The Number System
Chapter 4
Essential Question
Rational Numbers
WHAT happens when you add, subtract, multiply, and divide fractions?
Common Core State Standards Content Standards 7.NS.1, 7.NS.1b, 7.NS.1c, 7.NS.1d, 7.NS.2, 7.NS.2a, 7.NS.2b, 7.NS.2c, 7.NS.2d, 7.NS.3, 7.RP.3, 7.EE.3
Mathematical Practices 1, 3, 4, 5, 6, 7, 8
Math in the Real World Tennis T i About 70,000 tennis
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balls are used at the U.S. Open tennis tournament each year. This is only a small fraction of the 300,000,000 tennis balls produced each year. Write a fraction in simplest form that compares the number of tennis balls used at the U.S. Open to the number produced per year.
ldable on Cut out the Fo is book. page FL9 of th
Place your Fold able on page 338.
Use the Foldable apter throughout this ch about to help you learn rational numbers.
257
What Tools Do You Need? Vocab
Vocabulary bar notation
like fractions
terminating decimal
common denominator
rational numbers
unlike fractions
least common denominator
repeating decimal
Review Vocabulary An improper fraction is a fraction in which the numerator is greater than or equal to the 21 denominator, such as _ . A mixed number is a number composed of a whole number 4
1 . and a fraction, such as 5_ 4
In the organizer below, write each mixed number as an improper fraction and each improper fraction as a mixed number. The first one in each column is done for you.
Mixed Numbers and Improper Fractions Change Mixed Numbers
1
7
3_ =_ 2 2
Change Improper Fractions
1 41 _ = 10_ 4
1 3
16 _ =
8_ 5=
2
23 _ =
4 6_ = 9
90 _ =
3 10_ = 8
66 = _
3 7_ = 4
101 _ =
5 15_ = 6
87 _ =
5_ =
3 5
11 7
2
20
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258 Chapter 4 Rational Numbers
4
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When Will You Y Use U This?
Your Turn! connectED.mcgraw-hill.com
You will solve this problem in the chapter. 259
Are Yo Y You ou R Rea Ready? ea ea Quick Q uick k Review
Try the Quick Check below. Or, take the Online Readiness Quiz.
Check
Common Core Review 5.NF.3, 6.NS.6c
Example 1
Example 2
Write
Graph 3 2 on a number line.
25 _ in simplest form.
_
100
3
÷25
Find the two whole numbers between
25 1 _ = _ 4 100
2 which 3_ lies.
Divide the numerator and denominator by the GCF, 25.
3
2 3 < 3_
_7 8
3. 4. 5. 9.
_9
8 13 _ 8 15 _ 8
>
< or >
7
_3
–_ 8
8
_5 8
_3
8 13 _ 8
9 6. –_ 8 13 7. –_ 8 15 8. –_ 8
<
3 –_ 8 5 –_ 8 3 –_ 8 13
–_ 8
Identify Repeated Reasoning Compare and contrast the information in the tables.
Create On Your Own
10.
3 Use Math Tools Graph –_ and _ on a number line. Then use the graph 4 4 to explain how the representations of the two fractions differ.
11.
HOW can you graph negative fractions on the number line?
3
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262 Chapter 4 Rational Numbers
Thee Nu Th Number Num mber Sy System ste tem
Lesson 1
Terminating and Repeating Decimals What You'll Learn
Essential Question
Scan the lesson. List two headings you would use to make an outline of the lesson. •
WHAT happens when you add, subtract, multiply, and divide fractions? Vocab
Vocabulary
•
Vocab
Vocabulary Start-Up
Any fraction can be expressed as a decimal by dividing the numerator by the denominator.
repeating decimal bar notation terminating decimal
Common Core State Standards Content Standards 7.NS.2, 7.NS.2d, 7.EE.3
The decimal form of a fraction i`s called a repeating decimal. Repeating decimals can be represented using bar notation. In bar notation, a bar is drawn only over the digit(s) that repeat. − 0.3333… = 0.3
−− 0.1212… = 0.12
Mathematical Practices 1, 3, 4, 6, 7
−− 11.38585… = 11.385
If the repeating digit is zero, the decimal is a terminating decimal. − The terminating decimal 0.250 is typically written as 0.25. Match each repeating decimal to the correct bar notation. − 0.1111… 0.61 0.61111…
− 0.1
0.616161…
−− 0.61
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Real-World Link Jamie had two hits on her first nine times at bat. To find her batting “average,” she divided 2 by 9. 2 ÷ 9 = 0.2222… Write 0.2222… using bar notation. Round 0.2222… to the nearest thousandth.
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Lesson 1 Terminating and Repeating Decimals
263
Work Zone
Write Fractions as Decimals Our decimal system is based on powers of 10 such as 10, 100, and 1,000. If the denominator of a fraction is a power of 10, you can use place value to write the fraction as a decimal. Complete the table below. Write fractions in simplest form. Words
seven tenths
Fraction
Decimal
7 _
0.7
10
nineteen hundredths one-hundred five thousandths If the denominator of a fraction is a factor of 10, 100, 1,000, or any greater power of ten, you can use mental math and place value. Tutor
Examples Write each fraction or mixed number as a decimal.
1.
74 _ 100
Use place value to write the equivalent decimal. 74 _ = 0.74
74 Read _ as seventy-four hundredths.
100
100
74 So, _ = 0.74. 100
2.
_7
3.
20
×5 Show your work.
a.
Think
4
3 3 =5+_ 5_
35 7 _ =_ 20
_3
5
4
100
×5
7 = 0.35. So, _ 20
4
Think of it as a sum.
= 5 + 0.75
3 You know that _ = 0.75.
= 5.75
Add mentally.
4
3 So, 5_ = 5.75. 4
b.
Got It? 3 a. _ 10
264 Chapter 4 Rational Numbers
3 b. _ 25
1 c. -6_ 2
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c.
Do these problems to find out.
The Number System Tutor
Examples 4.
Write
_3 as a decimal. 8
0.375 8 3.000 - 24 60 - 56 40 - 40 0
Divide 3 by 8.
Division ends when the remainder is 0.
3 So, _ = 0.375. 8
5.
_1 as a decimal.
Write -
40
0.025 40 1.000 - 80 200 - 200 0
Divide 1 by 40.
So, -_ = -0.025. 1 40
6.
_
Write 7 as a decimal. 9
0.777… 9 7.000 - 63 70 - 63 70 - 63 7
Divide 7 by 9.
Notice that the division will never terminate in zero.
Show your . work
− 7 So, _ = 0.777… or 0.7.
d.
9
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Got It?
Do these problems to find out.
e.
Write each fraction or mixed number as a decimal. Use bar notation if needed. 7 d. -_ 8 3 f. -_ 11
1 e. 2_ 8 _ g. 8 1 3
f. g.
Lesson 1 Terminating and Repeating Decimals
265
Write Decimals as Fractions Every terminating decimal can be written as a fraction with a denominator of 10, 100, 1,000, or a greater power of ten. Use the place value of the final digit as the denominator. STOP
Watch Tutor
an d Re fl ec t
Example
h Suppose 0.6 of the fis is are goldfish. Write th in n tio decimal as a frac the space below.
7.
Find the fraction of the fish in the aquarium that are goldfish. Write in simplest form. 15 0.15 = _
Show your work.
h.
Got It?
i.
The digit 5 is in the hundredths place.
100 3 =_ Simplify. 20 3 So, _ of the fish are goldfish. 20
Fish
Amount
Angelfish
0.4
Goldfish
0.15
Guppy
0.25
Molly
0.2
Do these problems to find out.
Determine the fraction of the aquarium made up by each fish. Write the answer in simplest form. h. molly
j.
i. guppy
j. angelfish
Check
Guided Practice Write each fraction or mixed number as a decimal. Use bar notation if needed. (Examples 1–6) 2 1. _ = Show your . work
5
2. -_ = 9 10
4. During a hockey game, an ice resurfacer travels 0.75 mile. What fraction represents this distance? (Example 7)
5 3. _ = 9
Rate Yourself! Are you ready to move on? Shade the section that applies.
5.
For more help, go online to access a Personal Tutor.
266 Chapter 4 Rational Numbers
Tutor
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Building on the Essential Question How can you write a fraction as a decimal?
Name
My Homework eHelp
Independent Practice
Go online for Step-by-Step Solutions
Write each fraction or mixed number as a decimal. Use bar notation if needed. (Examples 1–6) 1 1. _ =
4 2. -4_ =
1 = 3 _ 8
3 4. _ =
5. -_ =
6. -_ =
7 7. 5_ =
3 8. 9_ =
9. -_ =
10. -_ =
11. -_ = 11
6 12. 2_ =
2
25
16
Show your . work
33 50
8 9
17 40
1 6
8
8
8
11
Write each decimal as a fraction or mixed number in simplest form.
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13. -0.2 =
14. 0.55 =
(Example 7)
15. 5.96 =
16. The screen on Brianna’s new phone is 2.85 centimeters long. What mixed number represents the length of the phone screen? (Example 7)
17
A praying mantis is an interesting insect that can rotate its head 180 degrees. Suppose the praying mantis at the right is 10.5 centimeters long. What mixed number represents this length? (Example 7)
Lesson 1 Terminating and Repeating Decimals
267
18.
Persevere with Problems Suppose you buy a 1.25-pound package of ham at $5.20 per pound. a. What fraction of a pound did you buy?
b. How much money did you spend?
H.O.T. Problems
Higher Order Thinking
19.
Identify Structure Write a fraction that is equivalent to a terminating decimal between 0.5 and 0.75.
20.
Persevere with Problems Fractions in simplest form that have denominators of 2, 4, 8, 16, and 32 produce terminating decimals. Fractions with denominators of 6, 12, 18, and 24 produce repeating decimals. What causes the difference? Explain.
21.
Persevere with Problems The value of pi (π) is 3.1415926… . The 10 1 mathematician Archimedes believed that π was between 3_ and 3_ . 7
71
Was Archimedes correct? Explain your reasoning.
Reason Inductively A unit fraction is a fraction that has 1 as its numerator. Write the four greatest unit fractions that are repeating decimals. Then write each fraction as a decimal.
23.
Model with Mathematics Write a real-world scenario in which it would be appropriate to write a value in fractional form.
268 Chapter 4 Rational Numbers
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22.
Name
My Homework
Extra Practice Write each fraction or mixed number as a decimal. Use bar notation if needed. 4 24. _ = 0.8 5
26. -_ =
1 25. -7_ =
1 27. 5_ =
4 9
20
3
x2 8 _4 = _
Homework Help
5
10
x2 4 So, _ 5 = 0.8.
12 . Write this 28. The fraction of a dime that is made up of copper is _ 16 fraction as a decimal.
Write each decimal as a fraction or mixed number in simplest form. 29. -0.9 =
30. 0.34 =
31. 2.66 =
Write each of the following as an improper fraction.
35.
1 33. 7_ = 3
34. -3.2 =
Be Precise Nicolás practiced playing the cello for 2 hours and 18 minutes. Write the time Nicolás spent practicing as a decimal.
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32. -13 =
Lesson 1 Terminating and Repeating Decimals
269
for Smarter Balanced 36. The table shows the lengths of four hiking trails. Select the appropriate decimal equivalent of each trail length. Hiking Trail
1.25 − 1. 6
1.6
− 1. 3
1.3 1.75
Trail Length Decimal Equivalents
Lakeview
1_ 4
Forest Lane
1_ 3
Sparrow Stroll
3 1_
Mountain Climb
1.2
1 1
10 2 1_ 3
37. Zoe went to lunch with a friend. After tax, her bill was $12.05. Which of the following rational numbers is equivalent to this amount? Select all that apply. 1 12_
25 _
20
241 _
2
5 12_
20
100
Common Core Spiral Review Round each decimal to the tenths place. 38. 5.69 ≈
5.NBT.4
39. 0.05 ≈
Graph and label each fraction on the number line below. 1 41. _
40. 98.99 ≈
6.NS.6
3 42. _ 4
2
0
44. The table shows the discount on athletic shoes at two stores selling sporting equipment. Which store is offering the greater discount? Explain. 6.NS.7
2 43. _ 3
1
Store
Discount
Good Sports
_1
Go Time
25%
5
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270
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Thee Nu Th Number Num mber Sy System ste tem
Lesson 2
Compare and Order Rational Numbers What You'll Learn
Essential Question
Scan the lesson. Write the definitions of common denominator and least common denominator (LCD). • Common Denominator
WHAT happens when you add, subtract, multiply, and divide fractions? Vocab
Vocabulary
• Least Common Denominator
Vocab
Vocabulary Start-Up
A rational number is a number that can be expressed as a ratio of two integers written as a fraction, in which the denominator is not zero. The Venn diagram below shows that the number 2 can be called many things. It is a whole number, integer, and rational number. The number -1.4444… is only a rational number.
rational number common denominator least common denominator
Common Core State Standards Content Standards 7.NS.2, 7.NS.2b, 7.EE.3
Mathematical Practices 1, 3, 4
Common fractions, terminating and repeating decimals, percents, and integers are all rational numbers. Write the numbers from the number bank on the diagram. agram. g Rational Numbers Integers 20%
–3
Whole Numbers
Number B a 1 2
2
nk
0.8 2.2− -1 1 13__2
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–1.4444...
