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CONNECTED MATHEMATICS® 3

Decimal OpsComputing With Decimals and Percents

Custom Edition

Glenda Lappan • Elizabeth Difanis Phillips • James T. Fey • Susan N. Friel

PEARSON Always Learning

Connected Mathematics 3 Decimal Ops

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Cover Art: Three Communication Design, Chicago.

Taken from:

Connected Mathematics Project 3, Decimal Ops: Computing With Decimals and Percents by Glenda Lappan, Elizabeth Difanis Phillips, James T. Fey, Susan N. Friel Copyright © 2014 by Michigan State University, Glenda Lappan, Elizabeth Difanis Phillips, James T. Fey, Susan N. Friel.

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Connected Mathematics 3 Decimal Ops

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24 24Investigation 2 Adding and Subtracting DecimalsThis Investigation has two goals. The first goal is to develop algorithms for adding and subtracting decimals. The second goal is to sharpen estimation skills and number sense. In each Problem, you will use what you know about operations with fractions to make sense of decimal operations.

2.1 Getting Things in the Right Place Adding Decimals

LAUNCH Sally Jane and her friend Zeke stopped at the Quick Shop to buy snacks. They picked out a bag of pretzels for $.89 and a bottle of cider for $1.97. At the checkout, the clerk said that the computers were down. He then calculated their bill himself. The clerk wrote some numbers on a piece of paper. He told Sally Jane and Zeke that they owed $10.87 for their snacks and $.66 more for tax.

Common Core State Standards

6.NS.B.3. Fluently add, subtract, multiply, anddivide multi-digit decimals using the standard algorithm for each operation.

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6.EE.B.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?… 6.EE.B.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number… 6.EE.B.7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Also 6.EE.A.2, 6.EE.A.2a

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Sally Jane pulled out her smart phone and got a total of $2.86 before tax. But she could not convince the clerk that he was wrong.

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• How do you think the clerk might have arrived at the total of $10.87? • How would you explain that the correct amount cannot be $10.87, without simply saying, “The calculator says $2.86”?

It seems as if the Quick Shop clerk did not understand place value for whole numbers and decimals.

Sally Jane explained to the clerk that she could tell by estimation that the answer could not be more than $3.

• Does Sally's estimate make sense? • How does place value play a role in the clerk's error?

26 26In this Problem, you will use equivalence of fractions and decimals to think about place value. You will use your understanding of place value to add and subtract decimals.

Some of the cards shown below have decimal numbers and some have fractions. Sorting these cards will help you think about place value.

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27 27ACTIV-e book Problem 2.1

A. 1. Sort the cards on the previous page into groups of numbers with equivalent values. 2. Order the groups of cards from least to greatest values. 3. Use blank cards to write at least one new card with an equivalent value that belongs in each group.

B. Isaac and Madeline are using decimal cards to practice finding sums of decimals. Isaac chose two cards and said they have a sum of 0.0778.

Madeline disagreed. She said, “If you use the equivalent expanded form for each decimal, the answer will be

0 + 0/10 + 14/100 + 4/1000 + 4/10000 .”

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1. Who is correct, Isaac or Madeline? Explain.2. What is the correct answer in decimal form? Infraction form?

continued on the next page

28 28C. Use your knowledge of place value and fractions to find the following sums.1. 70.4 + 7.042. 7.4 + 7.043. 0.704 + 0.7044. 0.704 + 7.45. Find two decimal cards that sum to each value.a. 14.44b. 77.8c. 7.744d. 0.778D. Sally Jane said, “You can add decimals the same way that you add whole numbers. You just need to line up the decimal points to make sure that numbers with the same place value are added.”

7.4 + 0.704 8.104

1. How did Sally Jane get each digit in the answer8.104?

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2. Lee Ann said, “It helps to put two zeros after the4 in 7.4 to be sure that everything lines up right.”

7.400 + 0.704 8.104

a. Why is 7.4 equal to 7.400?

b. Explain how Lee Ann's suggestion might behelpful.

E. Use your knowledge of place value and fractions to: • Estimate the answers for each decimal sum.

• Calculate exact values for each decimal sum.1. 1.1999 + 2.02

2. 1.762 + 6.9

3. 0.243 + 0.7

4. 3.47 + 8

29 29F. Four students calculated the sum 2.561 + 15.74 + 92.03 and got various answers. Study their work to see if any of their sums are correct. For those that are not correct, identify the errors they made.

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1. Mike

2.561 15.74 +92.03 133.38

2. Sam

2.561 15.740 + 92.030 109.231

3. Miley

2.561 15.740 + 92.030 110.331

4. Jackie

2.561 15.74 + 92.03

110.331

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2. Adam 25.638

- 17.450 8.188

3. Blanca 25.638 - 17.450

12.2884. Jonathan

25.638 - 17.45

23.993

D. Describe a systematic procedure or algorithm you can use for finding the difference of any two decimal numbers.

ACE Homework starts on page 34.

