Proportional Relationships

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Proportional Relationships ?

3

MODULE

LESSON 3.1

ESSENTIAL QUESTION

Representing Proportional Relationships

How can you use proportional relationships to solve real-world problems?

LESSON 3.2

Rate of Change and Slope LESSON 3.3

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Interpreting the Unit Rate as Slope

Real-World Video Speedboats can travel at fast rates while sailboats travel more slowly. If you graphed distance versus time for both types of boats, you could tell by the my.hrw.com steepness of the graph which boat was faster.

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Scan with your smart phone to jump directly to the online edition, video tutor, and more.

Interactively explore key concepts to see how math works.

Get immediate feedback and help as you work through practice sets.

67

Reading Start-Up Visualize Vocabulary

Vocabulary Review Words constant (constante) ✔ equivalent ratios (razones equivalentes) proportion (proporción) rate (tasa) ✔ ratios (razón) ✔ unit rates (tasas unitarias)

Use the ✔ words to complete the diagram. Reviewing Proportions

Preview Words 12 inches _______ , 1 foot

2:6, 3 to 4 5 __ 35 __ , 25, __ 10 50 70

$1.25 per ounce

Understand Vocabulary

constant of proportionality (constante de proporcionalidad) proportional relationship (relación proporcional) rate of change (tasa de cambio) slope (pendiente)

1. unit rate

A. A constant ratio of two variables related proportionally.

2. constant of proportionality

B. A rate in which the second quantity in the comparison is one unit.

3. proportional relationship

C. A relationship between two quantities in which the ratio of one quantity to the other quantity is constant.

Active Reading Key-Term Fold Before beginning the module, create a key-term fold to help you learn the vocabulary in this module. Write the highlighted vocabulary words on one side of the flap. Write the definition for each word on the other side of the flap. Use the key-term fold to quiz yourself on the definitions used in this module.

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Unit 2

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Match the term on the left to the definition on the right.

Are YOU Ready? Personal Math Trainer

Complete these exercises to review skills you will need for this module.

Write Fractions as Decimals EXAMPLE

1.7 ___ =? 2.5

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Online Assessment and Intervention

Multiply the numerator and the denominator by a power of 10 so that the denominator is a whole number.

1.7 × 10 17 ______ = __ 2.5 × 10 25

Write the fraction as a division problem. Write a decimal point and zeros in the dividend. Place a decimal point in the quotient. Divide as with whole numbers.

0.68 ⎯ 25⟌ 17.00 -15 0 2 00 -2 00 0

Write each fraction as a decimal.

1. _38

0.3 2. ___ 0.4

0.13 3. ____ 0.2

0.39 4. ____ 0.75

5. _45

0.1 6. ___ 2

3.5 7. ___ 14

7 8. __ 14

0.3 9. ___ 10

Solve Proportions

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EXAMPLE

5 x _ = __ 7 14 5 ×2 x ____ = __ 7×2 14

7 × 2 =14, so multiply the numerator and denominator by 2.

10 __ x __ = 14 14

5 × 2 =10

x = 10 Solve each proportion for x. 20 10 10. __ = __ x 18

x 30 11. __ = __ 12 72

4 12. _4x = __ 16

11 132 ___ 13. __ x = 120

36 14. __ = _4x 48

21 15. _9x = __ 27

24 16. __ = _2x 16

30 17. __ = _6x 15

18 18. _3x = __ 36

Module 3

69

Are YOU Ready? (cont'd) Complete these exercises to review skills you will need for this module.

Write Fractions as Decimals 1 19. Carlo invested in a mutual fund. The value of his investment increased by __ in one 16

1 month, and then decreased by __ the next month. Write these fractions as decimals. 25

Explain your method.

0.48 20. Kay knows that the first step for writing ____ in decimal form is to change the 6.4

denominator to a whole number. Explain how she can do so without changing the

Solve Proportions 21. An architect solved the proportion below in order to complete a scale drawing. Describe how to solve the proportion. Then find x. 36 12 __ = __ x 15

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Unit 2

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value of the fraction. Then use long division to find the decimal value.

LESSON

3.1 ?