Real-World Link Not all numbers are rational numbers. The Greek letter π (pi) represents the nonterminating and nonrepeating number whose first few digits are 3.14… . This number is an irrational number. Use the Internet to search for the digits of pi. Describe what you find.
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Lesson 2 Compare and Order Rational Numbers
271
Work Zone
Compare Rational Numbers A common denominator is a common multiple of the denominators of two or more fractions. The least common denominator or LCD is the LCM or least common multiple of the denominators. You can use the LCD to compare fractions. You can also use a number line. Tutor
Example 1.
_5
with , or = to make -1
Fill in the sentence.
_
-1 1 a true
6
6
Graph each rational number on a number line. 1 Mark off equal-size increments of _ between -2 and -1. 6
-1 _5
-2
6
-1 _4
-1 _3
6
-1 _2
6
-1 _1
6
-1
6
5 1 The number line shows that -1_ < -1_ . 6
Show your work.
Got It?
6
Do this problem to find out.
5 1 a. Use the number line to compare -5_ and -5_ . 9
a.
9
-6
-5 Tutor
Example 2.
Fill in the sentence.
_
with , or = to make 7
12
_8 a true 18
The LCD of the denominators 12 and 18 is 36. 7×3 7 _ =_
8 8×2 _ =_ 12 × 3 18 18 × 2 16 21 =_ =_ 36 36 8 21 _ 7 Since _ > 16 , _ >_ . 36 36 12 18 12
5 b. _ 6
272 Chapter 4 Rational Numbers
Do these problems to find out.
_7 9
1 c. _ 5
7 _ 50
d. -_ 9 16
-_ 7 10
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Got It?
The Number System
Tutor
Example 3.
In Mr. Huang’s class, 20% of students own roller shoes. In Mrs. Trevino’s class, 5 out of 29 students own roller shoes. In which class does a greater fraction of students own roller shoes? Express each number as a decimal and then compare. 20% = 0.2
5 _ = 5 ÷ 29 ≈ 0.1724 29
5 Since 0.2 > 0.1724, 20% > _ . 29
More students in Mr. Huang’s class own roller shoes.
Got It?
Show your . work
Do this problem to find out.
e.
e. In a second period class, 37.5% of students like to bowl. In a fifth period class, 12 out of 29 students like to bowl. In which class does a greater fraction of the students like to bowl?
Order Rational Numbers You can order rational numbers using place value. Tutor
Example 4.
− Order the set {3.44, π, 3.14, 3.4} from least to greatest. Line up the decimal points and compare using place value. 3.140
Annex a zero.
3.440
3.1415926…
π ≈ 3.1415926…
3.444… 3.−4 = 3.444…
Annex a zero.
− Since 0 < 4, 3.44 < 3.4.
Since 0 < 1, 3.14 < π.
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So, the order of the numbers from least to greatest is − 3.14, π, 3.44, and 3.4.
Got It?
Do this problem to find out.
1 _ f. Order the set {23%, 0.21, _ , 1 } from least to greatest. 4 5
Show your . work
f.
Lesson 2 Compare and Order Rational Numbers
273
Tutor
Example 5.
Nolan is the quarterback on the football team. He completed
_3
67% of his passes in the first game. He completed 0.64, , 5
and 69% of his passes in the next three games. List Nolan’s completed passing numbers from least to greatest. Express each number as a decimal and then compare. 3 67% = 0.67 0.64 = 64% _ = 0.6 69% = 0.69 5 Nolan’s completed passing numbers from least to greatest are
_3 , 0.64, 67%, and 69%. 5
Check
Guided Practice Fill in each if necessary. 1. -_ 5 4
Show your . work
with , or = to make a true sentence. Use a number line (Examples 1 and 2)
-_ 5
3 2. 1_
1
-1
5 1_
4
0
8
1
2
3. Elliot and Shanna are both soccer goalies. Elliot saves 3 goals out of 4. Shanna saves 7 goals out of 11. Who has the better average, Elliot or Shanna? Explain. (Example 3)
1 4. The lengths of four insects are 0.02 inch, _ inch, 8
2 0.1 inch, and _ inch. List the lengths in inches from 3 least to greatest. (Examples 4 and 5)
Rate Yourself! I understand how to compare and order rational numbers.
5.
Building on the Essential Question How can you compare two fractions?
Go online to access a Personal Tutor.
274 Chapter 4 Rational Numbers
Tutor
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I still have some questions about comparing and ordering rational numbers.
Darrin Klimek/Digital Vision/Getty Images
Great! You're ready to move on!
Name
My Homework eHelp
Independent Practice
Go online for Step-by-Step Solutions
Fill in each with , or = to make a true sentence. Use a number line if necessary. (Examples 1 and 2) 1. -_ 5 3
-_ 5 4
5 2. -7_
1 -7_
4. -_
-_
8
8
Show your . work
2 3. 6_ 3
1 6_
17 24
2
11 12
5 On her first quiz in social studies, Meg answered 92% of the questions correctly. On her second quiz, she answered 27 out of 30 questions correctly. On which quiz did Meg have the better score? (Example 3)
Order each set of numbers from least to greatest.
(Example 4 )
7. {-0.615, -_, -0.62}
1 6. {0.23, 19%, _ }
5 8
5
8. Liberty Middle School is holding a fundraiser. The sixth-graders have raised 52% of their goal amount. The seventh- and eighth-graders 2 have raised 0.57 and _ of their goal amounts, respectively. List the 5
classes in order from least to greatest of their goal amounts. (Example 5)
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Fill in each 7 gallons 9 1_ 12
with , or = to make a true sentence. 5 1_ gallons 8
5 10. 2_ hours 6
2.8 hours
Lesson 2 Compare and Order Rational Numbers
275
Model with Mathematics Refer to the graphic novel frame below. If
11.
1 the closet organizer has a total width of 69_ inches and the closet is 8
3 inches wide, will the organizer fit? Explain. 69_ 4
H.O.T. Problems 12.
Higher Order Thinking
Justify Conclusions Identify the ratio that does not have the same value as the other three. Explain your reasoning.
15 Persevere with Problems Explain how you know which number, 1_ ,
13.
16
17 63 _ , or _ , is closest to 2. 8
14.
32
_5 _5
5 5 Reason Inductively Are the fractions , _ , , and _ arranged in order 6 7 8
9
from least to greatest or from greatest to least? Explain.
Model with Mathematics Write a real-world problem in which you would compare and order rational numbers. Then solve the problem.
276 Chapter 4 Rational Numbers
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15.
Name
My Homework
Extra Practice Fill in each with , or = to make a true sentence. Use a number line if necessary. 16. -_ -_ 7 < 7 5
2 17. -3_
2
3
Mark off equal-size increments of _ 7 between –1 and 0.
Homework Help
4 -3_ 6
1
-1
-6 -5 -4 -3 -2 -1 7 7 7 7 7 7
0
4 _5 18. _ 7 < 8
3 19. 2_ 4
2 2_ 3
The LCD of the denominators 7 and 8 is 56. 5×7 4×8 32 35 5 _4 = _ =_ and _ =_ =_ 7 7×8 8 8×7 56 56 32 35 4 5 to make a true sentence.
Fill in each 26. -2
2
27. -4
29. -7
-8
30. -10
6.NS.7
-5
-1
32. Victoria, Cooper, and Diego are reading the same book for their language arts class. The table shows the fraction of the book each student has read. Which student has read the least amount? Explain your reasoning. 6.NS.7
28. -20
31. 50
Student
20
-100
Amount Read
Victoria
_2
Cooper
_1
Diego
_3
5 5
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5
The Nu The Number um mb ber Sys System tem
Inquiry Lab Add and Subtract on the Number Line Content Standards
HOW can you use a number line to add and subtract like fractions?
7.NS.1, 7.NS.1b, 7.NS.3
Mathematical Practices
In eight times at bat, Max hit 2 doubles, 5 singles, and struck out 1 time. Find the fraction of the times that Max hit either a single or a double.
1, 3, 5
Hands-On Activity 1 Step 1
Since there were 8 times at bat, create a vertical line that is divided into eighths.
number 1
0
Step 2
2 Graph the fraction of doubles, _ , on the 8 number line.
1
2 8
0
Copyright © McGraw-Hill Education
Comstock Images/Alamy
Step 3
5 2 From the _ point, count _ more on the
1
number line.
7 8
8
2 _ +5= So, _ 8
8
8
5 8
.
2 8
Max got a hit
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of the times he was at bat.
0
Inquiry Lab Add and Subtract on the Number Line
279
Hands-On Activity 2 Find
_3 - _4 . 6
6
Step 1
Divide a number line into sixths. Since we do not know if our answer is negative or positive, include fractions to the left and to the right of zero.
-2 -1
0
1 6
2 6
3 6
4 6
5 6
1
-2 -1
0
1 6
2 6
3 6
4 6
5 6
1
2 6
3 6
4 6
5 6
1
6
6
3 Graph _ on the number line.
Step 2
6
6
Step 3
Move 4 units to the
6
4 6
to
4 show taking away _ .
3 6
6
3 _ So, _ -4= 6
6
-2 -1
.
6
6
0
1 6
Hands-On Activity 3
_4 _
Find - - 2 . Fill in the missing numbers in the diagram below. 7
7
1 7
0 -1 7
-2 7
-3 7
-4 7
-5 7
-6 7
So, -_ -_ = 7 7 4
2
.
280 Chapter 4 Rational Numbers
Copyright © McGraw-Hill Education
-1
The Number System
Investigate
Collaborate
Work with a partner. Use a number line to add or subtract. Write in simplest form. 1 _ 1. _ +2= 5
5
Show your . work
0
2. - _ + -_ = 7 7 3
( 1)
0
5 3. - _ + _ = 3 8
8
0
8 4 4. _ -_ = 12
12
0
5 5. - _ + _ = 4 9
4 _ 6. _ -6= 7
9
7
0
Copyright © McGraw-Hill Education
0
Inquiry Lab Add and Subtract on the Number Line
281
Collaborate
Analyze and Reflect
Use Math Tools Work with a partner to complete the table. The first one is done for you. Expression
Use only the Numerators
Use a number line to add or subtract the fractions. 1 6
1 5 –_ – –_ 6 6
( )
6
–5 – (–1) = –4 1 -5-4 -3 -2 -1 6
5 _ –_ – 6 6 1
7.
-5
6
6
6
6
0
1 6
2 6
–5 – 1 = –6 0
_5 – _1
8.
6
6
5–1=4 0
5 _ –_ + 6 6 1
9.
–5 + 1 = –4 0
Create On Your Own
10.
11.
Reason Inductively Refer to the table above. Compare your results for using only the numerators with your results for using a number line. Write a rule for adding and subtracting like fractions.
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282 Chapter 4 Rational Numbers
Thee Nu Th Number Num mber Sy System ste tem
Lesson 3
Add and Subtract Like Fractions What You'll Learn
Essential Question
Scan the lesson. List two real-world scenarios in which you would add or subtract like fractions.
WHAT happens when you add, subtract, multiply, and divide fractions?
•
Vocab
Vocabulary
• like fractions
Common Core State Standards
Real-World Link
Content Standards
Shoes Sean surveyed ten classmates to find which type of tennis shoe they like to wear.
1. What fraction of students liked to wear cross trainers?
Shoe Type
Number
Cross Trainer
5
Running
3
High Top
2
7.NS.1, 7.NS.1c, 7.NS.1d, 7.NS.3, 7.EE.3
Mathematical Practices 1, 3, 4, 7
Number of students that wear cross trainers. Total number of students surveyed.
2. What fraction of students liked to wear high tops? Number of students that wear high tops. Total number of students surveyed.
3. What fraction of students liked to wear either cross trainers or high tops?
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Mark Andersen/Rubberball/Getty Images
Fraction of students that Fraction of students wear cross trainers. that wear high tops.
+
=
So, of the students liked to wear either cross trainers or high tops. 3 2 4. Explain how to find _ +_ . Then find the sum. 10
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10
Lesson 3 Add and Subtract Like Fractions
283
Key Concept
Add and Subtract Like Fractions Words
To add or subtract like fractions, add or subtract the numerators and write the result over the denominator.