32 322.3 Connecting Operations Fact Families

LAUNCH For whole numbers and fractions, addition and subtraction facts are related. For example, the numbers 4, 8, and 12 are related by four equations using addition and subtraction.

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Addition Sentence 4 + 8 = 12 Related Number Sentences

8 + 4 = 12 12 - 8 = 4 12 - 4 = 8

? How can you determine whether these fact-family relationships are true for decimal addition and subtraction?

ACTIVe-book Problem 2.3

TOOLS Use what you know about decimals to see whether fact-family ideas apply to operations with decimals. Use a calculator if necessary.

A .Write fact families for these equations. Check to see whether each related number sentence is true.

1. 0.02 + 0.103 = 0.1232. 1.82 − 0.103 = 1.717

B. Are the addition and subtraction relationships you see in whole-number fact families also true for addition and subtraction of decimals? Explain your reasoning.

C. Fact-family reasoning often helps when solving equations. Use the number line below to write two

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sentences relating 0.02, 0.123, and x. Then find the value of x.

TOOLS

33 33D. Use fact-family reasoning to find values of n that make these mathematical sentences true. Then check your answers.

1. n + 2.3 = 6.55

2. n − 6.88 = 7.21

3. 2n + 1.55 = 6.4

E. Suppose you have a five-dollar bill to buy fruit as shown in the photo below. You decide to buy three apples, four pears, and some bananas.

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1. How many bananas can you buy without goingover your total of $5?

2. After purchasing your fruit, how much changewill you get back from your five-dollar bill?

3. Write a sentence to show your thinking for part(2). Did you use fact families in answering the question? Explain.

ACE Homework starts on page 34.

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34 34ACE Applications

ACTIVE-e book For Exercises 1–4, find the sum of each pair of cards without using a calculator. Show your work and explain how an estimation strategy suggests that your answer is probably correct.1. 3.42 5.82. 5.012 0.933. 10.437 4.00344. 0.403 0.075. During the Memorial High School highway clean-up project, Ms. Palkowski cleaned 0.125 of a mile for the teachers' team and then 0.4 of a mile for the science club. How much did she clean altogether?

6. Multiple Choice Which is the correct decimalsum?A.

81.9 + 0.62 88.1

B. 81.9

+ 0.62 82.52

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C. 81.9 + 0.62 0.1439

D. 81.9 + 0.62 8.81

7. Find each sum.a. 4.9 + 3 3/4

b. 91.678 + 2.34 + 12.001

c. 2.75 + 3 2/58. Place decimal points in 102 and 19 so that thesum of the two numbers is 1.21. 9. Place decimal points in 34, 4, and 417 so that thesum of the three numbers is 7.97.

35 35For Exercises 10–15, find each difference without using a calculator. Show your work and explain how an estimation strategy suggests that your answer is probably correct.10. 5.2 − 0.12

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11. 4.54 − 2.912. 0.095 − 0.007113. 2.057 − 1.9914. 10 − 1.06815. 5.63 − 4.05 + 9.216. Place decimal points in 431 and 205 so thatthe difference of the two numbers is 16.19.

17. Find the value of n that make each sentencetrue. Then write the addition and subtraction fact family for the sentence.

a. 22.3 + 31.65 = nb. 18.7 − 4.24 = n

18. Find the value of n that makes each sentencetrue. Use fact families to help.

a. 2.3 + n = 3.42b. n − 11.6 = 3.75

19. Find the missing numbers.

a.

36.03 + n 45.218

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b.

n + 0.488 13.762

c. 0.45 + n + 0.4 = 2.62d. 75.4 − 10.801 + n = 77.781

20. Owen had a sinus infection. The thermometerbelow shows his temperature when he ran a fever.

After he took some medicine, his temperature dropped 3.4°F overnight.

a. What was Owen's temperature in the morning?b. How far above a normal temperature of 98.6°Fwas Owen's temperature even after medicine and a night's sleep?

36 36ACTIVe-book Connections

For Exercises 21–26, perform each calculation.

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21. 5(10 + 7)

22. 43 · 2

23. 10(42)

24. 24/6

25. 33

26. 2(10 − 8) + 8

For Exercises 27–30, write the calculation needed to answer the given question and explain how you know you've chosen the right operation.

27. A world-class athlete ran 200 meters in 19.19seconds. Suppose that he ran the first half of that race (around a curve) in 10.75 seconds. How long did it take him to run the second half of the race (on a straightaway)?

28. A female world-record-holder ran 200 metersin 22.58 seconds. She and the runner in Exercise 27 complete a two-person relay race of 400 meters. How long would it take them to run the relay race if they were to run at their individual winning speeds?

