After revisiting examples of proportional relationships in the previous lessons, this lesson is the first of four lessons that moves from proportional relationships to linear relationships with positive rates of change. The two activities use a situation where the height of a stack of styrofoam cups is not proportional to the number of cups in the stack. Students use the same tools they learned to represent proportional relationships in this situation—graphs, tables, and equations. They see that each cup increases the height of the stack by the same amount (unit rate becomes rate of change) and that they can use this to answer questions about the height for an unknown number of cups. They investigate and describe similarities and differences between linear relationships and proportional relationships in this context. They make connections between the rate of changeof the relationship and the slope of a line representing the relationship.
In this lesson, the focus is proportionality vs. linear relationships and rate of change. The meaning of the vertical intercept of the graph comes up briefly but will be revisited more fully in the next lesson.
The first two activities in this lesson use 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 could be modified so that students measure stacks of actual cups.
Lesson overview
 5.1 Warmup: Number Talk: Fraction Division (5 minutes)
 5.2 Activity: Stacking Cups (10 minutes)
 5.3 Activity: Connecting Slope to Rate of Change (15 minutes)
 Lesson Synthesis
 5.4 Cooldown: Stacking More Cups (5 minutes)
Learning goals:
 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 nonproportional linear relationship.
Learning goals (student facing):

Let’s explore some relationships between two variables.
Learning targets (student facing):
 I can find the rate of change of a linear relationship by figuring out the slope of the line representing the relationship.
Required materials:
 Graph paper
 Rulers
Required preparation:
 If students will use actual cups, gather cups and perform activities ahead of time so that you know measurements for your particular type of cup.
Glossary:

linear relationship  A linear relationship between two quantities means they are related like this: When one quantity changes by a certain amount, the other quantity always changes by a set amount. In a linear relationship, one quantity has a constant rate of change with respect to the other. The relationship is called linear because its graph is a line. The graph shows a 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, so the relationship is linear.
 Access the complete Grade 8 glossary.
Standards
 This lesson builds on the standards: CCSS.6.NS.ACCSS.7.RP.A.2.aMS.7.RP.2aMO.7.RP.A.2aMO.6.NS.A
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.
This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses.