2 Teach the Concept Lesson 3-5 Adding and Subtracting Like...

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Adding and Subtracting Like FractionsAdding and Subtracting Like Fractions

Lesson 3-5

Vocab

Essential

Question

What happens when you add, subtract, multiply, and divide rational numbers?

Common Core

State Standards

Content Standards 7.NS.1, 7.NS.1d, 7.NS.3, 7.EE.3

Mathematical Practices 1, 3, 4, 5, 6

Vocabulary

like fractions

What You’ll Learn

• Add rational numbers with common denominators.• Subtract rational numbers with common

denominators.

Real-World Link

Technology In a survey, users of E-readers were asked to describe why they prefer E-readers over books. One-eighth said that it was because they can change the font size and read faster. Five-eighths said that it was because the devices are portable—it’s like having a small library with them wherever they go!

Interactive

Study Guide

See pages 61–62 for:• Getting Started• Real-World Link• Notes

Like fractions are fractions with the same denominator.

Find each sum. Write in simplest form. a. 7 _

10 + 6 _

10 Estimate 1 + 1 _

2 = 1 1 _

2

7 _ 10 + 6 _ 10 = 7 + 6 _ 10 The denominators are the same. Add the numerators.

= 13 _ 10 or 1 3 _ 10 Simplify and rename as a mixed number. Is the answer reasonable?

b. 5 _ 8

+ (-

7 _ 8 ) Estimate 1 _ 2

+ (-1) = - 1

_ 2

5 _ 8 + (- 7 _ 8 ) = 5 + (-7)

_ 8 The denominators are the same. Add the numerators.

= -2 _ 8 or - 1 _ 4 Simplify. Compare to the estimate. Is it reasonable?

1a. 5 _ 6 + 4 _ 6 1 1

_ 2

1b. 4 _ 7 + (− 6 _ 7 ) - 2 _ 7

1c. 1 _ 5 + 4 _ 5 1 1d. - 5 _ 8 + 11 _ 8

3

_ 4

Tutor

Example 1

Do these problems to find out. Got It ?Got It ?

Words To add fractions with like denominators, add the numerators and write the sum over the denominator.

Symbols a _ c + b _ c = a + b _ c , where c ≠ 0

Example 2 _ 8 + 3 _ 8 = 2 + 3 _ 8 or 5 _ 8 0

28

38

58

Add Like FractionsAAKey Concept

s.

n id

ke

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Objectives add and subtract rational numbers with common denominators

Building on the Essential Question

At the end of the lesson, students should be able to answer “What types of problems might be solved by adding or subtracting fractions?”

Example 1

What’s the Math? add like fractions• In Example 1a, why isn’t the denominator of the sum

10 + 10, or 20? Only the numerators of like fractions are added to find the sum. The denominator is the same as the denominators of the addends.

• In Example 1b, why is the difference not 2 _ 0

? The fractions are like fractions, so you subtract the numerators, not the denominators.

Need Another Example?

Find each sum. Write in simplest form.

a. 3 _ 4 + 3 _ 4 1 1 _

2

b. 5 _ 9 + ( - 7 _ 9 ) -

2 _

9

2 Teach the Concept

120 Chapter 3 Operations with Rational Numbers

Rational Numbers and Exponents

Find 2 3 _ 8 + 3 7 _ 8 . Write in simplest form.

Estimate 2 + 4 = 6

2 3 _ 8 + 3 7 _ 8 = (2 + 3) + ( 3 _ 8 + 7 _ 8 ) Add the whole numbers and fractions separately.

= 5 + 10 _ 8 Add the numerators.

= 5 10 _ 8 or 6 1 _ 4 Simplify. Rename 5 10 _ 8

as 6 2 _ 8

or 6 1 _ 4

.

Check for Reasonableness 6 1 _ 4

≈ 6 �

Find each sum. Write in simplest form.

2a. 1 3 _ 4 + 4 3 _ 4 6 1

_ 2

2b. 3 2 _ 5 + 8 1 _ 5 11 3

_ 5

2c. -2 3 _ 7 + (-4 5 _ 7 )

Tutor

Example 2


−7 1

_ 7

The rule for subtracting fractions with like denominators is similar to the rule for addition.

Find 3 _ 10 - 9 _ 10 . Write in simplest form.

Estimate 1 _ 2

- 1 = - 1

_ 2

3 _ 10 - 9 _ 10 = 3 - 9 _ 10 The denominators are the same. Subtract the numerators.

