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.

If students compute or and use this as the increase per cup, consider asking:

• “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.

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.

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 -axis and interpret this value in terms of the situation: the height of the bottom part of the first cup, below the rim (MP2). 

Monitor for students who use different slope triangles when determining the slope of the line in the graph.

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 mean?” (It tells the height of the first cup not including the rim.)

• “How can you see on the graph the amount that one cup adds to the height?” (In a slope triangle with a horizontal distance of 1 on the graph, the vertical distance will show the amount each cup adds to the height.)

Display this image:

It is important to emphasize here that is not the constant of proportionality, since this is not a proportional relationship. This value is how much each cup adds to the height of the stack, and is called the “rate of change.” The rate of change in a linear relationship between and is the increase in when increases by 1. Note that the rate of change of the relationship has the same value as the slope of the line representing the relationship.