2-2 Equivalent Forms of 2-2 Rational Numbers 1....

57

1. Plan

2-2

Objective

To write equivalent fractions and decimals

Examples

1

Simplifying a Fraction

2

Writing an Equivalent Decimal

3

Writing an Equivalent Fraction

Math Understandings:

p. 50C

Math Background

The name for the set of rational numbers comes from the fact that a

ratio

nal number can be expressed as a

ratio

of integers. All decimals are rational if they end or repeat a group of digits forever. They can be written as fractions with integers in each numerator and denominator.

More Math Background:

p. 50C

Lesson Planning and

Resources

See p. 50E for a list of the resources that support this lesson.

Bell Ringer Practice

Check Skills You’ll Need

Use student page, transparency, or PowerPoint. For intervention, direct students to:

Factors

Lesson 2-1Extra Skills and Word Problems

Practice, Ch. 2

Special Needs

Review with students how to divide a greater number into a lesser one before they convert fractions to decimals.

L1

learning style: verbal

Below Level

Have students use long division to divide 1 by 3 2 by 3

;

and 2 by 9 Have students describe the quotients.

Sample: In each example, the same digit repeats forever.

L20 3. ;

0 6. 0 2. .


2-2 Equivalent Forms of Rational Numbers 57

2-2

2 2 5 5��

Equivalent Forms of Rational Numbers

1. Vocabulary Review Write the prime factorization of 100.

Find the GCF of each pair of numbers.

2. 6, 12 3. 8, 12

4. 25, 50 5. 36, 40

Lesson 2-1

Why Learn This?The baseball standings at the right use

both decimals and fractions. Decimals

and fractions are rational numbers.

A rational number is a number that can be

written in the form where a is an integer

and b is any nonzero integer. Two integers

a and b are relatively prime if 1 is their only

common factor. A fraction is in simplest

form when a and b are relatively prime.

Simplifying a Fraction

Write in simplest form using the GCF.

The GCF of 36 and 40 is 4.

Write in simplest form using prime factorization.

1. Write in simplest form using the GCF.

2. Write in simplest form using prime factorization.

PCT GBHoustonSt. LouisMilwaukeeChicago PittsburghCincinnati

.632

.500

.459

.459

.395

.359

WTeam Standings

L241917171514

141920202325

–– 5 6 6 910

12

12

12

ab,

ab

3640

910

d Divide the numerator and denominator by the GCF.

d Simplify.�

3640

36 � 440 � 4

�

5460

d Divide the common factors.

d Simplify.

d Write the prime factorizations of the numerator and denominator.

�

5460

2 � 3 � 3 � 32 � 2 � 3 � 5

2 � 3 � 3 � 3 2 � 2 � 3 � 5

�

� 910

1

1

1

1

1220

2745

6 4

425

12 420 4

35

��

�

3 3 33 3 5

35

��

�

New Vocabulary rational number, relatively prime, terminating

decimal, repeating decimal

To write equivalent fractions and decimals

What You’ll Learn

phm07c3_te_0202.fm Page 57 Thursday, May 25, 2006 12:22 PM

58

Activity Lab

Use before the lesson.

Teaching Resources

Activity Lab 2-2:

Converting Fractions to Decimals

Guided Instruction

Alternative Method

For Example 1, students can also rewrite as , which is

Error Prevention!

Explain that a decimal that goes on forever can have a pattern and still not be a repeating decimal. For instance, in

you can predict the next digit, but there is no single group of digits that repeats.

Additional Examples

Write in simplest form using the GCF.

Write in simplest form using prime factorization.

Write each batting average as a decimal.

a.

Joe made 4 hits in 20 times at bat.

.200

b.

Pat made 6 hits in 33 times at bat.

.182

Write 3.225 as a mixed number.

Teaching Resources

• Daily Notetaking Guide 2-2• Adapted Notetaking 2-2

Closure

•

Explain which types of decimals can be written as the ratio of two integers.

Sample: every terminating decimal and every non-terminating decimal that repeats one digit or the same group of digits

3640

4 94 10

��

44

910

910

910� �or 1 or .

0 121121112. . . .