= -6 _ 10 or - 3 _ 5 Simplify.

Check for Reasonableness - 3 _ 5

≈ -

1 _

2 �

Find each difference. Write in simplest form.

3a. 5 _ 15 - 10 _ 15 − 1 _ 3

3b. 3 _ 9 - 4 _ 9 − 1 _ 9

3c. 7 _ 8 - 3 _ 8 1

_ 2

Tutor

Example 3


Words To subtract fractions with like denominators, subtract the numerators and write the difference over the denominator.

Symbols a _ c - b _ c = a - b _ c , where c ≠ 0

Example 4 _ 9 - 3 _ 9 = 4 - 3 _ 9 or 1 _ 9

0 119

49

39

Subtract Like Fractions SSKey Concept

Alternative Method

When adding or subtracting mixed numbers, you can write them as improper fractions before adding or subtracting. If any of the numbers are negative, it is easier to use this method.2 3 _ 8 + 3 7 _ 8 = 19 _ 8 + 31 _ 8

= 50 _ 8 or 6 1 _ 4

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Example 2

What’s the Math? add mixed numbers • How can you estimate the sum? 2 3_

8 is about 2. 3 7_

8 is

about 4. So a good estimate is 2 + 4 or 6.


Find 3 4_9 + 8 2_9 . Write in simplest form. 11 2_3

Example 3

What’s the Math? subtract like fractions• How is the rule for subtracting like fractions like the

rule for adding like fractions? In both cases, the denominator of the answer is the same as the denominators of the fractions you are adding or subtracting. The numerator is the sum of the numerators if you are adding, and the difference of the numerators if you are subtracting.


Find 11_12 -

5_12 . Write in simplest form. 1_

2

Lesson 3-5 Adding and Subtracting Like Fractions 121

Watch Out!

When you subtract a negative number, you add the opposite. If the number you are subtracting from is negative, the result can be positive, negative, or zero.

Evaluate x - y when x = 3 _ 5 and y = -

4 _ 5 .

To subtract a negative number, add its additive inverse.

x - y = 3 _ 5 - (- 4 _ 5 ) Replace x with 3 _ 5

and y with - 4 _ 5

.

= 3 _ 5 + 4 _ 5 The additive inverse of - 4

_ 5 is 4 _ 5

.

= 3 + 4 _ 5 The denominators are the same. Add the numerators.

= 7 _ 5 or 1 2 _ 5 Simplify and rename as a mixed number.

Check Use a number line.

210

35

45

Evaluate each expression if a = 3 _ 8 , b = -

5 _ 8 , and c = 7 _ 8 .

4a. a -b 1 4b. b - c -1 1

_ 2

4c. c - a 1

_ 2

LaShaun has 5 1 _ 8 yards of ribbon to border scrapbook pages. If she uses 1 7 _ 8 yards on one page, how much ribbon is left?

Subtract the amount of ribbon she will use from the total amount of ribbon.

Estimate 5 1 _ 8

- 1 7 _ 8

≈ 5 - 2 or 3 yards

5 1 _ 8 - 1 7 _ 8 = 4 9 _ 8 - 1 7 _ 8 Rename 5 1 _ 8

as 4 9 _ 8

.

= 3 2 _ 8 Subtract the whole numbers and then the fractions.

= 3 2 _ 8 or 3 1 _ 4 Simplify.

LaShaun has 3 1 _ 4 yards of ribbon remaining.

Check for Reasonableness 3 1 _ 2

≈ 3 �

5. The Daytona International Speedway is one of the longest tracks used in NASCAR

races. It is 2 2 _ 4 miles long. Richmond International Speedway is 3 _ 4 mile long. How

much longer is the Daytona Speedway than the Richmond Speedway? 1 3

_ 4

mi

Tutor

Example 4

3 _ 5

- (- 4

_ 5 ) = 7 _ 5

or 1 2 _ 5

�


Tutor

Example 5

Do this problem to find out. Got It ?Got It ?


Example 4

What’s the Math? evaluate expressions with rational numbers• What is the first step to evaluate the expression?

Replace x with 3_5

and y with - 4_5 .

• What is 3_5 - ( -4_5 ) r ewritten as an addition

expression? 3_5

+4_5

• How would you change the number line graph if the problem were 3_5 -

4_5 ? The red arrow would stay the

same. The blue arrow would start where the red arrowends but extend 4_

5 unit to the left.


