Technology Assessment Management System

Journal page 268, Problem 4 See the iTLG.

762 Unit 9 Fractions, Decimals, and Percents

Teaching the Lesson materials

Key ActivitiesStudents use an estimation strategy for multiplying decimals. They solve a set of decimal multiplication problems that offers review and practice of the partial-products and lattice algorithms.

Key Concepts and Skills• Identify place value in decimals through hundredths. [Number and Numeration Goal 1]• Multiply decimals by whole numbers. [Operations and Computation Goal 4]• Round decimals and estimate products. [Operations and Computation Goal 6]• Use repeated addition to model multiplication. [Operations and Computation Goal 7]• Use a formula to calculate the area of a rectangle. [Measurement and Reference Frames Goal 2]• Use conventional notation to write number sentences. [Patterns, Functions, and Algebra Goal 2]

Ongoing Assessment: Recognizing Student Achievement Use journal page 268. [Operations and Computation Goal 6]

Ongoing Learning & Practice materialsStudents play Over and Up Squares to practice locating and plotting points on a coordinate grid.

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

Differentiation Options materials

Students multiply wholenumbers and estimate products.

Students use bills and coins to model multiplication number stories involvingwhole numbers and decimals.

Students estimate and compare the product of two decimal numbers with the product of twomixed numbers.

� Teaching Masters (Math Masters,pp. 114 and 297)

� Teaching Aid Masters (MathMasters, pp. 388 or 389 and 428)

� coins

ENRICHMENTREADINESSREADINESS

3

� Math Journal 2, p. 267� Student Reference Book, p. 257� Study Link Master (Math Masters,

p. 296)� Game Master (Math Masters,

p. 494)� per partnership: 2 six-sided dice;

colored pencils

2

� Math Journal 2, pp. 268 and 269� Study Link 9 �7 � Teaching Aid Masters (Math

Masters, pp. 404 and 434)� slate

1

Objectives To introduce multiplication of decimals by whole

numbers; and to reinforce the partial-products and lattice

methods for multiplication.

268

Multiplying DecimalsLESSON

9 �8

Date Time

Math Message

Toni has 8 blocks. Each block is 1.2 centimeters high. If she stacks the blocks, what will be the height of the stack? cm

1. Devon measured the length of the room by pacing it off. The length of his pace was 2.3 feet. He counted 14 paces. How long is the room? ft

2. Spiral notebooks are on sale for $0.35 each. How much will 25 spiral notebooks cost? $

3. Find the area of each rectangle below. Include the correct unit. Write a number model to show how you found the answer.

a.

Number model: Area �

b.

Number model: Area �

4. For each problem below, the multiplication has been done correctly, but the decimal point is missing in the answer. Write a number model to show how you estimated the answer. Then correctly place the decimal point in the answer.

a. 23 � 7.3 � 1 6 7 • 9 b. 6.91 � 82 � 5 6 6 • 6 2Number model: Number model:

c. 5,203 � 12.6 � 6 5 5 5 7 • 8 d. 0.38 � 51 � 1 9 • 3 8Number model: Number model:

15.4 in.

6 in.

30 cm1.5 cm

8.75

32.2

9.6

7 � 80 � 560

18 19184

�Sample answers for number models:

1.5 � 30 � 45 45 cm2

92.4 in26 � 15.4 � 92.4

20 � 10 � 200

5,200 � 10 � 52,000 0.4 � 50 � 20, or �12� of

50 � 25Math Journal 2, p. 268

Student Page

Lesson 9�8 763

Getting Started

Math MessageSolve the problem at the top of journal page 268.

Study Link 9�7 Follow-Up Have students share their answers and solutionstrategies. Some students may note that whenworking with populations rounded to the nearest ten thousand, they only have to consider the first two digits.

