Unit 3, Lesson 5
Introduction to Linear Relationships
Grade 8
5.1
Warm-up
5.2
Activity
Here is information about the two stacks of styrofoam cups in the photo.
- One stack has 6 cups, and its height is 15 cm.
- The other stack has 12 cups, and its height is 23 cm.
How many cups are needed for a stack with a height of 50 cm?
5.3
Activity
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If you didn’t create your own graph of the situation before, do so now.
- What are some ways to tell that the number of cups is not proportional to the height of the stack?
- What is the slope of the line in your graph? What does the slope mean in this situation?
- At what point does your line intersect the vertical axis? What do the coordinates of this point tell you about the cups?
Student Lesson Summary
A linear relationship is any relationship between two quantities where one quantity has a constant rate of change with respect to the other. For example, Andre babysits and charges a fee for traveling to and from the job, and then a set amount for every additional hour he works. Since the total amount he charges with respect to the number of hours he works changes at a constant rate, this is a linear relationship. But since Andre charges a fee for traveling, and the graph does not go through the point , this is not a proportional relationship. Here is a graph of how much Andre charges based on how many hours he works.
Graph, horizontal axis, time in hours, scale 0 to 9, by 1's. vertical axis, amount earned in dollars, scale 0 to 140, by 20's. line starting at 0 comma 10, passing through 2 comma 40 and 60 comma 100.
The rate of change can be calculated using the graph. Since the rate of change is constant, we can take any two points on the graph and divide the amount of vertical change by the amount of horizontal change. For example, the points and mean that Andre earns 40 dollars for working 2 hours and 100 dollars for working 6 hours. The rate of change is dollars per hour. Andre's earnings go up 15 dollars for each hour of babysitting.
Notice that this is the same way we calculate the slope of the line. That's why the graph is a line and why we call this a “linear relationship.” The rate of change of a linear relationship is the same as the slope of its graph.
Two quantities have a linear relationship when:
- One quantity changes by a set amount, whenever the other quantity changes by a set amount.
- The rate of change is constant.
- The graph of the relationship is a line.
This graph shows a linear relationship between number of days and number of pages read.
When the number of days increases by 2, the number of pages read always increases by 60. The rate of change is constant, 30 pages per day.
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.