Lesson 8

Practice Lesson 8 Solve Problems with Rational Numbers

Practice and Problem Solving ?Curriculum Associates, LLC Copying is not permitted.

Lesson 8

Solve Problems with Rational Numbers

Name:

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

?82?.

2

3 8

2

1 4

22

2148

21

2

4 8

0

B

1

Why

are

1 ?4?

and

3 ?8?

represented

with

negative

numbers

on the number line in the example?

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.

B

2

What do line that

you need to is measured

do to 2?14? to in eighths?

locate it Rewrite

on a number 2?14? as 2?28?.

B 3 You can also use an equation to solve the problem.

Complete the addition equation.

Change in oatmeal after breakfast 1

1 2?41? 2

1

Change in oatmeal after cookie recipe 5

1 2?38? 2

5

Total change ?

B 4 What is the total change in the number of

cups in the oatmeal container? Explain.

2?58?;

1

2?41?

2

1

1

2?38?

2

5

2?28?

2

3 ?8?

5

2?85?

cup

of

oatmeal

?Curriculum Associates, LLC Copying is not permitted.

Lesson 8 Solve Problems with Rational Numbers

6699

Solve.

M

5

Nien

has

a

flower

farm.

He

cuts

4

3 ?4?

square

yards

of

flowers

one

day

and

5

2 ?3?

square

yards

of

flowers

the

next.

Use addition to find the total change in the square yards

of flowers.

Show your work.

1

24

3 ?4?

2

1

1

25

2 ?3?

2

5

24

9 ?1?2?

2

5

8 ?1?2?

5

29

17 ?1?2?

5

210

5 ?1?2?

Solution:

The

total

change

is

210

5 ?1?2?

square

yards.

B 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 find the total change in the number of cups of detergent in the bottle.

20.4 1 (20.7) 5 21.1 cups

M

7

What

is

the

sum

27

1 ?6?

1

3

5 ?6?

in

simplest

terms?

Show your work.

27

1 ?6?

1

3

5 ?6?

5

26

7 ?6?

1

3

5 ?6?

5

23

2 ?6?

5

23

1 ?3?

Solution:

23

1 ?3?

C 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. 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

7700

Solution: 1.295 tons of glass were removed.

Lesson 8 Solve Problems with Rational Numbers

?Curriculum Associates, LLC Copying is not permitted.

Unit 1 The Number System

Unit 1

B Basic

Key

M Medium

C Challenge

27

Unit 1 The Number System ?Curriculum Associates, LLC Copying is not permitted.

Practice and Problem Solving

28

Lesson 8

Name:

Estimating with Decimals

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

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.

5.6 rounds up to 6. 2.8 rounds up to 3. 2.2 rounds down to 2.

22 21 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.

B 1 Based on the estimate, do you think the actual new

temperature will be above or below 0?F? Explain.

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

B 2 Write and solve an equation to find the new temperature

using exact values. What is the exact value of the new temperature?

5.6 2 2.8 2 2.2 5 0.6; 0.6?F

B 3 Peter got an exact new temperature of 11.6?F. Explain

how Peter can use the estimate to determine that his answer is wrong. Then find and correct Peter's error.

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.

M 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.

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

accurate the estimate will be.

?Curriculum Associates, LLC Copying is not permitted.

Lesson 8 Solve Problems with Rational Numbers

7711

Solve.

M 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 3(15) 5 45 lb

Joan 3(14.5) 5 43.5 lb

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

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

3(14.62) 5 43.86 lb

M 6 Estimate 55.8 4 (23.1). Then find the exact quotient.

Show your work. Possible work: estimate: 56 4 (24) 5 214 exact: 55.8 4 (23.1) 5 218 Solution: The estimate is 214, and the exact quotient is 218.

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.

M 7 Estimate the amount of money in Dusan's account after

he wrote the checks. Possible answers: $50 2 $24 2 $25 5 $1 or $50 2 $25 2 $25 5 $0

M 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.

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.

C 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.

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

7722

Lesson 8 Solve Problems with Rational Numbers

?Curriculum Associates, LLC Copying is not permitted.