Mental Math and ReflexesPose multiplication problems. Have students estimate the product and write a number model to show how they estimated.Discuss the strategies used. Suggestions:

Sample answers:9 � 18 9 � 20 � 18011 � 42 10 � 40 � 40022 � 76 20 � 75 � 1,50087 � 15 90 � 20 � 1,800

28 � 49 30 � 50 � 1,50053 � 78 50 � 80 � 4,00063 � 63 60 � 60 � 3,60098 � 59 100 � 59 � 5,900

81 � 119 80 � 100 � 8,00045 � 188 50 � 200 � 10,00072 � 414 70 � 400 � 28,000609 � 684 600 � 700 � 420,000

Links to the Future

� Math Message Follow-Up(Math Journal 2, p. 268)

Have students share their solution strategies.

� Use repeated addition: 1.2 � 1.2 � . . . � 1.2 � 9.6.

� Multiply 8 and 1 and then add up 0.2 eight times.

� Others may approach it as a multiplication problem. Havethose students explain how they decided where to place the decimal point. Ask: Why are 96 and 0.96 not reasonableanswers? Because 1.2 is less than 2, the answer must be lessthan 16, so 96 is not a reasonable answer. Because 1.2 isgreater than 0.96, 0.96 is too small, so the answer must be 9.6.

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

� Estimating Products of DecimalsTell students that in this lesson they will learn to find the productof decimals by multiplying the numbers as if they were wholenumbers and then using estimation to place the decimal point inthe answer.

WHOLE-CLASS

ACTIVITY

WHOLE-CLASS

DISCUSSION

1 Teaching the Lesson

764 Unit 9 Fractions, Decimals, and Percents

To practice estimating products, write the following problems on the board:

11 � 2.8 110 � 2.8 11 � 0.28

Ask students to estimate each product. Write some of theirresponses next to the problems, and discuss their estimates. For example, some students may round 11 � 2.8 to 11 � 3 and estimate 33. Others may multiply 10 � 3 � 30 or 10 � 2 � 20. The purpose of this estimate is to help them place the decimalpoint, so any of these estimates is satisfactory. Ask:● Which problem is most likely to have the answer 30.8? 11 � 2.8● How do you know? The estimates made for this problem were

about 20 or 30. The estimates for the other problems weremuch larger or smaller.

Write the number 308 next to the problem 110 � 2.8. Ask:● Where would you place the decimal point? After the 8● How do you know? Sample answer: The answer must be larger

than 110 � 1.

Write the number 308 next to the problem 11 � 0.28. Ask:● Where would you place the decimal point? Between the 3 and 0● How do you know? Sample answer: The answer is less than

11 � 1 � 11; so 308 and 30.8 are too large. The answer is muchlarger than 1 � 0.28; so 0.308 is too small. That leaves 3.08 as the answer.

Now write the following problem on the board:

Calculators are on sale for $9.29 each. How much will 5 of them cost?

Ask students to estimate the cost of 5 calculators.

5 � $9 � $45 and 5 � $10 � $50, so they will cost between 45 and50 dollars.

Have volunteers come to the board and multiply 5 � 929(9.29 without the decimal point). Ask some students to use the partial-products method and others to use the lattice method. (See margin.)

Finally, have students use their initial estimates of the total costto place the decimal. 5 � 929 � 4,645 and the estimate was about$50. So place the decimal point after the 6; the total cost is $46.45.

5

9 2

454

6 4 5

9

10

45

929� 54,500

100+ 454,645

268

Multiplying DecimalsLESSON

9 �8

Date Time

Math Message

Toni has 8 blocks. Each block is 1.2 centimeters high. If she stacks the blocks, what will be the height of the stack? cm

1. Devon measured the length of the room by pacing it off. The length of his pace was 2.3 feet. He counted 14 paces. How long is the room? ft

2. Spiral notebooks are on sale for $0.35 each. How much will 25 spiral notebooks cost? $

3. Find the area of each rectangle below. Include the correct unit. Write a number model to show how you found the answer.

a.

Number model: Area �

b.

Number model: Area �

4. For each problem below, the multiplication has been done correctly, but the decimal point is missing in the answer. Write a number model to show how you estimated the answer. Then correctly place the decimal point in the answer.

a. 23 � 7.3 � 1 6 7 • 9 b. 6.91 � 82 � 5 6 6 • 6 2Number model: Number model:

c. 5,203 � 12.6 � 6 5 5 5 7 • 8 d. 0.38 � 51 � 1 9 • 3 8Number model: Number model:

15.4 in.

