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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. .
learning style: verbal
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.
learning style: verbal
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.
learning style: verbal
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
phm07c3_te_0202.fm Page 58 Thursday, May 25, 2006 12:22 PM
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.
Write each decimal as a mixed number or fraction in simplest form.
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.
phm07c3_te_0202.fm Page 59 Thursday, May 25, 2006 12:22 PM
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.
Write each decimal as a mixed number or fraction in simplest form.
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
Write each decimal as a mixed number or fraction in simplest form.
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
•
phm07c3_te_0202.fm Page 60 Thursday, May 25, 2006 12:22 PM