Representing Proportional Relationships
10 min
Teacher Prep
Setup
Narrative
This activity gives students a chance to choose an appropriate scale when graphing a proportional relationship on a given set of blank axes (MP4). Monitor for students who create particularly clear graphs using situation appropriate scales. For example, since the problem is about a car wash, the scale for the axis showing the number of cars does not need to extend into the thousands.
Launch
Arrange students in groups of 2. Provide access to straightedges.
Ask students, “What are some different ways the communities you are a part of raise money for a cause?” (Walk-a-thon, put on an event and sell tickets, car wash, hold a raffle, sell coupon books). After a brief quiet think time, invite students to share their experiences.
Explain that an Origami Club wants to take a trip to see an origami exhibit at an art museum. Then read, or have a student read the Description in the Student Task Statement out loud. Explain that the same information is also shown in the table. Give students 3–4 minutes of quiet work time followed by a whole-class discussion.
Student Task
Here are two ways to represent a situation.
Description:
The Origami Club is doing a car wash fundraiser to raise money for a trip. They charge the same price for every car. After 11 cars, they raised a total of $93.50. After 23 cars, they raised a total of $195.50.
| number of cars |
amount raised in dollars |
|---|---|
| 11 | 93.50 |
| 23 | 195.50 |
Create a graph that represents this situation.
Sample Response
Activity Synthesis (Teacher Notes)
The purpose of this discussion is to introduce students to the term “rate of change.” Begin by inviting 2–3 students to share the graphs they created. Emphasize how different scales can be used, but in order to be helpful, the scale for the number of cars, c, on the horizontal axis should extend to at least 23 and the scale for the amount raised in dollars, m, on the vertical axis should extend to at least 200.
Next, tell students that an equation that represents this situation is m=8.5c, where c is the number of cars, and m is the total dollars raised. Display this equation for all to see, then discuss:
-
“What is the constant of proportionality and what does it mean?” (The constant of proportionality is 8.5 and it means that each car washed raised $8.50.)
-
“How can you see the constant of proportionality in the graph and the table? (Graph: The slope of the line is equivalent to 8.5. Table: For any given row, the amount raised in dollars divided by the number of cars washed equals 8.5.)
-
“Which representation do you think is more useful when calculating the constant of proportionality? Why?”
Explain that the constant of proportionality can be thought of as the rate of change: the amount one variable changes by when the other variable increases by 1. In the case of the Origami Club’s car wash, the rate of change of m, the amount they raise in dollars, with respect to c, the number of cars they wash, is 8.50 dollars per car.
Standards
- 8.EE.5·Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. <em>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</em>
- 8.EE.B.5·Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. <span>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</span>
25 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Representing Proportional Relationships
10 min
Teacher Prep
Setup
Narrative
This activity gives students a chance to choose an appropriate scale when graphing a proportional relationship on a given set of blank axes (MP4). Monitor for students who create particularly clear graphs using situation appropriate scales. For example, since the problem is about a car wash, the scale for the axis showing the number of cars does not need to extend into the thousands.
Launch
Arrange students in groups of 2. Provide access to straightedges.
Ask students, “What are some different ways the communities you are a part of raise money for a cause?” (Walk-a-thon, put on an event and sell tickets, car wash, hold a raffle, sell coupon books). After a brief quiet think time, invite students to share their experiences.
Explain that an Origami Club wants to take a trip to see an origami exhibit at an art museum. Then read, or have a student read the Description in the Student Task Statement out loud. Explain that the same information is also shown in the table. Give students 3–4 minutes of quiet work time followed by a whole-class discussion.
Student Task
Here are two ways to represent a situation.
Description:
The Origami Club is doing a car wash fundraiser to raise money for a trip. They charge the same price for every car. After 11 cars, they raised a total of $93.50. After 23 cars, they raised a total of $195.50.
| number of cars |
amount raised in dollars |
|---|---|
| 11 | 93.50 |
| 23 | 195.50 |
Create a graph that represents this situation.
Sample Response
Activity Synthesis (Teacher Notes)
The purpose of this discussion is to introduce students to the term “rate of change.” Begin by inviting 2–3 students to share the graphs they created. Emphasize how different scales can be used, but in order to be helpful, the scale for the number of cars, c, on the horizontal axis should extend to at least 23 and the scale for the amount raised in dollars, m, on the vertical axis should extend to at least 200.
Next, tell students that an equation that represents this situation is m=8.5c, where c is the number of cars, and m is the total dollars raised. Display this equation for all to see, then discuss:
-
“What is the constant of proportionality and what does it mean?” (The constant of proportionality is 8.5 and it means that each car washed raised $8.50.)
-
“How can you see the constant of proportionality in the graph and the table? (Graph: The slope of the line is equivalent to 8.5. Table: For any given row, the amount raised in dollars divided by the number of cars washed equals 8.5.)
-
“Which representation do you think is more useful when calculating the constant of proportionality? Why?”
Explain that the constant of proportionality can be thought of as the rate of change: the amount one variable changes by when the other variable increases by 1. In the case of the Origami Club’s car wash, the rate of change of m, the amount they raise in dollars, with respect to c, the number of cars they wash, is 8.50 dollars per car.
Standards
- 8.EE.5·Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. <em>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</em>
- 8.EE.B.5·Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. <span>For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.</span>