Representing Proportional Relationships

ESSENTIAL QUESTION

8.2.3.1 Students will use tables, graphs, and equations to represent proportional situations.

How can you use tables, graphs, and equations to represent proportional situations?

EXPLORE ACTIVITY

Representing Proportional Relationships with Tables

In 1870, the French writer Jules Verne published 20,000 Leagues Under the Sea, one of the most popular science fiction novels ever written. One definition of a league is a unit of measure equaling 3 miles.

A Complete the table. Distance (leagues) Distance (miles)

1

2

6

3

20,000 36

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B What relationships do you see among the numbers in the table?

8_MTXESE052888_31A C For each column of the table, find the ratio of the distance in miles to Kenneth Batelman the distance in leagues. Write each ratio in simplest form. 3 = __ 1

_____ =

2

_____ =

6

36 = _____

__________ =

20,000

D What do you notice about the ratios?

Reflect 1.

If you know the distance between two points in leagues, how can you find the distance in miles?

2.

If you know the distance between two points in miles, how can you find the distance in leagues? Lesson 3.1

71

Representing Proportional Relationships with Equations

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The ratio of the distance in miles to the distance in leagues is constant. This relationship is said to be proportional. A proportional relationship is a relationship between two quantities in which the ratio of one quantity to the other quantity is constant. A proportional relationship can be described by an equation of the form y = kx, where k is a number called the constant of proportionality. y

Sometimes it is useful to use another form of the equation, k = _x  .

EXAMPLE 1 Meghan earns $12 an hour at her part-time job. Show that the relationship between the amount she earned and the number of hours she worked is a proportional relationship. Then write an equation for the relationship.

STEP 2

Make a table relating amount earned to number of hours. Number of hours

1

2

4

8

Amount earned ($)

12

24

48

96

For each number of hours, write the relationship of the amount earned and the number of hours as a ratio in simplest form. amount earned ____________ number of hours

12 __ __ = 12 1 1

24 __ __ = 12 1 2

Math Talk

STEP 3

Describe two real-world quantities with a proportional relationship that can be described by the equation y = 25x.

Write an equation. Let x represent the number of hours. Let y represent the amount earned.

YOUR TURN

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Unit 2

First tell what the variables represent.

Use the ratio as the constant of proportionality in the equation y = kx. 12 The equation is y = __ 1 x or y = 12x.

Online Assessment and Intervention

96 __ __ = 12 1 8

12 Since the ratios for the two quantities are all equal to __ , the 1 relationship is proportional.

Mathematical Processes

Personal Math Trainer

48 __ __ = 12 4 1

3. Fifteen bicycles are produced each hour at the Speedy Bike Works. Show that the relationship between the number of bikes produced and the number of hours is a proportional relationship. Then write an equation for the relationship.

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STEP 1

For every hour Meghan works, she earns $12. So, for 8 hours of work, she earns 8 × $12 = $96.

Representing Proportional Relationships with Graphs You can represent a proportional relationship with a graph. The graph will be a line that passes through the origin (0, 0). The graph shows the relationship between distance measured in miles to distance measured in leagues.

Miles

10

5

(3,9)

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(2,6) (1,3)

O

5

Leagues

10

EXAMPL 2 EXAMPLE

STEP 1

Use the points on the graph to make a table.

10 8 6 4 2 O

Earth weight (lb)

6

12

18

30

Moon weight (lb)

1

2

3

5

6 12 18 24 30

Earth weight (lb)

Find the constant of proportionality. Moon weight __________ Earth weight

1 _ _ = 16 6

2 __ = _16 12

3 __ = _16 18

5 __ = _16 30

The constant of proportionality is 6_1 . STEP 3

Write an equation. Let x represent weight on Earth. Let y represent weight on the Moon. The equation is y = 6_1 x.

1 Replace k with __ in y = kx. 6

YOUR TURN The graph shows the relationship between the amount of time that a backpacker hikes and the distance traveled. 4. What does the point (5, 6) represent? 5. What is the equation of the relationship?

20

Distance (mi)

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STEP 2

Moon weight (lb)

The graph shows the relationship between the weight of an object on the Moon and its weight on Earth. Write an equation for this relationship.

