Warm Up
1. Traveling at a constant rate, a bus travels 14 miles in 15 minutes. At the same rate, what distance
will the bus travel in 50 minutes?
2. Working at a constant rate, a copy machine makes 210 copies in five minutes. At the same rate,
how many copies can the machine make in twelve minutes?
3. Each robotic operated arm at a candy factory packages 1,350 chocolate covered cherries every
15 minutes. Working at this rate, how long would it take one robotic operated arm to package
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10,800 chocolate covered cherries?
2.1 Introduction to Direct Variation • 67C
Follow Up
Assignment
Use the Assignment for Lesson 2.1 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 2.1 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 2.
Check for Students’ Understanding
Which of the following tables describes a direct variation?
1.
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2.
x
y
1
10
2
11
4
13
5
14
x
y
0
0
1
6
3
18
4
24
2.1 Introduction to Direct Variation • 74A
3.
y
0
4
1
8
2
12
3
16
x
y
1
30
2
15
4
10
5
5
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4.
x
74B • Chapter 2 Direct Variation and Constant of Proportionality
Warm Up
Explain why or why not each table describes two quantities that directly vary.
1.
2.
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3.
x
y
7
10
8
11
9
12
10
13
x
y
0
0
1
1
2
4
4
16
x
y
3
9
4
12
5
15
6
18
2.2 Determining Equivalent Ratios • 75C
Follow Up
Assignment
Use the Assignment for Lesson 2.2 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 2.2 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 2.
Check for Students’ Understanding
1. Create a table of values that describes two quantities that directly vary.
x
y
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2. How did you determine the values in the table for Question 1?
84 • Chapter 2 Direct Variation and Constant of Proportionality
3. Create a table of values that describes two quantities that do not directly vary.
x
y
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4. How did you determine the values in the table for Question 3?
2.2 Determining Equivalent Ratios • 84A
Warm Up
Washington Middle School collects canned food for a local community service food bank. Last year,
102 cans of food were contributed by 180 students.
1. Write the ratio representing the number of cans of food contributed to the total number of
students participating in the food collection.
2. This year, 210 students contributed cans of food for the annual food drive. Assume the number of
cans of food contributed per student for both years is the same. How many cans of food should
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the school expect to be to be contributed this year?
2.3 Determining and Applying the Constant of Proportionality • 85C
Follow Up
Assignment
Use the Assignment for Lesson 2.3 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 2.3 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 2.
Check for Students’ Understanding
Tyler can solve an average of 7.5 multiplication problems per minute.
1. Complete the table of values.
Time (minutes)
Number of problems
(problems)
1
2
3
4
5
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2. Define variables for the quantities that are changing in this problem situation.
3. Write a proportion showing the relationship between the number of minutes Tyler solves
multiplication problems, and the total number of problems Tyler solves.
2.3 Determining and Applying the Constant of Proportionality • 98A
4. Use the proportion to determine how many problems Tyler can solve in 15 minutes.
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5. Use the equation to determine the time it takes Tyler to solve 525 multiplication problems.
98B • Chapter 2 Direct Variation and Constant of Proportionality
Warm Up
Jeff’s new car can travel 14 miles per gallon of gas consumed.
1. Complete the table showing the proportional relationship between the gasoline consumption and
the resulting distance traveled.
Gasoline
(Gallons)
Distance Traveled
(Miles)
0
1
28
3
56
2. What is the constant of proportionality, and what does it represent in the situation?
3. Write an equation for the proportional relationship between gasoline consumed and distance
traveled and the constant of proportionality.
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4. Use the equation to determine the number of miles traveled on 12 gallons of gasoline.
2.4 Using the Constant of Proportionality to Solve Proportions • 99C
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5. Use the equation to determine the number of gallons of gasoline consumed on a 238 mile trip.
99D • Chapter 2 Direct Variation and Constant of Proportionality
Follow Up
Assignment
Use the Assignment for Lesson 2.4 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 2.4 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 2.
Check for Students’ Understanding
Jimmy is driving home for the holiday season. Traveling on the interstate, he was able to maintain a
constant speed of 60 miles per hour.
1. The table describes the time driven and the distance traveled.
Complete the table using the rate Jimmy was driving.
Time (hours)
Distance (miles)
0
60
2
180
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4
2. How did you determine the time it took Jimmy to drive 180 miles?
3. How did you determine the distance Jimmy drove in four hours?
2.4 Using the Constant of Proportionality to Solve Proportions • 104A
4. Is the relationship between the time and the distance an example of a direct proportion? Explain.
5. What is the constant of proportionality?
6. What does the constant of proportionality represent in this situation?
7. Write an equation representing the relationship between the time driven and the distance traveled.
Let d represent the distance driven, and t represent the time.
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104B • Chapter 2 Direct Variation and Constant of Proportionality
Warm Up
A baby elephant nurses for the first two years of its life. Shortly after birth, the baby elephant searches
for its mother’s milk. It drinks about 10 liters of milk every day.
