In previous lessons students saw that a proportional relationships can be viewed in two ways, depending on which quantity you regard as being proportional to the other. In this lesson they write equations for these two ways, and they see why the two constants of proportionality associated with each way are reciprocals of each other.
The activities in this lesson use familiar contexts, but not identical situations from previous lessons: measurement conversions and water flowing at a constant rate. Students are expected to use methods developed earlier: organize data in a table, write and solve an equation to determine the constant of proportionality, and generalize from repeated calculations to arrive at an equation (MP8). After students write or use an equation, they interpret their answers in the context of the situation (MP2).
Lesson overview
 5.1 Warmup: Missing Figures (5 minutes)

5.2 Activity: Meters and Centimeters (10 minutes)
 Includes "Are you Ready for More?" extension problem
 5.3 Activity: Filling a Water Cooler (15 minutes)
 5.4 Optional Activity: Feeding Shrimp (10 minutes)
 Lesson Synthesis
 5.5 Cooldown: Flight of the Albatross (5 minutes)
Learning goals:
 Use the word “reciprocal” to explain (orally and in writing) that there are two related constants of proportionality for a proportional relationship.
 Write two equations that represent the same proportional relationship, i.e., \(y=kx\) and \(x=(\frac{1}{k})y\), and explain (orally) what each equation means.
Learning goals (student facing):
 Let’s investigate the equations that represent proportional relationships.
Learning targets (student facing):
 I can find two constants of proportionality for a proportional relationship.
 I can write two equations representing a proportional relationship described by a table or story.
Glossary:
 Access the complete Grade 7 glossary.
Standards
 This lesson builds on the standard: CCSS.5.OA.BMO.5.RA.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.