Unit 5, Lesson 4
Introduction to Linear Relationships
Accelerated 7
Preparation
Lesson Narrative
In this lesson, students examine linear relationships with positive rates of change. They consider a situation where the height of a stack of styrofoam cups is not proportional to the number of cups in the stack. Using graphs, tables, and equations, they observe that each cup increases the height of the stack by the same amount. Since one quantity has a constant rate of change with respect to the other, this is an example of a linear relationship.
Using this context, students identify and describe similarities and differences between linear relationships and proportional relationships. They observe that the rate of change of the relationship and the slope of a line representing the relationship have the same value.
The meaning of the vertical intercept of the graph comes up briefly and will be revisited more fully in upcoming lessons, so it is not necessary for students to identify it at this time. In this lesson, the focus is on rate of change in linear relationships that are not proportional.
The activities in this lesson were written using a particular type of cup. Photos are included of all measurements needed, so this lesson can be used without any additional preparation. However, if desired, the lesson can be modified so that students measure stacks of actual cups. Teachers wanting to use real cups should measure the height of two stacks ahead of time, find the rate of change and make sure that it is approximately constant. Note that rounding error will likely play a role, so some flexibility in answers will be necessary if students measure actual stacks of cups.
- Compare and contrast (orally and in writing) proportional and nonproportional linear relationships.
- Interpret (orally and in writing) features of the graph (i.e., slope and y-intercept) of a non-proportional linear relationship.
Let’s explore some relationships between two variables.
Required Preparation
None
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.