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# Solving Proportional Relationship Problems

## Overview

Students watch a video showing three different ways to solve a problem involving a proportional relationship, and then they use each method to solve a similar problem. Students describe each approach, including the mathematical terms associated with each.

# Key Concepts

Three methods for solving problems involving proportional relationships include:

- Setting up a proportion and solving for the missing value
- Finding the unit rate and multiplying
- Writing and solving a formula using the constant of proportionality

# Goals and Learning Objectives

- Solve a problem involving a proportional relationship in three different ways: set up a proportion and solve for a missing value, use a unit rate, and use the constant of proportionality to write and solve a formula.

# Different Methods to Solve Problems

# Lesson Guide

- Have students watch the video.
- Briefly discuss the three approaches presented in the video.

# Mathematics

The video presents three different approaches to solving the same problem involving a proportional relationship between numbers of items and total cost for the items. Students will apply each approach, so they should understand the basic idea of each. Have students turn and talk or have a class discussion about the following questions:

- Did Jack’s method work for 13 pencils? Would it work for 23 pencils? Show me how to set up the problem with 23 pencils using Jack’s approach.
- Did Lucy’s method work for 13 pencils? Would it work for 23 pencils? Show me how to set up the problem with 23 pencils using Lucy’s approach.
- Did Karen’s method work for 13 pencils? Would it work for 23 pencils? Show me how to set up the problem with 23 pencils using Karen’s approach.

SWD: Allow students with language processing or attentional variability a preview of the video.

Cue students to “look out for” particular elements to support students’ understanding of the video’s contents. Provide a list of key ideas for students to review prior to viewing. Show the video as many times as needed, and pause it at key moments to allow for processing time.

ELL: Help students access the mathematics by reviewing key information before beginning the problem. Walk students through the problem and make sure that the students identify all of the relevant information in the problem.

## Opening

# Methods to Solve Problems

Watch the video.

Take notes about the methods that Jack, Lucy, and Karen use to solve the problem.

VIDEO: Pencils

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will use three different methods to solve a problem involving proportional relationships.

## Opening

Use three different methods to solve a problem involving proportional relationships.

# Marbles Problem

# Lesson Guide

Have students work in pairs for all problems.

# Mathematics

Students apply the three approaches presented in the video to a similar problem, and then they describe each approach, including the mathematical terms associated with each one.

# Interventions

**Student disagrees that all three approaches work or believes they are inconsistent with one another.**

- Try out each approach for 23 pencils, 49 pencils, and 150 pencils.
- Do the approaches lead to different results?

**Student doesn’t understand the concept of a unit rate.**

- The units in the video situation were pencils, and the unit rate was the cost for 1 pencil. What are the units in this problem situation? What is the unit rate?

**Student thinking is well-grounded in Jack’s approach, but not the other two.**

- What is especially useful about Karen’s approach?
- Using Jack’s approach, you have to set up a proportion each time you want to find a new value. What do you have to set up if you were going to use Lucy’s approach? Karen’s approach?

# Possible Answers

Jack’s method: $\frac{30\text{\hspace{0.17em}}\text{ounces}}{20\text{\hspace{0.17em}}\text{marbles}}=\frac{w}{36\text{\hspace{0.17em}}\text{marbles}}$

$\frac{30}{20}=\frac{w}{36}$

$\frac{30\cdot 36}{20}=w$

$\frac{\mathrm{1,080}}{20}=w$54 =

*w*A collection of 36 marbles weighs 54 ounces.

- Lucy’s method: To find the unit rate, divide 30 ounces by 20 marbles to get 1.5 ounces per marble. Then multiply 1.5 ounces per marble by 36 marbles to get 54 ounces.
- Karen’s method: The constant of proportionality is $\frac{30}{20}=\frac{3}{2}=1.5$. So, a formula for the weight in ounces,
*w*, in terms of the number of marbles,*n*, is*w*= 1.5*n*. For*n*= 36,*w*= 1.5(36) = 54, or 54 ounces.

## Work Time

# Marbles Problem

A collection of 20 marbles weighs 30 ounces. Each marble in the collection weighs the same amount.

How much does a collection of 36 marbles weigh? (Assume that each marble in the new collection weighs the same as each marble in the 20-marble collection.)

Solve the problem three different ways: Karen’s method, Jack’s method, and Lucy’s method.

## Hint:

- How can you set up the problem as a proportion?
- How can you find the weight of one marble? What would you call that weight?
- The formula for a proportional relationship is of the form

*y*=*kx*, where*k*is the constant of proportionality. How can you find*k*? - What are the two variables in this problem?