16 12 8

Personal Math Trainer

4 O

5 10 15 20 25

Time (h)

Online Assessment and Intervention

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Lesson 3.1

73

Guided Practice 1. Vocabulary A proportional relationship is a relationship between two quantities in which the ratio of one quantity to the other quantity is / is not

constant.

2. Vocabulary When writing an equation of a proportional relationship in the form y = kx, k represents the

.

3. Write an equation that describes the proportional relationship between the number of days and the number of weeks in a given length of time. (Explore Activity and Example 1) a. Complete the table. Time (weeks)

1

Time (days)

7

2

4

10 56

b. Let x represent

.

Let y represent

.

The equation that describes the relationship is

.

Each table or graph represents a proportional relationship. Write an equation that describes the relationship. (Example 1 and Example 2)

2 4

5

120 34

100 80 60 40 20 O

1

2

3

4

Distance (in.)

? ?

ESSENTIAL QUESTION CHECK-IN

6. If you know the equation of a proportional relationship, how can you draw the graph of the equation?

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Unit 2

5

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Oxygen atoms Hydrogen atoms

Map of Iowa

5. Actual distance (mi)

4. Physical Science The relationship between the numbers of oxygen atoms and hydrogen atoms in water

Name

Class

Date

3.1 Independent Practice

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Online Assessment and Intervention

The table shows the relationship between temperatures measured on the Celsius and Fahrenheit scales. Celsius temperature

0

10

20

30

40

50

Fahrenheit temperature

32

50

68

86

104

122

7. Is the relationship between the temperature scales proportional? Why or why not?

8. Describe the graph of the Celsius-Fahrenheit relationship.

9. Analyze Relationships Ralph opened a savings account with a deposit of $100. Every month after that, he deposited $20 more. a. Why is the relationship described not proportional?

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b. How could the situation be changed to make the situation proportional?

10. Represent Real-World Problems Describe a real-world situation that 1 can be modeled by the equation y = __ 20 x. Be sure to describe what each variable represents.

Look for a Pattern The variables x and y are related proportionally. 11. When x = 8, y = 20. Find y when x = 42. 12. When x = 12, y = 8. Find x when y = 12. Lesson 3.1

75

Snail Crawling

13. The graph shows the relationship between the distance that a snail crawls and the time that it crawls.

10

Time (min)

a. Use the points on the graph to make a table. Distance (in.) Time (min)

b. Write the equation for the relationship and tell what each variable represents.

8 6 4 2 O

10 20 30 40 50

Distance (in.)

c. How long does it take the snail to crawl 85 inches?

Work Area

FOCUS ON HIGHER ORDER THINKING

14. Communicate Mathematical Ideas Explain why all of the graphs in this lesson show the first quadrant but omit the other three quadrants.

15. Analyze Relationships Complete the table. Length of side of square

1

2

3

4

5

Perimeter of square

a. Are the length of a side of a square and the perimeter of the square related proportionally? Why or why not?

b. Are the length of a side of a square and the area of the square related proportionally? Why or why not?

16. Make a Conjecture A table shows a proportional relationship where k is the constant of proportionality. The rows are then switched. How does the new constant of proportionality relate to the original one?

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Area of square

LESSON

3.2 ?

Rate of Change and Slope

8.2.3.2 Students will find a rate of change or a slope.

ESSENTIAL QUESTION How do you find a rate of change or a slope?

Investigating Rates of Change

A rate of change is a ratio of the amount of change in the dependent variable, or output, to the amount of change in the independent variable, or input. Math On the Spot

EXAMPL 1 EXAMPLE

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Eve keeps a record of the number of lawns she has mowed and the money she has earned. Tell whether the rates of change are constant or variable. Day 1

Day 2

Day 3

Day 4

Number of lawns

1

3

6

8

Amount earned ($)

15

45

90

120

STEP 1

Identify the input and output variables. Input variable: number of lawns

STEP 2

Output variable: amount earned

Find the rates of change. change in $ - 15 30 ______ = 45 = __ = 15 Day 1 to Day 2: ____________ 3-1 2 change in lawns

Math Talk

Mathematical Processes

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change in $ - 45 45 ______ Day 2 to Day 3: ____________ = 90 = __ = 15 6-3 3 change in lawns

Would you expect the rates of change of a car’s speed during a drive through a city to be constant or variable? Explain.

change in $ - 90 30 _______ Day 3 to Day 4: ____________ = 120 = __ = 15 8-6 2 change in lawns The rates of change are constant: $15 per lawn.

