lesson 8 - drauden point middle schooldpms.psd202.org/documents/ppotock1/1503512818.pdf27 ©...

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©Curriculum Associates, LLC Copying is not permitted. 69

Solve Problems with Rational Numbers Name:

Lesson 8 Solve Problems with Rational Numbers

Lesson 8

Prerequisite: Add Negative Fractions

Study the example showing how to add negative fractions. Then solve problems 1–8.

Example

Marita has a container of oatmeal She uses 1 ·· 4 cup of the

oatmeal for breakfast and 3 ·· 8 cup for a cookie recipe What is

the total change in the amount of oatmeal she has?

You can use a number line to represent this problem Keep in

mind that 1 ·· 4 is equivalent to 2 ·· 8

2122 04821 4

82

1423

82

1 Why are 1 ·· 4 and 3 ·· 8 represented with negative numbers

on the number line in the example?

2 What do you need to do to 2 1 ·· 4 to locate it on a number

line that is measured in eighths?

3 You can also use an equation to solve the problem Complete the addition equation

Change in oatmeal 1

Change in oatmeal 5

Total change

after breakfast after cookie recipe

1

5 ?

4 What is the total change in the number of cups in the oatmeal container? Explain

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Rewrite 2 1 ·· 4 as 2 2 ·· 8 .

1 ·· 4 and 3 ·· 8 are represented with negative numbers on the number line because Marita uses

1 ·· 4 cup and 3 ·· 8 cup of oatmeal, so she is taking these amounts out of the container.

1 2 1 ·· 4 2 1 2 3 ·· 8 2

2 5 ·· 8 ; 1 2 1 ·· 4 2 1 1 2 3 ·· 8 2 5 2 2 ·· 8 2 3 ·· 8 5 2 5 ·· 8 cup of oatmeal

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©Curriculum Associates, LLC Copying is not permitted.70 Lesson 8 Solve Problems with Rational Numbers

Solve.

5 Nien has a fl ower farm He cuts 4 3 ·· 4 square yards of

fl owers one day and 5 2 ·· 3 square yards of fl owers the next

Use addition to fi nd the total change in the square yards of fl owers

Show your work.

Solution:

6 Carl has a new bottle of laundry detergent He uses 0 4 cup of laundry detergent while washing one load of laundry and 0 7 cup on another load Use addition to fi nd the total change in the number of cups of detergent in the bottle

7 What is the sum 27 1 ·· 6 1 3 5 ·· 6 in simplest terms?

Show your work.

Solution:

8 At a recycling center, items are sorted and removed according to the material they are made out of In one mixture of recycled materials, 2 625 tons of metal cans, 5 58 tons of paper, and some glass items are removed If the total change in weight of the recycled materials is 29 5 tons, how many tons of glass were removed?

Show your work.

Solution:

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1 24 3 ·· 4 2 1 1 25 2 ·· 3 2 5 24 9 ··· 12 2 5 8 ··· 12

5 29 17 ··· 12

5 210 5 ··· 12

20.4 1 (20.7) 5 21.1 cups

27 1 ·· 6 1 3 5 ·· 6 5 26 7 ·· 6 1 3 5 ·· 6

5 23 2 ·· 6

5 23 1 ·· 3

Possible work: 22.625 1 (25.58) 5 28.205

change in tons of glass 1 (28.205) 5 29.5

29.5 2 (28.205) 5 29.5 1 8.205 5 21.295

23 1 ·· 3

1.295 tons of glass were removed.

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The total change is 210 5 ··· 12 square yards.

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©Curriculum Associates, LLC Copying is not permitted. 71Lesson 8 Solve Problems with Rational Numbers

Name: Lesson 8

Estimating with Decimals

Study the example showing how to use estimation when computing with positive and negative decimals. Then solve problems 1–9.

1 Based on the estimate, do you think the actual new temperature will be above or below 0°F? Explain

2 Write and solve an equation to fi nd the new temperature using exact values What is the exact value of the new temperature?

