Understanding and Representing Ratios

Lesson 5

Math

Unit 1

6th Grade

Lesson 5 of 18

Objective


Represent ratios using double number lines and identify equivalent ratios. 

Common Core Standards


Core Standards

  • 6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Foundational Standards

  • 5.NF.B.5

Criteria for Success


  1. Understand the connection between a diagram for a ratio and a double number line for a ratio. Articulate benefits and/or uses for each representation.
  2. Create, with precision, a double number line to represent a ratio and equivalent ratios (MP.6).
  3. Identify equivalent ratios in a double number line and use them to solve problems.

Tips for Teachers


  • Lessons 5 and 6 introduce students to the double number line representation. The double number line is slightly more abstract than a discrete diagram because it does not show each individual item. Rather, it relies on the intervals to show the association between the quantities. Double number lines are more flexible than diagrams because larger quantities can be shown, but they are not as flexible as tables, which students will learn in later lessons.
  • When solving contextual problems using a double number line, students engage in MP.2. They de-contextualize a situation to represent it symbolically on a number line, and then re-contextualize as they identify equivalent ratios and interpret them in the context of the situation. 
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Anchor Problems


Problem 1

A Puerto Rican restaurant specializes in making arroz con gandules, or rice with pigeon peas. Chicken broth and rice are two of the ingredients used in this dish. For one batch, the restaurant uses a ratio of 3 cups of broth to 2 cups of rice. 

a.   How much broth and rice will the restaurant use to make 2 batches of this dish? To make 3 batches? Show your reasoning using a visual representation of your choice.

b.   On a busy Saturday, the restaurant uses 36 cups of broth for arroz con gandules. How much rice does the restaurant use, and how many batches is this? Show your reasoning using a visual representation of your choice.

Guiding Questions

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Problem 2

Arianna is making origami swans for her friend's birthday party. She wants to make 9 swans, one for each party guest. If Arianna takes 15 minutes to make each swan, will she be able to make 9 swans in 2 hours? Explain.

Guiding Questions

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References

Illustrative Mathematics Party Planning

Party Planning, accessed on July 18, 2017, 3:49 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 3

To make green-colored water, Brian mixes drops of green food dye and cups of water in a ratio of 4:3. 

a.   Draw a double number line to represent the ratio of drops of green food dye to cups of water.

b.   Use your double number line to find 2 equivalent ratios. 

c.   Brian’s friend, Evan, uses a ratio of 20 drops of green food dye to 15 cups of water. Will Evan’s water be the same color green as Brian’s? Explain your reasoning.

Guiding Questions

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Problem Set

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Target Task


The ratio of boys to girls in a sixth-grade class is 3:5.

a.   Draw a double number line to represent this ratio.

b.   If there were 15 girls in the class, how many boys were in the class? Explain or show your reasoning.

c.   Does the ratio of 18 boys to 30 girls represent an equivalent ratio to this sixth-grade class? Explain or show your reasoning.

Student Response

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Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Use examples from previous lessons where students drew diagrams to represent ratios. Have them represent the same ratios but with double number lines. Ask extension questions using the double number lines. 
  • Have students explore ratios in recipes. Students can draw double number lines to represent multiple batches of the recipe. An extension here can include more than two ingredients. 
  • Include problems where students determine if two ratios are equivalent and justify their answer using a double number line.
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Lesson 4

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Lesson 6

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Understanding & Describing Ratios

Topic B: Equivalent Ratios

Topic C: Representing Ratios in Tables

Topic D: Solving Part:Part:Whole Ratio Problems

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