Evaluate q - r if r = 7 1_5 and q = 9 3_5 . 2 2_5

Example 5

What’s the Math? subtract mixed numbers to solve a real-world problem• How could you solve the problem using improper

fractions instead of mixed numbers? Find the difference 41_

8-

15_8

.


Keora bought 3 3_8 pounds of apples. If she uses 1 7_8 pounds for a pie, how many pounds of apples does she have left? 1 1_

2


Rational Numbers and Exponents

3 Practice and ApplyHomework

The Independent Practice pages are meant to be used as the homework assignment. If you do not wish to assign the entire exercise set, you can use the table below to select appropriate exercises for your students’ needs.

Differentiated Homework Options

ALApproaching Level

12–33, 39, 41–69

OL On Level 13–35 odd, 36–39, 41–69

BL Beyond Level 34–69

You can use the same rules for finding the distance on the number line between two fractions as you did for finding the distance between two integers.

Find the distance between - 1 _ 7 and 5 _ 7 on a number line. Simplify if necessary.

Step 1Step 1 The common denominator is 7. Divide each unit between -1 and 1 into seven parts.

Step 2Step 2 Graph each fraction on the number line.

Step 3Step 3 Count the parts between the fractions. There are 6 between them.

So, the distance between - 1 _ 7 and 5 _

7 on the number line is 6 parts or 6 _ 7 unit.

Find the distance between each pair of points. Simplify if necessary.

6a. - 3 _ 4 and -

1 _ 4 1

_ 2

unit 6b. - 7 _ 20 and 9 _ 20

4

_ 5

unit 6c. - 11 _ 15 and 2 _ 15

13

_ 15

unit

Tutor

Example 6

10

6 parts

-1 17

57

-


Find each sum or difference. Write in simplest form. (Examples 1–3)

1. 3 _ 6 + 5 _ 6 1 1

_ 3

2. 2 _ 9 + (− 4 _ 9 ) - 2

_ 9

3. 4 _ 12 − 10 _ 12 - 1

_ 2

4. 8 _ 15 − 11 _ 15 - 1

_ 5

5. 3 3 _ 8 + 6 5 _ 8 10 6. 2 1 _ 6 + 8 3 _ 6 10 2

_ 3

Evaluate each expression if r = - 4 _ 7 , s = 2 _ 7 , and t = -

3 _ 7 . (Example 4)

7. s − t 5

_ 7

8. r − t - 1

_ 7

9. r - s - 6

_ 7

10. Mia is making a bookcase and has 92 5 _ 8 inches of wood. If she uses 30 7 _ 8 inches

of wood for the top and bottom, find the amount she has left for the sides. (Example 5) 61 3

_ 4

in.

11. Find the distance between - 7 _ 11 and 3 _ 11 on a number line. Simplify if necessary. (Example 6)

10

_ 11

unit

Find each sum or difference. Write in simplest form. (Examples 1–3) 17. 103

_

9 or 11

4

_ 9

22. 139

_ 13

or 10 9

_ 13

12. 5 _ 6 + (- 4 _ 6 ) 1 _ 6

13. − 11 _ 12 + 7 _ 12 - 1

_ 3

14. 3 _ 14 + (− 5 _ 14 ) - 1

_ 7

15. − 3 _ 7 +

6 _ 7 3

_ 7

16. 5 1 _ 4 + 5 1 _ 4 10 1

_ 2

17. 12 5 _ 9 + (−1 1 _ 9 ) 18. 2 _ 15 − 7 _ 15 - 1

_ 3

19. 5 _ 11 − 7 _ 11 - 2

_ 11

20. − 1 _ 5 − 4 _ 5 −1 −

7 _ 20 − 7 _ 20 - 7

_ 10

22. 2 12 _ 13 − (−7 10 _ 13 ) 23. −8 3 _ 10 − 4 9 _ 10

Evaluate each expression if x = - 5 _ 9 , y = 2 _ 9 , and z = -

4 _ 9 . (Example 4)

24. y − z 2

_ 3

25. y − x 7

_ 9

26. x − z - 1

_ 9

27. z − x 1

_ 9

Guided PracticeGuided PracticeCheck

Independent PracticeIndependent Practice Go online for Step-by-Step SolutionseHelp

21

- 66

_ 5

or -13 1

_ 5

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Example 6

What’s the Math? find the distance between two points on a number line• Why is the distance 6_7 unit rather than 6 units?

Each section of the number line represents 1_7

unit. There

are six sections between the points. 1_7

× 6 = 6_7


Find the distance between -3_5 and 2_5 on a number

line. 1 unit

Formative AssessmentGuided Practice Use these exercises to assess students’ understanding of the concept of the lesson.

