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A

A B C D E Multiply Decimals Multi-Part Lesson

PART

Objective

Students will estimate the product of decimals and judge the reasonableness of the results.

NGSSS

MA.6.A.5.3 Estimate the results of computations with fractions, decimals, and percents

and judge the reasonableness of the results. glencoe.com

1

Activity Review how to round decimals to the nearest whole number. Have students draw a number line from 1 to 10. Then ask them to locate and label 1.5, 2.5, 3.5, and so on through 9.5.

Is 1.2 closer to 1 or to 2? How can you tell? 1; The distance between 1.2 and 1 is less than the distance between 1.2 and 2.

Is 4.75 closer to 4 or to 5? How can you tell? 5; The distance between 4.75 and 5 is less than the distance between 4.75 and 5.

Use the number line to round the following numbers to the nearest whole number: 5.67; 3.41; 9.95. 6; 3; 10

Scaffolding Questions

Suppose James travels at an average of 5.4 miles per hour for 8.25 hours. How would you round the numbers to estimate how far he traveled? Round 5.4 to 5 and 8.25 to 8.

About how far would James travel?

40 miles

Would your estimate be higher or lower than the actual distance? Explain. Sample answer: Lower; both numbers are rounded down, so the estimated distance is less than the actual distance.

Tips

for New Teachers

Rounding If students have trouble remembering

the rules for rounding, teach them this rhyme: 5 and above, give it a shove.

4 and below, let it go.

Lesson 1-1 Multiply Decimals 27 glencoe.com

Multiply Decimals

A Multi-Part Lesson

1-1

PART B C D E

Lesson 1-1 Multiply Decimals 27

Estimate Products

SKATEBOARDING The record for the greatest distance traveled on skateboard in 24 hours was set by James Peters in 2007. He traveled about 7.6 miles per hour.

1. Round 7.6 to the nearest whole number. 8 2. Estimate how many miles James Peters

traveled in 24 hours. about 24 × 8 or 192 miles 3. Is your estimate higher or lower than the

actual distance he traveled? Explain. The estimate is higher than the actual distance he traveled because 7.6 was rounded up.

To estimate products of decimals, round each number. Then multiply.

Round Decimals

To round a decimal, first underline the digit to be rounded. Then look at the digit to the right of the place being rounded.

• If the digit is 4 or less, the underlined digit remains the same. • If the digit is 5 or greater, add 1 to the underlined digit.

• After rounding, change all digits after the underlined digit to zeros.

Estimate Products Using Rounding

Estimate 8.7 × 2.8.

Round to the nearest whole number to make it easier to compute mentally. 8.7 × 2.8 9 × 3 27 Round 8.7 to 9. Round 2.8 to 3.

The product is about 27.

Estimate each product.

a. 9.6 × 1.8 20 b. 8.7 × 2.9 27 c. 68.4 × 21.3 1,400 Main Idea

Estimate the product of decimals and judge the reasonableness of the results.

NGSSS MA.6.A.5.3 Estimate the results of computations with

fractions,decimals, and percentsand judge the reasonableness of the results.

027_033_C1L1_891639.indd 27 1/9/09 12:11:47 PM

Key Concept boxes highlight definitions, formulas, and other important ideas. Multiple

representations—words, symbols, examples, and models—help

students understand the concepts.

027_031_C1L1A_891640.indd 27

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Formative Assessment

Use the Check Your Progress exercises after each Example to determine students’ understanding of concepts.

us on Mathematical Content

Rounding decimals to the nearest whole number before multiplying is one way to estimate a product. Refer to the digit in the tenths place. If the digit in the tenths place is 5 or higher, the number is rounded to the next whole number. If the digit is less than 5, then the whole number remains the same.

Estimate 4.8 × 3.2. 5 × 3 =15

HELICOPTERS One of the first helicopters flew at a speed of 44.7 kilometers per hour. At this speed, about how far could the helicopter travel in 2.5 hours? 135 km

RECIPES Anna has \$20 to buy 8 pounds of tomatillos for a party. The tomatillos cost \$2.89 per pound. Does she have enough money to buy them? Explain your reasoning. No; Sample answer: \$2.89 is close to \$3. If you round \$2.89 to \$3, the tomatillos will cost about \$24.

IWB

28 Chapter 1 Multiply and Divide Decimals

28 Chapter 1 Multiply and Divide Decimals

DOGS A greyhound can travel 39.3 miles per hour. At this speed, about how far could a greyhound travel in 6.5 hours? 39.3 × 6.5 40 × 7 280 Round 39.3 to 40. Round 6.5 to 7.

The greyhound could travel about 280 miles in 6.5 hours.

d. MONEY Suppose one U.S. dollar is equal to 5.3 Egyptian pounds. About how many Egyptian pounds would you receive for \$48.50? about 250 Egyptian pounds

e. ASTRONOMY The Earth is rotating around the Sun about 18.6 miles per second. About how many miles does it travel in 4.8 seconds? about 100 mi

SCHOOL SUPPLIES Patrice has \$20 to buy 5 binders for her classes. She found binders that cost \$4.29 each. Does she have enough money to buy these binders? Explain your reasoning.

Estimate.

5 × \$4 = \$20 Estimate 4.29 as 4. 5 × \$5 = \$25 Estimate 4.29 as 5.

The actual cost of the binders is between \$20 and \$25. So, Patrice does not have enough money to buy the binders.

f. PROFIT The art club makes a profit of \$1.75 on every batch of cookies they sell at a bake sale. The goal is to earn at least \$50 selling batches of cookies. They estimate they will need to sell at least 30 batches. Is this estimate reasonable? Explain your reasoning.

f. Yes; \$1 × 30 = \$30 and \$2 × 30 = \$60. So, they will earn between \$30 and \$60.

Rounding Decimals Rounding Decimals When rounding decimals, such as 99.96 to the tenths, the 9 must round up. So, 99.96 rounded to the nearest tenth is 100.0.

027_033_C1L1_891639.indd 28 1/21/09 8:59:27 AM

Study Tips offer students helpful

information about the topics they are studying.

027_031_C1L1A_891640.indd 28

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Formative Assessment

Use Exercises 1–8 to check for understanding. Then use the table below to customize assignments for your students.

Odd/Even Assignments

Exercises 9–21 are structured so that students practice the same concepts whether they are assigned odd or even problems.

Differentiated Homework Options

Level Assignment Two-Day Option

AL Approaching Level 9–21, 36–38, 39–44 9–21 odd, 39–42 10–20 even, 36, 38, 39,

43, 44

OL On Level 9–25 odd, 26–29, 31, 35,

36, 38–44

9–21, 39–42 22–36, 38, 43, 44

BL Beyond Level 22–44

Lesson 1-1 Multiply Decimals 29 Common Error In Exercises 1–6, 9–11, and 15–17, students may have difficulty multiplying in horizontal form. Remind students that they may rewrite the problems in vertical form.

exercises are intended to be completed in

class. Example references show students where to look back for review.

In homework exercises in Practice and Problem Solving, the

Example references function in the same way.

