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630 Unit 8 Fractions and Ratios

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Key Concepts and Skills• Find equivalent names for mixed numbers. 

[Number and Numeration Goal 5]

• Convert between fractions and

mixed numbers. 

[Number and Numeration Goal 5]

• Subtract mixed numbers. 

[Operations and Computation Goal 4]

• Use benchmarks to estimate differences. 

[Operations and Computation Goal 6]

Key ActivitiesStudents subtract mixed numbers with like

denominators by renaming the minuend.

Students use benchmarks to estimate

differences.

Ongoing Assessment: Informing Instruction See page 632.

MaterialsMath Journal 2, p. 254

Math Masters, p. 414

Study Link 8�2

slate

Playing Mixed-Number SpinMath Journal 2, p. 255

Math Masters, pp. 488 and 489

per partnership: large paper clip

Students practice adding and

subtracting fractions and mixed

numbers with like and unlike

denominators.

Ongoing Assessment: Recognizing Student Achievement Use journal page 255. [Operations and Computation Goals 4 and 6]

Math Boxes 8�3Math Journal 2, p. 256

Geometry Template

Students practice and maintain skills

through Math Box problems.

Study Link 8�3Math Masters, p. 224

Students practice and maintain skills

through Study Link activities.

READINESS

Subtracting Mixed NumbersStudents use a counting-up algorithm to

explore mixed-number subtraction.

ENRICHMENTExploring a Pattern for Fraction Addition and SubtractionMath Masters, p. 225

Students explore patterns that result from

adding and subtracting unit fractions.

EXTRA PRACTICE

5-Minute Math5-Minute Math™, pp. 184 and 185

Students add mixed numbers.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

Subtracting Mixed Numbers

Objective To develop subtraction concepts related

to mixed numbers.t

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 142, 143

Common Core State Standards

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Subtracting Mixed NumbersLESSON

8 � 3

Date Time

Math Message

Subtract.

1. 3 3

_ 4 2. 4 4

_ 5 3. 7 5

_ 6

–1 1

_ 4 – 2 – 2 2

_ 6

Renaming and Subtracting Mixed Numbers

Fill in the missing numbers.

4. 5 1

_ 4 = 4 _ 4 5. 6 = 5

_ 3

6. 3 5

_ 6 = _ 6 7. 8

7

_ 9 = 16

_ 9

Subtract. Write your answers in simplest form. Show your work.

8. 8 – 1

_ 3 = 9. 5 – 2 3

_ 5 =

10. 7 1

_ 4 – 3 3

_ 4 = 11. 4 5

_ 8 – 3 7

_ 8 =

12. Isaac would like to practice the violin 7 1 _ 2 hours this week. So far, he has practiced

5 3

_ 4 hours. How many more hours does he need to practice this week?

2 1 _ 2 2 4

_ 5 5 1 _ 2

7 2 _ 3 2 2

_ 5

3 _ 4 3 1

_ 2

1 3 _ 4 hours

5 3

723

248-287_EMCS_S_MJ2_U08_576434.indd 254 2/15/11 2:44 PM

Math Journal 2, p. 254

Student Page

Lesson 8�3 631

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 2, p. 254)

Ask students to share their estimating strategies for Problem 2 using benchmarks. Sample answer: I know that 4 4 _ 5 is close to 5 and 5 – 2 = 3. I estimated my answer to be about 3.

Ask volunteers to write their solutions on the board and explain their strategies. Expect that most students will have subtracted the whole-number and fraction parts separately and renamed the differences in Problems 1 and 3 in simplest form.

▶ Subtracting Mixed Numbers

WHOLE-CLASS ACTIVITY

with RenamingWrite this problem on the board.

