 2,000 / 200 � 10

Math MessageThink of a number story that could besolved by dividing 4.2 by 7. Be prepared to discuss your answer.

Study Link 9�8Follow-UpHave small groups compare answers. Ask volunteers to share the strategies they used to determine where to place the decimal point in each product. Students indicate thumbs-up if they used a similar strategy.

Links to the Future

� Math Message Follow-UpDiscuss students’ answers. The goal of the Math Message andfollow-up activity is to increase students’ awareness of howdecimal division problems are based on real-life problems.

Suggest examples like the following, which illustrate differentuses for division:

� Partitioning into equal parts: A ribbon is 4.2 meters long. It must be cut into 7 pieces of the same length. 0.6 meters or 60 cm each

� Money and equal sharing: Think of 4.2 as $4.20, which is to be shared equally among 7 students. $0.60 or 60¢ each

� Calculating an average: Tom watched TV for a total of 4.2 hours in one week. His average viewing time per day was4.2 � 7 hours. 0.6 hours or 36 minutes

� Calculating a fraction or a percent: Alice and Dave took a 7-hour trip, and Alice drove for 4.2 hours. So �

47.2� is the fraction

of time that Alice drove, and (4.2 � 7) � 100 equals the percent of the time that she drove. 0.6 � 100 � 60 percent

Use of mental arithmetic, paper-and-pencil algorithms, and calculators to solveproblems involving the division of decimals by whole numbers is a Grade 5 Goal.

WHOLE-CLASS

DISCUSSION

1 Teaching the Lesson

770 Unit 9 Fractions, Decimals, and Percents

� Estimating Quotients of DecimalsAs they did with multiplication of decimals, students will use estimation to place decimal points in the answers to problemsinvolving division of decimals.

Write the following problems on the board. Write the 4-digitquotient shown for each problem, but do not include any decimal point.

Tell students that each problem shows the solution, but thedecimal point is missing in the quotient. Ask students to decidewhere each decimal point should go by estimating the quotient.Discuss students’ estimates and answers. Possible responses:

13.50 10 threes equals 30; this is a bit less than 40.5, 3 �4�0�.5� so the quotient is somewhat larger than 10. The

number 135.0 is too large, so the answer that is close to 10 is 13.50.

3 7.7 2 There are at least 30 fours in 150 (30 � 4 � 120).4�1�5�0�.8�8� There are not 40 fours in 150 (40 � 4 � 160). The

quotient must be between 30 and 40. So 37.72 is the answer.

.4 200 2.52 � 6 is about 3 � 6, which is �12�, or 0.5. So the

6�2�.5�2� only possible quotient is 0.4200. The other possible answers are all greater than 1.

.2200 There are many ways to round the numbers to 7�1�.5�4� create easy estimates: 2 � 10, 1 � 10, 2 � 5, and

1 � 5. All of these have answers from 0.1 to 0.4. Or, think of sharing $1.54 among 7 people: Each share will be less than $1.00 but more than $0.10. All estimates are between 0.1 and 1.0, so the answermust be 0.2200. Because the purpose of this estimateis to help students place the decimal point, any of these estimates is satisfactory.

Now write the following problem on the board:

� Bill paid $5.52 for 8 ballpoint pens. How much did 1 pen cost?Ask students to estimate the cost of 1 pen. Round $5.52 to $5 or to $6, and round 8 to 10. $5 � 10 � $0.50, or $6 � 10 � $0.60; so the cost of 1 pen is about 50 or 60 cents.

Have volunteers come to the board and divide 552 by 8 (ignoringthe decimal point). Remind students about the division algorithmthey have used before. If students are having difficulty gettingstarted, remind them that either of the estimates they just made(50 or 60) would be a good starting point.

13 50 3 7 72 4 2 00 2 2 003�4�0�.5� 4�1�5�0�.8�8� 6�2�.5�2� 7�1�.5�4�

WHOLE-CLASS

ACTIVITY

��

�270

Dividing DecimalsLESSON

9 � 9

Date Time

1. Janine is building a bookshelf. She has a board that is 3.75 meters long. She wants to cut it into 5 pieces of equal length. What will be the length of each piece?

meters

2. Three sisters set up a lemonade stand. On Wednesday they made $8.46. If they shared the money equally, how much did each girl get?

$

3. Alex and his three friends went out to lunch. The total bill, including tax and tip, was $42.52. They decided that each would pay the same amount. How much did each person pay?

$

4. Victor divides a 98.4 cm piece of string into 3 equal pieces. What is the length of each piece?

centimeters

For each problem below, the division has been done correctly, but the decimal point is missingin the answer. Write a number model to show how to estimate the answer. Use your estimateto correctly place the decimal point in the answer.