Lesson 1-1 Multiply Decimals 29

Estimate each product. 1–6. Sample answers given. 3. 60 × 30 = 1,800

Example 1 (p. 27) 1. 5.8 × 4 6 × 4 = 24 2. 27.3 × 9 27 × 10 = 270 3. 57.1 × 32 4. 3.5 × 1.8 4 × 2 = 8 5. 13.92 × 2.7 6. 94.89 × 3.11 14 × 3 = 42 90 × 3 = 270 Example 2 (p. 28)

7. MONEY A grocery store sells American cheese for \$3.89 per pound. About how much would 1.89 pounds of the cheese cost? about \$8

Example 3

(p. 28) 8. MEMORY Greg has 52 megabytes of free

space left on his MP3 player. He wants to download 7 songs that each use 7.9 megabytes of memory. He estimates that he will need 56 megabytes of memory. Explain why his estimate is reasonable. If 7.9 is rounded to 8, then the 7 songs will require about

56 megabytes of memory. Since 7.9 was rounded up, the actual amount of memory he will use will be less than 56 megabytes.

=Step-by-Step Solutions begin on page R1.

Extra Practice is on page EP2.

Estimate each product. 9–17. Sample answers given.

Example 1 (p. 27) 9. 9.7 × 3.3 10 × 3 = 3010. 3.4 × 5.6 3 × 6 = 18 1111 17.5 × 8.4 11. 20 × 8 = 160 13. 30 × 80 = 2,400 14. 100 × 11 = 1,100 12. 26.3 26 × 10 = 26013. 33.6 14. 99.1 × 9.7 ×82.1 × 11.2 15. 44.8 × 5.1 16. 28.21 × 8.02 17. 71.92 × 2.01 40 × 5 = 200 30 × 8 = 240 70 × 2 = 140 Example 2 (p. 28)

18. FRUIT On average, the U.S. produces 36.5 million tons of fruit each year. About how much fruit does it produce in 2.25 years?

about 80 million tons

19. SCIENCE A single year on Saturn is equal to 29.4 years on Earth. About

how many Earth-years are equal to 3.2 years on Saturn? about 90

Example 3

(p. 28)

20. CRAFTS Lisha is making headbands using ribbon. She would like to make 12 headbands. Each one requires 15.5 inches of ribbon. She estimates that she will need to buy 160 inches of ribbon. Is her estimate reasonable? Explain your reasoning.

21. GIFT CARDS Miguel received a \$50 gift card to a bookstore. He would

like to buy 3 books that cost \$15.75 each including tax. He estimates that he cannot buy all three books because each book costs about \$20, and all three books would cost \$60. Is his estimate reasonable? Explain your reasoning. His estimate is not reasonable. He rounded \$15.75 to \$20. If he rounded \$15.75 to \$16, the estimate would be \$48 for the three books.

B Use estimation to determine whether each answer is reasonable. If

the answer is reasonable, write yes. If not, write no and provide a reasonable estimate. 22. 22.8 × 4.7 = 107.16 yes 23. 2.1 × 4.9 × 7.2 = 105.84 no; 70 24. 7.8 × 1.1 × 4.2 = 50 no; 32 25. 43.8 × 2.8 × 3.1 = 371.8 yes 20. Her estimate is not reasonable. Since 12 rounds down to 10 and 15.5 rounds to 16 by a lesser amount, her estimate of 10 × 16 = 160 will be too low.

indicates multi-step problem

027_033_C1L1_891639.indd 29 10/1/10 3:19:41 PM

AL

If

students have trouble

estimating the product of decimals,

Then

use one of these reteach

options:

1

1 Quick Review Math Handbook,

p. 132

2

2 Have them use a number line to practice rounding decimals to the nearest whole number.

The Alternate Teaching Strategy provides

suggestions for remediation for students who did not

grasp the concept.

027_031_C1L1A_891640.indd 29

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30 Chapter 1 Multiply and Divide Decimals

Interpersonal

Present these questions to a small group.

Why is being able to estimate answers in mathematics important?

Sample answers: Sometimes you do not need to find an exact answer; estimates can help you check an answer.

Describe a situation in which you estimated an answer. How and why was estimation helpful? Sample answer: Before I solved a problem, I estimated the answer. It helped me decide whether my actual solution was reasonable.

When in real life is estimating more helpful than finding the exact answer? Sample answer: to determine how much I have spent when I am shopping

Name some real-world situations where estimating an answer is not helpful. Sample answer: In sports, such as a race, an exact time is needed to determine the winner.

AL OL

BL

Visit glencoe.com to rewatch the animated version of “Money Challenge.”

30 Chapter 1 Multiply and Divide Decimals

26. GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a–b.

I’ve already saved \$68 for the \$200 video game system.

a. How much more does Raj need until he has enough to buy the video game system? Raj needs to save \$132 more.

b. If he works for 25 hours, will he have enough to buy the video game system? Yes; if Raj works 25 hours, he will make \$137.50. 27. More; her wage

was rounded up, so the actual total is less than the estimate.

27

27 MONEY Hannah’s hourly wage at the ice cream

Day Hours Worked Monday 3.5 Tuesday 4.25 Wednesday 3.75 Thursday 2.5 Friday 4.75

shop is \$5.85. The table shows the number of hours she worked. She estimates her earnings to be \$120. Without calculating her actual earnings, determine if her estimate is more or less than her actual earnings. Explain your reasoning.

28. FUEL A car releases 19.6 pounds of carbon dioxide

for every 1 gallon of gasoline burned. Estimate the number of pounds of carbon dioxide released if 14.5 gallons is burned. about 300 pounds

REASONING The accuracy of an estimate depends on the place value the numbers are rounded to. For each exercise, round the first factor to the tens and then the ones before estimating. Which way gives an estimate that is closer to the actual product?

29. 35.1 × 8 30. 94.1 × 4 31. 58.8 × 6

32. NUTRITION The table shows some nutritional Orange Juice (1 cup)

Calories 112 Vitamin C 96.9 mg Carbohydrates 26.8 g Calcium 22.4 mg

facts about orange juice. Estimate each value for 1 quart of orange juice. (Hint: 4 cups is equal to 1 quart.)

33. TREES A King Palm can grow about 2.1 feet

a year. Estimate the height of the King Palm, in yards, after 15 years. about 10 yards

REASONING Estimate each product to determine if each calculator answer is correct. Explain your reasoning.

34. 46.85 9.75; 456.7875 35. 90.8 3.1; 217.18 yes; 46.85 × 9.75 ≈ 45 × 10 ≈ 450 no; 90.8 × 3.1 ≈ 90 × 3 ≈ 270 29. 35 × 8 = 280; 40 × 8 = 320; Rounding to the ones gives an estimate closer to the actual product. 30. 94 × 4 = 376; 90 × 4 = 360; Rounding to the ones gives an estimate closer to the actual product. 31. 59 × 6 = 354; 60 × 6 = 360; Rounding to the ones gives an estimate closer to the actual product. 32. Calories: 400; carbohydrates: 100 g; Vitamin C: 400 mg; calcium: 80 mg 027_033_C1L1_891639.indd 30 1/21/09 8:59:34 AM 027_031_C1L1A_891640.indd 30 027_031_C1L1A_891640.indd 30 3/2/09 11:03:52 AM3/2/09 11:03:52 AM

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4

Have students write how to estimate 8.7 × 2.2.

Are students continuing to struggle with estimating the product of decimals and judging the reasonableness of the results?

If Yes AL Reteach Worksheet

CRM (p. 12)

If No OL Skills Practice Worksheet

CRM (p. 13)

BL Enrich Worksheet

CRM (p. 16)

Follow-Up

Remind students to keep any notes regarding estimating products under the appropriate tab in their Foldables.