3 1 _ 3

- 1 2 _ 3

Ask: How does this problem differ from the Math Message problems? The fraction being subtracted from, 1 _ 3 , is smaller than the fraction that is being subtracted, 2 _ 3 . Ask volunteers to suggest solution strategies. Sample answer: Rename 3 1 _ 3 to make the fraction part larger. 2 4 _ 3 - 1 2 _ 3 = 1 2 _ 3

Remind students that when they add mixed numbers, some sums might have to be renamed to write them in simplest form. For example, a sum of 4 7 _ 4 could be renamed: 4 7 _ 4 = 4 + 4 _ 4 + 3 _ 4 , or 4 + 1 + 3 _ 4 = 5 3 _ 4 . Emphasize that the two mixed numbers are

equivalent; 4 7 _ 4 is another name for 5 3 _ 4 .

Mental Math and Reflexes Have students name a common denominator for the following:

halves and thirds sixths

fifths and thirds fifteenths

fourths and eighths eighths

fifths and eighths fortieths

halves, thirds, and fourths twelfths

halves, fifths, and sixths thirtieths

Math MessageSolve Problems 1–3 at the top of journal page 254. Use benchmarks to estimate the differences and to check the reasonableness of your solutions.

Study Link 8�2 Follow-UpHave partners compare answers and resolve differences. Ask volunteers how they know their answers are in simplest form. A fraction is in simplest form if the numerator and denominator have no common factors except 1.

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Ongoing Assessment: Informing Instruction

Watch for students who have difficulty

renaming mixed numbers for subtraction.

Have students practice the following steps:

1. Write the whole-number part of the

mixed number as an equivalent addition

expression that adds 1.

5 2

_ 5 = 4 + 1 +

2

_ 5

2. Use the denominator to rename the 1 as

a fraction.

5 2

_ 5 = 4 +

5

_ 5 +

2

_ 5

3. Combine the fraction parts.

5 2

_ 5 = 4 +

7

_ 5

632 Unit 8 Fractions and Ratios

Have students rename 3 1 _ 3 to an equivalent mixed number with a larger fraction part. 1 7 _ 3 , or 2 4 _ 3 Write their responses on the board or a transparency, and illustrate the mixed numbers with pictures.

3 1 _ 3 = 1 7

_ 3

3 1 _ 3 = 2 4

_ 3

Ask students to use each of the responses to solve the problem. Circulate and assist. When most students have finished, have volunteers explain which mixed-number name was more efficient for solving the subtraction problem. 2 4 _ 3 , because with 1 7 _ 3 you need to rename the fraction again in order to express the difference as a mixed number. Explain that deciding how to rename a fraction depends on the problem. Sometimes it might be more efficient to work with mixed numbers, other times it might be more efficient to work with improper fractions. In this case, the more efficient mixed-number name is a matter of personal preference.

Pose several problems. Encourage students to estimate each difference using what they know about benchmarks with fractions. They should use their estimates to check the reasonableness of their solutions. In each case, ask students first how they would rename the minuend and then how they would solve the problem. Suggestions:

● 8 - 3 2 _ 3 8 = 7 3 _ 3 ; solution: 4 1 _ 3

● 6 - 1 _ 4 6 = 5 4 _ 4 ; solution: 5 3 _ 4

● 4 3 _ 5 - 1 4 _ 5 4 3 _ 5 = 3 8 _ 5 ; solution: 2 4 _ 5

● 5 1 _ 6 - 2 5 _ 6 5 1 _ 6 = 4 7 _ 6 ; solution: 2 1 _ 3

● 6 5 _ 12 - 3 11 _ 12 6 5 _ 12 = 5 17 _ 12 ; solution: 2 1 _ 2

● Carlie purchased 10 feet of ribbon to make bows. She used 2 7 _ 8 feet to make one bow. How much ribbon remains to make additional bows? 7 1 _ 8 feet

● Sammy needs 3 1 _ 4 cups of sugar to make sugar cookies. He only has 1 3 _ 4 cups of sugar. How much more sugar does he need for the recipe? 1 1 _ 2 cups

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Mixed-Number SpinLESSON

8 �3

Date Time

Materials □ Math Masters, p. 488

□ large paper clip

Players 2

Directions

1. Each player writes his or her name in one of the boxes below.

2. Take turns spinning. When it is your turn, write the fraction or mixed number

you spin in one of the blanks below your name.