138150 23

25

60126 10

21

3 940

L3L1

Advanced Learners

Have students write each fraction as a decimal using a bar to indicate repeating digits.

L4

16 0 16.

19 0 1.

111 0 09.


English Language Learners

Review the math meaning of the word

prime

with students before defining

relatively prime.

Students define the word

ratio,

then connect it to the term

rational number.


58 Chapter 2 Rational Numbers

You can represent a rational number as a fraction. You can write a

fraction as a decimal by dividing the numerator by the denominator.

If the division results in a decimal that stops, the decimal is called a

terminating decimal. If the division results in a decimal that repeats the

same digit or group of digits forever, the decimal is a repeating decimal.

A bar indicates the repeating digits. So

Writing an Equivalent Decimal

Baseball In baseball, a player’s batting average is

A batting average is rounded to three decimal places and is written

without the leading 0.

a. Find the batting average of a hitter with 36 hits in 125 times at bat.

The player’s batting average is .288.

b. Find the batting average of a hitter with 27 hits in 99 times at bat.

The player’s batting average is about .273.

3. Find the batting average of a hitter with 39 hits in 85 times at bat.

You can write a terminating decimal as a fraction by multiplying both

the numerator and the denominator by the same power of 10.

Writing an Equivalent Fraction

Write 1.345 as a mixed number in simplest form.

4. Write 1.42 as a mixed number in simplest form.

0 3 0 3333. .� . . .

number of hitsnumber of times at bat.

d Write the batting average as a fraction.

d Divide. This is a terminating decimal.0.288

36125

d Divide. This is a repeating decimal.

d Write the batting average as a fraction.2799

0.27272727 � 0.27

d Write as a fraction with the denominator 1.

d Since there are 3 digits to the right of the decimal, multiply the numerator and the denominator by 1,000.

d Divide the numerator and the denominator by the GCF, 5.

�

1.3451

1,345 1,000

� 1,345 � 5 1,000 � 5

� � 1 269 200

69 200

�1.345

d Simplify. Write as a mixed number.

VVocabulary Tipocabulary TipThe fraction means You can write the fraction as

ab a b� .

ab

a b� .

.459

12150

2. Teach


59

2-2 Equivalent Forms of Rational Numbers 59

A. 0.5

B. 0.4

C. 0.3

D. 0.25

Check Your UnderstandingCheck Your Understanding

1. Vocabulary Since 123 is a rational number, it can be written in the

form

2. Number Sense A player has 15 hits in 34 times at bat and then

gets another hit. Did the batting average increase? Explain.

Match each fraction with its equivalent decimal.

3.

4.

5.

6.

123� .

14

310

1225

Homework ExercisesHomework ExercisesFor more exercises, see Extra Skills and Word Problems.

Write each fraction in simplest form.

7. 8. 9. 10.

11. 12. 13. 14.

Write each fraction as a decimal. Round to three decimal places.

15. 16. 17. 18.

19. 20. 21. 22.

23. Sports A baseball player has 34 hits in 102 times at bat. Another

baseball player has 24 hits in 96 times at bat. Write each player’s

batting average.

Write each decimal as a mixed number or fraction in simplest form.

24. 1.4 25. 0.33 26. 0.24 27. 4.44

28. 2.8 29. 0.05 30. 0.005 31. 7.32

32. Guided Problem Solving At a chili festival over the past few years,

Restaurant A won 56 out of 98 contests. Restaurant B won 84 out of

147 contests. Which restaurant has the better record?

• What fraction represents the wins for Restaurant A?

• What fraction represents the wins for Restaurant B?

33. Population In 2003, 0.219 of the people in the United States were

younger than 15 years old. Write the decimal as a fraction.

1520

4864

�4060

�1254

20100

1881

� 414

1260

23

825

1716

1617

�137

945

513

�2835

For Exercises See Examples

7–14 1 and 2

15–23 3

24–31 4

lesson quiz, PHSchool.com, Web Code: asa-0202

Yes; it increased from .441 to .457.

D

C

A

B

34

15

34

29

�23

�27

�29

15

0.667

�1 857.

0.320

0.200

1.063

0.385

0.941

�0 800.

