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# Units Resulting From Rate Calculations

## Overview

In this lesson, students focus on the units used with rates. Students are given calculations without units and must determine the correct units to use.

# Key Concepts

When dividing quantity A by quantity B to find a rate, the unit of the quotient is expressed in the form A per B.

When multiplying a B quantity by an A per B rate, you get an A quantity.

Some rates, while mathematically correct, are physically impossible in the real world.

# Goals and Learning Objectives

- Understand the units that result from rate calculations.

# What Do the Numbers Mean?

# Lesson Guide

A mature understanding of rate must include keeping track of the units being manipulated and sensibly interpreting the meaning of the quotient. This lesson focuses on the units, not the numbers.

Discuss the examples in the Opening with the students.

SWD: Some students may require prompts and/or reminders to use correct units when discussing measurements. It may be helpful for some students to use a sticky note reminder (paper or digital) to support consistency with this skill.

## Opening

# What Do the Numbers Mean?

You can get a new unit by dividing two other units. In fact, that is what defines a rate. For instance, if you divide 10 meters by 2 seconds, you get 5 $\frac{\text{meters}}{\mathrm{second}}$. The new unit is “meters per second.”

When multiplying and dividing units, you can cancel them out as shown here.

$\begin{array}{c}\text{inches}\xb7\text{}\frac{\text{centimeters}}{\text{inches}}=\text{centimeters}\\ 12\text{in}\text{.}\xb7\text{2}\text{.54}\frac{\text{cm}}{\text{in}\text{.}}=\text{30}\text{.48 cm}\end{array}$

- Discuss with your partner why units cancel out.
- What would be the correct unit for this equation?

30.48 cm • 0.3937$\frac{\text{in}}{\text{cm}}$ = 12 ___

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students must accurately place units within an equation and show how units are changed by multiplication and division.

Students will find the missing units in the calculations Emma's coach made about her run. Students will need to identify what units to use when dividing laps by minutes and when dividing minutes by laps.

## Opening

Accurately place units and show how units are changed when multiplied or divided.

# Focus On the Units

# Lesson Guide

Students have no calculations to do. They look at the given calculations and make sense of what units are appropriate for each calculation.

# Interventions

**Student has difficulty getting started.**

- Describe the Task in your own words to your partner.
- What two quantities does Emma's coach measure?
- What two ways can Emma's coach represent her speed given the situation?
- What units make sense given the situation?

**Student has an incorrect solution:**

- Have you checked your work?
- Does your answer make sense? Explain what your answer means.
- Can you try solving the problem a different way?

**Student has a solution.**

- Explain your strategy for solving the problem.
- Why are units important when using rates?

# Answers

- Answers will vary. The correct units for the calculations are:

30**minutes**÷ 15**laps**= 2**minutes per lap**

15**laps**÷ 30**minutes**= 0.5**laps per minute**

## Work Time

# Focus On the Units

Watch the video. Then read the problem below.

Emma has been running laps around the track. (A lap is a quarter mile.) Her coach recorded how many minutes she ran. After Emma’s run, her coach wrote these calculations on his clipboard:

30 ÷ 15 = 2 and 15 ÷ 30 = 0.5

The coach forgot to include the units with his calculations.

- Determine what the numbers mean. Explain your thinking.

VIDEO: Emma Running

## Hint:

- What two units were mentioned in the video?
- What is the relationship between those units?
- Which of those units goes with the larger number (30)?

# Prepare a Presentation

# Lesson Guide

Students will prepare a presentation on what the coach's numbers mean.

# Preparing for Ways of Thinking

Look for the following misconceptions. If these errors arise in Work Time, it will be important to discuss them during Ways of Thinking.

Look for a variety of solution strategies in which students made sense of the situations by modeling the mathematics using different tools.

Look for students who attach incorrect meanings to the quotients, such as:

- She ran 2 miles.
- She ran for 2 minutes.
- She ran 0.5 mile.
- She ran for 0.5 minute.
- It is the answer to the problem.

SWD: Students with disabilities may have difficulty articulating their thinking, constructing viable arguments, and critiquing each other's work. Consider pulling a small group of struggling students to provide guided modeling and more support. If pulling a small group is not feasible, strategically selecting partners that can articulate their thinking and are able to peer mentor will be beneficial to struggling students.

ELL: Encourage students to use the academic vocabulary they are learning. As they participate in the discussion, be sure to monitor for knowledge of the topic. Stay alert to follow up on statements that seem unclear or ambiguous. When ELLs contribute, focus on content, and don't allow grammar difficulties to distract you from understanding the meaning (as much as possible). Help ELLs who make grammar mistakes by rephrasing, as long as the rephrasing is not an interruption and does not interfere with their thinking.

# Challenge Problem

## Answers

- Answers will vary. Look for students who understand the concept of unit analysis.

## Work Time

# Prepare a Presentation

Prepare a presentation that explains your reasoning in finding the units. Which rate do you think the coach might use and when?

# Challenge Problem

Write an explanation about how to determine which units to use in the answer to a rate problem. Consider the unit of the quotient when you are dividing to find a rate and the unit of a product when you are using a rate.

# Make Connections

# Lesson Guide

In this discussion, students should understand each other's methods and make connections between methods.

Select a variety of work with an eye to making connections among different representations.

Display tables, number lines, and calculations so that you can move back and forth among the various representations. Encourage students to critique each method.