Have students explain how they would use an area model to find 3 _ 8 + 2 5 _ 8 . See students’ work.Have students exp

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Lesson 3-5 Adding and Subtracting Like Fractions 123

Watch Out!Watch Out!Find the Error Remind students that they can use a number line to check Xavier’s solution to the addition problem in Exercise 41.

Create Your Own Homework Online

can be used to create worksheets for the suggested assignments on page 123, or create your own worksheets for differentiated homework or review.

Fuse/Getty Im

ages

28. Be Precise Nan was 59 7 _ 8 inches tall at the end of summer. She was 62 1 _ 8 inches

by March. How much did she grow during that time? (Example 5) 2 1

_ 4

in.

29. Yahto needs 3 3 _ 4 cups of sugar to make cookies. He needs an additional 3 _ 4 cup for bread.

Find the total amount of sugar that Yahto needs. (Example 5) 4 1

_ 2

c

Find the distance between each pair of points. Simplify if necessary. (Example 6)

30. - 5 _ 8 and 1 _ 8

3

_ 4

unit 31. - 7 _ 15 and -

4 _ 15 1

_ 5

unit 32. - 1 _ 8 and 3 _ 8

1

_ 2

unit 33. - 3 _ 16 and 3 _ 16

3

_ 8

unit

Find each sum or difference. Write in simplest form.

34. −2 9 _ 10 + (−9 9 _ 10 ) + (−6 9 _ 10 ) −19 7

_ 10

35. 1 _ 9 - 2 4 _ 9 - 5 _ 9 −2

8

_ 9

36. A triathlon is a race with swimming, biking, and running. If an athlete swims for

18 2 _ 4 minutes, bikes for 59 1 _ 4 minutes, and runs for 37 3 _ 4 minutes, what is his total time?

The table shows the weight of Leon’s dog during its first 5 years.

Age (years) 1 2 3 4 5

Weight (pounds) 17 2 _ 8

18 5 _ 8

19 4 _ 8

18 3 _ 8

20 7 _ 8

a. How much weight did Leon’s dog gain or lose between years 3 and 4? between years 1 and 5?

b. If Leon’s dog gains 1 3 _ 8 pounds each year between years

5 and 7, how much will his dog weigh? 23 5

_ 8

lb

38. A lasagne recipe uses 1 2 _ 4 teaspoons basil, 1 _ 4 teaspoon pepper, and 4 teaspoons parsley.

If you double the recipe, how many teaspoons of seasoning will you use?

H.O.T. Problems H.O.T. Problems Higher Order Thinking

39. Use Math Tools Write a subtraction problem with a difference of − 2 _ 3 . Sample answer:

1

_ 3

− 1 = − 2

_ 3

40. Persevere with Problems Lopez Construction is replacing a window in a house. The window is currently 3 feet wide by 4 feet tall. The homeowner wants to add 9 inches to each side of the window. What is the new perimeter of the window in feet? Justify your reasoning.

41. Find the Error Xavier said the sum of −4 1 _ 9 and 1 7 _ 9 is -3 8 _ 9 . Is he correct?

Explain your reasoning. No; he did not add the fraction part of the mixed numbers correctly.

42. Use Math Tools Explain how you could use mental math to find the following sum. Then find the sum. Support your answer with a model. 42–43. See Answer Appendix.

1 1 _ 4 + 2 1 _ 3 + 3 2 _ 3 + 4 1 _ 2 + 5 1 _ 2 + 6 3 _ 4

43. Building on the Essential Question Write a real-world problem about cooking that can be solved by adding or subtracting fractions. Then solve the problem.

B

115 1

_ 2

min

37

5

0 7_8

1 1

_ 8

lb; 3 5

_ 8

lb

11 1

_ 2

t

C

17 feet; 3 9

_ 12

+ 3 9

_ 12

+ 4 9

_ 12

+ 4 9

_ 12

= 14 36

_ 12

or 17 feet


MATHEMATICAL PRACTICES

Emphasis On Exercise(s)

1 Make sense of problems and persevere in solving them.

40

3 Construct viable arguments and critique the reasoning of others.

41

5 Use appropriate tools. 39, 42

6 Attend to precision. 28

Mathematical Practices 1, 3, and 4 are aspects of mathematical thinking that are emphasized in every lesson. Students are given opportunities to be persistent in their problem solving, to express their reasoning, and to apply mathematics to real-world situations.


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