.333; .250

125

2 45

33100

120

625

1200

4 1125

7 825

They have the same record.

2191,000

1231

Adapted Practice 2-2 L1

Practice 2-2 Equivalent Forms of Rational Numbers

Write each number as a fraction or mixed number in simplest form.

1. �5 2. 0.63 3. �3.9 4.

5. 6. 7. 8.

9. A baseball player averaged 0.375 last season. Express the batting average as a fraction.

Write each fraction or mixed number as a decimal rounded to three places.

10. 11. 12.

13. 14. 15.

16. 17. 18.

19. 20. 21.


22. 0.006 23. �4.8 24. 0.97

25. 0.4 26. 9.05 27. �0.28

28. 3.082 29. �1.41 30. 4.23

31. 8.05 32. �3.02 33. 7.13

Solve.

34. The eighth grade held a magazine sale to raise money for theirspring trip. They wanted each student to sell subscriptions. Afterthe first day of the sale, 25 out of 125 students turned insubscription orders. Write a rational number in simplest form toexpress the student response on the first day.

35. Pete wanted to win the prize for selling the most subscriptions.Of 240 subscriptions sold, Pete sold 30. Write a rational numberin simplest form to express Pete’s part of the total sales.

24 14155

71522

79

21 7183

11824 7

11

3 111224

783

16

2232 9

21721

195105228

5221256

7799

4 56

� �3

� � 167

713

38

79

296

910

63100

51

0.333 �0.429 �0.667

3.167 �4.875 3.917

�4.636 3.056 �1.389

�2.778 5.467 �4.993

�4

9 �

3 �1 4

8 �3 7 13100

150

120

23100

41100

41500

725

120

25

97100

45

3500

15

18

38

L3

2-2 • Guided Problem Solving

Student Page 59, Exercise 33:

Population In 2003, 0.219 of the people in the United States wereyounger than 15 years old. Write the decimal as a fraction.

Understand

1. What are you being asked to do?

Plan and Carry Out

2. Write the fraction with a denominator of one.

3. How many digits are there to the right of the decimal point?

4. Multiply the numerator and denominator by 1,000.

5. Can the fraction be simplified? Why or why not?

Check

6. Is your answer reasonable? Write the decimal in word form.

Solve Another Problem

7. A group of teenagers is surveyed about their preference for musicperformers. Of the teenagers surveyed, 0.275 preferred individualartists. Express the decimal as a fraction.

GPS

Write 0.219 as a fraction.

1140

The fraction cannot be simplified, since 219

and 1,000 have no common factors.

Yes; 0.219 is read as two hundred nineteen

thousandths, which is what was written as a

fraction.

3

2191,000

0.2191

L3

3. Practice

Assignment Guide

Check Your Understanding

Go over Exercises 1–6 in class before assigning the Homework Exercises.

Homework Exercises

A

Practice by Example 7–31

B

Apply Your Skills 32–37

C

Challenge 38Test Prep and

Mixed Review 39–45

Homework Quick Check

To check students’ understanding of key skills and concepts, go over Exercises 16, 30, 33, 34, and 37.


60

60 Chapter 2 Rational Numbers

34. The circle graph at the right

shows the sizes of American

households. Write a decimal for

the fraction of households in

each category.

35. Algebra

Evaluate for

and Write your

answer in simplest form.

Math in the Media Refer to the cartoon below.

36. If Leroy Lockhorn missed one

of Loretta Lockhorn’s birthdays

in 25 years, what would his

“batting average” be?

37. Writing in Math

Explain why,

in 25 years of marriage, Leroy

Lockhorn could never have a

“batting average” of .980.

38. Challenge The number 77 is what fractional part of 7,777?

Test Prep and Mixed ReviewTest Prep and Mixed Review Practice

39. Tyler reads for 3 hours each day. There are 24 hours in a day. What

fraction of each day does he spend reading?

40. Which list shows the numbers in order from least to greatest?

41. Brandon simplified an expression as shown below.

Step 1:Step 2:Step 3:Step 4:Step 5: 0

In which step did Brandon make his first mistake?

Step 1 Step 2 Step 3 Step 4

Simplify each expression.