3. The first player to complete 10 true sentences is the winner.

Name Name

+

< 3

+

< 3

+

> 3

+

> 3

-

< 1

-

< 1

-

<

1

_ 2

-

<

1

_ 2

+

> 1

+

> 1

+

< 1

+

< 1

+

< 2

+

< 2

-

= 3

-

= 3

-

> 1

-

> 1

+

>

1

_ 2

+

>

1

_ 2

+

< 3

+

< 3

+

> 2

+

> 2

�

248-287_EMCS_S_MJ2_U08_576434.indd 255 2/16/11 1:01 PM

Math Journal 2, p. 255

Student Page

Lesson 8�3 633

Write this problem on the board.

3 1 _ 3

- 1 1 _ 2

Ask: How would you solve this problem? Expect that students will likely recognize 1 _ 3 is smaller than 1 _ 2 and might suggest renaming the fraction parts with common denominators. Ask volunteers to model their strategies on the board. Sample answer: Use 6 as a common denominator. Rename the mixed numbers. 3 2 _ 6 - 1 3 _ 6 Rename the minuend. 3 2 _ 6 = 2 8 _ 6 . Subtract. 2 8 _ 6 - 1 3 _ 6 = 5 _ 6 .

Pose a few mixed-number subtraction number stories. Suggestions:

● Daniel has a DVD set that provides 4 1 _ 2 hours of viewing. So far, he has watched 5 _ 6 of an hour. How many more hours of the DVD set does he still have to watch? 3 4 _ 6 , or 3 2 _ 3 hours

● Serena measured out 5 2 _ 3 feet of rope to make a jump rope. Her friends told her that it needed to be 6 1 _ 2 feet long. How much more rope did she need to measure out? 5 _ 6 ft more

▶ Subtracting Mixed Numbers PARTNER ACTIVITY

(Math Journal 2, p. 254; Math Masters, p. 414)

Ask students to complete the journal page. Remind them to use benchmarks to estimate the difference. Ask students to use an Exit Slip (Math Masters, page 414) to explain their strategy for solving Problem 10 on Math Journal 2, page 254.

2 Ongoing Learning & Practice

▶ Playing Mixed-Number Spin PARTNER ACTIVITY

(Math Journal 2, p. 255; Math Masters, pp. 488 and 489)

Algebraic Thinking Students practice estimating sums and differences of mixed numbers with like and unlike denominators by playing Mixed-Number Spin. Players use benchmarks to estimate sums and differences as they record number expressions that fit the parameters given on the Mixed-Number Spin record sheet, Math Journal 2, page 255. Each partnership makes a spinner using a large paper clip anchored by a pencil point to the center of the spinner on Math Masters, page 488. They use the numbers they spin to complete the number sentences on the record sheet.

Ask students to save their spinners for future use. Copy Math Masters, page 489 to provide additional record sheets.

Journal

Page 255 �

Ongoing Assessment: Recognizing Student Achievement

Use the Mixed-Number Spin record sheet on

journal page 255 to assess students’ ability

to make estimates for adding and subtracting

mixed numbers. Students are making

adequate progress if they correctly complete

number sentences using the benchmarks 1 _ 2

and 1. Some students may be able to make

the estimates and comparisons using 2 and 3

as benchmarks.

[Operations and Computation Goals 4 and 6]

Links to the FutureSubtraction of mixed numbers with unlike

denominators continues in Sixth Grade

Everyday Mathematics.

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Time Spent on Homework

Time Percent of Students

0–29 minutes 25%

30–59 minutes 48%

60–89 minutes 10%

90–119 minutes 12%

2 hours or more 5%

Math Boxes LESSON

8 � 3

Date Time

3. True or False? Write T or F.

a. All rectangles are

parallelograms. T

b. A kite is not

a parallelogram. T

c. Some parallelograms

are squares. T

d. Some trapezoids are

not quadrangles. F

2. Write each numeral in

number-and-word notation.

a. 56,000,000 56 million

b. 423,000 423 thousand

c. 18,000,000,000 18 billion

d. 9,500,000

4. Complete the “What’s My Rule?”

table and state the rule.