Lesson 1-1 Multiply Decimals 31

NGSSS

Diagnose Student Errors

Survey student responses for each item. Class trends may indicate common errors and misconceptions.

39. A. rounded both numbers down

instead of rounding up

B. correct

C. estimated beyond upper limit of range

D. guess

40. F. estimated pizza only

G. correct

H. estimated cost of all items

I. estimated cost of all items plus an extra slice of pizza

41. A. subtracted rounded numbers

incorrectly

B. correct

C. rounded \$5.85 to \$5 instead of \$6

D. subtracted rounded numbers

incorrectly

Lesson 1-1 Multiply Decimals 31

DIVING The table gives the scores for the Men’s Synchronized Country Points

China 468.18

Germany 450.42

Russian Federation 445.26 Australia 444.84 United States 440.64

10-Meter Diving at the 2008 Olympics. (Lesson 0-2)

43. How many more points did China earn than Germany? 17.76 44. What is the difference in the points China and the United

States earned? 27.54

Practice MA.6.A.5.3

NGSSS

39. Green peppers are on sale for \$2.89 per pound. Mrs. Moseley bought 1.75 pounds of peppers. About how much did she pay for the peppers? B

A. less than \$4

B. between \$5 and \$6

C. between \$6 and \$7

D. more than \$7

40. Medina’s school lunch menu is shown.

Pizza \$1.75 Fruit Punch \$0.75 Fish and Fries \$2.25 Milk \$0.80 Salad \$1.15 Pudding \$0.85

Friday

Which of the following is a reasonable estimate for the cost of two slices of pizza, a salad, and fruit punch? G

F. \$4 H. \$8

G. \$6 I. \$10

41. Mario and Andrew’s hourly charge for mowing lawns is shown.

Mario Andrew \$8.25/hr \$5.85/hr

Suppose Mario and Andrew each worked 20 hours. About how much more money did Mario earn? B

A. \$30

B. \$40

C. \$60

D. \$70

42. SHORT RESPONSE Javier bought 4 pencil toppers at the school store for \$3.69 each. He estimated how much he needs to pay and gave the cashier \$16. Is Javier’s estimation reasonable? Explain your reasoning. yes; 4 × \$3.69 ≈ 4 × \$4 = \$16

C 36. OPEN ENDED Name three decimals with a product that is about 40.

37. CHALLENGE A scooter can travel between 22 and 28 miles on each

gallon of gasoline. If one gallon of gasoline costs between \$3.75 and \$3.95 per gallon, about how much will it cost to travel 75 miles?

38. Suppose your friend multiplied 1.2 and 2.6 and got

31.2 as the product. Is your friend’s answer reasonable? Justify your response. No; using rounding, the answer should be close to 1 × 3 or 3. 36. Sample answer:

1.9 × 5.3 × 3.81

027_033_C1L1_891639.indd 31 1/9/09 12:12:07 PM

H.O.T. Problems require students to use Higher

Order Thinking skills to solve problems.

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Objective

Students will use models to multiply a decimal by a whole number.

NGSSS

MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently.

Resources

Manipulatives base-ten blocks Easy-to-Use Manipulatives

Hands-On Activity Tools and Resources

Base-Ten Models, p. 1

glencoe.com

1

Write these numbers on the board: 3.41 4.12 1.04

Which number has a 4 in the ones place? 4.12

Which number has no tenths? 1.04

Which number has a 1 in the hundredths place? 3.41

2

Activity 1 Have students work in pairs to model and check each step.

Interactive WhiteBoard

Demonstrate modeling decimals on an Interactive WhiteBoard. Have a student volunteer come to the IWB and use the highlighter tool to identify ones, tenths, and hundredths.

32 Chapter 1 Multiply and Divide Decimals

B

A B C D E Multiply Decimals Multi-Part Lesson

PART

Percents

A Multi-Part Lesson

1-1

PART B C D E glencoe.com

Multiply Decimals

A Multi-Part Lesson

1-1

PART B C D E

32 Chapter 1 Multiply and Divide Decimals

Main Idea

Use models to multiply a decimal by a whole number.

NGSSS MA.6.A.1.1 Explain and justify procedures for multiplying

and dividing fractions and

decimals.

MA.6.A.1.2 Multiply and divide fractions and

decimals efficiently.

Whole Numbers

The table shows the relationship between decimals and base-ten blocks. You can use base-ten blocks to multiply decimals.

Ones (1.0) Tenths (0.1) Hundredths (0.01)

One 10-by-10 flat represents 1 or 1.0.

One 1-by-10 rod represents 0.1.

One 1-by-1 cube represents 0.01.

WALKING Louisa walked her dog 0.2 mile on Monday, Wednesday, and Friday. Find the total distance Louisa walked all week.

What do you need to find? the total distance Louisa walked Just as 3 × 2 means 3 groups

3 groups

of 2, 3 × 0.2 means 3 groups of 2 tenths. Model three groups of two tenths using base-ten blocks.

Combine the tenths. There are six tenths.

So, 3 × 0.2 = 0.6. Louisa walked a total of 0.6 mile.

and Apply

1–8. Check students’ work for models.

Use base-ten blocks to find each product.

1. 3 × 0.3 0.9 2. 2 × 0.4 0.8 3. 4 × 0.02 0.08 4. 5 × 0.01 0.05 5. 3 × 0.5 1.5 6. 4 × 0.6 2.4 7. 2 × 0.08 0.16 8. 8 × 0.04 0.32

027_033_C1L1_891639.indd 32 1/9/09 12:12:13 PM

032_033_C1L1B_891640.indd 32

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Activity 2 Point out that 10 tenths can be combined to make one whole.

3

Use Exercises 9–16 to assess whether students know how to use models to multiply a decimal by a whole number.

From Concrete to Abstract Use Exercise 18 to bridge the gap between using models and using an algorithm to multiply a whole number and a decimal.

Extending the Concept Have students use base-ten blocks to compare the product of 3 × 12 with the product of 3 × 1.2. 3 × 12 =

36; The product of 3 × 1.2 has the same digits as 3 × 12, but with one decimal place. 3 ×

1.2 = 3.6

17. The factors have the same digits, but one factor in 2 × 1.6 is a decimal. The products, 32 and 3.2, have the same digits, but one is a decimal.

18. Sample answer: Multiply 5 by 0.4 and add to the product of 5 × 1. So, 5 × 1.4 is 7.0 or 7. Check students’ work for models.

19. Sample answer: The product is greater than the whole number. For example, 5 × 1.8 = 9 and 5 × 0.8 = 4.

20. Sample answer: Multiply the whole number by the decimal as with whole numbers and place the decimal point the same places to the left as the decimal number.

Lesson 1-1 Multiply Decimals 33

Lesson 1-1 Multiply Decimals 33 Model 2 × 1.6 to find the product.

Model two groups of one and six tenths.

2 groups

Combine the ones. Then combine the tenths.

one whole

Combine the 10 tenths into 1 whole.

So, 2 × 1.6 = 3.2.

and Apply

9–16. Check students’ work for models.

Use base-ten blocks to find each product.

