2.1B Activity: Constant of Proportionality, Unit Rate, and Slope*
Name: Period:
1. The graph below shows the distance a cat is from his bowl of milk over time. Which sentence represents
the graph?
A. The cat was 12 feet away from the milk and ran toward it reaching it after 4 seconds.
B. The cat was 4 feet away from the milk and ran towards it reaching it after 12 seconds.
C. The cat ran away from the milk at a rate of 3 feet per second.
D. The cat ran away from the milk at a rate of 4 feet per second.
E. The cat was 12 feet away from the milk and ran away from it at a rate of 4 feet per second.
2. Analyze the graph above and write down your observations. What is the cat’s rate of travel?
The cat starts at the bowl of milk. After 4 seconds he is 12 feet away from the milk.
He travels at a rate of 3 ft per second.
3. Create a table at the right to represent the graph and your observations.
4. Is this a proportional relationship? If so, what is the constant of proportionality?
Yes, this is a proportional relationship.
The constant of proportionality is 3 feet per second.
Time (seconds)
Distance (feet)
0 0
1 3
2 6
3 9
4 12
Remember that unit rate describes how many units of one quantity corresponds to one unit of a second quantity. Unit rate can also be called rate of change. In this case, the rate of change is the distance (feet) the cat travels per second.
5. Find the rate of change in this story. Write an equation.
The cat travels 3 feet per second. The equation is f = 3s, where f represents feet and s represents seconds.
(4, 12)
Creating the table helps students find the rate of change if they have not done so already.
Students can begin to think about direction in this problem. Ask students to interpret what it means if the line is going up, and if the line is going down.
Discuss the possibility of a negative rate of change.
feet
seconds
6. Sketch a graph for the other four stories from problem #1 (options A,B,D and E). Label the ordered pair
where your graph crosses the y–axis and label another ordered pair of your choice.
Determine if the
story is proportional or not. If proportional, state the constant of proportionality.
Story A: The cat was 12 feet away from the milk
and ran toward it, reaching it after 4
seconds.
This story is not proportional.
Story B: The cat was 4 feet away from the milk and
ran toward it, reaching it after 12 seconds.
This story is not proportional.
Story D: The cat ran away from the milk at a rate
of 4 feet per second.
This story is proportional. The constant of proportionality is 4.
Story E: The cat was 12 feet away from the milk
and ran away from it at a rate of 4 feet per second.
The story is not proportional.
7. Examine the graphs above that are not proportional. These graphs do not have a constant of proportionality, however, the change in the y-values divided by the change in the x-values
between any two points is constant. These graphs still have a rate of change or unit rate.
State the unit rate for each story.
Story A:
-3 feet per second
Story B:
-1/3 feet per second
Story D:
4 feet per second
Story E:
4 feet per second (0, 12)
(4, 0)
(0, 4)
(12, 0)
(3, 12)
(0, 0)
(1, 16)
(0, 12)
1 2 3 4 5 6 7 8 9101112131415 1
2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 seconds
feet
8. When a linear relationship is proportional, how does the unit rate/rate of change compare to the constant of proportionality? How do they compare in a non-proportional relationship?
When the relationship is proportional, the unit rate and constant of proportionality are the same value.
In a relationship that is not proportional, there is not a constant of proportionality. However, there is a unit rate that describes how many units of the first quantity correspond to one unit of the second quantity.
9. Create a story and graph for the given tables. Determine if the relationship is proportional and
find the unit rate.
a.
Time (seconds)
Distance (meters)
0 0
2 6
4 12
6 18
8 24
10 30
12 36
Story:
Sample answer: The dog ran away from the car. After 12 seconds, the dog was 36 meters away from the car.
Is it proportional? Why or why not?
Yes
Unit Rate:
3 meters per second
b.
Time (seconds)
Distance (meters)
0 24
1 20
2 16
3 12
4 8
5 4
6 0
Story:
Sample Answer: The leopard was 24 meters from the pond. After 6 seconds she reached the pond.
Is it proportional? Why or why not?:
No
Unit Rate:
-4 meters per second
10. DJ took 16 pounds of aluminum cans to the recycling center and received $12.00.
a.What rate did the center pay for the aluminum cans?
$0.75 per pound
b. Is the unit rate a constant of proportionality? Explain your reasoning.
Yes, the unit rate is a constant of proportionality. If he took 0 pounds of aluminum cans to the recycling center, he would receive no money. If the relationship were graphed, the line would go through (0,0) or the origin.
c. Without graphing, describe what the graph would look like for this situation.
The graph would be a line going through the origin.
(0,0)
(12,36) )
(0,24)
(6,0)
1 2 3 4 5 6 7 8 9 101112131415 2
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
0
seconds
meters
1 2 3 4 5 6 7 8 9101112131415 2
4 6 8 10 12 14 16 18 20 22 24
0
seconds
meters
11. Colby has $30 in his piggy bank. Each week, he takes out the same amount of money. After 5 weeks, he has no money left in his piggy bank.
a. How much money does Colby take out of his piggy bank each week? Explain your reasoning.
He took out $6 per week. If you divide 30 by 5, you get 6, which is the unit rate
b. Is the unit rate a constant of proportionality? Explain why or why not.
No, the unit rate is not a constant of proportionality. Colby starts with $30 in his
account. If you were to graph this relationship, the line would not go though the origin.
This is not a proportional relationship, so there is not a constant of proportionality.
c. Without graphing, describe what the graph would look like for this situation.
The graph would be a line starting at 30 and going down. It would intersect the x-axis at 5.
12. Examine the graphs below. Determine if the graphs show a proportional relationship and state the rate.
a.
Proportional? Why or why not? Unit Rate:
Yes 3
b.
Proportional? Why or why not? Unit Rate:
Yes (unit rate answers may vary)
c.
Proportional? Why or why not? Unit Rate:
No 2
d.
Proportional? Why or why not? Unit Rate:
Yes 3 These problems may be challenging. Encourage students to add missing intervals on the x-axis and y-axis if that helps them to better see a unit rate. Each of these graphs could be projected on a document camera so that students can show how they found the unit rate.
5 10
0 5 10
0
x y
2 4
0 2 4
0
x y
5 10
0 5 10 15
0 x
y
Students may struggle with finding the unit rate because rise/run has not been discussed yet.
13. How do you think the unit rate (or rate of change) is related to the steepness of the line?
What makes
the line steeper? What makes the line less steep?
The unit rate or rate of change shows the ratio of the difference in y-values to the difference in
x-values. A greater unit rate makes the line steeper, and a smaller unit rate makes the line less steep.
The unit rate is the ratio of the difference in y-values to the difference in x-values between any two points. This unit rate describes the steepness of a line and represents the slope of the line.
The slope, just like the unit rate, represents the incremental change that is occurring on the line.