Work Zone Examples
Numbers 5 5+2 2 7 _ +_ =_ or _
10 10 10 10 11 _ 11 - 4 7 _ - 4 =_ or _ 12 12 12 12
Algebra a+b _a + _b = _ , where c ≠ 0 c
c
c
a-b _a - _b = _ , where c ≠ 0 c
c
c
Fractions that have the same denominators are called like fractions. Tutor
Examples Add. Write in simplest form.
1.
_5 + _2
9 9 5+2 _5 + _2 = _ 9 9 9 7 _ = 9
1
Add the numerators.
7 9 2 9
Simplify.
8 9 6 9
5 9 4 9 5 9
3 9 2 9 1 9
0
2.
_3 ( _1 ) 5 5 3 1 -3 -1 _ _ - 5 + (- 5 ) = _ + (_ 5 5 ) - + -
Show your work.
-3 + (-1) 5 4 -4 _ = or -_ 5 5
=_
a. b.
d. 284 Chapter 4 Rational Numbers
Use the rules for adding integers.
Do these problems to find out.
1 _ a. _ +2
1 b. -_ +_ 7
2 2 c. -_ + -_ 5 ( 5)
1 1 d. -_ + _
3
3
3
7
4
4
Copyright © McGraw-Hill Education
c.
Got It?
Add the numerators.
The Number System Tutor
Example 3.
Sofia ate
_3 of a cheese pizza. Jack ate _1 of a cheese pizza 5
5
2 and _ of a pepperoni pizza. How much pizza did Sofia and Jack 5
eat altogether?
_3 + (_1 + _2 ) = _3 + (_2 + _1 )
Commutative Property of Addition
5 5 5 3 _ 1 2 _ = + +_ Associative Property of Addition 5 5 5 1 1 =1+_ or 1_ Simplify. 5 5 1 So, Sofia and Jack ate 1_ pizzas altogether. 5 5
5
5
)
(
Got It?
Show your . work
Do this problem to find out.
1 e. Eduardo used fabric to make three costumes. He used _ yard 4
3 2 for the first, _ yard for the second, and _ yard for the third 4
e.
4
costume. How much fabric did Eduardo use altogether?
Tutor
Examples 4.
_5 _3
Find - - .
8 8 5 3 5 3 -_ - _ = -_ + -_ 8 8 8 8 -5 + (-3) =_ 8 8 _ = - or -1 8
( )
5.
Find
Add -_. 3 8
Add the numerators. Simplify.
_5 - _7 .
8 8 5 -7 7 5 _-_=_ 8 8 8 2 1 = -_ or -_ 4 8
7 8 5 8
Subtract the numerators. Simplify.
-2 -1 0 1 2 3 4 5 6 8
Got It? Copyright © McGraw-Hill Education
8
8 8
9
g. -_ - _ 5 9
2 9
8
8
f.
Do these problems to find out.
5 _ f. _ -2 9
8
8
h. -_ - -_ 11 12
(
5 12
)
g. h.
Lesson 3 Add and Subtract Like Fractions
285
Choose an Operation You can add or subtract like fractions to solve real-world problems.
an d Re fl ec t
STOP
rd or In Example 6, what wo u yo words indicate that ve should subtract to sol ur yo the problem? Write answer below.
Tutor
Example 6.
6 _ of the population of the United States lives in 100 4 Florida. Another _ lives in Ohio. About what fraction more
About
100
of the U.S. population lives in Florida than in Ohio? 6 6-4 4 _ -_ =_ 100
100
100 1 _ = 2 or _ 100 50
Subtract the numerators. Simplify.
1 About _ more of the U.S. population lives in Florida than 50
in Ohio.
Check
Guided Practice Add or subtract. Write in simplest form.
(Examples 1–5)
3 _ 1. _ +1=
2 _ 2. _ +1=
4. -_ - -_ = 5 5
1 5 5. _ - -_ = 14 14
5
5
4
( 1)
7
5 _ 3 3. _ + 1 +_ =
(8
7
(
)
7. Of the 50 states in the United States, 14 have an Atlantic Ocean coastline and 5 have a Pacific Ocean coastline. What fraction of U.S. states have either an Atlantic Ocean or Pacific Ocean coastline? (Example 6)
)
8
2 _ 6. _ -6= 7
7
Rate Yourself! How confident are you about adding and subtracting like fractions? Check the box that applies.
Building on the Essential Question What is a simple rule for adding and subtracting like fractions? For more help, go online to access a Personal Tutor.
Tutor
Time to update your Foldable!
286 Chapter 4 Rational Numbers
Copyright © McGraw-Hill Education
8.
8
Name
My Homework eHelp
Independent Practice
Go online for Step-by-Step Solutions
Add or subtract. Write in simplest form. 5 _ 1. _ +6= 7
7
(Examples 1, 2, 4, and, 5)
7 3 2. _ + -_ =
3. -_ + -_ =
+ -_ = 5 -_ 4 4
2 6. -_ - _ =
1 9
( 8)
8
( 59 )
Show your . work
3 9 4. _ -_ = 10
3
10
( 3)
17 of the 7 In Mr. Navarro’s first period class, _ 28
5 9
1 8. To make a greeting card, Bryce used _ sheet 8
students got an A on their math test. In his
3 of red paper, _ sheet of green paper, and
11 second period class, _ of the students got
_7 sheet of white paper. How many sheets of
an A. What fraction more of the students got an A in Mr. Navarro’s first period class than in his second period class? Write in simplest form. (Example 6)
paper did Bryce use?
28
8
8
9. The table shows the Instant Messenger abbreviations students at Hillside Middle School use the most. a. What fraction of these students uses LOL or CUL8R
Ron Levine/Photodisc/Getty Images
when using Instant Messenger?
Copyright © McGraw-Hill Education
9
(Example 3)
Instant Messenger Abbreviations L8R (Later)
48 _
LOL (Laughing out loud)
26 _
BRB (Be right back)
19 _
CUL8R (See you later)
7 _
100 100 100
b. What fraction of these students uses L8R or BRB when
100
using Instant Messenger?
c. What fraction more of these students write L8R than CUL8R when using Instant Messenger?
Lesson 3 Add and Subtract Like Fractions
287
10.
Model with Mathematics Cross out the expression that does not belong. Explain your reasoning.
10 _3 _8 – _3 _ 7 +(– 7 ) 7 7
H.O.T. Problems
Higher Order Thinking
11.
1 Justify Conclusions Select two like fractions with a difference of _ 3 and with denominators that are not 3. Justify your selection.
12.
Persevere with Problems Simplify the following expression. 14 _ 10 3 12 _ 4 2 1 _ + 13 - _ + 11 - _ + ... - _ +_ -_ +_ 15
15
15
15
15
15
15
15
15
13.
Justify Conclusions Is the difference between a positive like fraction and a negative like fraction always, sometimes, or never positive? Justify your answer with an example.
14.
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. 3 1 1 2 1 1 1_ + 2_ + 3_ + 4_ + 5_ + 6_ 4
15.
3
3
2
2
4
288 Chapter 4 Rational Numbers
Copyright © McGraw-Hill Education
Persevere with Problems A construction company 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.
Name
My Homework
Extra Practice Add or subtract. Write in simplest form. _2 5 5 4 _ 16. _ + 3 = 15 17. -_ + -_ = 5
6
5
18. -_ + -_ = 15 16
( 6)
(
7 16
)
4+3 _4 + _3 = _ 5
Homework Help
5
5
7 2 = _ or 1_ 5
5
5 _ 19. _ -3= 8
7 2 20. _ -_ =
8
12
15 _ 21. _ - 13 =
12
18
18
5 13 22. Two nails are _ inch and _ inch long. How much shorter is 16 5 the _–inch nail? 16
16
Identify Structure Add. Write in simplest form. 81 19 47 23. (_ +_ = )+_ 100
100
_1
_2
6
6
3 3 24. _ +_ =
100
3 25. A recipe for Michigan blueberry pancakes calls for _ cup 4
1 1 flour, _ cup milk, and _ cup blueberries. How much more 4
4
Worldwide Volcano Eruptions
flour is needed than milk? Write in simplest form.
26. The graph shows the location of volcanic eruptions.
Copyright © McGraw-Hill Education
a. What fraction represents the volcanic eruptions for both North and South America?
b. How much larger is the section for Asia and South Pacific than for Europe? Write in simplest form.
North America 1 10
South America 2 10
Asia and South Pacific 6 10
Europe 1 10
Lesson 3 Add and Subtract Like Fractions
289
for Smarter Balanced First Pizza
27. A group of friends bought two large pizzas and ate only part of each pizza. The picture shows how much was left. How many pizzas did they eat?
28. The table shows the results of a survey on students’ favorite kind of movie. Select the appropriate values to complete the model to find the fraction of students that prefer comedy or action movies. +
Second Pizza
Type of Number of Movie Students
=
Action
29
Comedy
42
Drama
14
Horror
15
14
50
15
60
29
80
42
100
What fraction of the students who were surveyed prefers comedy or action movies?
Common Core Spiral Review Fill in each 7 29. _ 8
_3 4
with , or = to make a true sentence. 1 30. _ 3
_7 9
6.NS.7
5 31. _
_4
7
6 32. _
5
11
Find the least common denominator for each pair of fractions.
3 4 34. _ and _
1 7 and _ 35. _
13 7 and _ 36. _
5
3 6
7
14
6.NS.4
1 1 33. _ and _ 2
9 _
28
15
12
37. The results of a survey about favorite lunch choices are shown. Which lunch was chosen most often? 6.NS.7
Favorite Lunch Food
39 _
Hot Dogs
3 _
Need more practice? Download more Extra Practice at connectED.mcgraw-hill.com.
50
25 1 _ 10
Copyright © McGraw-Hill Education
Pizza
Grilled Cheese
290
Fraction of Students
Thee Nu Th Number Num mber Sy System ste tem
Lesson 4
Add and Subtract Unlike Fractions What You'll Learn
Essential Question
Scan the lesson. List two headings you would use to make an outline of the lesson.
WHAT happens when you add, subtract, multiply, and divide fractions?
•
Vocab
Vocabulary
• unlike fractions
Common Core State Standards
Real-World Link
Content Standards
Time The table shows the fractions of one hour for certain minutes.
7.NS.1, 7.NS.1d, 7.NS.3, 7.EE.3
Mathematical Practices
1. What fraction of one hour is equal to the sum of 15 minutes and 20 minutes? 15 minutes
1, 3, 4
Number Fraction of Simplified of Minutes One Hour Fraction
20 minutes
5 10
+
=
15 2. Write each fraction of an hour in simplest form in the third column of the table.
20 30
5 _ 60 10 _ 60 15 _ 60 20 _ 60 30 _ 60
1 1 1 3. Explain why _ hour + _ hour = _ hour. 3
2
1 1 7 4. Explain why _ hour + _ hour = _ hour. 12
2
12
Copyright © McGraw-Hill Education
C Squared Studios/Photodisc/Getty Images
6
Lesson 4 Add and Subtract Unlike Fractions
291
Key Concept
Add or Subtract Unlike Fractions A To add or subtract fractions with different denominators, • Rename the fractions using the least common denominator (LCD).
Work Zone
• Add or subtract as with like fractions. • If necessary, simplify the sum or difference.
Before you can add two unlike fractions, or fractions with different denominators, rename one or both of the fractions so that they have a common denominator. Watch Tutor
Example 1. STOP
an d Re fl ec t
_ _
Find 1 + 1 . 4
2
Method 1
actions Circle the pairs of fr actions. that are unlike fr_ 4 _ 1 _ 1 _ 5 _ _1 and 7 and 5 59 and 11 3
Use a number line. 1 2
3
1 4
0
Method 2
1 4 1 2
3 4
1
Divide the number line into fourths since the LCD is 4.
Use the LCD.
1 1 The least common denominator of _ and _ is 4. 4
2
Show your work.
a. b.
292 Chapter 4 Rational Numbers
Add the fractions. Simplify.
2
Got It?
4
4
Do these problems to find out.
Add. Write in simplest form. 1 _ a. _ +2 3
1 _ +3 c. _ 4
8
1 9 b. _ + -_
( 2) 1 1 d. -_ + (-_ 4) 3 10
Copyright © McGraw-Hill Education
d.
Rename using the LCD, 4.
4×1
3 1 _ Using either method, _ + 1 =_ .
6
c.
2×2 2 _ 1 =_ +4 4 3 =_ 4
4
2
1×1 1×2 _ _1 + _1 = _ +
The Number System Tutor
Example 2.
( _34 _59 ) + _74 .
Find - +
(-_34 + _59 ) + _74 = (_59 + (-_34 )) + _74 3 5 7 = _ + (-_ + _ 9 4 4)
Commutative Property of Addition Associative Property of Addition
5 = _ + 1 or 1_ 5 9
Got It?
Do these problems to find out.
4 _ 2 e. _ + _ +3 5
Show your . work
Simplify.
9
(7
5
e.
5 23 f. -_ + _ +_
(
)
3 10
8
)
10
f. Tutor
Example 3.