9. 2 × 1.3 2.6 10. 3 × 1.1 3.3 11. 4 × 1.02 4.08 12. 6 × 2.01 12.06 13. 5 × 2.5 12.5 14. 2 × 3.8 7.6 15. 5 × 1.09 5.45 16. 4 × 3.07 12.28

the Results

17–20. See margin.

17. Compare and contrast 2 × 16 and 2 × 1.6. How are the products the same? different?

18. MAKE A CONJECTURE Explain how to find 5 × 1.4 without using base-ten blocks. Justify your procedure. Then check your answer using base-ten blocks.

19. MAKE A CONJECTURE Suppose you multiply a whole number by a decimal greater than one. Is the product less than, greater than, or equal to the whole number? Explain your answer.

20. Write a rule that you can use to multiply a whole number by a decimal without using base-ten blocks.

027_033_C1L1_891639.indd 33 1/9/09 12:12:24 PM

Write Math features help students learn to use the

language of mathematics.

032_033_C1L1B_891640.indd 33

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Additional Answer 2. 4.92 + 4.92 + 4.92 = 14.76; 3 × 5 = 15; 3 × 4.92 = 14.76; 4.92 + 4.92 + 4.92 + 4.92 = 19.68; 4 × 5 = 20; 4 × 4.92 = 19.68; 4.92 + 4.92 + 4.92 + 4.92 + 4.92 = 24.6; 5 × 5 = 25; 5 × 4.92 = 24.6

34 Chapter 1 Multiply and Divide Decimals

Objective

Students will estimate and find the product of decimals and whole numbers.

NGSSS

MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently.

Also addresses MA.6.A.1.3, MA.6.A.5.3.

glencoe.com

1

Activity Provide each pair of students with a small object, such as a marker. Have them measure the length of their object to the nearest tenth using a metric ruler.

Have students estimate the length of two of their object.

Have students find the length of 3, 5, and 10 of their object.

Scaffolding Questions

Write the following on the board: 1. 429 × 11 = 793

2. 569 × 6 = 3,414 3. 13.3 × 5 = 665

How can you use rounding to tell whether the first answer is reasonable?

Sample answer: Round 429 to 400 and 11 to 10. Then multiply 400 × 10 = 4,000. The answer is not reasonable.

Is the second answer reasonable? How do you know? Sample answer: Yes; round 569 to 600 and multiply: 600 × 6 = 3,600.

How would you round to determine if the third answer is reasonable? Sample answer: Round 13.3 to 13 and then multiply by 5: 13 × 5 = 65. The answer is not reasonable.

What is the error in the third answer?

The decimal place was not shown.

C

A B C D E Multiply Decimals Multi-Part Lesson

PART glencoe.com

Multiply Decimals

A Multi-Part Lesson

1-1

PART B C D E

34 Chapter 1 Multiply and Divide Decimals

Main Idea

Estimate and find the product of decimals and whole numbers.

NGSSS MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals.

MA.6.A.1.2 Multiply and divide fractions and decimals efficiently. Also addresses MA.6.A.1.3, MA.6.A.5.3.

Whole Numbers

PLANTS Bamboo can grow about 4.92 feet in height per day. The table shows different ways to find the total height a bamboo plant can grow in two days. 2. See margin. 1. Use the addition problem and the

estimate to find 2 × 4.92. 9.84

2. Write an addition problem, an estimate, and a multiplication problem to find the total growth over 3 days, 4 days, and 5 days. 3. MAKE A CONJECTURE Explain how to find 6 × 4.92.

Using repeated addition can help you place the decimal point in the product of a whole number and a decimal. The whole number represents the number of times the decimal is used as an addend. So, place the decimal point in the product the same number of places from the right as the decimal factor.

Multiply Decimals

Find 4 × 0.83.

0.83 0.83 0.83 0.83

4 groups of 0.83 Estimate 4× 1 = 4

0.83 two decimal places

× 4

3.32 Place the decimal point two places from the right. Check for Reasonableness 3.324 ✓

Find 3 × 14.2.

14.2 14.2 14.2

3 groups of 14.2 Estimate 3× 14 = 42

14.2 one decimal place

× 3

42.6 Place the decimal point one place from the right. Check for Reasonableness 42.6 42 ✓

a. 5 × 0.25 1.25 b. 8 × 4.47 35.76 c. 9 × 2.63 23.67 Growth of Bamboo Over Two Days Add. 4.92 ft + 4.92 ft = 9.84 ft Estimate. 4.92 is about 5.2

× 5 = 10 Multiply. 2 × 4.92 ft =

3. Multiply 6 by 492. Place the decimal point two places from the right.

034_043_C1L1_891639.indd 34 1/9/09 12:12:49 PM

Some parts of lessons have been animated. Use these as a part of the lesson or as a review

for your students at a later time.

They are available online.

034_037_C1L1C_891640.indd 34

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us on Mathematical Content

Multiplying a decimal by a whole number is similar to multiplying two whole numbers. The only difference is determining where to place the decimal point in the product.

Formative Assessment

Use the Check Your Progress exercises after each Example to determine students’ understanding of concepts.

Subtracting Decimals

Remind students to annex zeros if needed to line up decimal places.

AL

If

students have trouble finding

the product of decimals and whole numbers,

Then

use the following reteach

option:

Remind them to estimate the product first. Have them write their estimates in the margin of their papers before calculating the actual products. Then they can compare their solutions with their estimates.

Lesson 1-1 Multiply Decimals 35

Find 2 × 0.45. 0.90 Find 4 × 18.9. 75.6 Find 3 × 0.016. 0.048

MUFFINS A berry muffin recipe calls for 0.75 cups of blueberries. Paul is making 5 batches. He already has 2.5 cups of blueberries. How many more cups does he need? 1.25 cups

IWB

Lesson 1-1 Multiply Decimals 35 If there are not enough decimal places in the product, you need to annex zeros to the left.

Annex Zeros in the Product

Find 2 × 0.018.

1

0.018 three decimal places

× 2 0.036

Annex a zero on the left of 36 to make three decimal places. Check by Adding 0.018+ 0.018 = 0.036 ✓

d. 3 × 0.02 0.06 e. 0.12 × 8 0.96 f. 11 × 0.045 0.495

TRAIL MIX A batch of trail mix calls for 1.2 pounds of dry cereal. Nigela is making 5 batches of trail mix. She already has 2.2 pounds of cereal. How many more pounds of dry cereal does she need?

Step 1 Multiply. Step 2 Subtract.

1.2 one decimal place 6.0

× 5 - 2.2

6.0 one decimal place 3.8

So, Nigela will need 3.8 more pounds of dry cereal.

g. BIRDS A bee hummingbird has a mass of 1.8 grams. How many grams are 6 hummingbirds and a 4-gram nest? 14.8 g h. MONEY Rafael has saved \$45 for a new computer screen. He

is saving \$8.75 a week for 5 weeks for the remaining amount. What is the total cost of the computer screen? \$88.75

Everyday Use

Math Use

Annexto annex a zero means to place a zero at the beginning or end of a decimal

Subtracting Decimals

When subtracting decimals, remember to line up the decimal points.

034_043_C1L1_891639.indd 35 1/9/09 12:12:58 PM

Examples illustrate all of the concepts taught in the lesson and mirror the exercises in

the exercise sets. Check Your Progress exercises give student

s an opportunity to try a similar

problem on their own.