6 in.

30 cm1.5 cm

8.75

32.2

9.6

7 � 80 � 560

18 19184

�Sample answers for number models:

1.5 � 30 � 45 45 cm2

92.4 in26 � 15.4 � 92.4

20 � 10 � 200

5,200 � 10 � 52,000 0.4 � 50 � 20, or �12� of

50 � 25Math Journal 2, p. 268

Student Page

Lesson 9�8 765

Multiplying Decimals continuedLESSON

9 �8

Date Time

Write a number model to estimate each product. Then multiply the factors as though they were whole numbers. Use your estimate to help you place the decimal in the answer.

5. 2.7 � 45 � 6. 8 � 5.7 �

Number model: Number model:

7. 5.08 � 27 � 8. 42 � 0.97 �

Number model: Number model:

9. 22 � 0.32 � 10. 0.02 � 333 �

Number model: Number model:

6.667.04

137.16

45.6121.5

40.74

Try This

Sample answers for number models:

Sample answers for number models:

3 � 50 � 150

5 � 30 � 150

8 � 6 � 48

40 � 1 � 40

20 � 0.3 � 6, or �13�

of 21 � 70.02 � 300 � 6, or �1

200�

of 300 � 6

Math Journal 2, p. 269

Student Page

Adjusting the Activity

2 6

30

00

82

8 48 4

6 8

4

112 6

30

00

82

8 48 4

6 8

4

11

Decimal multiplied by whole number Decimal multiplied by decimal

Help students summarize the use of estimation to place thedecimal point in the answer when multiplying decimals.

Example: 6 � 3.7 � ?

1. Estimate the product.

6 � 3.7 is about 6 � 4, or 24.

2. Multiply the factors as though they were whole numbers.

6 � 37 � 222

3. Use the estimate to place the decimal point in the answer.

22.2 is close to the estimate of 24.

� Multiplying Decimals(Math Journal 2, pp. 268 and 269; Math Masters, pp. 404 and 434)

Ask students to complete journal pages 268 and 269 and compare answers.

Have students use lattice multiplication adapted for decimals. Showthem how to find the intersection of the decimal points along the horizontal andvertical lines; then slide down the diagonal. Encourage students to still makeestimates in order to check their work.

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

Ongoing Assessment:Recognizing Student Achievement

Use journal page 268, Problem 4 to assess students’ ability to estimate theproduct of a whole number and a decimal. Students are making adequateprogress if they are able to correctly place the decimal points and write numbermodels for Problems 4a–4c. Some students may be able to solve Problem 4d,which involves a decimal less than 1.

[Operations and Computation Goal 6]

Journalpage 268 �Problem 4

PARTNERACTIVITY

766 Unit 9 Fractions, Decimals, and Percents

� Playing Over and Up Squares(Student Reference Book, p. 257; Math Masters, p. 494)

Students play Over and Up Squares to practice locating and plotting points on a coordinate grid. See Lesson 6-9 for additional information.

� Math Boxes 9�8(Math Journal 2, p. 267)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 9-6. The skill in Problem 6 previews Unit 10 content.

Writing/Reasoning Have students write a response to the following: Explain how you solved Problem 2b. Sample answer:�34� is equivalent to 75%. �

35� is equivalent to 60%. I named a percent

value between 60% and 75%.

� Study Link 9�8(Math Masters, p. 296)

Home Connection Students estimate products of decimals and whole numbers. They multiply decimals and whole numbers.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

PARTNER

ACTIVITY

2 Ongoing Learning & Practice

267

Math Boxes LESSON

9 �8

Date Time

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

Number model:

Area �

�12� � (11� 5)� 27.5

27.5 in2

6. a. Which is warmer, �7 �C or �3.5�C?

How many degrees warmer?

b. Which is colder, �18�C or �9.6�C?