YOUR TURN 1.

The table shows the approximate height of a football after it is kicked. Tell whether the rates of change are constant or variable. Find the rates of change:

The rates of change are

constant / variable.

Time (s)

Height (ft)

0

0

0.5

18

1.5

31

2

26

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Lesson 3.2

77

EXPLORE ACTIVITY

Using Graphs to Find Rates of Change You can also use a graph to find rates of change.

The graph shows the distance Nathan bicycled over time. What is Nathan’s rate of change?



change in distance 30 ​ ________________          =  ​  = ​ _____  ​   ​= ​ __________ change in time 1 2-1

miles per hour

0change in distance  6       ​ ________________  ​= ​__________   ​= _____ ​   ​ = change in time  4 -

(4,60)

50

(3,45)

40 30

(2,30)

20 10 O

B Find the rate of change from 1 hour to 4 hours.

Distance (mi)

A Find the rate of change from 1 hour to 2 hours.

60

(1,15) 2

4

6

Time (h)

miles per hour

C Find the rate of change from 2 hours to 4 hours.

change in distance 60 ​ ________________        ​= ​ __________   ​= _____ ​   ​ = change in time  4 -

miles per hour

Reflect 2. Make a Conjecture  Does a proportional relationship have a constant rate of change?

3. Does it matter what interval you use when you find the rate of change of a proportional relationship? Explain.

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D Recall that the graph of a proportional relationship is a line through the origin. Explain whether the relationship between Nathan’s time and distance is a proportional relationship.

Calculating Slope m

y

When the rate of change of a relationship is constant, any segment of its graph has the same steepness. The constant rate of change is called the slope of the line.

Run = 3

Rise = 2 x O

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Slope Formula The slope of a line is the ratio of the change in y-values (rise) for a segment of the graph to the corresponding change in x-values (run). y2 - y1 m = _____ x -x 2

1

EXAMPL 2 EXAMPLE

My Notes

Find m, the slope of the line. STEP 1

Choose two points on the line. P1(x1, y1) = (-3, 2) P2(x2, y2) = (-6, 4)

STEP 2

Find the change in y-values (rise = y2 - y1) and the change in x-values (run = x2 - x1) as you move from one point to the other. run = x2 - x1 = -6 - (-3) = -3

rise = y2 - y1 =4-2 =2

Rise = 2

P1 (-3, 2) O

y -y

rise 1 2 ______ m = ___ run = x - x

1

2

2 = ___ -3 = -_2 3

YOUR TURN 4. The graph shows the rate at which water is leaking from a tank. The slope of the line gives the leaking rate in gallons per minute. Find the slope of the line. Rise = Slope =

Run =

y

Amount (gal)

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STEP 3

P2 (-6, 4)

If you move up or right, the change is positive. If you move down or left, the change is negative.

5

Run = -3

Leaking tank

6 4 2 O

x 2

4

6

8 10

Time (min)

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Lesson 3.2

79

Guided Practice Tell whether the rates of change are constant or variable. (Example 1) 1. building measurements

2. computers sold

Feet

3

12

27

75

Week

2

4

9

20

Yards

1

4

9

25

Number Sold

6

12

25

60

3. distance an object falls

4. cost of sweaters

Distance (ft)

16

64

144

256

Number

2

4

7

9

Time (s)

1

2

3

4

Cost ($)

38

76

133

171

5. Find the rate of change from 1 minute to 2 minutes. 400 change in distance ___________ _____ ________________ = = = change in time 2-

ft per min

1000

Distance (ft)

Erica walks to her friend Philip’s house. The graph shows Erica’s distance from home over time. (Explore Activity)

800 600 400 200 O

2

4

6

Time (min)

6. Find the rate of change from 1 minute to 4 minutes. Find the slope of each line. (Example 2) y

y

8.

5

5

x

x -5

O

5

-5

-5

slope =

? ?

O

-5

slope =

ESSENTIAL QUESTION CHECK-IN

9. If you know two points on a line, how can you find the rate of change of the variables being graphed?

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Unit 2

5 © Houghton Mifflin Harcourt Publishing Company

7.

