Unit 3, Lesson 3
Representing Proportional Relationships
Grade 8
3.1
Warm-up
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.
Table:
| number of cars |
amount raised in dollars |
|---|---|
| 11 | 93.50 |
| 23 | 195.50 |
Create a graph that represents this situation.
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.
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.