# Mathematics

Ask students to describe how they approached the problems. Ask questions such as the following to get students to think about the different solution strategies:

- Compare how [Names] made sense of the problem.
- How did each student model the mathematics?
- Which method is clearer and shows the structure of the mathematics?
- What tools, if any, did different students use? Were the tools appropriate?
- Why doesn't an answer of 2 laps per minute make sense in the given situation?

If Emma ran 2 laps per minute, she would be running 0.5 miles per minute (a lap is 0.25 miles) or 1 mile in 2 minutes. Since the top runners run a mile in about 4 minutes, that would be impossible.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about other students’ presentations. Make sure you are able to understand each solution in terms of the units rather than just the numbers.

## Hint:

As students present, ask questions such as:

- Where do you see the number of laps, the time, and the speed in your solution to the problem about Emma?
- Why couldn’t you change the numbers for laps and time?
- How did you make sense of Emma’s situation?

# Working With Units

# Lesson Guide

Students will use what they know about rates to determine the units for three situations.

# Mathematical Practices

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

- Focus the discussion on the meaning of each situation. Identify students who can logically express their reasoning and justify their answers mathematically. For example, in the third problem, look for students who understand that when they divide miles by hours, they will get a rate that has a unit of miles per hour. When they multiply that rate by the number of hours, they will get the number of miles.

# Answers

- The unit is minutes per lap.
- The unit is beats per minute.
- The unit is miles per an hour.
- The unit is miles.

## Work Time

# Working With Units

Understanding what the units mean when you find a rate or use a rate is just as important as calculating correctly.

Write the appropriate unit for each statement.

- Divide a quantity in minutes by a quantity in laps.
- Divide a quantity in beats by a quantity in minutes.
- Divide a quantity in miles by a quantity in hours.
- Multiply the answer from (c) by a quantity in hours.

## Hint:

- When you divide to find a rate, the unit of the quotient is expressed in the form A per B .
- Think of 30 minutes ÷ 5 laps. What would the unit in the answer be?

# Units of Rates

# A Possible Summary

When you divide to find a rate, you also divide the units. When you multiply by a rate, only one unit cancels out.

ELL: When writing the summary, be sure that ELLs have access to a dictionary and that they have some time to discuss their summary with a partner before writing, to help them organize their thoughts. Allow ELLs who share the same language of origin to discuss in their preferred language.

# Additional Discussion Points

When dividing an A quantity by a B quantity to find a rate, the unit of the quotient is expressed in the form of A per B.

- When multiplying a B quantity by an A per B rate, you get an A quantity.
- Some rates, while mathematically correct, are physically impossible in the real world.

## Formative Assessment

# Summary of the Math: Units of Rates

Summarize what you have learned about rate problems by concentrating on the units.

## Hint:

Check your summary

- Does your summary explain how to express the unit of the quotient when dividing to find a rate?
- Does your summary explain how to express the unit of the product when multiplying by a rate?

# Check Your Knowledge of Rates

# Lesson Guide

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and which students would benefit from review before the assessment. Have students work on the Self Check individually.

# Assessment

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

# Interventions

**Student has difficulty getting started.**

- Describe the Task in your own words to your partner.
- What units make sense given the situation?

**Student has an incorrect solution:**

- Have you checked your work?
- Does your answer make sense? Explain what your answer means.
- Can you try solving the problem a different way?

**Student has a solution.**

- Explain your strategy for solving the problem.
- Why are units important when using rates?
- ELL: It might be very helpful for ELLs and other students to have a sample or model for the concepts, strategies, and applications that will be addressed in the quiz and the format you want them to follow. Be prepared to address and explicitly re-teach or review vocabulary, concepts, strategies, and applications.
- SWD: Students with disabilities may benefit from having the Self Check presented in a variety of ways (as auditory and visual information). Provide students the option of listening to the content for this task. Some students may need additional time to complete the Self Check. Be sure to make provisions for the additional time or consider reducing the number of tasks required for students to demonstrate mastery of skill.

# Answers

- He can multiply the rate $3.00 per pound by 5.75 pounds to find the price in dollars.
- She can divide 12 miles by 1.5 hours to find her speed in miles per hour.
- He can divide 325 square feet of wall by 85 square feet per quart to find the number of quarts.
- She can divide 13.1 miles by 2.6 hours to find her speed in miles per hour.

## Formative Assessment

# Check Your Knowledge of Rates

Now take some time to check how much you know about rates. For these problems, do not calculate the exact answers. Instead, do the following for each problem:

- Explain what calculation the person can perform to find what he or she wants to know.
- Explain what the unit of the answer is.

- At the store, Jason sees that the price of cherries is $3.00 per pound. What calculation can he perform to find the price of 5.75 pounds of cherries?
- Rosa knows that she can skate to the park in 1.5 hours. The park is 12 miles away. What calculation can she perform to find her speed?
- Denzel knows that 1 quart of paint will cover about 85 square feet of wall. He needs enough paint to cover 325 square feet of wall. What calculation can Denzel perform to find how many quarts of paint he will need?
- Mina ran a half marathon (13.1 miles) this weekend. She completed the run in 2.6 hours. What calculation can Mina perform to find her speed during this run?

## Hint:

Make sure you write the unit of the answer.

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class.

## 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.

**Something I still wonder about rates and units is …**