Students used physical objects to learn about ratios in the previous lesson. Here they use diagrams to represent situations involving ratios and continue to develop ratio language. The use of diagrams to represent ratios involves some care so that students can make strategic choices about the tools they use to solve problems. Both the visual and verbal descriptions of ratios demand careful interpretation and use of language (MP6).
Students should see diagrams as a useful and efficient ways to represent ratios. There is not really a right or wrong way to draw a diagram; what is important is that it represents the mathematics and makes sense to the student, and the student can explain how the diagram is being used. However, a goal of this lesson is to help students draw useful diagrams efficiently.
For example, here is a diagram to show 6 cups of juice and 3 cups of soda water in a recipe.
When students are asked to draw diagrams, they often include unnecessary details such as making each cup look like an actual cup, which makes the diagrams inefficient to use for solving problems. Examples of very simple diagrams help guide students toward more abstract representations while still relying on visual or spatial cues to support reasoning.
Diagrams can also help students see associations between quantities in different ways. For example, we can see there are 2 cups of juice for 1 cup of soda water by grouping the items as shown below.
While students may say “for every 2 cups of juice there is 1 cup of soda,” note that for now, we will not suggest writing the association as \(2:1\). Equivalent ratios will be carefully developed in upcoming lessons. Diagrams like the one above are referred to as “discrete diagrams” in these materials, but students do not need to know this term. In studentfacing materials they are simply called “diagrams.”
The discrete diagrams in this lesson are meant to reflect the parallel structure of double number lines that students will learn later in the unit. But for now, students do not need to draw them this way as long as they can explain their diagrams and interpret discrete diagrams like the ones shown in the lesson.
Lesson overview
 2.1 Warmup: Number Talk: Dividing by 4 and Multiplying by \(\frac14\) (10 minutes)
 2.2 Optional Activity: A Collection of Snap Cubes (10 minutes)
 2.3 Activity: Blue Paint and Art Paste (10 minutes)

2.4 Activity: Card Sort: Spaghetti Sauce (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 2.5 Cooldown: Paws, Ears, and Tails (5 minutes)
Learning goals:
 Coordinate discrete diagrams and multiple written sentences describing the same ratios.
 Draw and label discrete diagrams to represent situations involving ratios.
 Practice reading and writing sentences describing ratios, e.g., “The ratio of these to those is \(a:b\). The ratio of these to those is \(a\) to \(b\). For every \(a\) of these, there are \(b\) of those.”
Learning goals (student facing):
 Let’s use diagrams to represent ratios.
Learning targets (student facing):
 I include labels when I draw a diagram representing a ratio, so that the meaning of the diagram is clear.
 I can draw a diagram that represents a ratio and explain what the diagram means.
Required materials:
 copies of blackline master
 colored pencils
 tools for creating a visual display
 preprinted slips, cut from copies of the blackline master
Required preparation:
 For the Card Sort: Spaghetti Sauce activity, make 1 copy of the blackline master for each group of 2 students, plus a few extras.
 The blackline master shows the correct matches.
 Find the blackline master in the Additional Materials section of this lesson.
 Keep the extra copies whole to serve as answer keys.
 Cut up the rest of the slips for students to use, and throw away the cut slips that say “The above diagram also matches this sentence.”
 It may be helpful to copy each group's slips on a different color of paper, so that misplaced slips can quickly be put back.
Glossary:
 Access the complete Grade 6 glossary.
Standards
 This lesson builds on the standard: CCSS.5.NF.B.3MS.5.NF.3
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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