5. Find the area of the rectangle.

Area: 72 m2

(unit)

25%

12%

10%

48%5%

Time Spent on Homeworktitle

12 m

6 m

in out

48

40 5

1 1 _ 8

0

16 2

4 146

189

126 127

231 232

Area = b ∗ h

1. Make a circle graph of the survey results.

Rule

95 hundred-

thousand

out = in ÷ 8

6

0

248-287_EMCS_S_MJ2_G5_U08_576434.indd 256 3/16/11 2:29 PM

Math Journal 2, p. 256

Student Page

Fill in the missing numbers.

1. 3 3

_ 8 = 2 —8 2. 4

5

_ 6 = 11

_ 6

3. 2 1

_ 9 = 1 —9 4. 6

3

_ 7 = 10

_ 7

5. 4 3

_ 5 = 3 —5 6. 7

2

_ 3 = —

3

Subtract. Write your answers in simplest form.

7. 5 3

_ 4 8. 6 2

_ 3 9. 5 4

_ 5

- 3 1

_ 4 - 4 1

_ 3 - 3 3

_ 5

10. 4 - 3

_ 8 = 11. 6 - 5

_ 9 =

12. 5 - 2 3

_ 10 = 13. 7 - 4 3

_ 4 =

14. 3 2

_ 5 - 1 3

_ 5 = 15. 4 3

_ 8 - 3 7

_ 8 =

STUDY LINK

8�3 Subtracting Mixed Numbers

71 72

Name Date Time

3

5

56

11

10

8

16. 654 * 205 = 17. 654 * 502 =

18. 654 * 250 = 19. 654 * 520 = 163,500

328,308134,070

340,080

Practice

2 1

_ 2 2 1

_ 3

5 4

_ 9

2 1

_ 4

1

_ 2 1 4

_ 5

2 7

_ 10

3 5

_ 8

2 1

_ 5

221-253_434_EMCS_B_MM_G5_U08_576973.indd 224 2/22/11 4:08 PM

Math Masters, p. 224

Study Link Master

634 Unit 8 Fractions and Ratios

▶ Math Boxes 8�3

INDEPENDENT ACTIVITY

(Math Journal 2, p. 256)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-1. The skill in Problem 5 previews Unit 9 content.

Writing/Reasoning Have students write a response to the following: Explain one advantage and one disadvantage to using number and word notation in Problem 2. Sample

answer: One advantage is that the number is written the same way that it is said aloud. One disadvantage is that you cannot perform operations on them as easily.

Writing/Reasoning Have students write a response to the following: Explain how you determined your answer to Problem 3c. Sample answer: The answer is true because a

parallelogram has two pairs of parallel sides with opposite sides congruent. When all sides of a parallelogram are the same length and form right angles, the parallelogram is also a square.

▶ Study Link 8�3

INDEPENDENT ACTIVITY

(Math Masters, p. 224)

Home Connection Students practice renaming and subtracting mixed numbers.

3 Differentiation Options

READINESS

SMALL-GROUP ACTIVITY

▶ Subtracting Mixed Numbers 15–30 Min

To explore mixed-number subtraction, have students use a counting-up algorithm. Remind students that another way to subtract is to count up. With whole numbers, they could solve 5 - 3 by thinking 3 plus what equals 5? Discuss how they would count up to solve this problem: 5 2 _ 5 - 3 4 _ 5 . Expect that students will

suggest “3 4 _ 5 plus what equals 5 2 _ 5 ?” Explain that counting up is one way to answer plus what?

Demonstrate how to count up from 3 4 _ 5 to 5 2 _ 5 .

1. Count up, keeping track of the amounts used.

Count up 1 _ 5 : 3 4 _ 5 + 1 _ 5 = 4 1 _ 5

Count up 1: 4 + 1 = 5 1

Count up 2 _ 5 : 5 + 2 _ 5 = 5 2 _ 5 2 _ 5

2. Add. 1 _ 5 + 1 + 2 _ 5 = 1 3 _ 5

So the difference between 5 2 _ 5 and 3 4 _ 5 is 1 3 _ 5 .

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Add.