5. 6.

3��4� 3�.8� 6��2� 9� 8�.9� 2�

Number model: Number model:

7. 8.

4��6�.4� 6� 5�� 5��4�.3� 5�

Number model: Number model: 5 / 5 � 110 / 5 � 2

300 / 6 � 5045 / 3 � 15

32.8

10.63

2.82

0.75

1 4•6

1•6 15 8 7•

4 9•8 2

22 23233

Sample answers for number models:

Math Journal 2, p. 270

Student Page

Dividing Decimals continuedLESSON

9 � 9

Date Time

Write a number model to estimate each quotient. Then divide the numbers as though they were whole numbers. Use the estimate to help you place the decimal point in the answer.

9. � 9.44 / 4 10. 89.6 / 4 �

Number model: Number model:

11. 46.8 � 12 � 12. 253.8 / 6 �

Number model: Number model:

13. 2.96 / 8 � 14. � 3.65 � 5

Number model: Number model:

0.730.37

42.33.9

22.42.36

Try This

5 / 5 � 1

Sample answers for number models:

Sample answers for number models:

�39� � �

13� � 0.33œ

250 / 5 � 50

100 / 5 � 25

50 / 10 � 5

10 / 5 � 2

271

Math Journal 2, p. 271

Student Page

Lesson 9�9 771

Adjusting the Activity

Finally, have students use their initial estimates of the total cost toplace the decimal point in the answer. 552 � 8 � 69; the estimatewas about $0.50 or $0.60, so place the decimal point before the 6.The cost for 1 pen is $0.69.

Have students rename quotients involving amounts in dollars and centsas cents. Use the amount in cents to estimate and then rename the estimate asa dollars-and-cents amount. For example, in the problem above, think of $5.52as 552 cents. Then 1 pen costs between 50 cents and 60 cents, or between$0.50 and $0.60.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

Help students summarize the use of estimation to place thedecimal point when dividing decimals.

Example:

3.66 � 6 � ?

1. Estimate the quotient. 3.66 � 6 is about 3 � 6, or 0.5.

2. Divide the numbers as though they were whole numbers. 366 � 6 � 61

3. Use the estimate to place the decimal point. 0.61 is close to the estimate of 0.5.

� Dividing Decimals(Math Journal 2, pp. 270 and 271; Math Masters, p. 404)

Students complete journal pages 270 and 271. Encourage them toexplain the strategies they used to place the decimal points in thequotients.

Ongoing Assessment:Recognizing Student Achievement

Use journal page 270, Problems 5–7 to assess students’ ability to estimate thequotient of a decimal divided by a whole number. Students are making adequateprogress if they are able to correctly place the decimal points and write numbermodels for Problems 5–7. Some students may be able to solve Problem 8,which involves a quotient less than 1.

[Operations and Computation Goal 6]

Journal

page 270 �Problems 5–7

PARTNER

ACTIVITY

8�5�5�2� 8�5�5�2�� 400 50 � 480 60

152 72� 80 10 � 72 9

72 0 69� 72 9

0 69

772 Unit 9 Fractions, Decimals, and Percents

STUDY LINK

9 � 9 Dividing Decimals

Name Date Time

For each problem below, the division has been done correctly, but the decimal point is missing in the answer. Correctly place the decimal point in the answer.

1. 88.8 / 6 � 1 4•8 2. 1.35 / 5 �•2 7 0 0

3. 99.84 / 4 � 2 4•9 6 4. 2.58 / 3 �•8 6 0

5. 163.8 / 7 � 2 3•4 6. 233.28 / 4 � 5 8 •3 2

7. Explain how you decided where to place the decimal point in Problem 3.

Sample answer: I rounded 99.84 up andthen divided to get an estimate. 100 � 4 = 25,and 24.96 is close to 25.

8. 6��2�5�.2�

Answer:

9. 4�1�5�4�.8�

Answer:

10. 9��5�.8�5�

Answer: 4.2 38.7 0.65

Try This

11. � �58� � �

28� 12. �

59� � �

13� � 13. � �1

70� � �1

20� 14. �1

90� � �

12� � �1

40��1

90��

29��

78�

Practice

Divide. Show your work.