29. Yvonne plans to frame a poster she bought ata concert. The frame will be rectangular with base

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0.45 meters and height 0.95 meters. What total length of frame material will she need?

30. Multiple Choice The average number of pointsper game for two star high-school basketball players are shown below.

Average Points per Game

Player Points27 19.6324 17.54

The high-school basketball team averages 70.9 points per game. What is the average number of points per game scored by all the other players together?A. 14.18B. 17.54C. 19.63D. 33.73

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37 3731. Find the missing lengths in the figurebelow. Then find the perimeter of the figure. All measurements are in inches.

32. The perimeter of a parallelogram is 15.42 cm.The length of one of its sides is 2.93 cm. What are the lengths of its other sides?

33. Write these times as decimal parts of an hour.

a. 45 minutes

b. 24 minutes

c. 90 minutes

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For Exercises 34–39, find each sum or difference in fraction form.34. 4/5 + 3/4

35. 2 3/5 + 3/8

36. 1 2/3 + 3 5/6

37. 8/3 - 8/3

38. 1 4/5 - 1/2

39. 4 2/8 - 1 3/4

40. Rewrite Exercises 34–39 with decimal numbersand find the results of the operations using the decimal equivalents. Compare your decimal answers to the fraction answers.

38 3841. Will likes to keep track of his Memorial HighSchool friends during the highway clean-up project. He thinks of himself as being at 0 on the number line. He pictures friends who are on the road ahead of him to the right on the number line. Friends who are on the road behind him, he pictures to the left.

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Sample: Tamika is 3 km ahead, at +3. Cyrus is 2 km behind, at −2.

a. Justin is located the same distance from Willas Will is from Tamika. Justin and Tamika are on opposite sides of Will. Mark Justin's position on a copy of the number line above.

b. Carlos is at the location shown on the numberline. Is Carlos ahead of or behind Will? By how many kilometers?

c. Write three numbers that show locations behindWill.

d. Write three numbers that show locations aheadof Will.

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39 39ACTIVe-book Extensions

42. a. Use the numbers 2, 9, 7, and 4 only once each. Write the greatest possible number of the form 3. __ __ __ __.

b. Use the numbers 2, 9, 7, and 4 only once each.Write the least possible number of the form 3. __ __ __ __.

c. Write all numbers of the form 3. __ __ __ __that are greater than 3.795.

d. Write all numbers of the form 3. __ __ __ __that are less than 3.73 but greater than 3.4399.

For Exercises 43–46, use 1, 2, 3, or 4 to form decimal numbers so that each sum or difference is as close as possible to the given number. You may use the same digit more than once. For example, you may write 0.33. The symbol ≈ means “is approximately equal to.”43. 0.__ __ + 0.__ __≈ 1/3

44. 0.__ __ + 0.__ __≈ 0.9

45. 0.__ __ + 0.__ __≈ 2/7

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46. 0.__ __ + 0.__ __ ≈ 0.125For Exercises 47 and 48, use 1, 2, 3, or 4 to form decimal numbers so that each calculation is correct. You may use the same digit more than once in a number.

47. 0.__ __ + 0.__ __ = 0.75

48. 0.__ __ + 0.__ __ = 0.3

40 402 Mathematical Reflections

In this Investigation, you used your understanding of fractions and place value to develop algorithms for adding and subtracting decimals. The following questions will help you summarize what you have learned.

Think about these questions. Discuss your ideas with other students and your teacher. Then write a summary of your findings in your notebook.

1. How does interpreting decimals as fractions help you make sense of adding and subtracting decimals? Give an example to show your thinking.

2. How does the place-value interpretation of decimals help you add and subtract decimals? Give an example to show your thinking.

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3. Describe algorithms for adding and subtracting any two decimal numbers.

41 41ACTIVe-book Common Core Mathematical Practices

As you worked on the Problems in this Investigation, you used prior knowledge to make sense of them. You also applied Mathematical Practices to solve the Problems. Think back over your work, the ways you thought about the Problems, and how you used Mathematical Practices.

Tori described her thoughts in the following way:

In Problem 2.3, we used fact families to find some unknown values. These families show how the same numbers are related to each other in different ways. It is easy to see this with whole numbers. 8 + 4 = 12 and 12 − 4 = 8 show the same relationship between 4, 8, and 12.

But this fact-family idea works for all numbers.

When we had to find N in N − 6.88 = 7.21, we could write 7.21 + 6.88 N. So, N = 14.09.

Common Core Standards for Mathematical Practice MP7 Look for and make use of structure

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? • What other Mathematical Practices can you identify in Tori's reasoning? • Describe a Mathematical Practice that you and your classmates used to solve a different Problem in this Investigation.