_2 _1
Find - - . 3
Method 1
2
Use a number line. -1
-2
2
3
- 7 -1 - 5 - 4 - 3 - 2 - 1 6
6
Method 2
6
6
6
6
1 2
0
Use the LCD.
2×2 3×2 3 4 = -_ - _ 6 6 3 -4 =_-_ 6 6 -4 3 = _ or 6
1×3 2×3
-_ - _ = - _ - _ 2 3
Divide the number line into sixths since the LCD is 6.
Rename using the LCD, 6. Simplify. -4 Rewrite -_ as _ . 4 6
-7 _ 6
6
Subtract the numerators. Simplify.
7 7 4 2 3 1 Check by adding -_ + _ = -_ + _ = -_ or -_ 6 6 6 3 6 2
1 1 Using either method, -_ - _ = -_ or -1_ . 2 3
2
7 6
6
Copyright © McGraw-Hill Education
g.
Got It?
Do these problems to find out.
h.
Subtract. Write in simplest form. 5 _ g. _ -1 8
4
1 3 _ h. _ 4
3
2 1 i. _ - -_ 5 2
( )
i. Lesson 4 Add and Subtract Unlike Fractions
293
Choose an Operation Add or subtract unlike fractions to solve real-world problems. Tutor
Example 4.
Use the table to find the Blood Type Frequencies fraction of the total population A B AB ABO Type O that has type A or type B blood. 11 21 1 1 _ _ _ _ Fraction 25 50 10 25 To find the fraction of the total 21 1 population, add _ and _ . 50 10 1 × 5 21 1 21 × 1 _+_=_+_ 50 10 50 × 1 10 × 5 21 _ =_ + 5 50 50 13 26 = _ or _ 50 25
Rename using the LCD, 50. Add the fractions. Simplify.
13 So, _ of the population has type A or type B blood. 25
Check
Guided Practice Add or subtract. Write in simplest form. 3 _ 1. _ + 1 = 5
(Examples 1–3)
2. -_ + -_ =
7 _ 1 3. _ + 3 +_ =
1 3 5. _ - -_ = 4 8
3 _ 6. _ -1=
5 6
10
( 49 )
(8
11
)
8
Show your . work
4 _ 4. _ - 3 = 5
( )
10
4
3
5 7. Cassandra cuts _ inch off the top of a photo and 16
_3 inch off the bottom. How much shorter is the total 8
height of the photo now? Explain. (Example 4)
Rate Yourself! Are you ready to move on? Shade the section that applies. Cre8tive Studios/Alamy
8.
For more help, go online to access a Personal Tutor.
Tutor
Time to update your Foldable!
294 Chapter 4 Rational Numbers
Copyright © McGraw-Hill Education
Building on the Essential Question Compare adding unlike fractions and adding like fractions.
Name
My Homework eHelp
Independent Practice
Go online for Step-by-Step Solutions
Add or subtract. Write in simplest form.
(Examples 1–3)
2. -_ + -_ = 15 5
7 15 _ 3. _ + 2 + -_ =
2 4. -_ - _ =
7 _ 5. _ -1=
7 6. -_ + _ =
2 7. -_ - _ =
5 _ 8. _ + 11 =
7 _ 9. _ +5=
1 _ +3= 1 _ 6 8
(
7 10
)
4 9
( 3)
1
5
9
15
8
(8
5
7 12
3
12
9
) ( 8)
10
6
Justify Conclusions Choose an operation to solve each problem. Explain your reasoning. Then solve the problem. Write in simplest form. (Example 4) 10. Mrs. Escalante was riding a bicycle on a bike
11 Four students were scheduled to give book
2 path. After riding _ of a mile, she discovered 3 3 that she still needed to travel _ of a mile to
reports in 1 hour. After the first report,
reach the end of the path. How long is the bike
_2 hour remained. The next two reports took 3 _1 hour and _1 hour. What fraction of the hour
path?
remained?
4
4
6
12. One hundred sixty cell phone owners were surveyed.
Copyright © McGraw-Hill Education
a. What fraction of owners prefers using their cell phone for text messaging or playing games? Explain.
b. What fraction of owners prefers using their phone to take pictures or text message?
How do you use a cell phone? Playing games 1 4
Taking pictures 3 8 3 8
Text messaging
Lesson 4 Add and Subtract Unlike Fractions
295
13. Pepita and Francisco each spend an equal amount of time on homework. The table shows the fraction of time they spend on each subject. Complete the table by determining the missing fraction for each student.
Homework
Fraction of Time Pepita
Francisco
_1
Math English Science
2
_2 3 _1 6
_3 8
1 2 14. Chelsie saves _ of her allowance and spends _ of her allowance at 5
3
the mall. What fraction of her allowance remains? Explain.
H.O.T. Problems
Higher Order Thinking
Persevere with Problems Fractions whose numerators are 1, such
15.
1 1 as _ or _ , are called unit fractions. Describe a method you can use to add 2
3
two unit fractions mentally.
16.
Use a Counterexample Provide a counterexample to the following statement. 1 The sum of three fractions with odd numerators is never _ . 2
17.
Reason Inductively Suppose a bucket is placed under two faucets. If one faucet is turned on alone, the bucket will be filled in 6 minutes. If the other faucet is turned on alone, the bucket will be filled in 4 minutes. What fraction of the bucket will be filled in 1 minute if both faucets are turned on at the same time? Explain. Copyright © McGraw-Hill Education
296 Chapter 4 Rational Numbers
Name
My Homework
Extra Practice Add or subtract. Write in simplest form. _7 5 _ 4 _ 18. _ +1= 8 19. _ -1= 4 5 6 8 1 1 × 2 5 5 _+_=_+_ 8 8 4 4×2 k or w e Hom 2 5 Help _ _ =8+8 7 =_ 8 1 3 21. _ - -_ = 4 10
2 5 20. _ - -_ =
3 _ 5 2 22. -_ + _ + = 4
( )
3
(
( 3)
6
3
1 23. -_ + _ = 7 8
)
3
Choose an operation to solve each problem. Explain your reasoning. Then solve the problem. Write in simplest form. 24. Ebony is building a shelf to hold the two boxes shown. What is the least width she should make the shelf? 4 5 ft
3 4 ft
5 1 25. Makayla bought _ pound of ham and _ pound of turkey. How much 4
8
more turkey did she buy?
_3
26.
_1
3 4 Persevere with Problems Find the sum of _ and _ . Write in 4 8
simplest form.
27.
1 _ Find the Error Theresa is finding _ + 3 . Find 4 5 her mistake and correct it. Explain your answer.
1+3 _1 + _3 = _ 5
4+5
Copyright © McGraw-Hill Education
4
Lesson 4 Add and Subtract Unlike Fractions
297
for Smarter Balanced 28. The table shows the number of hours Orlando spent at football practice last week. Select the appropriate numbers below to complete the model to find the number of hours Orlando spent practicing on Tuesday and Friday. 1
+
=
+
3
=
How many hours did Orlando spend practicing on
9 10
4
12
5
16
6
19
Day
Time (h)
Monday
_1
Tuesday
_3
Thursday
_1
Friday
_5
2 4 3 6
Tuesday and Friday? 5 1 29. Brett has _ of his monthly income left to spend. He has budgeted _ of his 6 8 1 _ income for a new video game and of his income for savings. Determine if 3 each statement is true or false. 7 a. Brett will have _ of his income left if he only buys the video game.
8 1 of his income left if he only puts money in savings. b. Brett will have _ 2 3 of his income left after buying the video game c. Brett will have _ 8
True
False
True
False
True
False
and putting money in savings.
Common Core Spiral Review Write each improper fraction as a mixed number.
5.NF.3
7 30. _ =
14 31. _ =
101 32. _ =
22 33. _ =
77 34. _ =
23 35. _ =
5
9
3
10
100
8
Copyright © McGraw-Hill Education
298 Need more practice? Download more Extra Practice at connectED.mcgraw-hill.com.
Thee Nu Th Number Num mber Sy System ste tem
Lesson 5
Add and Subtract Mixed Numbers What You'll Learn
Essential Question
Scan the lesson. List two real-world scenarios in which you would add or subtract mixed numbers. •
WHAT happens when you add, subtract, multiply, and divide fractions?
Common Core State Standards
•
Content Standards 7.NS.1, 7.NS.1d, 7.NS.3, 7.EE.3
Mathematical Practices 1, 3, 4
Real-World Link Hockey Junior and adult hockey sticks are shown below. Junior 2
length 3 3 ft
Adult 5
length 4 6 ft
5 2 1. Use the expression 4_ - 3_ to find 6 3 how much longer the adult hockey stick is than the junior hockey stick. Rename the fractions using the LCD, 6.
5 4_ -
=
7 2 2. Explain how to find 3_ - 2_ . Then use your conjecture 5 10 to find the difference.
Copyright © McGraw-Hill Education
Bruce Bennett/Getty Images
6
Subtract the fractions. Then subtract the whole numbers.
connectED.mcgraw-hill.com
Lesson 5 Add and Subtract Mixed Numbers
299
Work Zone
Add and Subtract Mixed Numbers To add or subtract mixed numbers, first add or subtract the fractions. If necessary, rename them using the LCD. Then add or subtract the whole numbers and simplify if necessary. Sometimes when you subtract mixed numbers, the fraction in the first mixed number is less than the fraction in the second mixed number. In this case, rename one or both fractions in order to subtract.
Tutor
Examples 1.
_
_
Find 7 4 + 10 2 . Write in simplest form. 9
9
Estimate 7 + 10 = 17
_ 9 _ + 10 2 9 ______ 6 2 17_ or 17_ 74
Add the whole numbers and fractions separately.
Simplify.
3
9
2 Check for Reasonableness 17_ ≈ 17 3
2.
_5 _
Find 8 - 2 1 . Write in simplest form. 6
3
Estimate 9 - 2 = 7 Show your work.
a. b. c.
5 8_
5 8_
6 1 _ -2 3 _____
6
2 -2_
Rename the fraction using the LCD. Then subtract.
6 _____ 3 1 or 6 _ 6_ 6
2
1 Check for Reasonableness 6_ ≈ 7 2
Got It?
Simplify.
Do these problems to find out.
Add or subtract. Write in simplest form.
d.
5 1 a. 6 _ + 2_
3 1 b. 5_ + 2_
5 1 c. 1_ + 4_
3 4 - 1_ d. 5_
3 7 e. 13 _ - 9_
2 1 f. 8_ - 2_
8
5
f. 300 Chapter 4 Rational Numbers
10
5
10
8
4
9
3
6
2
Copyright © McGraw-Hill Education
e.
8
The Number System
Tutor
Example 3.
_ _
Find 2 1 - 1 2 . 3
3
Method 1
Rename Mixed Numbers
1 1 Estimate 2 - 1_ = _ 2 2
1 2 1 Since _ is less than _ , rename 2_ before subtracting. 3
3
3
_3
Change 1 to . 3
1 2_ 3
1 2_
3 _ 4 1_ + 1 or 1_
=
3
4 1_
3 2 -1_ 3 _____
Method 2 1 2_
3
3
Subtract the whole numbers and then the fractions.
_2 ≈ _1 3 2 Show your . work
Write as Improper Fractions
_7
3 2 _ -1 3 _____
3
1 4 Rename 2_ as 1_ .
3 2 -1_ 3 _____ _2 3
Check for Reasonableness
3
1 7 Write 2_ as _ .
3 5 _ 3 _____ _2 3 1 2 2 So, 2_ - 1_ = _. 3 3 3
3
g.
3
5 2 Write 1_ as _ . 3 3
h.
Simplify.
i.
2 Using either method, the answer is _ . 3
Got It?
Do these problems to find out.
j.
Subtract. Write in simplest form. 1 g. 7 - 1 _
3 11 h. 5_ - 4_
3 2 i. 11 _ - 2_
k.
3 j. 8 - 3 _
3 1 k. 3_ - 1_
5 l. 16 - 5 _
l.
Copyright © McGraw-Hill Education
2
4
8
4
12
4
5
5
6
Lesson 5 Add and Subtract Mixed Numbers
301
Choose an Operation Add or subtract unlike fractions to solve real-world problems. Tutor
Example 4.
An urban planner is designing a skateboard park. The length
_
of the skateboard park is 120 1 feet. The length of the parking
_
2
lot is 40 1 feet. What will be the length of the park and the 3
parking lot combined? 3 1 1 2 120_ + 40_ = 120_ + 40_ 2
3
6
5 = 160 + _
6
6
5 = 160_ 6
1 _ 1 _ Rename _ as 3 and _ as 2 . 2
6
3
6
Add the whole numbers and fractions separately. Simplify.
