# Calculating The Unit Rate of A Ratio Situation

## Overview

Discuss the important ways that listeners contribute to mathematical discussions during Ways of Thinking presentations. Then use ratio and rate reasoning to solve a problem about ingredients in a stew.

# Key Concepts

Students find the unit rate of a ratio situation.

# Goals and Learning Objectives

- Contribute as listeners during the Ways of Thinking discussion.
- Understand the concept of a unit rate that is associated with a ratio.
- Use rate reasoning to solve real-world problems.

# Participate in Mathematical Discussions

# Lesson Guide

During mathematical discussions, students will sometimes present their work to their classmates, while other times they will listen to their classmates present. Discuss that listeners’ contributions during mathematical discussions are equally as important as presenters’ contributions.

A good classroom math community focuses on the importance of learning, not simply on getting the right answer. Classmates help each other learn, but ultimately students must take responsibility for their own learning.

Ask students:

- How can listeners contribute during presentations to improve the classroom math community?

Have students think about the question for a moment, and then ask them to discuss their ideas with their partner.

Ask students to share their ideas with the class. Record each idea and share with the class. When all the ideas have been listed, review each one for clarity.

The Hints for students list important ways listeners contribute to math discussions:

- Clarify—Tell the presenter when you do not understand. “I did not understand. What do you mean by …?”
- Critique—Challenge reasoning that is flawed. “How do you know that …?”
- Connect—Explore how different strategies result in the same answer. “What you said is like ….”
- Compare—Describe similarities and differences. “Is that different from …?”
- Notice Structure—Ask whether a conclusion is always, sometimes, or never true. “Is your conclusion always true for …?”

Suggest these ways students can engage in their own learning while watching their classmates’ presentations:

- Listen carefully so you understand the work and its relation to the mathematics in the lesson and unit.
- Notice how your way of thinking is related to your classmates’ ways of thinking.
- Ask questions.

## Opening

# Participate in Mathematical Discussions

Listening actively and asking questions during discussions are as important as the presentations.

To create a good math community, you should:

- Focus on learning, not getting the answer right.
- Help each other learn.
- Be responsible for your own learning.

View the Hints during Ways of Thinking to read questions you can ask the presenter. Pick one or ask your own question.

Discuss the following with your classmates.

- What can you as a listener contribute to improve your classroom math community?

## Hint:

Here are ways listeners contribute to math discussions:

- Clarify—Tell the presenter when you do not understand.

“I did not understand. What do you mean by …?” - Critique—Challenge reasoning that is flawed.

“How do you know that …?” - Connect—Explore how different strategies result in the same answer.

“What you said is like … “ - Compare—Describe similarities and differences.

“Is that different from …?” - Notice Structure—Ask whether a conclusion is always, sometimes, or never true.

“Is your conclusion always true for …?”

# Rate Situations

# Lesson Guide

Students think about rate as a special kind of ratio. All the processes used in earlier lessons when working with ratios apply here.

Have students think about the rate situations on their own, and then have them discuss their ideas with their partner. Then ask students to share their ideas with the class.

The questions get students to think about examples of rates from everyday life.

You can have students think about other examples of rates, such as interest rates, inflation rate, speed, and so on. Record and share these examples with the class and make the point that all are single values. They are ratios, however, because they represent the ratio of some quantity to one unit of another quantity. For example, the rate of 60 mph can be represented as the ratio of 60 miles for every one hour. Rates are unit ratios because the second quantity is always 1.

# Possible Answers

Answers will vary. Possible answers:

- Heart rate or drum rhythm
- Purchase groceries, such as produce or bulk goods, by weight
- The flow of a liquid into a tank

## Opening

# Rate Situations

Discuss the following with your classmates.

In what types of situations would you see these rates?

- Beats per minute
- Dollars per kilogram
- Gallons per hour

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will explain how to calculate rates.

## Opening

Explain how to calculate rates.

# Cost of Groceries

# Lesson Guide

Have students work on their own for several minutes before interacting with their partner. The time students spend working on their own will help them persevere though difficulty.

