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# Egginess Part 2

# Equivalent Ratios

## Overview

Students are asked to fix a botched mixture that does not follow a given recipe. To fix the mixture, students must find a ratio of eggs to flour that is equivalent to 2:3, but without explicit instruction on the concept of equivalent ratios.

# Key Concepts

Students are invited to investigate the underlying idea of equivalent ratios by “correcting” the ratio between two ingredients in a botched mixture that does not follow a given recipe.

# Goals and Learning Objectives

- Explore a problem based on a recipe with two ingredients.
- Share approaches, clarify reasoning, and develop clear explanations of how to know a mixture has the right balance of ingredients.

# Egginess Problem

# Lesson Guide

Tell students that today they will whip up a recipe using eggs and flour and in the process learn about ratios. Have students watch the video about mixing eggs and flour.

# Mathematics

The video presents a problem situation that students will investigate in this lesson. The situation involves finding a ratio of eggs to flour (measured in tablespoons) that is equivalent to 2:3 and can be created with a minimum of 4 tablespoons of flour. Note that the concept of an equivalent ratio should not be directly taught during this lesson.

SWD: Students with disabilities may not as readily connect new information to previously learned skills and concepts. Activate background knowledge about fractions to promote connections between this and the new topic being investigated here.

ELL: When showing the video, be sure that ELLs can follow the explanations contained therein 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.

## Opening

# Egginess Problem

Watch the video.

- What is the problem presented in the video?

VIDEO: Egginess Part 1

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will fix a mixture to get the right ratio of eggs to flour (i.e., the right amount of “egginess.”)

## Opening

Determine how to fix the egg and flour mixture to get the right amount of egginess.

# Fix the Mixture

# Lesson Guide

Have students work in pairs on the problem.

ELL: Ensure students use precise mathematical language. Allow ELLs to use diagrams along with their language of choice. Provide students with a safe learning environment where they can attempt new mathematical concepts.

# Mathematical Practices

**Mathematical Practice 2: Reason abstractly and quantitatively.**

The problem situation invites students to move back and forth between sense-making about the amounts of egg and flour and how they relate to one another in the recipe and the mixture.

# Interventions

**Student has difficulty getting started.**

- The recipe is for a small amount of the mixture, but Jan’s dad is going to have to make a larger amount in order to get the right amount of “egginess” since he put too much flour in.
- If he adds 1 egg, would he have the right egginess?
- Jan’s dad can add flour, eggs, or both to the mixture. According to the recipe, which ingredient does he need to add to get the right amount of egginess?

**Student thinks the difference between the amount of egg and flour needs to be constant (e.g., adding 1 egg fixes the mixture because the difference between the number of tablespoons of flour and eggs is 1).**

- Does there always have to be 1 more tablespoon of flour than the number of eggs? What if there were 100 eggs and 101 tablespoons of flour. Would this mixture be the right amount of egginess?
- How much flour is needed per egg, according to the recipe? How much flour is there per egg in the mixture?
- If he adds 1 egg, how much flour would there be per egg?

**Student has a solution.**

- What if her dad added an extra egg instead of an extra tablespoon of flour? How could he fix his mixture to get the right amount of egginess?
- What if he needed to make a lot more of the mixture and he started with 60 tablespoons of flour. How many eggs would he need to get the right amount of egginess?
- Prepare an explanation for how you know you have the right amount of egginess for any amount of flour.

# Possible Answers

- To get a whole number value for both quantities, he needs to add 2 eggs and 2 tablespoons of flour to have an eggs-to-flour ratio of 4:6. He could add 0.5 tablespoon and 1 egg and have the right egginess, with an eggs-to-flour ratio of 3:4.5.
- Her dad needs to add more eggs, but since eggs can only be added as wholes, he is going to have to add more flour also. For every 2 eggs, there must be 3 tablespoons of flour, so for 1 egg there must be half of that amount of flour, which is 1.5 tablespoons.

## Work Time

# Fix the Mixture

There are 2 eggs and 4 tablespoons of flour already in the bowl.

Discuss:

- How can the you fix the egg and flour mixture to get the right amount of egginess?
- Explain why your changes to the mixture will result in the right amount of egginess.

Ask yourself:

- Nothing can be taken away from the mixture because the eggs and flour are mixed together.
- What could you add to the mixture to get the right egginess?

# 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 use a variety of approaches when thinking about the problem, for example, drawing pictures, thinking abstractly about the relationships among the values, and staying grounded in the problem situation by talking about the consistency of the mixture
- Students who use a range of approaches when solving the problem, for example, finding a unit ratio (e.g., amount of flour per single egg), using a number line, using a table, or using an additive approach

## Work Time

# Prepare a Presentation

Write an explanation of your strategy to solve the egginess problem.

# Make Connections

# Lesson Guide

Begin the discussion by having students share a range of strategies for thinking about and solving the problem. As a variety of strategies are being presented and clarified by comments and questions from students, focus the discussion by asking these questions:

- How can you be sure you have the right amount of egginess?
- What operations did you use to figure out how to fix the mixture?
- What are other amounts of eggs and flour you could use to get the right amount of egginess? (Possible answers: 6 eggs and 9 tablespoons of flour; 8 eggs and 12 tablespoons of flour; 20 eggs and 30 tablespoons of flour)

Have the class compare, contrast, and synthesize the responses to the first question into a tentative explanation that is clear and mathematically correct. The explanation need not include the term *equivalent ratio*, but should express that the balance of eggs to flour is the same in the recipe and in the new mixture with the right amount of egginess. Students will have opportunities to revisit the idea of equivalent ratios throughout this unit.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about the strategies that your classmates used to solve the problem.

As your classmates present, ask questions such as:

- How can you prove that your changes will result in the right amount of egginess (in case Nana is skeptical)?
- What operations did you use to figure out how to fix the mixture?
- What other amounts of eggs and flour could you use to make a mixture with the right amount of egginess (in case you need to make more or less of the mixture)?

# Make Connections

# Lesson Guide

After sharing their strategies, have students watch the video to see how the mixture was fixed to get the right egginess. Then discuss how students’ strategies were similar to or different from the one in the video.

SWD: Some students may need to watch the video more than once before they are able to justify and articulate their opinion.

## Performance Task

# Ways of Thinking: Make Connections

Watch the video.

- How does your strategy for fixing the mixture compare with the one in the video?

VIDEO: Egginess Part 2

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out what strategies students would use to solve a problem like the “egginess” problem.

## 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.

**When I encounter a problem like the egginess problem, one strategy I use for finding the answer is …**