5 The total length is 160_ feet. 6
Check
Guided Practice Add or subtract. Write in simplest form. 1 4 1. 8_ + 3_ = 5 2
(Examples 1–3)
5 1 2. 7_ - 3_ = 6 6
3 3. 11 - 6_ = 8
9 4. A hybrid car’s gas tank can hold 11_ gallons of gasoline. 10
3 gallons of gasoline. How much more It contains 8_ 4
gasoline is needed to fill the tank?
How confident are you about adding and subtracting mixed numbers? Shade the ring on the target.
Building on the Essential Question How can you subtract mixed numbers when the fraction in the first mixed number is less than the fraction in the second
Heath Korvola/Getty Images
5.
(Example 4)
Rate Yourself!
mixed number?
302 Chapter 4 Rational Numbers
Tutor
Copyright © McGraw-Hill Education
For more help, go online to access a Personal Tutor.
Name
My Homework eHelp
Independent Practice
Go online for Step-by-Step Solutions
Add or subtract. Write in simplest form.
_5 + 11_1 =
1 4 1. 2_ + 7_ = 9
(Examples 1–3)
2. 8
9
4 1 3. 10_ - 2_ =
4
12
5
5
Show your . work
3 4 4.4 9_ - 2_ =
3 1 - 4_ = 5 11_ 4 3
3 1 6. 9_ - 2_ =
3 2 7. 6_ - 1_ =
1 1 8. 14_ - 7_ =
2 9. 8 - 3_ =
5
5
10
6
3
5
3
5
3
Justify Conclusions For Exercises 10 and 11, choose an operation to solve. Explain your reasoning. Then solve the problem. Write your answer in simplest form. (Example 4) 10. If Juliana and Brody hiked both of the trails listed in the table, how far did they hike?
Trail
Length (mi)
Woodland Park
2 3_
Mill Creek Way
5 2_
3 6
5 feet. Find the width of Kasey’s 11 The length of Kasey’s garden is 4_ 8 7 _ garden if it is 2 feet shorter than the length. 8
1 12. Karen wakes up at 6:00 A.M. It takes her 1_ hours to shower, get 4
Copyright © McGraw-Hill Education
1 hour to eat breakfast, dressed, and comb her hair. It takes her _ 2
brush her teeth, and make her bed. At what time will she be ready for school?
Lesson 5 Add and Subtract Mixed Numbers
303
Add or subtract. Write in simplest form.
_1
_2
3 1 13. -3_ + (-1_ )= 4 4
4 3 3 2 14. _ +_ = 5 2
5 1 2 15. 6_ + 1_ + 5_ =
5 1 1 16. 3_ + 2_ − 4_ =
3
3
H.O.T. Problems 17.
4
9
6
3
Higher Order Thinking
Model with Mathematics Write a real-world problem that could be 1 7 represented by the expression 5_ - 3_ . Then solve your problem. 2
8
18.
Persevere with Problems A string is cut in half. One of the halves is thrown away. One fifth of the remaining half is cut away and the piece left is 8 feet long. How long was the string initially? Justify your answer.
19.
Model with Mathematics Using three mixed numbers as side lengths, 1 draw an equilateral triangle with a perimeter of 8_ feet. 4
304 Chapter 4 Rational Numbers
Copyright © McGraw-Hill Education
Show your . work
Name
My Homework
Extra Practice Add or subtract. Write in simplest form. 1 3 3 1 1 20. 6_ - 2_ = 32 21. 8_ + 10_ = 4 4 8 3 1 3 3 5 6_ - 2_ = 5_ - 2_ 4 4 4 4 2 = 3_ 4 1 _ = 32
_
Homework Help
3 2 23. 3_ + 4_ = 7
5 22. 13 - 5_ = 6
3 3 24. 4_ - 1_ =
7
10
5 1 25. 12_ - 6_ =
4
2
8
Justify Conclusions Choose an operation to solve. Explain your reasoning. Then solve the problem. Write your answer in simplest form. 3 26. The length of Alana's hair was 9_ inches.
1 27. Emeril used a total of 7_ cups of flour to
4
4
1 After her haircut, the length was 6_ inches.
1 cups of make three pastries. He used 2_
How many inches did she have cut?
1 cups for the flour for the first and 2_
2
4
3
second. How much flour did Emeril use for the third pastry?
5 28. Margarite made the jewelry shown. If the necklace is 10_ inches longer 8 than the bracelet, how long is the necklace? 1
7 4 in. bracelet
necklace
29. Find the perimeter of the figure. Write your answer in simplest form. 3
3
Copyright © McGraw-Hill Education
2 8 yd
1 30. Suppose you want to place a shelf that is 30_ inches long in the center 3
3 inches wide. About how far from each edge of the of a wall that is 45_
2 8 yd 3
2 8 yd
4
wall should you place the shelf? Lesson 5 Add and Subtract Mixed Numbers
305
for Smarter Balanced 3 31. A recipe for snack mix calls for 4_ cups of cereal. The amount of peanuts 4 2 _ needed is 1 cups less than the amount of cereal needed. Complete each box 3 below to make a true statement.
The recipe calls for
cups of peanuts. A total of
cups of peanuts and cereal are needed in all. 3 1 32. Maria practiced the piano for 2_ hours last week and 1_ hours this week. 4 2 Use the bar diagram sections to construct a bar diagram that represents how many hours Maria practiced in the past 2 weeks.
How many hours did Maria practice the piano in the past 2 weeks?
Common Core Spiral Review Round each mixed number to its nearest whole number. Then estimate each product. 5.NF.4 1 2 33. 5_ × 7_ ≈ 4
3
×
1 14 34. 1_ × 8_ ≈
≈
11
15
×
≈
4 35. Zoe’s average running speed is about 6_ miles per hour. Suppose Zoe runs 4
5.NF.4
306 Need more practice? Download more Extra Practice at connectED.mcgraw-hill.com.
Copyright © McGraw-Hill Education
5
3 for 1_ hours. About how far will she have run? Explain.
Th T he N he Number Num mber mb m ber SSy System Sys yste ste tem em
Problem-Solving Investigation
Draw a Diagram Content Standards
Case #1 Science Experiment
7.NS.3, 7.EE.3
the ground Casey drops a ball from a height of 12 feet. It hits for each and bounces up half as high as it fell. This is true successive bounce.
1, 4, 6
What is the height the ball reaches after the bounce?
1 2 3
Understand
Mathematical Practices
fourth
What are the facts?
Casey dropped the ball from a height of 12 feet. It bounces up half as high for each successive bounce.
Plan
What is your strategy to solve this problem?
Draw a diagram to show the height of the ball after each bounce.
Solve
How can you apply the strategy?
The ball reaches a height of fourth bounce.
foot after the
6 ft 12 ft ft 11 ft
ft
Copyright © McGraw-Hill Education
fStop/SuperStock
2
4
1
Check
2
3
4
Does the answer make sense?
Use division to check. 12 ÷ 2 = 6, 6 ÷ 2 = 3, 3 ÷ 2 = 1.5, 1.5 ÷ 2 = 0.75.
Analyze the Strategy
Watch Tutor
2 Be Precise If the ball is dropped from 12 feet and bounces up _ as high 3
on each successive bounce, what is the height of the fourth bounce?
connectED.mcgraw-hill.com
Problem-Solving Investigation Draw a Diagram
307
el Case #2 Trav
s, driven 60 mile Mr. Garcia has er’s house. way to his sist
_2
which is 3 of
the
e
v he have to dri s oe d r e h rt a How much f ster’s house? si is h to t e g to
1
Understand Read the problem. What are you being asked to find?
I need to find
.
What information do you know?
Mr. Garcia has driven
of the way to his sister’s house. This is
equal to
.
Is there any information that you do not need to know?
I do not need to know
2 3
.
Plan Choose a problem-solving strategy.
I will use the
strategy.
Solve Use your problem-solving strategy to solve the problem.
Use the bar diagram that represents the distance to his sister’s house. 2 Fill in two of the sections to represent _ . 3
60 miles
of the 3 parts = 60. Each part is
miles. The
distance to his sister’s house is 60 + So, Mr. Garcia has
miles left to drive.
Check Use information from the problem to check your answer.
308 Chapter 4 Rational Numbers
.
Copyright © McGraw-Hill Education
4
=
The Number System y
Collaborate
Work with a small group to solve the following cases. Show your work on a separate piece of paper.
Case #3 Fractions
_1
of what was left. Marta ate a quarter of a whole pie. Edwin ate 4 _1 Cristina then ate 3 of what was left.
What fraction of the pie remains?
Case #4 Games Eight members of a ch ess club are having a to urnament. In the first every player will play round, a chess game against every other player.
How many games will be
in the first round of th
e tournament?
Case #5 Distance bikes to Alejandro and Pedro are riding their _5 they are 8 of the way there.
How much farther do they have to
school. After 1 mile,
go?
Copyright © McGraw-Hill Education
©Brand X Pictures/PunchStock
Case #6 Seats The number of seats in the first row of a concert h hallll is 6. The second row has 9 seats, the third row has 12 seats, and the fourth row has 15 seats.
Use anygy! strate
How many seats will be in the eighth row?
Problem-Solving Investigation Draw a Diagram
309
Mid-Chapter Check Vocab
Vocabulary Check 1. Define rational number. Give some examples of rational numbers written in different forms. (Lessons 3 and 4)
2. Fill in the blank in the sentence below with the correct term.
(Lesson 1)
Repeating decimals can be represented using
.
Skills Check and Problem Solving Add or subtract. Write in simplest form. 5 _ 3. _ +3= 8
2 4. -_ + _ =
1 5. -_ -_ = 15 15
3 5 7. 8_ - 2_ =
1 1 8. 5_ - 1_ =
1 9
8
(Lessons 3–5)
9
11
Shuow yo r . work
5 2 6. 2_ + 1_ = 9
4
3
12
6
9. The table at the right shows the fraction of each state that is water. Order the states from least to greatest fraction of water. (Lesson 2)
10. The maximum height of an Asian elephant is 9.8 feet. What mixed
3
What Part is Water? Alaska
3 _
Michigan
40 _
Wisconsin
number represents this height?
(Lesson 1)
41 97 _1 6
Month Weight (lb)
11.
Persevere with Problems The table shows the weight of a newborn infant for its first year. During which three-month period was the infant’s (Lesson 5)
3 6 9 12
310 Chapter 4 Rational Numbers
1 7_
4 1 12_ 2 _ 16 5 8 _ 19 4 5 _ 23 3 20
Copyright © McGraw-Hill Education
weight gain the greatest?
0
Thee Nu Th Number Num mber Sy System ste tem
Lesson 6
Multiply Fractions What You'll Learn
Scan the lesson. Predict two things you will learn about multiplying fractions. •
Essential Question WHAT happens when you add, subtract, multiply, and divide fractions?
Common Core State Standards
•
Content Standards 7.NS.2, 7.NS.2a, 7.NS.2c, 7.NS.3, 7.EE.3
Real-World Link
Mathematical Practices 1, 3, 4
Lunch There are 12 students at the lunch table. Two thirds of the students ordered a hamburger for lunch. One half of those students that ordered a hamburger put cheese on it.
Step 1
Step 2
Draw an X through the students that did not order a hamburger.
Didn't I order cheese with that?
Draw a C on the students that ordered cheese on their hamburger.
1. What fraction of the students at the lunch table
Copyright © McGraw-Hill Education
Elena Elisseeva/age fotostock
ordered a cheeseburger? Write in simplest form. 1 2 of _ ? Write in simplest form. 2. What is _ 2
3
3. Write your own word problem that involves fractions that can be solved using a diagram like the one above.
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Lesson 6 Multiply Fractions
311
Key Concept
Multiply Fractions M Words
To multiply fractions, multiply the numerators and multiply the denominators.
Work Zone Examples
Numbers
Algebra
1×2 2 _1 × _2 = _ or _ 2
3
a·c ac _a · _c = _ or _ , where b, d ≠ 0
6
2×3
b
d
b·d
bd
When multiplying two fractions, write the product in simplest form. The numerator and denominator of either fraction may have common factors. If this is the case, you can simplify before multiplying. Watch Tutor
Examples Multiply. Write in simplest form.
1.
_1 × _1 2
3
1×1 _1 × _1 = _ 2
2.
3
2×3 1 =_ 6
Multiply the numerators. Multiply the denominators. Simplify.
( _34 ) 3 -3 2 2 × (-_ =_ × _ 4) 1 ( 4 )
2× -
-3 2 Write 2 as _ and -_ as _ . 4 3
1
2 × (-3) 1×4 -6 1 =_ or -1_ 4 2
=_
3.
_2 × (-_3 ) 8
7
Multiply the numerators. Multiply the denominators. Simplify.
1
_2 × (-_3 ) = _2 × (-_3 ) 8
7
4
7
Divide 2 and 8 by their GCF, 2.
8 4
= _ or -_
Show your work.
1 × (-3) 7×4
a.
