Lesson 5: Defining Equivalent Ratios
About this lesson
Previously, students understood equivalent ratios through physical perception of different batches of recipes. In this lesson, they work with equivalent ratios more abstractly, both in the context of recipes and in the context of abstract ratios of numbers. They understand and articulate that all ratios that are equivalent to \(a:b\) can be generated by multiplying both \(a\) and \(b\) by the same number (MP6).
By connecting concrete quantitative experiences to abstract representations that are independent of a context, students develop their skills in reasoning abstractly and quantitatively (MP2). They continue to use diagrams, words, or a combination of both for their explanations. The goal in subsequent lessons is to develop a general definition of equivalent ratios.
Lesson overview
 5.1 Warmup: Dots and Half Dots (10 minutes)

5.2 Activity: Tuna Casserole (15 minutes)
 Includes "Are you Ready for More?" extension problem
 5.3 Activity: What Are Equivalent Ratios? (15 minutes)
 Lesson Synthesis
 5.4 Cooldown: Why Are They Equivalent? (5 minutes)
Learning goals:
 Generate equivalent ratios and justify that they are equivalent.
 Present (in words and through other representations) a definition of equivalent ratios, including examples and nonexamples.
Learning goals (student facing):
 Let’s investigate equivalent ratios some more.
Learning targets (student facing):
 If I have a ratio, I can create a new ratio that is equivalent to it.
 If I have two ratios, I can decide whether they are equivalent to each other.
Required materials:
 tools for creating a visual display
Glossary:
 equivalent ratios  Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\cdot\frac12 = 4\) and \(6\cdot\frac12 = 3\). A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because and are equivalent ratios.
 Access the complete Grade 6 glossary.
Standards
 This lesson builds on the standard: CCSS.3.OA
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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