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The Differentiated Homework

Options provide suggestions for the

exercises that are appropriate for basic, core, or advanced students. Many of

the homework exercises are paired so

that students could do the od ds one

day and the evens the next day.

Formative Assessment

Use Exercises 1–9 to check for understanding. Then use the table below to customize assignments for your students.

Odd/Even Assignments

Exercises 10–25 are structured so that students practice the same concepts whether they are assigned odd or even problems.

Common Error Students may have trouble placing the decimal point when there are leading or trailing zeros. In Exercise 21, remind students that the trailing zero in the product may be deleted.

Differentiated Homework Options

Level Assignment Two-Day Option

AL Approaching Level 10–24, 27, 29–37 11–23 odd, 31, 32 10–24 even, 27, 29, 30,

33–37

OL On Level 11–23 odd, 25–27, 29–37 10–24, 31, 32 25–27, 29, 30, 33–37

BL Beyond Level 25–37

36 Chapter 1 Multiply and Divide Decimals

Finding Area

In Exercise 22, remind students to find the area of the poster by multiplying its base by its height.

Visual/Spatial

Have students use grid paper to align their columns and keep track of the decimal point. Their work might resemble the following.

AL OL ELL 2 1 . . 6 2 6 4 8 6 6 2 1 6 2 2 8 1 0 1 3 X

36 Chapter 1 Multiply and Divide Decimals Multiply. 4. \$4.98 Examples 1–3 (pp. 34–35) 1. 2.7 × 6 16.2 2. 1.4 × 4 5.6 3. 0.52 × 3 1.56 4. \$0.83 × 6 5. 5 × 0.09 0.45 6. 4 × 0.012 0.048 7. 0.065 × 18 1.17 8. 0.015 × 23 0.345 Example 4 (p. 35) 9

9 VACATIONS The table shows the number 14.7 69.389

Fuel 12 4.89 17 4.72 15 5.09 Number of Gallons Cost per Gallon (\$) of gallons of gasoline the Beckleys

purchased on their road trip. What was the total cost for gas for the trip? Justify your procedure. \$215.27; (12 × \$4.89) + (17 × \$4.72) + (15 × \$5.09)

= Step-by-Step Solutions begin on page R1.

Extra Practice begins on page EP2.

Multiply. Examples 1–3 (pp. 34–35) 10. 1.2 × 7 8.4 11. 1.7 × 5 8.5 12. 0.7 × 9 6.3 13. 0.9 × 4 3.6 14. 2 × 1.3 2.6 15. 2.4 × 8 19.2 16. 0.8 × 9 7.2 17. 3 × 0.5 1.5 18. 3 × 0.02 0.06 19. 7 × 0.012 0.084 20. 0.0036 × 19 21. 0.0198 × 75 0.0684 1.485

22. MEASUREMENT Asher recently bought the

3.2 ft 4 ft

poster shown at the right. What is its area? (Hint: Use A=bh.) 12.8 ft2

23. MEASUREMENT The height of Mount Everest, in

meters, can be found by multiplying 8.85 by 1,000. Find the height of Mount Everest. Explain your answer. 8,850 m; 8.85 rounds to 9 and 9 × 1,000 = 9,000

Example 4

(p. 35)

24. SCHOOL SUPPLIES Sharon buys 14 folders for \$0.75 each. How much

change will she receive if she pays with \$15? \$4.50 25

25 TEMPERATURE The hottest temperature recorded in the world, in degrees Fahrenheit, can be found by multiplying 13.46 by 10. Find this

temperature. Justify your procedure. 134.6° F; 1,346 × 10 = 13,460. Since 13.46 has 2 decimal places, 13.46 × 10 = 134.60

B 26. MEASUREMENT The thickness of

Coin Thickness (mm) penny 1.55 nickel 1.95 dime 1.35 quarter 1.75 each type of coin is shown in

the table. How much thicker is a stack of a dollar’s worth of nickels than a dollar’s worth of quarters? Explain

your answer. 32 mm; 20 × 1.95 = 39; 4 × 1.75 = 7; 39 - 7 = 32

indicates multi-step problem

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4

Ask students to explain the method they used to solve Exercise 15.

Follow-Up

Remind students to summarize what they learned in this lesson under the Multiply Decimals by Whole Numbers tab of their Foldables. Encourage them to include several original examples.

Are students continuing to struggle with multiplying decimals by whole numbers?

If Yes AL Reteach Worksheet

CRM (p. 18)

AL Differentiated Instruction Option 1 (p. 27c)

If No OL Skills Practice Worksheet

CRM (p. 19)

BL Enrich Worksheet

CRM (p. 22)

NGSSS

Diagnose Student Errors

Survey student responses for each item. Class trends may indicate common errors and misconceptions.

32. A. found the cost of one-day passes for 2 adults and 2 children

B. correct

C. found the cost of 3 adult and 2 children’s one-day passes

D. found for two-day passes instead of one-day passes

Tips

for New Teachers

Multiplying Decimals When students multiply by decimals such as 15 × 0.052, they may write the product as 00.780. Explain that only one zero is needed to the left of the decimal point, and that they should omit all zeros after the last nonzero digit to the right of the decimal point.

Lesson 1-1 Multiply Decimals 37 Quick Check provides

reteaching suggestions for students who continue to struggle.

Lesson 1-1 Multiply Decimals 37 C 27. OPEN ENDED Write a real-world problem involving multiplication by

a decimal factor. Then solve the problem.

28. CHALLENGE Discuss two different ways to find the value of the

expression 5.4 × 1.17 × 100 that do not require you to first multiply

5.4 × 1.17.

29. REASONING Use the product of 123 × 47 to find the product of

123 × 0.47. Explain the difference in the two products.

30. Your friend thinks that 1.5 × 8 = 1.20 because you

do not count the zero when placing the decimal point. Is your friend

correct? Justify your reasoning. No; zero represents the number of

hundredths and should be counted.

Estimate. (Lesson 1-1A) 33–35. Sample answers given.

33. 47.2 × 1.8 100 34. 3.86 × 5.19 20 35. 108.39 × 72.9 7,000

36. SPORTS Kaitlyn recorded the number of hours that she Day Hours Practiced Monday 2 _13 Wednesday 1 _1

3

Friday 1 _34

practiced soccer drills. What was the total time she

practiced? (Lesson 0-5) 5 _ 5

12 h

37. GEOMETRY Find the perimeter of a triangle with side lengths 2.12 centimeters, 2.03 centimeters, and

1.98 centimeters. (Lesson 0-2) 6.13 cm

Practice MA.6.A.1.3

NGSSS

31. GRIDDED RESPONSE The school store is selling the following items.

Item Price Pennant \$2.49 Bumper Sticker \$1.79 Magnet \$0.89

If Miguel buys two pennants, two bumper stickers, and four magnets, how much in dollars will he spend for

all the items? 12.12

32. The table shows the admission prices

to an amusement park. Admission Prices One-Day Pass Two-Day Pass Adult \$39.59 \$43.99 Child (ages 3–9) \$30.59 \$33.99

What is the total price of one-day passes for two adults and three

children? B A. \$140.36 C. \$179.95 B. \$170.95 D. \$189.95 27–29. See Ch. 1 Answer Appendix. 034_043_C1L1_891639.indd 37 1/21/09 9:00:47 AM New teachers, or teachers new to teaching

mathematics, may especially appreciate the

Tips for New Teachers.

