planes, 3 trains, and paychecks - kyrene school district€¦ · a cryptarithm is a puzzle which...

LESSON 3: Planes, Trains, and Paychecks • M3-193

LEARNING GOALS• Write and solve equations that represent

relationships given in tables, graphs, and situations.

• Identify independent and dependent quantities represented in tables, graphs, and scenarios.

• Analyze the relationship between the independent and dependent quantities in a situation using graphs, tables, and equations.

• Determine whether the data represented in a graph of an equation are discrete or continuous.

You have identified independent and dependent quantities in relationships and have expressed these relationships using equations. How can you relate independent and dependent quantities in a variety of different situations, using a variety of different representations?

WARM UPIdentify two quantities in each situation. Then state which quantity depends on the other.

1. Snow is falling at a rate of 3 inches per hour.

2. The outside temperature is increasing at an average of 10 degrees each day.

3. The car wash generated an average profit of $8.00 per car.

4. The income from the sale of movie tickets was $8.50 per person.

5. A dog groomer charges $35 for every dog.

Planes, Trains, and PaychecksMultiple Representations of Equations

3

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M3-194 • TOPIC 3: Graphing Quantitative Relationships

Getting Started

To the Equation-Mobile!

A mobile (MO-beel) is hanging art. It features all kinds of different objects suspended from string or wire. Balance is important to the visual effect of mobiles.

Determine what value each shape represents in each mobile.

1. 36

2. 30



Equations from TablesACTIVIT Y

3.1

SnapSmart charges the same price for each 3 in. by 4 in. picture print.

1. The table shows a few orders and the cost of each.

a. What is the cost of one print? Explain how you determined the cost.

b. Define variables for the quantities in this situation. Then write an equation that models the relationship between these quantities.

c. Create a graph for this situation.

020 40 60 80 100 120

Number of Prints

x

y

Cos

t (d

olla

rs)

2

4

6

8

10

12

14

16

d. Tell whether the quantities in the SnapSmart scenario are discrete or continuous. Explain your reasoning.

e. You can draw a line to show the shape of the graph. Do all the points on the line make sense in this scenario?

Number of Prints

Cost ($)

10 1.20

32 3.84

50 6.00

110 13.20



2. The table shows the cost of a particular item.

a. Describe how the cost is related to the number of items.

b. Define variables for the quantities in this situation. Then write an equation that models the relationship between these quantities.

c. Explain whether the quantities in this situation are discrete or continuous.

d. Create a graph for this situation.

64

Cos

t (d

olla

rs)

10 12 14 1682

10

0

20

30

40

50

60

70

80

90

100

y

x

Number of Items

e. What do you notice about the shape of the graph? You can connect the points to see the shape.

Number of Items

Cost ($)

1 6

8 48

16 96

5 30



3. Read each situation and analyze the corresponding table of values. Identify the independent and dependent quantities in each. Then, write an equation that models the relationship between the quantities.

a. The total profit made on cutting lawns and the profit made by each person are represented in the table shown.

Total Profit Made ($)

Profit Made by Each Person ($)

21 7.00

25.50 8.50

45 15.00

b. The number of boxes of cookies sold and the total profit are represented in the table shown.

Boxes of Cookies Sold Total Profit ($)

3 7.50

5 12.50

7 17.50

c. The number of tiles required to complete a job and the number of tiles ordered are represented in the table shown.

Number of Tiles Required 75 95 115

Number of Tiles Ordered 90 110 130

A table of values

can be represented

vertically or

horizontally.



Equations from GraphsACTIVIT Y

3.2

The graph shows the relationship between the distance of a train from the station and the time in minutes.

1. Complete the table using the points from the graph.

2. Define variables and write an equation to represent the relationship between the quantities.

3. How far is the train from the station after 20 minutes?

4. Connect the points to show the shape of the graph. Do all of the values on the line make sense in this situation?

32Time (minutes)

Dis

tanc

e fr

om S

tatio

n (m

iles)

5 6 7 8x

41

2

0

4

6

8

10

12

14

16

18

20

22

24

y



The graph shows the relationship between the height of a plane and its distance from the airport in miles.