Got It?
c. 312 Chapter 4 Rational Numbers
Multiply.
Do these problems to find out.
Multiply. Write in simplest form. 3 _ a. _ ×1 5
2
2 ( ) b. _ × -4 3
c. -_ × -_ 7 3 1
( 3)
Copyright © McGraw-Hill Education
b.
3 28
The Number System
Multiply Mixed Numbers When multiplying by a mixed number, you can rename the mixed number as an improper fraction. You can also multiply mixed numbers using the Distributive Property and mental math.
Watch Tools Tutor
Example 4.
_
_
Find 1 × 4 2 . Write in simplest form. 5
2
_1 × 4 = 2
Estimate
2
Method 1
_1 × 4_2 = _1 × 5
2
2
Rename the mixed number. 11 2 22 Rename 4_ as an improper fraction, _ . 22 _ 5
5
1
1 × 11 =_
Multiply.
11 =_
Simplify.
1 = 2_
Simplify.
1×5 5
5
Method 2
5
Divide 2 and 22 by their GCF, 2.
Use mental math.
2 2 is equal to 4 + _ . The mixed number 4_ 5
5
1 2 _ 2 So, _ × 4_ = 1 (4 + _ ). Use the Distributive Property to 5
2
5
2
multiply, then add mentally.
_1 (4 + _2 ) = 2 + _1 5
2
5
1 = 2_
Think Half of 4 is 2 and half of 2 fifths is 1 fifth. Rewrite the sum as a mixed number.
5
1 Check for Reasonableness 2_ ≈ 2 5
1 2 1 So, _ × 4_ = 2_ . 5
2
5
1 Using either method, the answer is 2_ . 5
Got It?
Show your . work
Do these problems to find out.
d.
Copyright © McGraw-Hill Education
Multiply. Write in simplest form. 4 1 d. _ × 8_ 4
9
1 e. 5_ ×3 3
7 2 f. -1_ × -2_ 8
(
5
)
e. f. Lesson 6 Multiply Fractions
313
Tutor
Example 5.
_
Humans sleep about 1 of each day. Let each year equal
_
3
365 1 days. Determine the number of days in a year the 4
average human sleeps. 1 1 Find _ × 365_ . 4
3
1 Estimate _ × 360 = 120 3
1,461 _1 × 365_1 = _1 × _ 4
3
3
Rename the mixed number as an improper fraction.
4
487
1,461 1 =_ ×_ 4 3
Divide 3 and 1,461 by their GCF, 3.
1
487 3 =_ or 121_ 4
4
Multiply. Then rename as a mixed number.
3 Check for Reasonableness 121_ ≈ 120 4
3 days each year. The average human sleeps 121_ 4
Check
Guided Practice Multiply. Write in simplest form.
2. -_ × -_ = 4 9
2 _ 1. _ ×1= 3
(Examples 1–4)
1
3
1 _ 3. 2_ ×2=
( 8)
4
3
Show your . work
4.
2 The weight of an object on Mars is about _ its 5
1 weight on Earth. How much would an 80_ -pound dog 2
weigh on Mars?
5.
(Example 5)
Rate Yourself! How well do you understand multiplying fractions? Circle the image that applies.
Building on the Essential Question How is the process of multiplying fractions different from the process of adding fractions? Clear
Somewhat Clear
Not So Clear Tutor
Time to update your Foldable!
314 Chapter 4 Rational Numbers
Copyright © McGraw-Hill Education
For more help, go online to access a Personal Tutor.
Name
My Homework eHelp
Independent Practice Multiply. Write in simplest form.
Go online for Step-by-Step Solutions
(Examples 1–4)
3 _ 1. _ ×1=
2 _ 2. _ ×2=
1 3. -9 × _ =
4. -_ × -_ = 5 6
2 _ 5. _ ×1=
2 6. -_ × _ =
2 _ 7 _ × 15 =
7 4 _ 8. _ × =
1 2 9. -1_ ×_ =
4
5
8
3
2
Show your . work
1
5
( 5)
3
7
16
1 12
4
(
8
2
5
)
3
1 times its length. 10. The width of a vegetable garden is _ 3
3 feet, what is the width If the length of the garden is 7_ 4
in simplest form? (Example 5)
2 of the students in Rick’s class watched television. 11 One evening, _ 3 8
1 of them recorded the show. What fraction of watched the show, _ 4
the students in Rick’s class watched and recorded a reality TV show?
Write each numerical expression. Then evaluate the expression. 12. one half of negative five eighths
13. one third of eleven sixteenths
Copyright © McGraw-Hill Education
Burke/Triolo Productions/Brand X Pictures/Getty Images
3 watched a reality show. Of the students that Of those students, _
Lesson 6 Multiply Fractions
315
14.
Model with Mathematics Refer to the graphic novel frame below.
a. The height of the closet is 96 inches, and Aisha would like to have 4 rows of cube organizers. What is the most the height of each cube organizer can be? b. Aisha would like to stack 3 shoe boxes on top of each other at the bottom 1 of the closet. The height of each shoe box is 4_ inches. What is the total 2 height of the 3 boxes?
H.O.T. Problems 15.
Higher Order Thinking
Model with Mathematics Write a real-world problem that involves 3 1 and _ . finding the product of _ 4
8
16.
Persevere with Problems Two positive improper fractions are multiplied. Is the product sometimes, always, or never less than 1? Explain.
17.
Reason Inductively Find two fractions that satisify each of the following. 2 a. each greater than _ with a product less than 2 5
_ 5
2
316 Chapter 4 Rational Numbers
2
Copyright © McGraw-Hill Education
1 1 with a product greater than _ b. each greater than _
Name
My Homework
Extra Practice Multiply. Write in simplest form. _4 4 1 4 ( ) 18. _ × -6 = –4 5 19. -_ × -_ = 4 9 5 _4 × (–6) = _4 × –_6 5 5 1 ) ( k or 4 × –6 w e m _ Ho = Help 5×1 –24 _4 =_ 5 or –4 5
1 1 20. 3_ × -_ = 5 3
( )
( )
( )
1 _ 21. _ ×3= 3
4
1 4 22. _ × -_ =
5 3 23. _ × 2_ =
( 8)
9
1 24. Each DVD storage case is about _ inch thick 5
What will be the height in simplest form of
3 25. Mark left _ of a pizza in the refrigerator. On 8
1 Friday, he ate _ of what was left of the pizza. 2
What fraction of the entire pizza did he eat on Friday?
12 cases sold together?
Multiply. Write in simplest form. 1 2 26. (_ )= 4
5
6
2 27. (-_) 3 = 3
_1
_2
_1
_1
1 5 3 28. _ ×_= 2
4
Copyright © McGraw-Hill Education
29.
Justify Conclusions Alano wants to make one and a half batches of the pasta salad recipe shown at the right. How much of each ingredient will Alano need? Explain how you solved the problem.
1 30. Philip rode his bicycle at 9_ miles per hour. If he rode 2 3 _ for of an hour, how many miles in simplest form did 4
Pasta Salad Recipe Ingredient
Amount
Broccoli
1 1_ c
Cooked pasta
3 3_ c
Salad dressing
_2 c
Cheese
1 1_ c
4 4
3
3
he cover? Lesson 6 Multiply Fractions
317
for Smarter Balanced 2 1 31. Of the dolls in Marjorie’s doll collection, _ have red hair. Of these, _ have green 5 4 1 2 _ eyes, have blue eyes, and have brown eyes. Determine if each statement 3 12 is true or false.
_
1 a. _ of Marjorie’s doll collection has red hair and green eyes.
10 4 of Marjorie’s doll collection has red hair and blue eyes. b. _ 15 29 of Marjorie’s doll collection has red hair and brown eyes. c. _ 60
32. The table shows the number of teaspoons of vanilla needed to make different batches of cookies. Select one box from each row to describe how to find the number of teaspoons of vanilla needed to make n batches of cookies. Row 1
Subtract
Add
Multiply
Row 2
4
n
_1
Row 3
to
by
from
Row 4
4
n
_1
True
False
True
False
True
False
Batches
1
2
3
4
5
Vanilla (tsp)
_1
_1
_3
1
1 1_
4
2
4
n
4
Divide
4
4
1 batches of cookies? How many teaspoons of vanilla are needed to make 6_ 2
Common Core Spiral Review For each multiplication sentence, write two related division sentences. 1 _ 1 34. _ × 1 =_
33. 3 × 4 = 12
6
2 1 4 35. 2_ × 4_ = 10_ 2
5
3
18
5 3 1 36. 5_ × 1_ = 6_ 8
5
4
318 Need more practice? Download more Extra Practice at connectED.mcgraw-hill.com.
Copyright © McGraw-Hill Education
5
5.NBT.5
Thee Nu Th Number Num mber Sy System ste tem
Lesson 7
Convert Between Systems What You'll Learn
Essential Question
Scan the lesson. List two real-world scenerios in which you would convert measurments.
HOW do you convert between measurement systems?
•
Common Core State Standards Content Standards
•
7.RP.3, 7.NS.2, 7.NS.3
Mathematical Practices 1, 3, 4, 5, 6 Watch
Real-World Link
5K Race To raise money for a health organization, the Matthews family is participating in a 5K race. A 5K race is 5 kilometers. 1. How many meters long is the race? 5 kilometers =
meters
2. One mile is approximately 1.6 kilometers. About how many miles is the race? 5 kilometers ≈
miles
3. A kilometer is a unit of length in the metric measurement system. A mile is a measure of length in the customary measurement system. Write the following units of length under the correct measurement system.
Metric
Customary
kilometer
mile
Copyright © McGraw-Hill Education
John Lund/Blend Images/Getty Images
centimeter, foot, inch, meter, millimeter, yard
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Lesson 7 Convert Between Systems
319
Work Zone
Convert Between Measurement Systems You can multiply by fractions to convert between customary and metric units. The table below lists common customary and metric relationships. Customary and Metric Relationships Type of Measure
Metric
1 inch (in.) 1 foot (ft) 1 yard (yd) 1 mile (mi)
≈ ≈ ≈ ≈
2.54 centimeters (cm) 0.30 meter (m) 0.91 meter (m) 1.61 kilometers (km)
Weight/Mass
1 pound (lb) 1 pound (lb) 1 ton (T)
≈ ≈ ≈
453.6 grams (g) 0.4536 kilogram (kg) 907.2 kilograms (kg)
Capacity
1 cup (c) 1 pint (pt) 1 quart (qt) 1 gallon (gal)
≈ ≈ ≈ ≈
236.59 milliliters (mL) 473.18 milliliters (mL) 946.35 milliliters (mL) 3.79 liters (L)
Length
STOP
Customary
Tutor
Examples
an d Re fl ec t
1.
What metric unit of s to measure correspond miles? to pounds? Write your answers below.
Convert 17.22 inches to centimeters. Round to the nearest hundredth if necessary. 2.54 cm Since 2.54 centimeters ≈ 1 inch, multiply by _ . 1 in. 2.54 cm 2.54 cm Multiply by _ . Divide out 17.22 ≈ 17.22 in. · _ 1 in. 1 in.
≈ 43.7388 cm
common units. Simplify.
So, 17.22 inches is approximately 43.74 centimeters.
2.
Convert 5 kilometers to miles. Round to the nearest hundredth if necessary. 1 mi Since 1 mile ≈ 1.61 kilometers, multiply by _ .
Show your work.
1.61 km
1 mi 5 km ≈ 5 km · _ 1.61 km
5 mi ≈_ or 3.11 mi
a.
1.61
1 mi Multiply by _ . Divide out common units. 1.61 km
Simplify.
So, 5 kilometers is approximately 3.11 miles.
c.
Got It?
Do these problems to find out.
Complete. Round to the nearest hundredth if necessary. a. 6 yd ≈ m
320 Chapter 4 Rational Numbers
b. 1.6 cm ≈ in.
c. 17 m ≈ yd
Copyright © McGraw-Hill Education
b.
The Number System
Tutor
Examples 3.
Convert 828.5 milliliters to cups. Round to the nearest hundredth if necessary. 1c Since 1 cup ≈ 236.59 milliliters, multiply by _ . 236.59 mL
1c 828.5 mL ≈ 828.5 mL · _
1c Multiply by _ and 236.59 mL
236.59 9 mL
divide out common units. Simplify.
828.5 c ≈_ or 3.50 c 236.59
So, 828.5 milliliters is approximately 3.50 cups.
4.
Convert 3.4 quarts to milliliters. Round to the nearest hundredth if necessary. 946.35 mL Since 946.35 milliliters ≈ 1 quart, multiply by _ . 1 qt
946.35 mL 3.4 qt ≈ 3.4 4 qt · _ 1 qt
≈ 3,217.59 mL
946.35 Multiply by _ . Divide out common units. 1 qt
Simplify.
So, 3.4 quarts is approximately 3,217.59 milliliters.