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Objective

Students will use decimal models to multiply decimals.

NGSSS

MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and

decimals efficiently.

Resources

Materials 10-by-10 grids Easy-to-Use Manipulatives

Hands-On Activity Tools and Resources

Decimal Models, p. 2

glencoe.com

1

Have each student explain how to use models to multiply a decimal by a whole number. Then review by demonstrating how to use base-ten blocks to model 2 × 1.23.

2

Activity 1 Have students read the

multiplication in words: “Seven tenths times six tenths.” Be sure students understand that each small square represents 0.01, which is one hundredth of the 10-by-10 decimal model.

Virtual Manipulatives

Use a 10-by-10 grid on the Interactive WhiteBoard and the Virtual Base-Ten Blocks to model the steps students should follow to multiply decimals in the lab. Be sure they

understand that each small square in the grid is a hundredth, and each row or column is a tenth.

0.2 0.3

6. Sample answer: There is 1 decimal place in each factor, so the product will have 2 decimal places, and you can multiply normally, but add the decimals from right to left; 0.12

38 Chapter 1 Multiply and Divide Decimals

D

A B C D E Multiply Decimals Multi-Part Lesson

PART glencoe.com

Multipy Decimals

A Multi-Part Lesson

1-1

PART B C D E

38 Chapter 1 Multiply and Divide Decimals

Multiply Decimals by Decimals

You can also model decimals on a 10-by-10 grid.

Find 0.7 × 0.6. Use decimal models. Draw a 10-by-10

decimal model. Recall that each small square represents 0.01.

Shade a rectangle that is 7 tenths units wide and 6 tenths units long.

0.6 0.7

Count the number of shaded squares.

There are forty-two hundredths in the shaded region. So, 0.7 × 0.6 = 0.42.

and Apply

1–3. See Ch. 1 Answer Appendix for models. Use decimal models to show each product.

1. 0.3 × 0.3 0.09 2. 0.4 × 0.9 0.36 3. 0.9 × 0.5 0.45

the Results

4. Tell how many decimal places are in each factor and in each product of Exercises 1–3 above.

5. Is 0.2 × 0.3 equal to 0.6 or 0.06? Justify your reasoning with a model. 0.06; See margin for model.

6. MAKE A CONJECTURE Use the pattern you discovered in Exercise 4 to find 0.6 × 0.2. Check your conjecture with a model or a calculator. See margin.

7. Find two decimals with a product of 0.24. Sample answer: 0.6 × 0.4 Main Idea

Use decimal models to multiply decimals. NGSSS

MA.6.A.1.1 Explain and justify procedures for multiplying

and dividing fractions and

decimals.

MA.6.A.1.2 Multiply and divide fractions and

decimals efficiently.

4. 1 in each factor and 2 in the product; 1 in each factor and 2 in the product; 1 in each factor and 2 in the product

034_043_C1L1_891639.indd 38 1/9/09 12:13:15 PM

Every effort is made to show answers on the reduced Student Edit

ion page or in the margin of the Teacher

Edition. However, answers that do not fit in either of these places can be

found in Answer Appendix

pages at the end of each chapter.

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3

Activity 2 Make sure students understand why they need a 10-by-100 grid that shows thousandths. Reinforce to students that the model is not to scale.

How many decimal places are in the factors? 3

How many decimal places will be in the product? 3

What is the place value of a decimal with three places? thousandths

Use Exercises 1 and 8 to determine whether students understand how to use models to multiply decimals.

From Concrete to Abstract Use Exercise 13 to bridge the gap between using models to find the product of decimals and using rules to determine where to place the decimal point in the product.

Extending the Concept Have students discuss how to estimate the product of 0.84 × 0.9. Sample answer: 0.84 is approximately 1 and 0.9 is approximately 1. 1 × 1 = 1

13. Sample answer: The total number of decimals in the factors is the same as the number of places the decimal is placed from right to left in the product.

14a. Sample answer: The fi rst product is less than the multiplicand because the multiplier is less than one. You don’t want all of what you started with, only a part of a whole, so the product is less than the multiplicand.

14b. Sample answer: The second product is equal to the multiplicand because of the Identity Property. Any number times one is equal to that number itself.

14c. Sample answer: The third product is greater than the multiplicand because the multiplier is more than one. So, the product is more than what you started with.

Lesson 1-1 Multiply Decimals 39

Lesson 1-1 Multiply Decimals 39 Model 0.8 × 0.16. Use an area model.

Since you are multiplying tenths by hundredths, think about a 10-by-100 grid. There are 1,000 squares in all, but you will not draw them.

100 units

10 units

Draw and shade 8 tenths of the height to model 0.8 and shade 16 hundredths across to model 0.16.

100 units

10 units 0.8

0.16

Find the number of squares shaded. Each square represents one thousandth of the full rectangle.

There are 128 out of 1,000 squares, or 128 thousandths, shaded. So, 0.8 × 0.16 = 0.128.

and Apply

Use decimal models to show each product. 8–10. See students’ work for models. 8. 0.3 × 0.14 0.042 9. 0.8 × 0.03 0.024 10. 0.5 × 0.04 0.020

the Results

11.MAKE A CONJECTURE Find the product of 0.04 and 0.08 without using a model. 0.04 × 0.08 = 0.0032

12. Explain why 0.4 × 0.5 = 0.2. Since 4 rows of 5 will be shaded this represents 0.20 or 0.2. 13.MAKE A CONJECTURE How does the number of decimal places

in the product relate to the number of decimal places in the factors?

14. Analyze each product. 13–14. See margin. First Second Product

Factor Factor

0.9 × 0.6 = 0.54 1.0 × 0.6 = 0.60 1.5 × 0.6 = 0.90

a. Explain why the first product is less than 0.6. b. Explain why the second product is equal to 0.6. c. Explain why the third product is greater than 0.6.

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How would you find the estimated cost of 8 unlimited play MP3 downloads?

Sample answer: Round \$1.49 to \$1 and multiply 1 by 8. So 8 unlimited play downloads costs about \$8.

Is your estimate for 8 unlimited play downloads greater or less than the actual cost? Explain. Sample answer: less, since the actual cost was rounded down

40 Chapter 1 Multiply and Divide Decimals

Objective

Students will multiply decimals by decimals.

NGSSS

MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently.

Also addresses MA.6.A.1.3, MA.6.A.5.3.

glencoe.com

1

Activity Write the following on the board. Ask students to evaluate each expression and describe the pattern in the products.

100 × 7

10 × 7

1 × 7

0.1 × 7

0.01 × 7

700; 70; 7; 0.7; 0.07; Sample answer: Each product is one tenth of the previous product.

Scaffolding Questions

Downloads Price MP3 \$0.99 Unlimited play MP3 \$1.49 Music video \$2.89 Ask:

How could you estimate the cost of 3 MP3 downloads? Sample answer: Round \$0.99 to \$1, and multiply 1 by 3. So 3 downloads cost about \$3.

How would you estimate the cost of 6 music video downloads? Sample answer: Round \$2.89 to \$3 and multiply 3 by 6. So 6 video downloads cost about \$18.

E

A C D E PART A B C D E Multiply Decimals Multi-Part Lesson

PART

Internet Research

Show students a video on multiplying decimals. You can find videos on education video-sharing Web sites. Have students work in small groups to take notes on what they learned. Tell them they will make their own video in a later lesson.

glencoe.com

Multipy Decimals

A Multi-Part Lesson

1-1

PART B C D E

40 Chapter 1 Multiply and Divide Decimals

Main Idea

Multiply decimals by decimals.