Name

Class

Date

3.2 Independent Practice

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Online Assessment and Intervention

10. Rectangle EFGH is graphed on a coordinate plane with vertices at E(-3, 5), F(6, 2), G(4, -4), and H(-5, -1). a. Find the slopes of each side.

b. What do you notice about the slopes of opposite sides?

c. What do you notice about the slopes of adjacent sides?

11. A bicyclist started riding at 8:00 A.M. The diagram below shows the distance the bicyclist had traveled at different times. What was the bicyclist’s average rate of speed in miles per hour?

8:00 A.M.

4.5 miles

8:18 A.M.

7.5 miles

8:48 A.M.

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12. Multistep A line passes through (6, 3), (8, 4), and (n, -2). Find the value of n.

13. A large container holds 5 gallons of water. It begins leaking at a constant rate. After 10 minutes, the container has 3 gallons of water left. a. At what rate is the water leaking?

b. After how many minutes will the container be empty?

14. Critique Reasoning Billy found the slope of the line through the 2 - (-2) points (2, 5) and (-2, -5) using the equation ______ = _25. What mistake 5 - (-5) did he make?

Lesson 3.2

81

15. Multiple Representations Graph parallelogram ABCD on a coordinate plane with vertices at A(3, 4), B(6, 1), C(0, -2), and D(-3, 1).

10

y

6

a. Find the slope of each side.

2 -6

b. What do you notice about the slopes?

-2 O -2

2

6

x 10

-6

c. Draw another parallelogram on the coordinate plane. Do the slopes have the same characteristics?

-10

FOCUS ON HIGHER ORDER THINKING

Work Area

16. Communicate Mathematical Ideas Ben and Phoebe are finding the slope of a line. Ben chose two points on the line and used them to find the slope. Phoebe used two different points to find the slope. Did they get the same answer? Explain.

18. Reason Abstractly What is the slope of the x-axis? Explain.

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Unit 2

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17. Analyze Relationships Two lines pass through the origin. The lines have slopes that are opposites. Compare and contrast the lines.

LESSON

3.3 ?

Interpreting the Unit Rate as Slope

8.2.3.3 Students will interpret the unit rate as slope.

ESSENTIAL QUESTION How do you interpret the unit rate as slope?

EXPLORE ACTIVITY

Relating the Unit Rate to Slope

A rate is a comparison of two quantities that have different units, such as miles and hours. A unit rate is a rate in which the second quantity in the comparison is one unit.

A Find the slope of the graph using the points (1, 2) and (5, 10). Remember that the slope is the constant rate of change.

Misty Mountain Storm Snowfall (in.)

A storm is raging on Misty Mountain. The graph shows the constant rate of change of the snow level on the mountain.

10

5

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Cavan Images/ Getty Images

O

5

Time (h)

10

B Find the unit rate of snowfall in inches per hour. Explain your method.

C Compare the slope of the graph and the unit rate of change in the snow level. What do you notice?

D Which unique point on this graph gives you the slope of the graph and the unit rate of change in the snow level? Explain how you found the point.

Lesson 3.3

83

Graphing Proportional Relationships Math On the Spot my.hrw.com

You can use a table and a graph to find the unit rate and slope that describe a real-world proportional relationship. The constant of proportionality for a proportional relationship is the same as the slope.

EXAMPLE 1 Every 3 seconds, 4 cubic feet of water pass over a dam. Draw a graph of the situation. Find the unit rate of this proportional relationship. Make a table. Time (s) Volume (ft ) 3

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STEP 2

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STEP 3

Math Talk

In a proportional relationship, how are the constant of proportionality, the unit rate, and the slope of the graph of the relationship related?

6

9

12

15

4

8

12

16

20

Water Over the Dam

Draw a graph. Find the slope. 8 rise _ slope = ___ run = 6

Mathematical Processes

3

= _43

Amount (cu ft)

STEP 1

20 8

10 6 O

10

Time (sec)

20

The unit rate of water passing over the dam and the slope of the graph of the relationship are equal, _34 cubic feet per second.

Reflect What If? Without referring to the graph, how do you know that the point ( 1, _43 ) is on the graph?