50 100 150 200 250 300Distance (miles)

00

10,000

20,000

Hei

ght

(fee

t)y

x

5. Complete the table using the points from the graph.

6. Write an equation to represent the relationship between the quantities.

7. What is the height of the plane at 150 miles?

8. Connect the points to show the shape of the graph. Do all of the values on the line make sense in this situation?


NOTES


Equations from ScenariosACTIVIT Y

3.3

Crystal got a job working at the local hardware store making $8.76 per hour.

1. Write an equation that models the relationship between the number of hours Crystal worked and how much she earned.

2. How much would Crystal earn if she worked the given times.

a. 5 hours

b. 2 1 __ 2 hours

c. 5 hours and 30 minutes

d. 10 hours and 15 minutes

3. Use your equation to calculate the number of hours Crystal worked given her total pay.

a. $218.75

b. $293.46

c. $203.67



4. Complete the table.

Time Worked (hours) Earnings ($)

5

2.5

5.5

10.25

218.75

293.46

203.67

5. Use the table to complete the graph.

1510Time Worked (hours)

Earn

ing

s (d

olla

rs)

25 30 35 40 45205

25

0

50

75

100

125

150

175

200

225

y

x

6. Connect the points to show the shape of the graph. Do all of the points on the line make sense in this situation?



7. Jake’s dog eats an average of 40 pounds of dry dog food in one month.

a. Write an equation to model the relationship between the number of pounds of dog food and the number of months.

b. Complete the table and graph.

c. Connect the points to show the shape of the graph. Do all of the points on the line make sense in this situation?

1Time (months)

Dog

Foo

d (p

ound

s)

3 4 520

0

50

100

150

y

x

200

Time (months)

Amount of Dog

Food (lb)

0

1

2

120

160

200


NOTESTALK the TALK

Let Us Organize the Ways

Multiple representations—including drawings or diagrams, verbal descriptions, tables, graphs, and equations—can be useful in analyzing and solving problems.

1. Complete the graphic organizer by describing the advantages of each representation.

• verbal• table• graph• equation




VERBAL TABLE

MULTIPLE REPRESENTATIONS

GRAPH EQUATION


Assignment


Practice1. Lashawna works at the local candy shop. The

bulk candy is sold by the pound. Customers

place the candy they would like to buy in a

plastic bucket, and then Lashawna weighs it

to determine how much the customer owes.

Before calculating the price, Lashawna must

subtract the weight of the plastic bucket. The

candy bucket weighs 0.72 pound.

a. Complete the table.

b. Write an equation that models the

relationship between the quantities in

this situation.

c. Use the table to create a graph of the relationship.

d. Explain whether all points on the line make sense.

2. Lashawna is packaging some bulk candy for

a sale. The price is $3.98 per pound.

a. Write an equation to model the

relationship between the total cost and

the weight of the candy.

b. Complete the table.

c. Use the table to create a graph of

the relationship.

d. Explain whether it makes sense to connect

the points on your graph.

RememberWhen a quantity can have values that are only counting numbers,

it is called a discrete quantity. When a quantity can have any value,

it is called a continuous quantity.

WriteCreate an algebraic equation.

Represent the equation using

a word problem, a table, and

a graph.

Total Weight (lb) Weight of Candy (lb)

2.84

3.00

0.71

0.98

1.71

1.13

Total Cost ($) Weight of Candy (lb)

2.50

3.20

4.98

1.97

9.47

13.93



ReviewUse the graph to estimate each solution.

1. How long did it take Serena to travel 70 miles?

Dis

tanc

e (m

iles)

Time (hours)

200

0

50

100

150

x

y

321 4 5

Distance Traveled by Serena

2. How long did it take to burn 100 calories?

Cal

orie

s B

urne

d

Time (minutes)

Calories Burned Exercising200

0

50

100

150

x

y

3015 45 60

Write each statement as an algebraic expression.

3. five less than twice a number

4. seven and one half more than a number

Solve each equation.

5. 20 5 6x

6. 15.5 1 p 5 44

StretchA cryptarithm is a puzzle which replaces digits with letters. Your job is to use reasoning to determine what

digits the letters stand for. When two letters are the same, they represent the same digit. When the letters

are different, they represent different digits.

In this famous cryptarithm, the sum is correct. Can you solve it?

S E N D1 M O R EM O N E Y


planes, 3 trains, and paychecks - kyrene school district€¦ · a cryptarithm is a puzzle which...

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