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.
Table:
number of
cars
amount raised
in dollars
11
93.50
23
195.50
Create a graph that represents this situation.
Are You Ready for More?
3.2
Activity
Info Gap: Graphing Proportional Relationships
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because . . . .”
Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
If your teacher gives you the data card:
Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.
Student Lesson Summary
Proportional relationships can be represented in multiple ways. Which representation we choose depends on the purpose. And when we create representations we can choose helpful values by paying attention to the context. For example, a stew recipe calls for 3 carrots for every 2 potatoes. One way to represent this is using an equation. If there are potatoes and carrots, then .
Suppose we want to make a large batch of this recipe for a family gathering, using 150 potatoes. To find the number of carrots, we could just use the equation: carrots.
Now suppose the recipe is used in a restaurant that makes the stew in large batches of different sizes depending on how busy of a day it is, using up to 300 potatoes at a time.
Then we might make a graph to show how many carrots are needed for different amounts of potatoes. We set up a pair of coordinate axes with a scale from 0 to 300 along the horizontal axis and 0 to 450 on the vertical axis, because . Then we can read how many carrots are needed for any number of potatoes up to 300.
graph, horizontal axis, number of potatoes, scale 0 to 300, by 50's. vertical axis, number of carrots, scale 0 to 450, by 50's. line passes through origin, 100 comma 150, 150 comma 225, 200 comma 300, 250 comma 375, and 300 comma 450.
number of potatoes
number of carrots
150
225
300
450
450
675
600
900
Or if the recipe is used in a food factory that produces very large quantities and where the potatoes come in bags of 150, we might just make a table of values showing the number of carrots needed for different multiples of 150.
No matter the representation or the scale used, the constant of proportionality, , is evident in each. In the equation, it is the number we multiply by. In the graph, it is the slope. In the table, it is the number we multiply values in the left column to get values in the right column. We can think of the constant of proportionality as a rate of change:the amount one variable changes by when the other variable increases by 1. In this case, the rate of change of with respect to is carrots per potato.
Glossary
rate of change (in a linear relationship)
The rate of change is the amount changes when increases by 1. On a graph, the rate of change is the slope of the line.
In this graph, increases by 15 dollars when increases by 1 hour. The rate of change is 15 dollars per hour.
Graph, horizontal axis, x, amount earned in hours. vertical axis, y, amount earned in dollars. line starting at 0 comma 10, passing through 2 comma 40 and 60 comma 100.
