Unit 5, Lesson 4
Introduction to Linear Relationships
Accelerated 7
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Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the context image. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
MLR7: Compare and Connect
This activity presents students with a situation that is not proportional since there is a non-zero starting amount, and it leads to the definition of a linear relationship. Students explore rate of change, which in this situation is the increase in the height of the stack each time a cup is added to the existing first cup. Students can use any method or representation to solve this problem (MP1).
Monitor for students who use different strategies such as making tables or graphs or writing an equation.
Arrange students in groups of 2–4. Provide access to straightedges and graph paper. Allow 2 minutes of individual think time before students work together in groups.
Encourage students to approach this problem using any method. If needed, ask students to consider the following:
The height each cup adds
If the first cup is different from the others
Creating a graph or table to help reason about the problem
Here is information about the two stacks of styrofoam cups in the photo.
How many cups are needed for a stack with a height of 50 cm?
Poll the class about the number of cups they came up with to reach a total height of 50 cm. If necessary, discuss the meaning of non-whole number answers and whether they are appropriate in this situation.
Display this photo. If the class used real cups, stack enough of them to get to a height of 50 cm.
Invite students who used the different strategies mentioned in the Activity Narrative to share their reasoning with the class.
Ask students if they think the relationship is proportional. If there is disagreement, encourage students to explain their reasoning, but come to the conclusion that it is not proportional. If necessary, show how doubling the number of cups from 6 to 12 did not double the height.
Then explain to students that this situation is an example of a linear relationship: when one quantity has a constant rate of change with respect to the other. In this situation, the total height of the stack has a constant rate of change with respect to the number of cups added. In other words, when the quantity of cups changes by a certain amount, the total height of the stack changes by a set amount.
To Gather
Straightedges
In this activity, students continue to examine the relationship between the number of cups and the height of the stack by creating a graph of the relationship.
Note that students’ graphs may consist of discrete points corresponding to coordinate pairs (number of cups, height) or of the entire line as shown in the solution. It is understood that only the points that represent a whole number of cups have a valid interpretation in the context. This continuous graphical representation of a linear relationship, whether the context is continuous or discrete, is very common and will be seen throughout this unit.
Students connect the slope of the line in their graph with the rate of change, which is the additional height of the stack per cup added. They also find where their graph intersects the
Monitor for students who use different slope triangles when determining the slope of the line in the graph.
Give students 8–10 minutes of group work time and follow with a whole-class discussion.
If students need a reminder of how to find the slope of a line, encourage them to draw a slope triangle.
If you didn’t create your own graph of the situation before, do so now.
The goal of this discussion is for students to see that the rate of change in a linear relationship has the same value as the slope of the line representing the relationship.
Begin by inviting 1–2 previously selected students to share their graph, including slope triangles, for all to see. Discuss with students:
“What is the same and what is different about these graphs?” (The graphs all show the same line. The graphs used different slope triangles to calculate slope, but they all came up with the same value of
“What does the point on the graph
Display this image:
It is important to emphasize here that
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
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
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.
Help us improve by sharing suggestions or reporting issues.
If students compute
“How did you determine the increase per cup?”
“Does your strategy match if you find the increase per cup of the other stack?”
Students may also be looking for exact numbers, even though measurement is approximate. Reassure them that numbers that approximately agree are close enough.