NGSSS MA.6.A.1.1 Explain and justify procedures for multiplying

and dividing fractions and

decimals.

MA.6.A.1.2 Multiply and divide fractions and

decimals efficiently. Also addresses MA.6.A.1.3, MA.6.A.5.3.

Multiply Decimals by Decimals

PLANETS The table shows the

Mercury 0.3 Venus 0.9 Mars 0.3 Earth 1 Jupiter 2.3 Saturn 1 Uranus 0.8 Neptune 1.1 Planet Weight (pounds)

weight of a 1-pound object on each planet.

1. A 0.5-pound object weighs one half as much as a 1-pound object. What would this object weigh on Jupiter? 1.15 lb

2. What would an object that weighs 5 pounds on Earth weigh on Jupiter? 11.5 lb

3. MAKE A CONJECTURE How can you use the results of Exercises 1 and 2 to find 0.5 × 2.3? Find 5 × 23 and place the decimal point two places from the right.

When multiplying a decimal by a decimal, multiply as with whole numbers. To place the decimal point, find the sum of the number of decimal places in each factor. The product has the same number of decimal places.

Multiply Decimals

Find 4.2 × 6.7. Estimate 4.2× 6.7 4× 7 or 28

4.2 one decimal place

× 6.7 one decimal place

294 + 252

28.14 two decimal places

The product is 28.14. Compared to the estimate, the product is reasonable.

Find 3.6 × 0.05. Estimate 3.6× 0.05 4 × 0 or 0

3.6 one decimal place

× 0.05 two decimal places

0.180 three decimal places

The product is 0.180 or 0.18. Once you place the decimal point, you can drop the zero at the right.

a. 5.7 × 2.8 15.96 b. 4.12 × 0.05 0.206 c. 0.014 × 3.7 0.0518

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AL

If

students have trouble

multiplying decimals by decimals,

Then

use one of these reteach

options:

1

1 Quick Review Math Handbook,

pp. 131 –132

2

2 Use grid paper to line up the numbers and keep track of decimal placement. Some students may benefit from removing the decimals. Their work might resemble the following. Have students highlight the squares in each factor that are after the decimal. Then have them count the highlighted squares, and use that number to place the decimal.

Formative Assessment

Use the Check Your Progress exercises after each Example to determine students’ understanding of concepts.

Formative Assessment

Use Exercises 1–10 to check for understanding. Then use the table on the next page to

customize assignments for your students.

us on Mathematical Content

In adding decimals, the decimal points must be aligned. In multiplying decimals, the decimal points are not aligned. The decimals are multiplied as if they were whole numbers. The position of the decimal point in the product is based on the sum of the number of decimal places in each factor.

Lesson 1-1 Multiply Decimals 41

Find 8.3 × 2.9. 24.07 Find 5.3 × 0.12. 0.636 Find 2.4 × 0.013. 0.0312

MONEY Jennifer makes \$8.79 an hour walking dogs. She walks dogs for 6.75 hours per week. How much money does she earn each week? \$59.33 Justify your answer. Since \$59.3325 is about 9 × 7, or 63, the answer is reasonable.

IWB

Annex a Zero

Find 1.4 × 0.067.

0.067 three decimal places

× 1.4 one decimal place

268 + 67

0.0938 Annex a zero to make four decimal places.

d. 0.04 × 0.32 0.0128 e. 0.26 × 0.205 0.0533 f. 1.33 × 0.06 0.0798

CARS A certain car can travel 28.45 miles with one gallon of gasoline. The gasoline tank can hold 11.5 gallons. How many miles can this car travel on a full tank of gas? Justify your answer.

Estimate 28.45 × 11.5 30 × 12 or 360

28.45 two decimal places

× 11.5 one decimal place

14225 2845 + 2845

327.175 The product has three decimal places.

The car could travel 327.175 miles. Since 327.175 ≈ 360, the answer is reasonable.

g. NUTRITION FACTS A nutrition label indicates that one serving of apple crisp oatmeal has 2.5 grams of fat. How many grams of fat are there in 3.75 servings? Justify your answer.

Multiply. 3. 29.87127 Examples 1–3 (pp. 40–41) 1. 0.6 × 0.5 0.3 2. 1.4 × 2.56 3.584 3. 27.43 × 1.089 4. 0.3 × 2.4 0.72 55 0.52 × 2.1 1.092 6. 0.45 × 0.053 0.02385 7. 2.7 × 1.35 3.645 8. 0.03 × 0.09 0.0027 9. 0.04 × 2.12 0.0848 Example 4 (p. 41)

10. MEASUREMENT A mile is equal to approximately 1.609 kilometers. How many kilometers is 2.5 miles? Justify your answer. 4.0225 km; 1.609 ≈ 1.5 and 1.5 × 2.5 = 3.75; 3.75 ≈ 4.0225

Real-World Link A car that can travel 21 miles on one gallon of gasoline will cost about \$925 per year more, in gasoline costs alone, than a car that can travel 31 miles on one gallon of gasoline.

indicates multi-step problem

g. 9.375 g; 25 × 375 = 9,375, and there are 3 total decimal places, so the decimal point is placed 3 places from the right.

Lesson 1-1 Multiply Decimals 41

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Odd/Even Assignments

Exercises 11–24 are structured so that students practice the same concepts whether they are assigned odd or even problems.

Common Error Students may have trouble remembering the order of operations in Exercises 29–31. Remind them to multiply before adding and subtracting.

FIND THE DATA

In Exercise 36, students should use the Data File on pp. 2–5 to write a real-world problem.

Focus Question Follow-Up

What do you know about multiplying a whole number by a decimal greater than 1? Sample answer: The product is greater than the number.

What do you know about multiplying a whole number by a decimal between 0 and 1? Sample answer: The product is less than the number.

Do you think those same rules apply to dividing by a decimal? Sample answer: I think they will be the opposite.

Intrapersonal

Have students work through at least one problem in this lesson on their own. Ask them to describe the steps they used to solve the problem in a short paragraph. Encourage students to include an explanation of why their answers are reasonable when compared to their original estimates.

AL OL

ELL

33. 30.39 in2; Since 3.1 × 6.9 = 21.39 and 3(6.1 - 3.1) = 9, then the area of the fi gure is 21.39 + 9 or 30.39 in2.

Differentiated Homework Options

Level Assignment Two-Day Option

AL Approaching Level 11–24, 39–42, 46–57 11–23 odd, 47, 48 12–24 even, 39–42, 46,

49–57

OL On Level 11–27 odd, 28, 29, 31,

33–37, 39–42, 44–57

11–24, 47, 48 25–42, 46, 49–57

BL Beyond Level 25–57

42 Chapter 1 Multiply and Divide Decimals

42 Chapter 1 Multiply and Divide Decimals

= Step-by-Step Solutions begin on page R1.

Extra Practice begins on page EP2.

