Up to this point, students have worked with ratios of quantities where the units are the same (e.g., cups to cups) and ratios of quantities where the units are different (e.g., miles to hours). Sometimes in the first case, the sum of the quantities makes sense in the context, and we can ask questions about the total amount as well as the component parts. For example, when mixing 3 cups of yellow paint with 2 cups of blue paint, we get a total of 5 cups of green paint. (Notice that this does not always work; 3 cups of water mixed with 2 cups of dry oatmeal will not make 5 cups of soggy oatmeal.) In the paint scenario, the ratio of yellow paint to blue paint to green paint is \(3:2:5\). Furthermore, if we double the amount of both yellow and blue paint, we will double the amount of green paint. In general, if the ratio of yellow to blue paint is equivalent, the ratio of yellow to blue to green paint will also be the equivalent. We can see this is always true because of the distributive property:
\(a:b:(a+b)\) is equivalent to \(2a:2b:(2a+2b)\) because \(2a+2b=2(a+b)\).
These ratios are sometimes called “partpartwhole” ratios.
In this lesson, students learn about tape diagrams as a handy tool to represent ratios with the same units and as a way to reason about individual quantities (the parts) and the total quantity (the whole). Here students also see ratios expressed not in terms of specific units (milliliters, cups, square feet, etc.) but in terms of “parts” (e.g., the recipe calls for 2 parts of glue to 1 part of water).
Lesson overview
 15.1 Warmup: True or False: Multiplying by a Unit Fraction (10 minutes)
 15.2 Activity: Cubes of Paint (10 minutes)

15.3 Activity: Sneakers, Chicken, and Fruit Juice (20 minutes)
 Includes "Are you Ready for More?" extension problem
 15.4 Optional Activity: Invent Your Own Ratio Problem (10 minutes)
 Lesson Synthesis
 15.5 Cooldown: Room Sizes (5 minutes)
Learning goals:
 Comprehend the word “parts” as an unspecified unit in sentences (written and spoken) describing ratios.
 Draw and label a tape diagram to solve problems involving ratios and the total amount. Explain (orally) the solution method.
Learning goals (student facing):
 Let’s look at situations where you can add the quantities in a ratio together.
Learning targets (student facing):
 I can solve problems when I know a ratio and a total amount.
 I can create tape diagrams to help me reason about problems involving a ratio and a total amount.
Required materials:
 snap cubes
 graph paper
 tools for creating a visual display
Required preparation:
 Prepare a set of 50 red snap cubes and 30 blue snap cubes for each group of students.
Glossary:

tape diagram  A tape diagram is a group of rectangles put together to represent a relationship between quantities. For example, this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint.
If each rectangle were labeled 5, instead of 10, then the same picture could represent the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint.
 Access the complete Grade 6 glossary.
Standards

This lesson builds on the standards: CCSS.3.OA.B.5MS.3.OA.5 CCSS.5.NF.B.7MS.5.NF.7
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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