42. 43. 44. 45.

Oneperson

Twopeople

Threepeople

Four ormore people

Households by Size

SOURCE: U.S. Census Bureau

�14

�14

�16�

131

2� ab

a � 3 b � �5.

18

121

127

172

1 5 11 3, , ,� � � �3, 11, 1, 5

1 3 5 11, , ,� � � �11 3 1 5, , ,

4 16 4 2 6 2 1� � � � � �

4 16 8 6 2 1� � � � �

4 16 8 6 1� � � �

4 16 8 6� � �

4 2 6� �

� � �19 6( ) � �11 20 � �25 25 19 15� �( )

nline

Visit: PHSchool.comWeb Code: ase-0202

Multiple Choice

34. 0 25. ; 0.3; 0.16; 0.25

�25

.960

37. Answers may vary. Sample: He cannot forget just of a birthday.1

2

1101

A

J

A

�25 9 �50 34

For Exercises See Lesson

42–45 1-3

A fraction is in simplest form when the greatest common factor(GCF) of the numerator and denominator is 1.

Example 1: Write in simplest form.

Use prime factorization and circle the common factors.

24 � 2 � 2 � 2 � 336 � 2 � 2 � 3 � 3

So, .

Write each fraction in simplest form.

1. 2. 3.

4. 5. 6.

Write each fraction or mixed number as a decimal rounded to three decimal places.

7. 8. 9.

10. 11. 12.


13. 0.2 14. 0.16 15. 0.3

16. 2.75 17. 4.52 18. 0.36

111

43537

592327

79

1827

12228

23240

4263

23048

1664

2436 5

23

2436

Reteaching 2-2 Equivalent Forms of Rational Numbers

To write a fraction as a decimal:

Divide numerator by denominator.

Divide until the remainder is 0 or until theremainder repeats.

Use a bar to show digits repeating.

Example 2: Write as a decimal.

; Remainder repeats.

So = 0.833. . . , or .

To write a decimal as a fraction:

Example 3: Write 0.375 as a fraction.

Write as a fraction 0.375 �

with the denominator 1.

Since there are 3 digits �

to the right of the decimal point, multiplythe numerator and thedenominator by 1,000.

Divide the numerator �

and denominator bythe GCF.

Simplify. �

So 0.375 � .38

384

375 � 1251000 � 1253

37510002

0.37511

0.83356

56

3

2

1

-4820-18

20-18

2

0.8336q5.000

0.778 �3.286

14

2 4 925

1325

34

310

425

15

�45 �3

723

�58

23

0.556

0.0915.429 1.333

L2

Place each number shown below in the correct column(s) in the table.Be careful: Some numbers might go into more than one column.

, , 4, �50, � , , 2 , � , 0.36, �1, 16, 0.8, 0.3, 2.7, � , 25, �

1.

Whole Numbers Integers Rational Numbers

4 4 4

16 16 16

25 25 25 �

�1 �1

�50 �50 2

0.36 �3

0.8 �3

0.3 �

2.7

2. There are 6 pairs of equivalent rational numbers in the table. Listthem below in order from smallest to largest.

a.

b.

c.

d.

e.

f.

2505

810

45

710

620

12

925

45

25053 8

10345710

620

12

925

45

Enrichment 2-2 Equivalent Forms of Rational Numbers

Critical Thinking

� , �50

� , �

, 0.3

, 0.36

, 0.8

2 , 2.7710

45

925

620

3 81034

5

2505

L4

Alternative Assessment

In small groups, students choose a digit from 1–9. Members use the digits to form a decimal to 3 places, such as 15.124. Then students work together to write the decimal as a fraction in simplest form. 15 31

250

4. Assess & Reteach

Lesson Quiz

Write each as a fraction in simplest form.

1.

2.

3.

Write as a decimal.

0.125


4.

2.75

5.

0.4

3042

57

�1218 �

23

216

234

25

Test Prep

Resources

For additional practice with a variety of test item formats:• Test-Taking Strategies, p. 97• Test Prep, p. 101• Test-Taking Strategies with Transparencies

•


2-2 Equivalent Forms of 2-2 Rational Numbers 1....

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