5.
Convert 4.25 kilograms to pounds. Round to the nearest hundredth if necessary. 1 lb Since 1 pound ≈ 0.4536 kilogram, multiply by _ . 1 lb 4.25 kg ≈ 4.25 5 kg · _ 0.4536 6 kg
4.25 lb ≈_ or 9.37 lb 0.4536
0.4536 kg
1 lb Multiply by _ . Divide out common
units.
0.4536 kg
Simplify.
So, 4.25 kilograms is approximately 9.37 pounds.
Got It?
Show your . work
d.
Do these problems to find out.
Complete. Round to the nearest hundredth if necessary. d. 7.44 c ≈ mL e. 22.09 lb ≈ kg
e. f.
Copyright © McGraw-Hill Education
f. 35.85 L ≈ gal
Lesson 7 Convert Between Systems
321
Tutor
Example 6.
An Olympic-size swimming pool is 50 meters long. About how many feet long is the pool? 1 ft Since 1 foot ≈ 0.30 meter, use the ratio _ . 0.30 m
1 ft 50 m ≈ 50 m · _
1 ft Multiply by _ .
0.30 m 1 ft ≈ 50 m · _ 0.30 m 50 ft ≈_ or 166.67 ft 0.30
0.30m
Divide out common units, leaving the desired unit, feet. Divide.
An Olympic-size swimming pool is about 166.67 feet long. Check
Guided Practice Complete. Round to the nearest hundredth if necessary. 1. 3.7 yd ≈
m
2. 11.07 pt ≈
(Examples 1 – 5)
mL
3. 650 lb ≈
kg
Show your . work
4. About how many feet does a team of athletes run in a 1,600-meter relay race? (Example 6)
5. Raheem bought 3 pounds of bananas. About how many kilograms did he buy? (Example 6)
Building on the Essential Question How can you use dimensional analysis to convert between measurement systems?
Rate Yourself! Are you ready to move on? Shade the section that applies.
322 Chapter 4 Rational Numbers
Tutor
Copyright © McGraw-Hill Education
For more help, go online to access a Personal Tutor.
JupiterImages/Comstock Premium/Alamy
6.
Name
My Homework eHelp
Independent Practice
Go online for Step-by-Step Solutions
Complete. Round to the nearest hundredth if necessary. 1. 5 in. ≈
cm
2. 2 qt ≈
(Examples 1 – 5)
mL
3 58.14 kg ≈
lb
Show your . work
4. 4 L ≈
7. 4.725 m ≈
gal
5. 10 mL ≈
ft
8. 3 T ≈
kg
6. 63.5 T ≈
9. 680.4 g ≈
kg
lb
11. A glass bottle holds 3.75 cups of water. About how many milliliters of water can the bottle hold? (Example 6)
12. A Cabbage Palmetto has a height of 80 feet. What is the approximate height of the tree in meters? (Example 6)
Copyright © McGraw-Hill Education
Jill Ferry/Flickr/Getty Images
10. A notebook computer has a mass of 2.25 kilograms. About how many pounds does the notebook weigh? (Example 6)
c
Lesson 7 Convert Between Systems
323
Persevere with Problems Determine the greater amount for each situation. 13 Which box is greater, a 1.5-pound box of raisins or a 650-gram box of raisins?
H.O.T. Problems
14. Which is greater a 2.75-gallon container of juice or a 12-liter container of juice?
Higher Order Thinking
15.
Reason Inductively One gram of water has a volume of 1 milliliter. What is the volume of the water if it has a mass of 1 kilogram?
16.
Persevere with Problems The distance from Earth to the Sun is approximately 93 million miles. About how many gigameters is this? Round to the nearest hundredth. (Hint: In 1 gigameter there are about 621,118.01 miles.)
Be Precise Order each set of measures from greatest to least. 17. 1.2 cm, 0.6 in., 0.031 m, 0.1 ft
18. 2 lb, 891 g, 1 kg, 0.02 T
1 19. 1_ c, 0.4 L, 950 mL, 0.7 gal
20. 4.5 ft, 48 in., 1.3 m, 120 cm
4
21.
_
5 Model with Mathematics Convert 2 1 inches and 2_ inches to 8
8
centimeters. Round to the nearest tenth. Then draw a segment whose length is between those two measures. Show your . work Copyright © McGraw-Hill Education
324 Chapter 4 Rational Numbers
Name
My Homework
Extra Practice Complete. Round to the nearest hundredth if necessary. 22. 15 cm ≈ 5.91
in.
23. 350 lb ≈
≈ 15 cm · _ 2.54 cm 1 in.
≈ 5.91 in. ≈_ 2.54 15 in.
L
km
27. 19 kg ≈
lb
≈ 158.76 kg 26. 50 mL ≈
28. The Willis Tower has a height of 1,451 feet. What is the estimated height of the building in meters?
30.
24. 17 mi ≈
350 lb≈ 350 lb · _
1 in.
25. 32 gal ≈
kg
0.4536 kg 1 lb 0.4536 kg ≈ 350 lb · _ 1 lb
15 cm≈ 15 cm · _ 2.54 cm Homework Help
158.76
fl oz
29. Which is greater, a bottle containing 64 fluid ounces or a bottle containing 2 liters of water?
Use Math Tools A bakery uses 900 grams of peaches in a cobbler. About how many pounds of peaches does the bakery use in a cobbler?
Determine which quantity is greater.
Copyright © McGraw-Hill Education
Car Culture/Car Culture® Collection/Getty Images
31. 3 gal, 10 L
32. 14 oz, 0.4 kg
33. 4 mi, 6.2 km
34. Velocity is a rate usually expressed in feet per second or meters per second. How can the units help you calculate velocity using the distance a car traveled and the time recorded?
Lesson 7 Convert Between Systems
325
for Smarter Balanced 35. The diagram shows the length of a fork from the cafeteria. Which measurements are approximately equal to the length of the fork? Select all that apply. 15.2 cm
0.152 m
6 in.
152 cm
1.52 m
36. The masses of 4 different animals from a zoo are shown in the table. Convert each measure to pounds. Then sort the animals from least to greatest weight.
Animal Brown Bear Giraffe
Animal
Weight (lb)
Lion
Least
Rhinoceros
Mass (kg) 272.16 1,134.0 226.8 1,587.6
Greatest
How many pounds greater is the heaviest animal than the lightest animal?
Explain how you can use units to be sure you are multiplying by the correct fraction when converting between measurement systems. Give an example.
Common Core Spiral Review Convert. Round to the nearest tenth if necessary. 37. 17 ft =
yd
38. 82 in. =
5.MD.1
ft
39. 3 mi =
in meters?
5.MD.1
326 Need more practice? Download more Extra Practice at connectED.mcgraw-hill.com.
Copyright © McGraw-Hill Education
40. A skyscraper is 0.484 kilometer tall. What is the height of the skyscraper
ft
Thee Nu Th Number Num mber Sy System ste tem
Lesson 8
Divide Fractions What You'll Learn
Essential Question
Scan the lesson. Predict two things you will learn about dividing fractions. •
WHAT happens when you add, subtract, multiply, and divide fractions?
Common Core State Standards
•
Content Standards 7.NS.2, 7.NS.2c, 7.NS.3, 7.EE.3
Mathematical Practices 1, 3, 4, 5
Real-World Link Oranges Deandre has three oranges and each orange is divided 1 evenly into fourths. Complete the steps below to find 3 ÷ _ .
Copyright © McGraw-Hill Education
Pam McLean/The Image Bank/Getty Images
4
Step 1
Draw three oranges. The first one is drawn for you.
Step 2
Imagine you cut each orange into fourths. Draw the slices for each orange.
1 So 3 ÷ _ = 12. Deandre will have 4
orange slices.
1 . Use a diagram. 1. Find 3 ÷ _ 2
1 and 3 × 2? 2. What is true about 3 ÷ _ 2
connectED.mcgraw-hill.com
Lesson 8 Divide Fractions
327
Key Concept
Watch
Divide Fractions D Words
To divide by a fraction, multiply by its multiplicative inverse, or reciprocal.
Examples
Numbers
Work Zone
Algebra
_a ÷ _c = _a · _d , where b, c, d ≠ 0
_7 ÷ _3 = _7 · _4 4
8
8
3
b
d
b
c
1 1 Dividing 3 by _ is the same as multiplying 3 by the reciprocal of _ , 4 4 which is 4. reciprocals
_1 = 12
3÷
3 · 4 = 12
4
same result
STOP
an d Re fl ec t
al What is the recip4roc 2 of 15? of –_ ? Write _ of 3 ? 9 your answers below.
Is this pattern true for any division expression?
_7
3 8 7 ÷ _, which can be rewritten as _ . Consider _
_3
4
8
_7
_7 × _4
_3
_3 × _4
3 8 8 _ =_ 4
3
_7
_4
4
× 8 3 =_ 1
4
Multiply the numerator and denominator by the 3 4 , which is _ . reciprocal of _ 4
3
_3 × _4 = 1 4
3
7 _ =_ ×4 8
3
7 _ 7 _ So, _ ÷ 3 =_ × 4 . The pattern is true in this case. 8
4
8
3
Tutor
Examples 1.
_
Find 1 ÷ 5.
_1 ÷ 53= _1 ÷ _5 1 3 3
5 1 Multiply by the reciprocal of _ , which is _ .
1 =_
Multiply.
15
5
1
5
Copyright © McGraw-Hill Education
1 _ =_ ×1 3
328 Chapter 4 Rational Numbers
A whole number can be written as a fraction over 1.
The Number System
2.
Find
_3 ÷ (-_1 ). Write in simplest form. 4
2
_1 Estimate 1 ÷ - 2 =
( )
_3 ÷ (-_1 ) = _3 · (-_2 ) 2
4
Multiply by the reciprocal of -_, which is -_ . 1 1 2
1
4
2
1
3 2 = _ · (-_) 4
Divide 4 and 2 by their GCF, 2.
1
2
1 = -_ or -1_ 3 2
2
1 Check for Reasonableness -1_ ≈ -2 2
Got It?
Show your . work
Multiply.
a.
Do these problems to find out.
b.
Divide. Write in simplest form. b. -_ ÷_ 5 9
3 _ a. _ ÷1 4
4
4
c. -_ ÷ -_
8
5 6
( 23 )
c.
Divide Mixed Numbers To divide by a mixed number, first rename the mixed number as a fraction greater than one. Then multiply the first fraction by the reciprocal, or multiplicative inverse, of the second fraction. Tutor
Example 3.
_ _
Find 2 ÷ 3 1 . Write in simplest form. 3
3
10 _2 ÷ 3_1 = _2 ÷ _ 3
3
3
3
2 _ =_ · 3 3
10
1
1
3
10
1
5
2 _ =_ · 3 1 =_ 5
Copyright © McGraw-Hill Education
Got It?
1 Rename 3_ a fraction greater than one. 3
10 3 Multiply by the reciprocal of _ , which is _ . 3
Divide out common factors.
Multiply.
d.
Do these problems to find out.
Divide. Write in simplest form. 1 d. 5 ÷ 1_ 3
10
3 1 e. -_ ÷ 1_ 4 2
e. 1 f. 2_ ÷5 3
f. Lesson 8 Divide Fractions
329
Watch Tutor
Example 4.
_
The side pieces of a butterfly house are 8 1 inches long. 4 How many side pieces can be cut from a board measuring 49 1 inches long?
_ 2
1 1 by 8_ . To find how many side pieces can be cut, divide 49_ 2
4
Estimate Use compatible numbers. 48 ÷ 8 = 6
1 1 _ _ 49_ ÷ 8_ = ÷ 33 2
99 2
4
4
99 _ =_ · 4 2
33
3
2
2
33
1
1
99 _ =_ · 4
=_ or 6 1 6
Rename the mixed numbers as fractions greater than one. 33 4 Multiply by the reciprocal of _ , which is _ . 4
33
Divide out common factors.
Multiply.
So, 6 side pieces can be cut. Check for Reasonableness Compare to the estimate. 6 = 6
Check
Guided Practice Divide. Write in simplest form.
(Examples 1 – 3)
1 1 1. _ ÷_= 8
2. -3 ÷ -_ = 7
( 6)
3
3 3. -_ ÷ _ = 7 8
4
Show your . work
1 2 miles in 1_ hours. 4. On Saturday, Lindsay walked 3_ 2
5
What was her walking pace in miles per hour? Write in
Rate Yourself! Are you ready to move on? Shade the section that applies.
simplest form. (Example 4)
Creatas/PunchStock
5.
Bulding on the Essential Question How is dividing fractions related to multiplying? Tutor
Time to update your Foldable!
330 Chapter 4 Rational Numbers
Copyright © McGraw-Hill Education
For more help, go online to access a Personal Tutor.
Name
My Homework eHelp
Independent Practice Divide. Write in simplest form.
