The purpose of this lesson is to introduce the concept of a proportional relationship by looking at tables of equivalent ratios. Students learn that all entries in one column of the table can be obtained by multiplying entries in the other column by the same number. This number is called the constant of proportionality. The activities use contexts that make using the constant of proportionality the more convenient approach, rather than reasoning about equivalent ratios.
In any proportional relationship between two quantities \(x\) and \(y\), there are two ways of viewing the relationship; \(y\) is proportional to \(x\), or \(x\) is proportional to \(y\). For example, the two tables below represent the same relationship between time elapsed and distance traveled for someone running at a constant rate. The first table shows that distance is proportional to time, with constant of proportionality 6, and the second table, representing the same information, shows that time is proportional to distance, with constant of proportionality \(\frac16\).
These tables illustrate the convention that when we say “\(y\) is proportional to \(x\)” we usually put \(x\) in the left hand column and \(y\) in the right hand column, so that multiplication by the constant of proportionality always goes from left to right. This is not a hard and fast rule, but it prepares students for later work on functions, where they will think of \(x\) as the independent variable and \(y\) as the dependent variable.
Lesson overview
 2.1 Warmup: Notice and Wonder: Paper Towels by the Case (5 minutes)
 2.2 Activity: Feeding a Crowd (15 minutes)
 2.3 Activity: Making Bread Dough (10 minutes)

2.4 Optional Activity: Quarters and Dimes (10 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 2.5 Cooldown: Green Paint (5 minutes)
Learning goals:
 Comprehend that the phrase “proportional relationship” (in spoken and written language) refers to when two quantities are related by multiplying by a “constant of proportionality.”
 Describe (orally and in writing) relationships between rows or between columns in a table that represents a proportional relationship.
 Explain (orally) how to calculate missing values in a table that represents a proportional relationship.
Learning goals (student facing):
 Let’s solve problems involving proportional relationships using tables.
Learning targets (student facing):
 I understand the terms proportional relationship and constant of proportionality.
 I can use a table to reason about two quantities that are in a proportional relationship.
Required materials:
 measuring spoons
 measuring cup
Required preparation:
 A measuring cup and a tablespoon is optional—they may be handy for showing students who are unfamiliar with these kitchen tools.
Glossary:
 constant of proportionality  In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality. In this example, the constant of proportionality is 3, because \(2 \cdot 3 = 6\), \(3 \cdot 3 = 9\), and \(5 \cdot 3 = 15\). This means that there are 3 apples for every 1 orange in the fruit salad.
 proportional relationship  In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. For example, in this table every value of \(p\) is equal to 4 times the value of \(s\) on the same row. We can write this relationship as \(p = 4s\). This equation shows that \(s\) is proportional to \(p\).
 Access the complete Grade 7 glossary.
Standards
 This lesson builds on the standard: CCSS.6.RP.A.3MO.6.RP.A.3aMO.6.RP.A.3b
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).
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