Unit 1

Practice Lesson 8 Solve Problems with Rational Numbers

Practice and Problem Solving ?Curriculum Associates, LLC Copying is not permitted.

29

Unit 1 The Number System

Lesson 8

Name:

Estimating with Fractions

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

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

9

5 ?6?

23?41?

Nearest Integer 10 23

24 23 22 21 0 1 2 3 4 5 6 7 8 9 10

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

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

B 2 What is the exact distance of the jump?

Show your work. To find the distance, I'd add the distance of each point from 0.

9

5 ?6?

1

3

1 ?4?

5

9

10 ?1?2?

1

3

3 ?1?2?

5

12

13 ?1?2?

5

13

1 ?1?2?

Solution:

The

diver

jumped

a

distance

of

13

1 ?1?2?

feet.

B 3 Compare the estimated distance of the jump to the actual

distance? Is the actual distance reasonable?

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

?Curriculum Associates, LLC Copying is not permitted.

Lesson 8 Solve Problems with Rational Numbers

7733

Solve.

M

4

Think

about

the

problem

26

3 ?5?

2

1

27

?145?2

1

2

?51?.

a. Estimate the answer.

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

b. Find the actual answer.

26

3 ?5 ?

2

1

27

4 ?1?5?

2

1

2

1 ?5?

5

26

9 ?1?5?

1

7

4 ?1?5?

1

2

3 ?1?5?

5

2

13 ?1?5?

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

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

M

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.

5 1 5 5 10 dozen

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

5

1 ?4?

1

4

5 ?6?

5

5

3 ?1?2?

1

4

10 ?1?2?

5

10

1 ?1?2?

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

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

C 6 What is the smallest integer that makes the difference

x

2

1

3 ?4?

1

4 ?5?

2

greater

than

1?

Use

estimation

to

find

the

answer, and then find the exact difference to show that

your answer is correct.

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 ?2?0?

1

16 ?2?0?

2

5

1

?29?0?.

Because

1

9 ?2?0?

is

greater

than

1,

I

know

that

3

is

the

makes the difference greater than 1.

7744

Lesson 8 Solve Problems with Rational Numbers

?Curriculum Associates, LLC Copying is not permitted.

Unit 1

Practice Lesson 8 Solve Problems with Rational Numbers

30

Practice Lesson 8 Solve Problems with Rational Numbers

Practice and Problem Solving

Lesson 8

Solve Problems with Rational Numbers

Solve the problems.

Name:

B

1 Which expression is the best estimate of 1.25 1 (22.69)?

How might a number

A 1 1 (23)

C 2 1 (23)

line help you

B 1 1 (22)

D 2 1 (22)

estimate decimals?

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

Sue estimated 22.69 as 22 instead of 23, which is

actually closer to 22.69 on the number line.

B

2

Which

expression

is

the

best

estimate

of

1

1 ?5?

3

3

?32??

A 234

B 134

C 233

D 133

Should you round each value up or down?

M

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.

How can you estimate decimal amounts?

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.

Solve.

M

4 Tell whether each statement is True or False.

a.

23

3 ?4?

1

1

5

23

2 ?3?

2

(21)

b. 2.855 2 1.375 5 1.48

u u3 True

False

u3 u True

False

c.

212

7 ?12?

1

1

29

5 ?6?

2

5

3

7 ?12?

d.

6

2 ?5?

2

1

27

1 ?10?

2

5

13

1 ?2?

u u3 True

False

u3 u True

False

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

M

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

An estimate might help you determine whether Justin is correct.

balance of $105.58. Was his balance correct? If it was,

write an equation to support the answer. If not, find the

correct balance.

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

C

6 Write a sum, difference, 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.

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

Sum: Possible answer: 1?43? 1 3?56?

Difference: Possible answer: 12.2 2 5.9

Product: Possible answer: 22.9(22.1)

Quotient: Possible answer: 18.1 4 3.2

Unit 1 The Number System ?Curriculum Associates, LLC Copying is not permitted.

?Curriculum Associates, LLC Copying is not permitted.

Lesson 8 Solve Problems with Rational Numbers

7755

7766

Lesson 8 Solve Problems with Rational Numbers

?Curriculum Associates, LLC Copying is not permitted.

Unit 1

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download