Go online for Step-by-Step Solutions
(Examples 1 – 3)
3 _ 1. _ ÷6=
2. -_ ÷ -_ =
1 1 ÷7_ = 3 _ 2 2
4. 6 ÷ -_ =
5. -_ ÷ (-2) =
2 1 6. _ ÷ 2_ =
8
2 3
7
( 12 )
( 12 )
4 9
3
7 Cheryl is organizing her movie collection. If each movie case is _3 inch wide, how many movies can fit on a shelf 5_1 feet wide? 4
4
8. Use the table to solve. Write your answers in simplest form. a. How many times as heavy is the Golden Eagle as the Red-Tailed Hawk?
2
(Example 4)
Bird
Maximun Weight (lb)
Golden Eagle
9.
10
Northern Bald Eagle
9 9_
Red-Tailed Hawk
1 3_
b. How many times as heavy is the Golden Eagle as the Northern Bald Eagle?
9 13_
10 2
Model with Mathematics Draw a model of the verbal expression below and then evaluate the expression. Explain how the model shows the division process.
Copyright © McGraw-Hill Education
Photodisc/Getty Images
one half divided by two fifths
Show your . work
Lesson 8 Divide Fractions
331
Copy and Solve For Exercises 10 and 11, show your work on a separate piece of paper. 10.
Multiple Representations Jorge recorded the distance that five of his friends live from his house in the table shown. a. Numbers Tye lives about how many times farther away than Jamal? b. Algebra The mean is the sum of the data divided by the number of items in the data set. Write and solve an equation to find the mean number of miles that Jorge’s friends live from his house. Write your answer in simplest form.
Student
Miles
Lucia
1 5_
Lon
2 8_
Sam
5 12_
Jamal Tye
2 3
6
7 2_ 9
13 17_ 18
c. Model Draw a bar diagram that can be used to find how many more miles Lon travels than Lucia to get to Jorge’s house.
1 11. Tara bought a dozen folders. She took _ of the dozen and then divided 3
the remaining folders equally among her four friends. What fraction of the dozen did each of her four friends receive? How many folders was this per person?
H.O.T. Problems 12.
4 _ Find the Error Blake is finding _ ÷ 6 . Find his 5 7 mistake and correct it.
_4 ÷ _6 = _5 ∙ _6 5
13.
5 a Persevere with Problems If _ is divided by a certain fraction _ , 6
a 1 . What is the fraction _ ? the result is _ 4
14.
7 4 7 30 1 =_ or 1_ 28 14
b
b
1 Reason Inductively So far, the Rabun family has traveled 30 miles in _ 2 hour. If it is currently 3:00 P.M. and their destination is 75 miles away from them, at what time will the Rabun family reach their destination? Explain how you solved the problem Copyright © McGraw-Hill Education
332 Chapter 4 Rational Numbers
Name
My Homework
Extra Practice Divide. Write in simplest form. _2 5 _ 15. _ ÷5= 3 6 9 _5 ÷ _5 = _5 × _6 9 6 9 5 1
2 1 16. -5_ ÷ -2_ = 7
(
7
)
1 _ 17. -5_ ÷2= 5
3
2
5 _ =_ ×6
Homework Help
9 5 3
1
=_ 1×2 3 ×1 2 =_ 3
1 18. Vinh bought 4_ gallons of ice cream to 2
1 19. William has 8_ cups of fruit juice. If he 4
1 serve. If a pint is _ of a gallon, how many
3 divides the juice into _ -cup servings, how
pint-sized servings can be made?
many servings will he have?
8
4
1 Justify Conclusions So far, a storm has traveled 35 miles in _ hour.
20.
2
If it is currently 5:00 P.M. and the storm is 105 miles away from you, at what time will the storm reach you? Explain how you solved the problem.
_2
_1
9
3
1 1 3 9 ÷_ . Write in simplest form. 21. Find _
22.
Use Math Tools Write the letter of each statement below in the section of any operation to which the statement applies.
Addition
Copyright © McGraw-Hill Education
Subtraction
A
Use a common denominator.
B
Multiply by the multiplicative inverse.
C
Write the result in simplest form.
Multiplication
Division
Lesson 8 Divide Fractions
333
for Smarter Balanced 1 23. Tracy has 94_ inches of string that she uses for making bracelets. She uses 4 1 7_ inches of string to make each bracelet. How many bracelets can Tracy make? 4
24. A grocery store offers 4 different size boxes of peanuts as shown below. Large
Medium Small
3
6 4 lb
1
2 4 lb
1
1 8 lb
Write large, medium, or small in each box to make a true statement. The
box is 3 times larger than the
box.
The
box is 6 times larger than the
box.
The
box is 2 times larger than the
box.
Common Core Spiral Review Add or subtract. Write in simplest form. 1 _ 25. _ +1= 5
1 _ 26. _ -1=
4
3
6
5.NF.2
4 _ 27. _ +2 = 9
7
11 _ 28. _ - 3 = 15
29. The cheerleaders made spirit buttons for the basketball team. They used blue and red ribbons. How much total ribbon did they use?
5.NF.2
20
Ribbon Blue Red
_3 ft 8
2
of string?
5
5.NF.2
334 Need more practice? Download more Extra Practice at connectED.mcgraw-hill.com.
8
Copyright © McGraw-Hill Education
1 2 30. How much longer is a 2_ -inch-long piece of string than a _ -inch-long piece
_3 ft
Thee Nu Th Num Num mbeerr Sys mb SSyys yst ste teem tem m
Fashion Designer
Copyright © McGraw-Hill Education
(t)Justin Pumfrey/The Image Bank/Getty Images; (b)DreamPictures/Blend Images/Getty Images
Do you enjoy reading fashion magazines, keeping up with the latest trends, and creating your own unique sense of style? You might want to consider a career in fashion design. Fashion designers create new designs for clothing, accessories, and shoes. In addition to being creative and knowledgeable about current fashion trends, fashion designers need to be able to take accurate measurements and calculate fit by adding, subtracting, and dividing measurements.
Explore college and careers at ccr.mcgraw-hill.com
Is This the Career for You? Are you interested in a career as a fashion designer? Take some of the following courses in high school. Algebra Art Digital Design Geometry Find out how math relates to a career in Fashion Design.
335
A Flair for Fashion! Use the information in the table to solve each problem. Write in simplest form. 1. For size 8, does Dress Style A or B require
4. For Style B, how much more fabric is
more fabric? Explain.
required for size 14 than for size 12? 5. A designer has half the amount of fabric needed to make Style A in size 10. How
2. How many yards of fabric are needed to
much fabric does she have?
make Style A in sizes 8 and 14? 3. Estimate how many yards of fabric are needed to make Style B in each of the sizes shown. Then find the actual amount
1 yards of fabric left on it. 6. A bolt has 12_ 8
How many dresses in Style B size 12 could be made? How much fabric is left over?
of fabric.
Amount of Fabric Needed (yards) Dress Style
Size 8
Size 10
Size 12
Size 14
A
3 3_
1 3_
3 3_
7 3_
B
8 _ 31 4
2 _ 31 2
4 _ 37 8
8
4
Career Project
Copyright © McGraw-Hill Education
336 Chapter 4 Rational Numbers
•
Siede Preis/Photodisc/Getty Images
It’s time to update your career portfolio! Use blogs and webpages of fashion designers to answer some of these questions: Where did they go to school? What was their first job? What do they say is the most difficult part about being a fashion designer? What inspires them to create their designs? What advice do they have for new designers?
Suppose you are a n employer hiring a fa shion designer. What qu estion would you ask a potential employee? •
The Th T hee Num Number Num mbe ber Sy System ste tem
Chapter Review
Check
Vocab
Vocabulary Check
Unscramble each of the clue words. After unscrambling each of the terms, use the numbered letters to find a vocabulary term that relates to all of the other terms. RAB TONNOTIA
TAMTINRINGE
GIEPEATNR
KIEL STAFCOIRN
LUKIEN
NOMMOC NIOAREOMNDT
Complete each sentence using one of the unscrambled words above. 1. The process of using a line over the repeating digits of a decimal is called
.
2. Fractions with different denominators are called
fractions.
3. The least common multiple of the denominators is called the
Copyright © McGraw-Hill Education
least
.
4. The decimal form of a fraction is a(n) 5. A
decimal.
decimal is a decimal in which the repeating digit is zero.
6. Fractions with the same denominator are called
.
Chapter Review
337
Key Concept Check Use Your Use your Foldable to help review the chapter. Tape here
Rule
Rule
Rule
Tab 2
Rule
Tab 1
Operations wit h Fractions
Tape here
Got it? Circle the correct term or number to complete each sentence. 1 1 _ and _ , 3 are like fractions. 1. _ 5
(3 5)
2. To add like fractions, add the (numerators, denominators). 3. To add unlike fractions, rename the fractions using the least common (numerator, denominator). 3
5. To divide by a fraction, (multiply, divide) by its reciprocal. 1 1 6. The least common denominator of _ and _ is (10, 50). 5
338 Chapter 4 Rational Numbers
10
Copyright © McGraw-Hill Education
1 4. The reciprocal of _ is (−3, 3).
The h Number b SSystem Syy tem
Performance Task Managing M i M Money Tamiko has recently started managing her own finances. She tracks her debts and income, as well as any gifts that she receives from her family members. Some of her recent transactions are listed below. Transaction
Amount ($)
Borrowed money from a friend
43.75
Received a gift from Dad
50.00
Spent money on lunches
62.50
Received allowance
20.00
Write your answers on another piece of paper. Show all of your work to receive full credit. Part A What rational number represents the net result of the transactions shown in the table? Explain what your answer represents.
Part B The following week, Tamiko receives a check for $109.60 for working at a local fast food restaurant. Determine the net result of her transactions using the result from Part A. Explain what your answer represents.
Part C 3 Tamiko gets a small bonus check for $32.50. She wants to save _ of this amount. 5 How much will she save?
Copyright © McGraw-Hill Education
Part D Tamiko wants to buy a dress with some of her money. At the store, all dresses are on sale. One dress in particular is on sale for $60 and the original price was $90. What part of the original price does she save? Express your answer as a fraction.
Chapter Review
339
Reflect Answering the Essential Question Use what you learned about operations with rational numbers to complete the graphic organizer. Describe a process to perform each operation.
Add
Subtract
Essential Question WHAT happens when you add, subtract, multiply, and divide fractions?
Multiply
Divide
340 Chapter 4 Rational Numbers
Copyright © McGraw-Hill Education
Answer the Essential Question. WHAT happens when you add, subtract, multipy, and divide fractions?
The Number System
T C E J O UNIT PR Watch
Explore the Ocean Depths For this project, imagine that your dream job is to become an oceanographer. In this project you will: • Collaborate with your classmates as you research information about the ocean. • Share the results of your research in a creative way.
Reflect on how mathematical ideas can be
•
represented.
Collaborate Collaborate
Go Online Work with your group to research and complete each activity. You will use your results in the Share section on the following page. 2 1. About _ of Earth is covered by ocean. 3 Research the five oceans of the world and create a table that shows about 2 what fraction each ocean is of that _ .
2. What is the greatest ocean depth? Find out and then display it on a vertical number line along with other facts about what you can find at different ocean depths.
3. Coral reefs are the home of many ocean creatures. Look up some facts about the state of coral reefs in the world today and display them in a creative way.
4. Choose three different types of whales that live in the ocean. Compare things like their size, the amount of food they eat, or the climate in which they live. Organize the information in a table or graph.
5. Research one of the larger icebergs in the Arctic Ocean. Sketch an image of the iceberg next to a vertical number line that shows the approximate top and bottom 7 of the iceberg. Remember, about _ of an iceberg is under water. 8
Copyright © McGraw-Hill Education
Gerald Nowak/Westend61/Photolibrary
3
Unit 2 Project
341
Share Collaborate
With your group, decide on a way to share what you have learned about ocean depths. Some suggestions are listed below, but you could also think of other creative ways to present your information. Remember to show how you used mathematics in your project! • Use presentation software to organize what you have learned in this project. Share your presentation with the class. • Imagine you need to apply for funds to go on a deep sea exploration. Write a persuasive letter or speech that highlights the importance of studying ocean depths.
connec
t
with Science
Environmental Li teracy Research an animal that lives in the ocean that is on the enda ngered species list. Give a presentation to your class that answers the following questions: • What are some of the causes for the animals being on the endangered species list? • What efforts are currently being made to protect the animal you chose?
Check out the note on the right to connect this project with other subjects.
Reflect On Your Own
6.
Answer the Essential Question How can mathematical ideas be represented? a. How were mathematical ideas involving integers represented in the information you discovered about oceans?
b. How were mathematical ideas involving rational numbers represented in the information you discovered about oceans? Copyright © McGraw-Hill Education
342 Unit 2 Project