YOUR TURN

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84

Unit 2

Tomas rides his bike at a steady rate of 2 miles every 10 minutes. Graph the situation. Find the unit rate of this proportional relationship.

Tomas’s Ride 10

Distance (mi)

2.

5

O

5

Time (min)

10

© Houghton Mifflin Harcourt Publishing Company

1.

Using Slopes to Compare Unit Rates You can compare proportional relationships presented in different ways.

EXAMPL 2 EXAMPLE

Math On the Spot

The equation y = 2.75x represents the rate, in barrels per hour, that oil is pumped from Well A. The graph represents the rate that oil is pumped from Well B. Which well pumped oil at a faster rate? Use the equation y = 2.75x to make a table for Well A’s pumping rate, in barrels per hour. Time (h) Quantity (barrels) STEP 2

STEP 3

Amount (barrels)

STEP 1

Well B Pumping Rate

1

2

3

4

2.75

5.5

8.25

11

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20

10

O

10

Time (h)

20

Use the table to find the slope of the graph of Well A. - 2.75 2.75 ________ = ____ = 2.75 barrels/hour slope = unit rate = 5.5 1 2-1

Use the graph to find the slope of the graph of Well B. rise __ 10 slope = unit rate = ___ run = 4 = 2.5 barrels/hour

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STEP 4

Compare the unit rates. 2.75 > 2.5, so Well A’s rate, 2.75 barrels/hour, is faster.

Reflect 3.

Describe the relationships among the slope of the graph of Well A’s rate, the equation representing Well A’s rate, and the constant of proportionality.

YOUR TURN 4.

The equation y = 375x represents the relationship between x, the time that a plane flies in hours, and y, the distance the plane flies in miles for Plane A. The table represents the relationship for Plane B. Find the slope of the graph for each plane and the plane’s rate of speed. Determine which plane is flying at a faster rate of speed. Time (h) Distance (mi)

1

2

3

4

425

850

1275

1700

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Lesson 3.3

85

Guided Practice Give the slope of the graph and the unit rate. (Explore Activity and Example 1) 1. Jorge: 5 miles every 6 hours

2. Akiko

Jorge

4

8

12

16

Distance (mi)

5

10

15

20

Akiko

O

10

Time (h)

20

20 10

O

10

Time (h)

3. The equation y = 0.5x represents the distance Henry hikes, in miles, over time, in hours. The graph represents the rate that Clark hikes. Determine which hiker is faster. Explain. (Example 2)

20

Clark Distance (mi)

20

10

O

Write an equation relating the variables in each table. (Example 2) 4.

? ?

Time (x)

1

2

4

6

Distance (y)

15

30

60

90

5.

Time (x)

16

32

48

64

Distance (y)

6

12

18

24

ESSENTIAL QUESTION CHECK-IN

6. Describe methods you can use to show a proportional relationship between two variables, x and y. For each method, explain how you can find the unit rate and the slope.

86

Unit 2

10

Time (h)

20

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10

Distance (mi)

Distance (mi)

20

Time (h)

Name

Class

Date

3.3 Independent Practice

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Online Assessment and Intervention

7. A Canadian goose migrated at a steady rate of 3 miles every 4 minutes. a. Fill in the table to describe the relationship. Time (min)

4

8

20

Distance (mi)

9

12

b. Graph the relationship.

c. Find the slope of the graph and describe what it means in the context of this problem.

Migration Flight Distance (mi)

20

10

O

10

Time (min)

20

8. Vocabulary A unit rate is a rate in which the first quantity / second quantity in the comparison is one unit. 9. The table and the graph represent the rate at which two machines are bottling milk in gallons per second. Machine 2 Machine 1 Time (s) Amount (gal)

1

2

3

4

0.6

1.2

1.8

2.4

Amount (gal)

© Houghton Mifflin Harcourt Publishing Company

20

10

O

10

Time (sec)

20

a. Determine the slope and unit rate of each machine.

b. Determine which machine is working at a faster rate.

