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# Protein Shake Mistake

# Double Number Line for Modeling

## Overview

Students watch a video in which a double number line is used to solve a problem about getting the right amount of protein mix. Using the double number line is an example of modeling with mathematics, which is Mathematical Practice 4.

# Key Concepts

A double number line shows corresponding values for two variable quantities with a constant ratio between them. Each pair of tick marks that go together shows a ratio equivalent to all of the other ratios between corresponding tick marks.

# Goals and Learning Objectives

- Watch an example of students using mathematics to model a relationship between quantities (MP4).
- Use a double number line to solve a problem.
- Use a double number line to deepen understanding of equivalence in the context of a relationship between quantities with a constant ratio.

SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. Students can highlight the critical features and/or concepts and will help them to pay close attention to salient information.

# Model With Mathematics

# Mathematical Practices in Action

Have students watch the video and listen to the dialogue between Carlos, Jan, and Martin. This video shows students engaged in Mathematical Practice 4: Model with mathematics.

After students watch the video, hold a brief discussion about what “modeling” means in the context of getting the right amount of protein mix. Emphasize Carlos’s, Denzel's, and Jan’s use of the double number line as a way of representing the two quantities and modeling the relationship between values for milk and protein powder.

ELL: When showing the video, be sure that ELLs can follow the explanations by “chunking” the video. Pause the video at key times to allow ELLs time to process the information. Ask students if they need to watch it a second time. Check for understanding by asking questions before moving on. The second video models academic discussion, which helps students see how they should discuss the problem.

## Opening

# Model With Mathematics

- Watch the first video, which introduces the problem in Protein Shake Mistake.
- Can you explain what the problem is asking?

- Then watch the second video, in which students use ratios to model the mathematics in the problem.
- How did Carlos, Denzel, and Jan model the protein shake problem?
- How did they adjust the model to determine the amount they needed?

VIDEO: Protein Shake Mistake

VIDEO: Model with Mathematics

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will represent equivalent ratios on a double number line.

## Opening

Represent equivalent ratios on a double number line.

# Stars to Triangles

# Lesson Guide

Have students work in pairs on the problems.

# Mathematical Practices

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

Students may argue about how to construct and interpret the double number lines, which presents opportunities to practice critiquing one another’s reasoning and developing more coherent arguments in order to communicate successfully.

**Mathematical Practice 4: Model with mathematics.**

As students work creating a double number line to show that two ratios are equivalent, remind them that their work is an example of using mathematics to model a relationship between two variable quantities.

# Interventions

**Student has difficulty translating from words to numbers or from the picture to numbers.**

- The ratio of stars to triangles is different from the ratio of triangles to stars. Be careful which number you put first.
- The ratio is a comparison of the number of stars to the number of triangles. The value of the ratio depends on which number you put first.
- The ratio between these numbers can be simplified or left as one actual number to the other actual number.

**Student doesn’t interpret “difference” and/or “ratio” correctly.**

- To find the difference between two numbers, subtract the lesser number from the greater number.
- The value of a ratio is the first number divided by the second number.

# Possible Answers

- The ratio of the number of stars to the number of triangles is 15:3 or 5:1.
- The ratio of the number of triangles to the number of stars is 3:15 or 1:5.

## Work Time

# Stars to Triangles

Refer to the diagram and answer the following questions.

- What is the ratio of the number of stars to the number of triangles?
- What is the ratio of the number of triangles to the number of stars?

Ask yourself:

- What does the first number in a ratio represent?
- What does the second number in a ratio represent?

# Sketch It!

# Lesson Guide

Have students work in pairs on the problem.

# Mathematical Practices

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

- Students may argue about how to construct and interpret the double number lines, which presents opportunities to practice critiquing one another’s reasoning and developing more coherent arguments in order to communicate successfully.

**Mathematical Practice in Action 4: Model with mathematics.**

- As students work creating a double number line to show that two ratios are equivalent, remind them that their work is an example of using mathematics to model a relationship between two variable quantities.

# Interventions

**Student has difficulty constructing a double number line.**

- Use the strategy Denzel, Jan, and Carlos used in the Opening video.
- The double number line shows you which values of one quantity (milk) correspond to which values of another quantity (protein mix). The tick marks that line up with one another must be labeled with values that correspond to one another.
- Draw two number lines with equally spaced tick marks. On the number line showing triangles, label the third tick mark “3”; on the number line showing stars, label the third tick mark “15 .” Now label the points for 1 triangle and 5 stars. What other points can you label?

**Student successfully constructs a double number line but interprets it incorrectly.**

- The number lines show the pairs of values that go together: the tick marks that line up are the values that go together. What does “go together” mean?
- Each pair of tick marks that go together shows a ratio, and all those ratios are equivalent.
- If two values “go together” (line up), it means they have a ratio equivalent to other pairs of values that go together in this situation.

# Possible Answers

- Answers will vary. Students should draw another set of objects so that the ratio is 5:1. Possible answer: 30 stars and 6 triangles
- Answers will vary. Possible answer:
- Answers will vary. Possible answer:

## Work Time

# Sketch It!

Use the ratio of stars to triangles that you found in Task 3.

- Sketch a set of objects that has the same ratio but not the same number of objects.
- Show that the ratio that you made in your picture is equivalent to the ratio in task 3. Draw a double number line to show this; or, use the Double Number Line Tool.
- On your double number line, find three pairs of numbers that represent the same ratio of stars to triangles.

Ask yourself:

- In your diagram, what should the ratio of stars to triangles be equivalent to?
- What pairs of numbers do you need to line up on your double number line?