These problems are solved using division to convert the given ratio to a unit rate. Watch for errors, such as dividing $8.94 by $\frac{1}{2}$ instead of converting $\frac{1}{2}$ dozen to 6 cans and then dividing $8.94 by 6. Allow the use of a calculator, as appropriate.

Have partners discuss their work with their classmates. Make sure students focus on explaining how they calculated the unit rates.

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Students will need to make sense of the problem and decide on the best entry point. For example, students need to realize that $\frac{1}{2}$ dozen cans of tomatoes is 6 cans. 6 is the number they’ll divide by to find the rate per can of tomatoes.

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

Students’ primary task in this problem is to explain the reasoning behind their calculations. To construct a cohesive argument, they need to make conjectures and follow logic to verify the truth of their conjectures.

**Mathematical Practice 5: Use appropriate tools strategically.**

Students will need to decide what tools are useful in solving the problem. Should they use a ratio table or equations as in the prior lessons? When dividing by fractions, students will likely need to use a calculator. Regardless of the tool used, they should always check their answers to make sure the answers are reasonable for the problem situation.

# Interventions

**Student gets stuck on a problem.**

- What did you find most difficult? What can you do to continue working through the problem?
- If you have trouble, look for ideas in the items your teacher has previously shared.

**Student is unsure about what a rate is.**

- Talk to your partner about whether it would it make sense to find the unit rate per gallon, per penny, or per dollar.

**Student spends a lot of time on an unproductive strategy.**

- How would you set up an equation to solve this problem? Would a ratio table or another strategy make sense?

**Student has difficulty dividing fractional numbers by fractional numbers.**

- Can you make sense of the problem? You can use a calculator.

# Possible Answers

- Strategies can vary. $3.10 ÷ 5 = $0.62; 1 lb of potatoes is $0.62.

---|---

$8.94 ÷ 6 = $1.49; 1 can of tomatoes is $1.49

$1.29 ÷ 0.75 = $1.72; 1 lb of carrots is $1.72

$0.66 ÷ 34 = $0.88; 1 lb of celery is $0.88

$2.80 ÷ 2 = $1.40; 1 lb of dried beans is $1.40 - Cost per serving:
- $3.10 ÷ 12 ≈ $0.26 per serving of potatoes
- $8.94 ÷ 12 ≈ $0.75 per serving of tomatoes
- $1.29 ÷ 12 ≈ $0.11 per serving of carrots
- $0.66 ÷ 12 ≈ $0.01 per serving of celery
- $2.80 ÷ 12 ≈ $0.23 per serving of dried beans
- Stew cost: $3.10 + $8.94 + $1.29 + $0.66 + $2.80 = $16.79
- $16.79 ÷ 12 servings ≈ $1.40 per serving
- Answers will vary. It is important to include measurement units, because different units can be used to express drastically different quantities even if the numbers used are the same. For example, there is a significant difference between a recipe that calls for 2 teaspoons of salt per pound of vegetables and a recipe that calls for 2 cups of salt per pound of vegetables—even though both rates use the number 2.

## Work Time

# Cost of Groceries

Copy the top image.

- Write a rate for the unit cost of each ingredient: 1 lb of potatoes, 1 can of tomatoes, 1 lb of carrots, 1 lb of celery, and 1 lb of dried beans.
- The ingredients are used to make a stew that serves 12 people. What is the cost of each ingredient per serving? What does the stew cost per serving?
- It is important you express your rates with a measurement unit? Explain why.

## Hint:

- What do you divide to find the unit cost of each ingredient?
- What do you divide to find the rates per serving?
- What is the measurement unit for each rate?
- Are your quantities in the correct order?

# Prepare a Presentation

# Lesson Guide

Have partners work on the presentation together.

# Preparing for Ways of Thinking

Have students prepare a presentation even if they have not finished all parts of the problem. Tell students to focus their presentation on the way they calculated the rates.