Lesson 3.3

87

10. Cycling The equation y = _19 x represents the distance y, in kilometers, that Patrick traveled in x minutes while training for the cycling portion of a triathlon. The table shows the distance y Jennifer traveled in x minutes in her training. Who has the faster training rate? Time (min)

40

64

80

96

Distance (km)

5

8

10

12

Work Area

FOCUS ON HIGHER ORDER THINKING

11. Analyze Relationships There is a proportional relationship between minutes and dollars per minute, shown on a graph of printing expenses. The graph passes through the point (1, 4.75). What is the slope of the graph? What is the unit rate? Explain.

13. Critical Thinking The table shows the rate at which water is being pumped into a swimming pool.

Time (min)

2

5

Amount (gal)

36

90

7

126 216

Use the unit rate and the amount of water pumped after 12 minutes to find how much water will have been pumped into the pool after 13_12 minutes. Explain your reasoning.

88

Unit 2

12

© Houghton Mifflin Harcourt Publishing Company

12. Draw Conclusions Two cars start at the same time and travel at different constant rates. A graph for Car A passes through the point (0.5, 27.5), and a graph for Car B passes through (4, 240). Both graphs show distance in miles and time in hours. Which car is traveling faster? Explain.

MODULE QUIZ

Ready

Personal Math Trainer

3.1 Representing Proportional Relationships 1. Find the constant of proportionality for the table of values.

Online Assessment and Intervention

x

2

3

4

5

y

3

4.5

6

7.5

my.hrw.com

2. Phil is riding his bike. He rides 25 miles in 2 hours, 37.5 miles in 3 hours, and 50 miles in 4 hours. Find the constant of proportionality and write an equation to describe the situation.

3.2 Rate of Change and Slope Find the slope of each line. y

3.

y

4.

4

4

2

2 x

x -4

-2

O

2

4

-2

-4

-4

-2

O

2

4

-2 -4

© Houghton Mifflin Harcourt Publishing Company

3.3 Interpreting the Unit Rate as Slope 5. The distance Train A travels is represented by Time (hours) Distance (km) d = 70t, where d is the distance in kilometers and 2 150 t is the time in hours. The distance Train B travels 4 300 at various times is shown in the table. What is the 5 375 unit rate of each train? Which train is going faster?

ESSENTIAL QUESTION 6. What is the relationship among proportional relationships, lines, rates of change, and slope?

Module 3

89

MODULE 3 MIXED REVIEW

Personal Math Trainer

Assessment Readiness my.hrw.com

Selected Response

5. What is the slope of the line below? y

1. Which of the following is equivalent to 5–1? C

2

D -5

x

2. Prasert earns $9 an hour. Which table represents this proportional relationship? A

Hours Earnings ($)

4 36

6 54

8 72

B

Hours Earnings ($)

4 36

6 45

8 54

C

Hours Earnings ($)

2 9

3 18

4 27

D

Hours Earnings ($)

2 18

3 27

4 54

3. A factory produces widgets at a constant rate. After 4 hours, 3,120 widgets have been produced. At what rate are the widgets being produced? A 630 widgets per hour B 708 widgets per hour C

4

-_15

780 widgets per hour

D 1,365 widgets per hour

4. A full lake begins dropping at a constant rate. After 4 weeks it has dropped 3 feet. What is the unit rate of change in the lake’s level compared to its full level? A 0.75 feet per week

-4

-2

D −1.33 feet per week

1 B -_ 2

_ 1 2

D 2

6. Jim earns $41.25 in 5 hours. Susan earns $30.00 in 4 hours. Pierre’s hourly rate is less than Jim’s, but more than Susan’s. What is his hourly rate? A $6.50

C

B $7.75

D $8.25

$7.35

Mini-Task 7. Joelle can read 3 pages in 4 minutes, 4.5 pages in 6 minutes, and 6 pages in 8 minutes. a. Make a table of the data. Minutes Pages

b. Use the values in the table to find the unit rate.

c. Graph the relationship between minutes and pages read. y 6 4 2 O

x 2

4

6

Minutes

90

Unit 2

4

-2

C

Pages

−0.75 feet per week

2

A -2

B 1.33 feet per week C

O

© Houghton Mifflin Harcourt Publishing Company

A 4 1 B _ 5

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Proportional Relationships

Proportional Relationships ? 3 MODULE LESSON 3.1 ESSENTIAL QUESTION Representing Proportional Relationships How can you use proportional relatio...

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