3 Peter got an exact new temperature of 11 6ºF Explain how Peter can use the estimate to determine that his answer is wrong Then fi nd and correct Peter’s error

4 Janet estimated to the nearest half degree before she found her exact answer Is Janet’s estimate more or less accurate than the estimate in the example problem? Explain

Example

The temperature at noon was 5 6°F During the afternoon the temperature dropped 2 8°F and then dropped another 2 2°F in the evening Estimate the new temperature

You can use a number line to understand the problem by approximating, or rounding, each temperature to the nearest degree

2122 0 1 2 3 4 5 6 7

Now, you can estimate the new temperature by finding 6 2 3 2 2 So, using estimation, the new temperature will be about 1°F

5 6 rounds up to 6 2 8 rounds up to 3 2 2 rounds down to 2

71

above; The estimate is 1°F, so the actual new temperature should be just a little above 0°F.

5.6 2 2.8 2 2.2 5 0.6; 0.6°F

Peter’s answer is much higher than the estimate. It should be pretty close to the estimate.

Peter added 2.8 and 2.2 to 5.6. He should have subtracted 2.8 and 2.2 from 5.6.

more; In general, the more precise that the estimate of each temperature is, the more

accurate the estimate will be.

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Solve.

5 Kyle and Joan are estimating the total weight of three boxes that each weigh 14 62 pounds Each student’s estimate is shown below

Kyle Joan

3(15) 5 45 lb 3(14 5) 5 43 5 lb

Which estimate is more accurate? Explain how you know Then fi nd the actual total weight of the boxes

6 Estimate 55 8 4 (23 1) Then fi nd the exact quotient

Show your work.

Solution:

Use the following situation to solve problems 7–9.

Dusan had $49 60 in his checking account He wrote one check for $24 40 and another check for $25 49

7 Estimate the amount of money in Dusan’s account after he wrote the checks

8 Explain why the estimate might not be enough information to show Dusan whether he has enough money in his checking account to cover the checks

9 Dusan says that because he always rounds to the nearest dollar, he will always have more money in his account than his estimate Is Dusan correct? Explain

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Joan’s estimate should be more accurate because she used a more precise estimate;

3(14.62) 5 43.86 lb

Possible answers: $50 2 $24 2 $25 5 $1 or $50 2 $25 2 $25 5 $0

The estimate shows that Dusan has very little in his account, but it is not accurate enough

to say for sure if he has enough in his account to cover the checks.

No; rounding to the nearest dollar does not mean that all numbers will be rounded up.

Possible work: estimate: 56 4 (24) 5 214

exact: 55.8 4 (23.1) 5 218

The estimate is 214, and the exact quotient is 218.

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Solve.

4 Think about the problem 26 3 ·· 5 2 1 27 4 ·· 15 2 1 2 1 ·· 5

a. Estimate the answer

b. Find the actual answer

c. Compare the estimate and the actual answer Is your actual answer reasonable?

5 James baked 5 1 ·· 4 dozen peanut butter cookies and 4 5 ·· 6

dozen oatmeal cookies for a bake sale

a. Estimate the number of dozens of cookies he baked

b. Find the actual number of dozens of cookies he baked

c. How do the numbers compare? Is the actual number of dozens you found reasonable?

6 What is the smallest integer that makes the diff erence

x 2 1 3 ·· 4 1 4 ·· 5 2 greater than 1? Use estimation to fi nd the

answer, and then fi nd the exact diff erence to show that your answer is correct

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27 2 (27) 1 2 5 27 1 7 1 2 5 2

The actual answer is close to the estimate, so the actual answer is reasonable.

5 1 5 5 10 dozen

5 1 ·· 4 1 4 5 ·· 6 5 5 3 ··· 12 1 4 10 ··· 12 5 10 1 ··· 12

The actual number is close to the estimate, so the actual number is reasonable.

Possible answer: Both 3 ·· 4 and 4 ·· 5 round to 1, so the sum is approximately equal to

x 2 (1 1 1), or x 2 2. Because 2 2 2 5 0 and 3 2 2 5 1, I know 3 would be the smallest

integer that would give a sum greater than 1.

3 2 1 3 ·· 4 1 4 ·· 5 2 5 3 2 1 15 ··· 20 1 16 ··· 20 2 5 1 9 ··· 20 . Because 1 9 ··· 20 is greater than 1, I know that 3 is the

makes the difference greater than 1.

