The purpose of this lesson is to develop students’ ability to work with a table of equivalent ratios. It also provides opportunities to compare and contrast different ways of solving equivalent ratio problems.
Students see that a table accommodates different ways of reasoning about equivalent ratios, with some being more direct than others. They notice (MP8) that to find an unknown quantity, they can:
 Find the multiplier that relates two corresponding values in different rows (e.g., “What times 5 equals 8?”) and use that multiplier to find unknown values. (This follows the multiplicative thinking developed in previous lessons.)
 Find an equivalent ratio with one quantity having a value of 1 and use that ratio to find missing values.
All tasks in the lesson aim to strengthen students’ understanding of the multiplicative relationships between equivalent ratios—that given a ratio \(a:b\), an equivalent ratio may be found by multiplying both \(a\) and \(b\) by the same factor. They also aim to build students’ awareness of how a table can facilitate this reasoning to varying degrees of efficiency, depending on one’s approach.
Ultimately, the goal of this unit is to prepare students to make sense of situations involving equivalent ratios and solve problems flexibly and strategically, rather than to rely on a procedure (such as “set up a proportion and cross multiply”) without an understanding of the underlying mathematics.
To reason using ratios in which one of the quantities is 1, students are likely to use division. In the example above, they are likely to divide the 90 by 5 to obtain the amount earned per hour. Remind students that dividing by a whole number is the same as multiplying by its reciprocal (a unit fraction) and encourage the use of multiplication (as shown in the activity about hourly wages) whenever possible. Doing so will better prepare students to: 1) scale down, i.e., to find equivalent ratios involving values that are smaller than the given ones, 2) relate fractions to percentages later in the course, and 3) understand division of fractions (including the “invert and multiply” rule) in a later unit.
Lesson overview
 12.1 Warmup: Number Talk: Multiplying by a Unit Fraction (10 minutes)
 12.2 Activity: Comparing Taco Prices (10 minutes)
 12.3 Activity: Hourly Wages (10 minutes)

12.4 Optional Activity: Zeno’s Memory Card (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 12.5 Cooldown: Price of Bagels (5 minutes)
Learning goals:
 Choose multipliers strategically while solving multistep problems involving equivalent ratios.
 Describe (orally and in writing) how a table of equivalent ratios was used to solve a problem about prices and quantities.
 Remember that dividing by a whole number is the same as multiplying by an associated unit fraction.
Learning goals (student facing):
 Let’s use a table of equivalent ratios like a pro.
Learning targets (student facing):
 I can solve problems about situations happening at the same rate by using a table and finding a “1” row.
 I can use a table of equivalent ratios to solve problems about unit price.
Glossary:
 Access the complete Grade 6 glossary.
Standards

This lesson builds on the standard: CCSS.5.NF
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.