# Prepare a Presentation

# Preparing for Ways of Thinking

Listen and look for students who:

- Work through each approach systematically, making connections between them in the process
- Have a solid grasp of how (and why) to set up a proportion, as in Jack’s approach
- Have a solid grasp of the concept and usefulness of the unit rate in Lucy’s approach
- Have a solid grasp of the concept and usefulness of the constant of proportionality in Karen’s approach
- Discuss and debate about whether the approaches are consistent with one another
- Discuss and debate about which approach is most useful

ELL: When listening to students’ responses, give students advance notice they will be presenting their work on a specific problem during the Ways of Thinking section. This will give them ample time to prepare a thoughtful response.

# Answer

Using Jack’s method, in order to set up a proportion and solve it, I used the fact that the ratio of the weight of any collection of these marbles to the number of marbles in the collection will have the same value as the ratio of the weight of the collection of 20 marbles to the number in that collection $\left(\frac{30}{20}\right)$.

Using Lucy’s method, I found the weight of a single marble—which is a unit rate—in order to then multiply that weight by the number of marbles in the new collection.

Using Karen’s method, I found the constant of proportionality between the weight of a collection of marbles and the number in the collection, and then I wrote a formula for the proportional relationship between weight and number using this constant of proportionality.

# Challenge Problem

## Possible Answer

## Work Time

# Prepare a Presentation

- Describe each method you used to solve the marble problem.
- Think about the words—particularly the mathematical vocabulary—you used to describe each method. What do the words mean? Are there any words that you used to describe one method but not the other two?
- Be prepared to share your description of each method with the class.

# Challenge Problem

Graph the formula that you wrote when you used Karen's method to solve the marble problem.

# Make Connections

# Lesson Guide

Highlight the usefulness of each approach, and invite students to make connections between the approaches as they share their work. Then have students who worked on the Challenge Problem share their work, and get feedback from the class on the graphs. Ask the following questions about the graphs:

- Should the graph consist of discrete points or a continuous line?
- Where should the graph begin? Where should it end?
- What are the units on each of the axes?
- How does Jack’s approach relate to the graph? If you were to make a table of values, how would it relate to Jack’s approach?
- How does Lucy’s approach relate to the graph? How is the unit rate shown by the graph?
- How can you tell just from looking at the graph whether the weight is proportional to the number of marbles?

SWD: As students present their solutions, make connections between different solutions to the same problem. This allows students to see the multiple ways to analyze and solve a problem. Write down all important connections, and have students record them in their Notebook.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates' explanations of how they used the three methods to solve the marble problem.

## Hint:

As your classmates present, ask questions such as:

- What is the unit rate in the marble problem?
- What is the constant of proportionality in the problem?
- Why would you write a formula to help you solve a proportion problem?
- When would you set up a proportion and solve for a single value?
- Describe the form of the formula for a proportion problem.
- Which of the three methods do you like best? Which do you think is easiest? Which do you think is most difficult?

# Solve Proportion Problems

# A Possible Summary

Writing a formula using the constant of proportionality to represent a proportional relationship between two quantities is an efficient and general way to find values of interest. Setting up a proportion works well for finding a single missing value, and once you’ve set it up, you can easily find the unit rate and the constant of proportionality.

# Additional Discussion Points

Review the following points if time allows:

- Show the three different methods to use for finding a solution to a problem involving proportional relationships.
- Show how a unit rate and the constant of proportionality are connected.
- Discuss why using a formula with the constant of proportionality in it might be more efficient than setting up and solving proportions (when solving a group of problems with the same underlying proportional relationship).

ELL: Write the key points on a poster so that students can refer back to them throughout the unit. When working with ELLs, provide supplementary materials, such as graphic organizers, to illustrate new concepts and vocabulary necessary for mathematical learning. Make sure the poster is placed in a prominent location in the classroom.

## Formative Assessment

# Summary of the Math: Solve Proportion Problems

Write a summary about the different ways to solve proportion problems.

## Hint:

Check your summary.

- Do you describe three different methods for solving problems involving proportional relationships?
- Do you describe the form of the formula for a proportional relationship? Do you explain what each part of the formula represents?
- Do you show how the unit rate and the constant of proportionality are connected?
- Do you explain why using a formula with a constant of proportionality might be more efficient than setting up and solving proportions when solving a group of problems with the same underlying proportional relationship.

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to learn each student’s favorite method for solving a proportion problem and why that method is his or her favorite.

## 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.

**My favorite method for solving a proportion problem is ___ because …**