26 3 ·· 5 2 1 27 4 ··· 15 2 1 2 1 ·· 5 5 26 9 ··· 15 1 7 4 ··· 15 1 2 3 ··· 15 5 2 13 ··· 15

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Name: Lesson 8

Estimating with Fractions

Study the example showing how to use estimation when computing with positive and negative fractions. Then solve problems 1–6.

1 To the nearest integer, what is the distance of the jump?

2 What is the exact distance of the jump?

Show your work.

Solution:

3 Compare the estimated distance of the jump to the actual distance? Is the actual distance reasonable?

Example

A high diving board is about 9 5 ·· 6 feet above the water’s

surface A diver jumps off the diving board and ends up

3 1 ·· 4 feet below the surface of the water What is the total

distance of the jump?

You can use rounding and a number line to help you estimate the answer

Actual Distance Nearest Integer

9 5 ·· 6 10

23 1 ·· 4 23

21222324 0 1 2 3 4 5 6 7 8 9 10

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Based on the number line, the distance is about 13 feet.

The distances are close, so the actual distance is reasonable.

The diver jumped a distance of 13 1 ··· 12 feet.

To find the distance, I’d add the distance of each point from 0.

9 5 ·· 6 1 3 1 ·· 4 5 9 10 ··· 12 1 3 3 ··· 12 5 12 13 ··· 12 5 13 1 ··· 12

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6 Write a sum, diff erence, product, and quotient whose answers could all have an estimate of about 6 One expression should include only mixed numbers, one expression should include only decimals, one expression should include only negative rational numbers, and one expression should include only positive rational numbers

Sum:

Diff erence:

Product:

Quotient:

Solve.

4 Tell whether each statement is True or False

a. 23 3 ·· 4 1 1 5 23 2 ·· 3 2 (21) u True u False

b. 2 855 2 1 375 5 1 48 u True u False

c. 212 7 ·· 12 1 1 29 5 ·· 6 2 5 3 7 ·· 12 u True u False

d. 6 2 ·· 5 2 1 27 1 ·· 10 2 5 13 1 ·· 2 u True u False

5 Justin earned $25 67 one week and $37 85 the next week He also purchased a DVD for $19 99 and gas for his car for $22 07 He started with a zero balance in his checking account Justin recorded all of these transactions in his checkbook and ended up with a balance of $105 58 Was his balance correct? If it was, write an equation to support the answer If not, fi nd the correct balance

An estimate might help you determine whether Justin is correct.

When do you round a rational number up? When do you round a rational number down?

Can you tell if the sum or difference will be positive or negative?

76

33

33

Justin is not correct. Justin should have subtracted

his total purchases (19.99 1 22.07 5 42.06) from

his earnings (25.67 1 37.85 5 63.52). His balance is

$63.52 2 $42.06 5 $21.46

Possible answer: 1 3 ·· 4 1 3 5 ·· 6

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Possible answer: 12.2 2 5.9

Possible answer: 22.9(22.1)

Possible answer: 18.1 4 3.2


Name:

1 Which expression is the best estimate of 1 25 1 (22 69)?

A 1 1 (23) C 2 1 (23)

B 1 1 (22) D 2 1 (22)

Sue chose B as the correct answer How did she get that answer?

Solve Problems with Rational Numbers

Solve the problems.

3 The answer to which situation is best estimated by 27?

A The temperature falls 1 9°F from 4 3°F

B A hot-air balloon rises 4 1 meters from the ground and then rises 2 3 meters more

C A submarine goes 3 8 feet below the surface of the ocean and then goes 2 5 feet farther down

D The amount of money Sarah has after she buys milk for $3 85 and bread for $2 45

2 Which expression is the best estimate of 1 1 ·· 5 3 3 2 ·· 3 ?

A 2 3 4

B 1 3 4

C 2 3 3

D 1 3 3

How might a number line help you estimate decimals?

Should you round each value up or down?

How can you estimate decimal amounts?

Lesson 8

75

Sue estimated 22.69 as 22 instead of 23, which is

actually closer to 22.69 on the number line.

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