How many degrees colder?8.4�C

�18�C

3.5�C

�3.5�C

1. a. If you threw a 6-sided die 48 times,about how many times would youexpect it to land on a number greaterthan or equal to 4?

times

b. If you threw a 6-sided die 54 times,about how many times would youexpect it to land on a number greaterthan 4?

times18

24

3. Homer’s is selling roller blades at 25% offthe regular price of $52.00. Martin’s isselling them for �

13� off the regular price of

$60. Which store is offering the better buy?

Show how you solved the problem.

Homer’s: 25% � �14�, $52 / 4�

$13, and $52 � $13 � $39.Martin’s: $60 / 3 � $20,and $60 � $20 � $40.

Homer’s

4. If 1 centimeter on a map represents 300kilometers, then 2.5 centimeters represents

kilometers. Choose the best answer.

600

650

350

75038 39

59

136 60 139

145

2. Name a percent value

a. greater than �15� and less than �

12�.

b. less than �34� and greater than �

35�.

70%

30%

61 6281

Sample answer:

11"

5"

Sample answers:

Math Journal 2, p. 267

Student Page

STUDY LINK

9 � 8 Multiplying Decimals

Name Date Time

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

1. 6 º 4.3 � 2 5•8 2. 72 º 6.8 � 4 8 9• 6

3. 0.96 º 47 � 4 5•1 2 4. 5.12 º 22 � 1 1 2•6 4

5. 8,457 º 9.8 � 8 2 8 7 8 •6 6. 0.04 º 140 � 5•6

7. Explain how you decided where to place the decimal point in Problem 4.

I rounded the numbers to 5 and 20 and thenmultiplied to get 100. So, the product should be close to 100, and 112.64 is.Try This

8. 5.9 º 36 � 9. 0.46 º 84 � 10. � 7.21 º 53212.4 38.64 382.13

Practice

11. � 96 � 6 12. 4�6� 7� �

13. � 411 / 3 14. 9�9� 0� 3� � 100 R3, or 100�39�137

16 R3, or 16�34�16

Multiply. Show your work.

Sampleanswer:

Math Masters, p. 296

Study Link Master

� Multiplying Whole Numbers and Estimating Products(Math Masters, p. 297)

To provide experience with whole-number multiplication and estimating products, have students complete Math Masters, page 297.

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

To explore multiplication of whole numbers by decimals using amoney context, have students use the items on Math Masters,page 114 and dollars and cents to model, write, and solve multiplication number stories. For example:

Max bought 5 packs of light bulbs.

� Comparing Products(Math Masters, p. 388 or 389)

To apply students’ understanding of decimal multiplication and decimal/fraction equivalencies, have students estimate andcompare the product of two mixed numbers and the product of two decimals. Ask students to record their responses to thefollowing in a Math Log or on an Exit Slip:

Think about these two multiplication problems: 5�

12� � 2�

13� 2.36 � 5.206

Without using a paper-and-pencil algorithm or a calculator, whichproduct do you think is greater? Explain. Sample answer: 5�

12� � 2�

13�;

5�12� � 5.5 and 5.5 � 5.206. 2�

13� � 2.33�, which is only a bit less than

2.36. So the first product is greater.

5–15 Min

PARTNER

ACTIVITYENRICHMENT

About how much money did he spend?5 � $1.09 � $5.45

5–15 Min

PARTNER

ACTIVITYREADINESS

5–15 Min

INDEPENDENT

ACTIVITYREADINESS

3 Differentiation Options

Lesson 9�8 767

LESSON

9�8

Name Date Time

Multiplying Whole Numbers

Write a number model to estimate each product. Then multiplywith a paper-and-pencil algorithm. Show your work.

1. 7 º 68 � 2. 534 º 6 �Number model: Number model:

3. � 58 º 67 4. 33 º 275 �Number model: Number model:

30 º 300 � 9,00060 º 70 � 4,200Sample answer:Sample answer:

9,0753,886

500 º 6 � 3,0007 º 70 � 490Sample answer:Sample answer:

3,20447618 19

Try This

5. Margo’s favorite socks are on sale for $2.89 per pair. She has $25. Can she buy 6 pairs?

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

Sample answer: Round $2.89 to $3.00, then multiply by 6, which gives $18. She has $25, so she has more than enoughmoney.

yes

Math Masters, p. 297

Teaching Master

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