1. a. 1

_ 1 + 1

_ 2 =

b. 1

_ 2 + 1

_ 3 =

c. 1

_ 3 + 1

_ 4 =

d. 1

_ 4 + 1

_ 5 =

e. 1

_ 5 + 1

_ 6 =

2. What pattern do you notice in Problems 1a–1e?

3. Use the pattern above to solve these problems.

a. 1

_ 6 + 1

_ 7 =

b. 1

_ 10 + 1

_ 11 =

c. 1

_ 99 + 1

_ 100 =

4. Do you think this pattern also works for problems like 1

_ 8 + 1

_ 3 ? Explain.

5. The plus signs in Problem 1 have been replaced with minus signs. Find each answer.

a. 1

_ 1 - 1

_ 2 = b. 1

_ 2 - 1

_ 3 =

c. 1

_ 3 - 1

_ 4 =

d. 1

_ 4 - 1

_ 5 =

e. 1

_ 5 - 1

_ 6 =

f. Describe the pattern.

LESSON

8�3

Name Date Time

Addition and Subtraction Patterns

Sample answer:

In each problem, the answer is the sum of the denominators

Sample answer: Yes. This pattern will work whenever two

over the product of the denominators.

unit fractions are added.

21

_ 110

1

_ 6

13

_ 42

1

_ 2

199

_ 9,900

1

_ 12

5

_ 6

11

_ 30

9

_ 20

7

_ 12 1 1

_ 2 , or 3

_ 2

1

_ 20 1

_ 30

Sample answer: The numerator is always 1.

The denominator of the result is the product of the

denominators of the subtracted fractions.

221-253_434_EMCS_B_MM_G5_U08_576973.indd 225 2/22/11 4:08 PM

Math Masters, p. 225

Teaching Master

Lesson 8�3 635

Pose problems for students to solve by counting up. Suggestions:

● 4 2 _ 8 - 1 7 _ 8 2 3 _ 8 ● 7 - 2 8 _ 10 4 1 _ 5

● 3 3 _ 5 - 1 4 _ 5 1 4 _ 5 ● 5 1 _ 4 - 2 3 _ 4 2 1 _ 2

ENRICHMENT PARTNER ACTIVITY

▶ Exploring a Pattern for Fraction 15–30 Min

Addition and Subtraction(Math Masters, p. 225)

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Algebraic Thinking To apply students’ understanding of addition and subtraction of fractions, have them find patterns in unit fraction addition and subtraction problems. When most students have completed the problems, discuss the pattern in Problem 1. In each problem, the answer is the sum of the denominators over the product of the denominators.

Problem 4 asks whether this pattern works for the sum of any two unit fractions 1 _ a + 1 _ b . It does. Add 1 _ a and 1 _ b using the QCD method:

1 _ a + 1 _ b = 1 ∗ b _ a ∗ b + 1 ∗ a _ b ∗ a = b _ a ∗ b + a _ b ∗ a = b + a _ a ∗ b

Have students verify that this pattern holds true for other unit fractions, for example: 1 _ 3 + 1 _ 5 .

The pattern for Problem 5 is similar. The denominator of the result is the product of the denominators of the unit fractions being subtracted. The numerator is always 1.

Problems 5a–5e involve the subtraction of two unit fractions where the difference between the denominators is 1. The pattern described in Problem 5f reflects that type of problem. You may want to extend the Math Master activity by having students find the results of the following subtraction problems (where the difference between the denominators of the unit fractions is greater than 1):

1 _ 2 - 1 _ 5 3 _ 10 1 _ 3 - 1 _ 8 5 _ 24 1 _ 5 - 1 _ 12 7 _ 60

Ask students to describe a pattern for subtracting any two unit fractions. Sample answer: The numerator of the result is found by subtracting the denominator of the first fraction from the denominator of the second fraction. The denominator of the result is the product of the denominators of the two unit fractions.

EXTRA PRACTICE

SMALL-GROUP ACTIVITY

▶ 5-Minute Math 5–15 Min

To offer students more experience with adding mixed numbers, see 5-Minute Math, pages 184 and 185.

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