Encourage students to include in their presentation any mistakes they made along the way rather than only their final solution.

Look for students who have different explanations for the same unit rate and different rates.

# Challenge Problem

## Answers

- Answers will vary. Possible answers:

3.75 ÷ 45 ≈ 0.083 miles for each minute

0.10 ÷ 1 = 0.10 miles for last minute 0.1 > 0.083 - Karen ran faster in the last minute than the average rate of the first 45 minutes. To get average rate, the distance in miles is divided by the time in minutes.

## Work Time

# Prepare a Presentation

- Explain how you calculated the rates.
- Explain why it is important to use measurement units with your rates.
- Identify any mistakes you made and what you learned from them.
- Describe what you found most difficult about calculating rates.

# Challenge Problem

Karen ran for 46 min. At the end of 45 min, her total distance was 3.75 mi. At the end of 46 min, her total distance was 3.85 mi.

- Compare Karen’s average speed during the last minute of her run to her average speed during the first 45 min.
- Did Karen run the last minute faster, slower, or at the same speed? Justify your answer mathematically.

# Make Connections

# Lesson Guide

Help students compare different explanations for the same rate. Ask students:

- What makes one explanation more convincing than another?

Facilitate a discussion between students who calculated different rates to help the class understand the mistaken reasoning.

The following three key concepts should be brought out in students’ discussion of their work:

- A rate is the value of one quantity with respect to another quantity, written [something] per [something]. Example: miles per hour
- The units of both quantities in a rate are different. Example: miles measure distance, hours measure time
- A rate is expressed as a single value. The ratio relationship in a rate is always the quantity per one of the other quantity. Example: the car went 25 miles per [1] hour, or traveled at the rate of 25 miles per every one hour.

While you are facilitating the discussion, keep track of the types of questions students ask presenters. At the end of the discussion, thank students who asked presenters questions from the listed Hints as well as students who formulated their own questions.

Ask a few students who completed the Challenge Problem to present their way of thinking about the speed problem.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about how your classmates’ strategies for calculating rates are similar to and different from yours.

## Hint:

As your classmates present, ask questions such as:

- What did you find most difficult?
- What was the difference between finding the unit cost and finding the cost of each ingredient per serving?
- Which ingredient was the most expensive? Least expensive?

# Explain Rates

# Lesson Guide

Have each student write a Summary of the Math in the lesson, and then work together to write a class summary. When done, if the class summary is helpful, record and share the summary with the class.

# A Possible Summary

A unit rate is a ratio in which the value of the second quantity is 1. The price for each pound of potatoes is $0.62 per lb. The price for each serving of stew is $1.40 per serving. Both of these examples are unit rates.

Calculating a rate involves comparing two quantities by division, just like ratios—for example, the price of potatoes divided by the number of pounds of potatoes. Rates always have quotient units: dollars/pound or dollars/serving.

# Mathematics

Have students read the Summary of the Math in this lesson. Ask students:

- What is an unknown quantity?
- How do the properties of operations help in determining the value?
- What is a variable?

Include any additional details that students thought were important.

# Additional Discussion Points

Additional things you might want to discuss are:

- The term
*per*means "for each." - The Work Time problem was about unit price. These are rates of dollars per unit of ingredient, or the amount of what you are buying.
- There are other common types of rates:
- Time rates have time in the denominator and describe how quickly things change as time passes. Speed is an example of time rate.
- Conversion rates are used to convert from one type of unit to anther. For example, 12 inches per foot is the rate that can be used to convert the length measured in feet to the same length measured in inches. Another example is the dollars to euros exchange rate.

## Formative Assessment

# Summary of the Math: Explain Rates

Write a summary that explains what a rate is and why it always has a measurement unit.

## Hint:

Check your summary.

- Do you define rate?
- Do you explain how to calculate a rate?
- Do you explain why measurement units are important for rates?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to see students' understanding of how ratios and rates are related.

## Work Time

# Reflection

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

**Rates are related to ratios because…**