Math Masters, p. 298

Study Link Master

272

Math Boxes LESSON

9 � 9

Date Time

1. Calculate.

a. 10% of 90 �

b. 5% of 140 �

c. % of 30 � 24

d. % of 48 � 36

e. 20% of � 9457580

79

3. Complete the table with equivalent names. 4. Divide. Use a paper-and-pencil algorithm.

268 � 12 �

or 22�13�

22 R4, 22�142�,

150

106 109

2. Insert parentheses to make each numbersentence true.

a. 4 �(6 � 3)� 3 � 10

b. 57 �(24 � 15)� 18

c. 40 �(30 � 60)� 100 � 20

d. 56 /(7 � 3)� 1438 39

Fraction Decimal Percent

�17000� 0.7 70%

�17050� 0.75 75%

�35� 0.6 60%

�17020� 0.72 72%

61 62 129

5. What is the area of the parallelogram?Include the correct unit.

Number model:

Area � 16 in28 � 2 � 16

6. Study the figure. Draw the other half alongthe vertical line of symmetry.

"2

"8

135

Math Journal 2, p. 272

Student Page

� Playing Polygon Pair-Up(Student Reference Book, p. 258; Math Masters, pp. 496 and 497)

Students play Polygon Pair-Up to practice identifying properties ofpolygons. See Lesson 1-6 for additional information.

� Math Boxes 9�9(Math Journal 2, p. 272)

Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons 9-5 and 9-7. The skill inProblem 6 previews Unit 10 content.

Writing/Reasoning Have students write a response to the following: Draw another parallelogram that has thesame area as the parallelogram in Problem 5. Does the

parallelogram you drew have the same perimeter as the one inProblem 5? Explain your answer. Sample answer: The perimeter of the parallelogram in Problem 5 is about 20 inches. I cannot tellexactly because the length of the shorter side is not given. Theperimeter of the parallelogram that I drew is about 16 inches. Theyhave the same area, but they do not have the same perimeter.

� Study Link 9�9(Math Masters, p. 298)

Home Connection Students use estimation to place thedecimal point in the quotient of division problems. Theydivide decimals by whole numbers.

INDEPENDENT

ACTIVITY

4"

4"

INDEPENDENT

ACTIVITY

PARTNER

ACTIVITY

2 Ongoing Learning & Practice

� Dividing Whole Numbers and Estimating Quotients(Math Masters, p. 299)

To provide experience with whole-number division and estimatingquotients, have students complete Math Masters, page 299.

� Solving Division Number Stories Involving Money(Math Masters, pp. 114 and 428)

To explore division of decimals by whole numbers using a moneycontext, have students use the items on Math Masters, page 114and dollars and cents to model, write, and solve division numberstories. Be sure to have students discuss how they handled anyremainders. For example:

Jen and Carmen bought:

They shared the cost equally. How much did each girl spend?($3.59 � $2.99) � 2 � $3.29

� Writing and Solving Division Number Stories with Decimals

To apply students’ understanding of division of decimals,have them write and solve number stories involving thedivision of a decimal by a whole number. Suggest that

students write one of each of the following types of division number stories:

� Partitioning into equal parts

� Money and equal sharing

� Calculating an average

� Calculating a fraction or a percent

15–30 Min

PARTNER

ACTIVITYENRICHMENT

5–15 Min

PARTNER

ACTIVITYREADINESS

15–30 Min

INDEPENDENT

ACTIVITYREADINESS

3 Differentiation Options LESSON

9�9

Name Date Time

Dividing Whole Numbers

Write a number model to estimate each quotient. Then divide with a paper-and-pencil algorithm. Show your work.

1. 79 / 6 � 2. 92 / 3 �Number model: Number model:

3. � 573 / 4 4. 945 / 18 �Number model: Number model:

1,000 / 20 � 50600 / 4 � 150

52 R9, 52�198�, or 52�

12�143 R1, or 143�

14�

90 / 3 � 3080 / 8 � 10

30 R2, or 30�23�13 R1, or 13�

16�

22 23

5. The school has $357 to spend on new science books.If the books cost $9 each, how many books can they buy? books

Explain how to solve this problem without using a paper-and-pencil algorithm.

Sample answer: Round 357 to 360, then divide 360 by 9, which gives 40. I rounded the amount of money up, so thereis only enough for 39 books.

39

Try This

Sample answer: Sample answer:

Sample answer: Sample answer:

Math Masters, p. 299

Teaching Master

Lesson 9�9 773

LESSON

4 �4

Name Date Time

Items to Purchase

light bulbs4-pack

$1.09

VCR tape$3.25

tissues$0.73

transparent tape$0.84

batteries4-pack

$3.59

toothpaste$1.39

ballpoint pen$0.39

tennis ballscan of 3

$2.59

paperback book$2.99

Math Masters, p. 114

Teaching Master