# Prepare a Presentation

# Preparing for Ways of Thinking

Listen and look for the following student thinking to highlight during the Ways of Thinking discussion:

- Students who have difficulty translating between the pictures, descriptions, and numbers
- Students who reflect on the significance of the order of the numbers when writing ratios
- Students who reflect on what the difference between numbers of stars and triangles says versus what the ratios between these numbers says
- Students who discuss the spacing of tick marks on their double number lines
- Students who debate about how to construct and interpret the double number lines

# Challenge Problem

## Possible Answers

- Answers will vary. Possible answer: The differences are 5 − 1 = 4, 15 − 3 = 12, 30 − 6 = 24, and 40 − 8 = 32. As the values increase, the differences increase.

## Work Time

# Prepare a Presentation

Explain how you made a set of stars and triangles that has the same ratio of stars to triangles as the ratio in the given diagram.

Show how you made a double number line that represents equivalent ratios of stars to triangles.

Suppose you find the difference between the number of stars and triangles in the original set. Explain why making a set of stars and triangles with the same difference will not result in sets with equivalent ratios.

# Challenge Problem

Find the difference between each pair of aligned numbers on your double number line.

- What do you observe about the differences?

# Make Connections

# Lesson Guide

As students give their presentations, ask them to talk about the strategies they used to find equivalent ratios. Invite into the conversation students who used double number lines directly to help them think about the problems and students who used other strategies first, before connecting their thinking to the double number line.

# Mathematical Practice

**Mathematical Practice in Action 4: Model with mathematics.**

Point out to students that the double number line is an efficient way of modeling the relationship between stars and triangles.

Ask:

- What does this kind of model (the double number line) help you see or understand about these quantities?
- Think about the double number line Carlos, Denzel, and Jan used in the start of the lesson. What did this kind of model help them see or understand about the relationship between milk and protein powder?

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates' strategies for making sets of stars and triangles and for representing equivalent ratios on a double number line.

As your classmates present, ask questions such as:

- What is the ratio in your diagram?
- Why is this ratio equivalent to the ratio in the given diagram?
- How do you know that the ratios on your double number line are equivalent?
- Can you give an example of sets of stars and triangles that have the same difference but not equivalent ratios?

# Find Equivalent Ratios

# Lesson Guide

Watch for students who have the following errors and help students overcome any misunderstandings:

- Students who do not set up a double number line correctly, for example, lining up the parts of the ratio on the same side instead of opposite each other
- Students who write ratios as fractions by switching the numerator and denominator, for example, writing the ratio 14:35 as $\frac{35}{14}$ instead of $\frac{14}{35}$
- Students who find equivalent ratios by adding the same quantity to each part of the ratio, for example, finding an equivalent ratio for 2:3 by adding 4 to 2 and 4 to 3 to get the ratio 6:7

ELL: Be sure to encourage ELLs to share, even if their pace might be slower than their native counterparts. Give ELLs ample wait time to explain their thinking.

# Possible Answers

- Explanations will vary. Possible answer: Both the ratio 3:4 and the ratio 15:20 are equivalent because the tick marks for the corresponding values line up with one another. All the ratios between corresponding values are equivalent on a double number line.
- The ratio 14:35 expressed as a fraction in simplest form is $\frac{2}{5}$.
- The ratio 15:18 expressed as a fraction in simplest form is $\frac{5}{6}$.
- The ratio 26:39 expressed as a fraction in simplest form is $\frac{2}{3}$.

- Answers will vary. Possible answer: The ratio 2:3 is equivalent to the ratios 4:6, 6:9, and 8:12.
- Answers will vary. Possible answer: You can write the ratio 2:5 as the fraction $\frac{2}{5}$, which is another way of saying 2 ÷ 5 and 2 ÷ 5 = 0.4.

## Work Time

# Find Equivalent Ratios

- The ratio 15:20 is equivalent to the ratio 3:4. Explain why and make a double number line to support your explanation.
- Express each of these ratios as a fraction in simplest form:
- 14:35
- 15:18
- 26:39

- Show the ratio 2:3 and three other equivalent ratios on a double number line.
- The ratio 2:5 can be expressed as the decimal 0.4. Explain why.

Ask yourself:

- When you make a double number line, what pairs of numbers do you need to line up?
- What operation do you use to express a ratio as a decimal?

# Overview of Equivalent Ratios

# Lesson Guide

Have pairs quietly discuss the information about ratios. Have them give examples of other equivalent ratios. As student pairs work together, listen for students who may still have misconceptions so you can address them in the class discussion.

After a few minutes, discuss the Summary as a class. Have a volunteer give a definition of equivalent ratios. Ask another student to model using a double number line to show and find equivalent ratios.

ELL: Ask students to write their explanations for equivalent ratios on a chart so students can see the information in written form in addition to hearing it.

## Formative Assessment

# Summary of the Math: Overview of Equivalent Ratios

**Read and Discuss**

- A ratio is often represented as a pair of numbers, either in colon notation (280:210 or 4:3) or as a fraction $(\frac{280}{210}=\frac{4}{3})$.
- You can determine whether two ratios are equivalent in the following way:

The ratios 2:3 and 4:6 are equivalent if $\frac{2}{3}=\frac{4}{6}$. - You can represent equivalent ratios using a double number line. A double number line shows the relationship between two quantities by lining up the quantities at the same tick marks. Each of these pairs of numbers represents a ratio, and all of the ratios on a given double number line are equivalent; that is, the ratios all have the same value.

Can you:

- Find equivalent ratios?
- Represent equivalent ratios on a double number line?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out how a double number line helps students understand equivalent ratios.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**The double number line helps me understand equivalent ratios because ...**