Multiply. Examples 1–3 (pp. 40–41) 11. 0.7 × 0.4 0.28 12. 1.5 × 2.7 4.05 13. 0.4 × 3.7 1.48 14. 3.1 × 0.8 2.48 15. 0.98 × 7.3 7.154 16. 2.4 × 3.48 8.352 17. 6.2 × 0.03 0.186 18. 5.04 × 3.2 16.128 19. 14.7 × 11.36 166.992 20. 27.4 × 33.68 922.832 21. 0.28 × 0.08 0.0224 22. 0.45 × 0.05 0.0225 Example 4 (p. 41)

23. ANIMALS A giraffe can run up to 46.93 feet per second. How far could a giraffe run in 1.8 seconds? Justify your answer. 84.474 ft; 46.93 × 1.8 ≈ 45 × 2 = 90

24. MEASUREMENT Katelyn has a vegetable garden that measures

16.75 feet in length and 5.8 feet in width. Find the area of

the garden. Justify your answer. 97.15 ft2; 16.75 × 5.8 17 × 6 = 102

Multiply.

B 25. 25.04 × 3.005 75.2452 26. 1.03 × 1.005 1.03515 27. 5.12 × 4.001 20.48512

28. WALKING Junnie walked for 2.5 hours at a speed of 3.2 miles per hour.

Maurice walked for 1.8 hours at a speed of 4.1 miles per hour. (Hint: distance equals speed times time.)

a. Who walked farther? Junnie

b. How much farther did that person walk? 0.62 mile farther

ALGEBRA Evaluate each expression if a= 1.3, b= 0.042, and c= 2.01.

29. ab+c 2.0646 30. 6.023a-c 5.8199 31. 3.25c+b 6.5745 32. abc 0.109746

33. MEASUREMENT Find the area of the figure at

3 in. 6.9 in.

3.1 in. 6.1 in.

the right. Justify your procedure. See margin. 34. ALGEBRA Which of the three numbers

9.2, 9.5, or 9.7 is the correct solution of 2.65t= 25.705? Explain your answer. 9.7; 2.65 × 9.7 = 25.705

35

35 GROCERY SHOPPING Pears cost \$0.98 per pound and apples cost \$1.05 per pound.

Mr. Bonilla bought 3.75 pounds of pears and 2.1 pounds of apples. How much did he pay for the pears and apples? Explain your answer.

36. FIND THE DATA Refer to the Data File on pages 2–5. Choose some

data and write a real-world problem in which you would multiply decimals. See students’ work.

For each statement below, find two decimals a and b that make the

statement true. Then find two decimals a and b that make the

statement false. Explain your reasoning.

37. If a> 1 and b< 1, then ab< 1. 38. If ab< 1, then a< 1 and b< 1. 37. Sample answer: If a= 1.5 and b= 0.2, then ab= 0.3, which is less than 1; if a= 1.5 and b= 0.8, then ab= 1.2, which is not less than 1. 38. Sample answer: If ab= 0.63, then a could be 0.9 and b could be 0.7, which are both less than 1; if ab= 0.5, then a could be 0.5, which is less than 1, but b would be 1.0, which is not less than 1. AlgebraTo show multiplication with variables, the variables and/or numbers are placed next to each other.

ab means a×b 6.023a means 6.023×a.

\$5.88; Each price is about \$1. He bought about 6 pounds of fruit. 6 × 1 = 6 ≈ \$5.88

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4

Ask students to write a short paragraph explaining how the previous lesson on multiplying decimals by whole numbers helped them in understanding how to multiply decimals by decimals.

Formative Assessment

Check for student understanding of concepts in Multi-Part Lesson 1-1.

CRM Chapter Quiz 1, p. 64

Are students continuing to struggle with multiplying decimals by decimals?

If Yes AL Reteach Worksheet

CRM (p. 24)

If No OL Skills Practice Worksheet

CRM (p. 25) BL Enrich Worksheet CRM (p. 29) BL Differentiated Instruction Option 1 (p. 27c)

NGSSS

Diagnose Student Errors

Survey student responses for each item. Class trends may indicate common errors and misconceptions.

47. A. found perimeter, not area

B. made mistake in computation

C. made mistake in computation

D. correct

Lesson 1-1 Multiply Decimals 43

Ask students to compare and contrast multiplying whole numbers and decimals. Sample answer: Multiplying whole numbers and decimals is similar because you can multiply decimals as if they were whole numbers. Multiply each digit of one factor by the ones, tens, hundreds, and so on of the other factor. Multiplying decimals is different from multiplying whole numbers because you need to place the decimal point in the product. You can count the number of decimal places in each factor and use this number for the number of decimal places in the product, annexing zeros to the left if needed.

Multiply. (Lesson 1-1C)

49. 45 × 0.27 12.15 50. 3.2 × 109 348.8 51. 24 × 5.6 134.4 52. 2.94 × 16 47.04 53. The distance around Earth at the equator is 24,805.94 mi

24,889.78 mi

about 24,889.78 miles. The distance around Earth through the North Pole and South Pole is

about 24,805.94 miles. (Lesson 1-1A) a–b. Sample answers given.

a. Estimate how many miles you would travel if you circled the equator 3 times. about 75,000 mi

b. Estimate how many more miles you would travel if you

circled the equator 10 times rather than 10 times around the poles. about 1,000 mi Add or subtract. (Lesson 0-3)

54. _ 3 10+ 5 _ 10 4

_

5 55. _ 2 9+ 7 _ 9 1 56. 5 _ 8- 2 _ 8 3

_

8 57. _ 10 11- 7 _ 11 3

_

11

C 39. OPEN ENDED Write a multiplication problem in which the product is between 0.05 and 0.75. Sample answer: 0.1 × 0.6

40. NUMBER SENSE Place the decimal point in the answer to make it correct. Explain your reasoning. 3.9853 × 8.032856 = 32013341...

41. REASONING Determine whether the following statement is always,

sometimes, or never true. Give examples to justify your answer. The product of two decimals less than 1

is less than either of the factors.

42. NUMBER SENSE Is the product of 0.4 × 1.8 greater than or less than 0.4? Explain your reasoning.

CHALLENGE Evaluate each expression.

43. 0.3(3 - 0.5) 0.75 44. 0.16(7 - 2.8) 0.672 45. 1.06(2 + 0.58) 2.7348

46. Describe two methods for determining where to

place the decimal point in the product of two decimals. See Ch. 1 Answer Appendix.

40. 32.013341...; Sample answer: 3.9853 × 8.032856 rounds to 4 × 8 = 32, so the answer must be about 32. 41. always; Sample answer: 0.3 × 0.5 = 0.15; 0.75 × 0.6 = 0.45 42. greater than 0.4; It is being multiplied by a decimal greater than 1.

48. 0.448 kilogram; The golf balls’ total mass is 12 × 0.046 or 0.552 kilogram. Practice MA.6.A.1.1, MA.6.A.1.2

NGSSS

47. What is the area of the rectangle? D

5.62 cm

1.4 cm

A. 14.04 cm2 C. 8.992 cm2

B. 10.248 cm2 D. 7.868 cm2

48. SHORT RESPONSE A soccer ball and 12 golf balls weigh a total of 1 kilogram. Each golf ball weighs about 0.046 kilogram. What is the weight of the soccer ball? Explain your reasoning.

Lesson 1-1 Multiply Decimals 43

034_043_C1L1_891639.indd 43 10/1/10 3:21:42 PM

NGSSS Practice exercises help students

solidify their knowledge using exercises in a format

similar to the state test.

040_043_C1L1E_891640.indd 43

References

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