1. Define variables and write an equation representing the relationship between the amounts of milk
the baby elephant consumes and the time it spends consuming the milk. Assume the elephant
maintains the same rate of consumption.
2. Name the constant of proportionality and describe what it represents in this problem situation.
3. Use your equation to complete the table of values showing the amount of time spent consuming
milk, and the amount of milk consumed.
Time
(days)
Amount of Milk
(liters)
1
20
3
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4
50
2.5 Graphing Direct Proportions • 105C
4. Graph the values in the table you completed on the coordinate plane shown. Graph the time, in
days, on the x-axis, and graph the amount of milk consumed, in liters, on the y-axis.
55
y
50
Amount of Milk (liters)
45
40
35
30
25
20
15
10
5
0
1
2
3
Time (days)
4
5
x
5. What do you notice about the points on the graph?
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6. Did you connect the points on your graph? Why or why not?
105D • Chapter 2 Direct Variation and Constant of Proportionality
Follow Up
Assignment
Use the Assignment for Lesson 2.5 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 2.5 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 2.
Check for Students’ Understanding
On an average, Emma can swim 20 laps in 25 minutes.
1. How long does it take Emma to swim one lap?
2. Define the variables and write an equation to represent the relationship between the number of
laps Emma swims and the time. Assume the rate of swimming is constant.
3. Name the constant of proportionality and describe what it represents in this problem situation.
4. Complete the table of values showing the amount of time spent swimming, and the number of
Number of Laps
(laps)
Time
(minutes)
1
2.5
3
4
6.25
118 • Chapter 2 Direct Variation and Constant of Proportionality
© 2011 Carnegie Learning
laps swam using your equation.
5. Graph the values in the table you completed on the coordinate plane shown. Graph the number
of laps on the x-axis, and graph the time on the y-axis.
7
y
Time (minutes)
6
5
4
3
2
1
0
1
2
3
4
Number of Laps (laps)
5
x
6. What do you notice about the points on the graph?
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7. Did you connect the points on your graph? Why or why not?
2.5 Graphing Direct Proportions • 118A
Warm Up
Months
Milk
(Liters)
0
0
1
8
2
16
3
24
4
32
5
40
6
48
1. Describe one possible situation that could be represented by this table of values.
2. Does the time in terms of months vary directly with the number of liters of milk? Explain.
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3. What is the constant of proportionality and what does it represent in the situation?
4. If the table values were used to create a graph, would the points appear to be linear? Explain.
5. If the table values were used to create a graph, would the line begin at the origin? Explain.
2.6 Using Direct Proportions • 119C
Follow Up
Assignment
Use the Assignment for Lesson 2.6 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 2.6 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 2.
Check for Students’ Understanding
1. Complete the table in a way that shows the gasoline consumption varies directly with the
distance traveled.
Gasoline
(Liters)
Distance Traveled
(Kilometers)
0
1
2
3
4
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2. Explain how you determined the unknown values in the table.
3. What is the constant of proportionality and what does it represent in the situation?
2.6 Using Direct Proportions • 126A
4. Complete the table in a way that shows the gasoline consumption does not vary directly with the
distance traveled.
Gasoline
(Liters)
Distance Traveled
(Kilometers)
0
1
2
3
4
5. Explain how you determined the unknown values in the table.
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6. What is the constant of proportionality and what does it represent in the situation?
126B • Chapter 2 Direct Variation and Constant of Proportionality
Warm Up
Determine if each describes a directly proportional relationship between the time and the distance.
Explain your reasoning.
If the relationship between the time and the distance is not directly proportional, make changes in the
data such that the result is a directly proportional relationship.
Time
Distance
1
2
2
4
3
8
4
16
5
32
© 2011 Carnegie Learning
1.
2.7 Interpreting Multiple Representations of Direct Proportions • 127C
2.
60
y
55
50
45
Distance
40
35
30
25
20
15
10
5
0
1
2
3
4
5
x
© 2011 Carnegie Learning
Time
127D • Chapter 2 Direct Variation and Constant of Proportionality
Follow Up
Assignment
Use the Assignment for Lesson 2.7 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 2.7 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 2.
Check for Students’ Understanding
An ant can lift objects that weigh 20 times its weight. This is the equivalent of a 220 pound (100
kilogram) man lifting over 4,400 pounds.
1. Use the information to write a statement that models a direct proportion relationship.
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2. Use your statement to create a table of values.
3. Use your table of values to determine the constant of proportionality and describe what it
represents in this problem situation.
2.7 Interpreting Multiple Representations of Direct Proportions • 136A
4. Use the constant of proportionality to write an equation that represents the relationship between
the two quantities.
5. Design a question that can be answered by using your equation.
6. Use your equation to answer your question.
8. Design a question that can be answered using your graph.
9. Use your graph to answer your question.
136B • Chapter 2 Direct Variation and Constant of Proportionality
© 2011 Carnegie Learning
7. Create a graph using your table of values.