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# Comparing Numbers with Ratios

## Overview

This lesson formally introduces and defines a ratio as a way of comparing numbers to one another.

# Key Concepts

A *ratio* is defined by the following characteristics:

- A ratio is a pair of numbers (
*a*:*b*). - Ratios are used to compare two numbers.
- The value of a ratio
*a*:*b*is the quotient*a*÷*b*, or the result of dividing*a*by*b*.

Other important features of ratios include the following:

- A ratio does not always tell you the values of quantities being compared.
- The order of values in a ratio matters.

# Goals and Learning Objectives

- Introduce a formal definition of ratio.
- Use the definition of ratio to solve problems related to comparing quantities.
- Understand that ratios do not always tell you the values of the quantities being compared.
- Understand that the order of values in a ratio matters.

# Introduction to Ratios

# Lesson Guide

In this lesson, students learn the formal definition of a ratio and then use it to solve problems. The start of the lesson introduces the concept of a ratio as a way of comparing numbers of objects using division, which is an alternative to comparing numbers using subtraction. Have students look at the picture of stars and triangles and then read the text.

Ask:

- The ratio of triangles to stars is 3:10. What is the ratio of stars to triangles? (Answer: 10:3)

ELL: Keep in mind that some students may not feel comfortable reading aloud. Be prepared to support them in this task.

## Opening

# Introduction to Ratios

Look at the picture of stars and triangles and read the following information.

- One way that you can compare the number of stars and the number of triangles is to say that there are 7 more stars than there are triangles. This comparison looks at the difference between two quantities; it uses the operation of subtraction.
- Another way that you can compare the number of stars and the number of triangles is to say that for every 3 triangles there are 10 stars. You can say that the
*ratio*of triangles to stars is 3 to 10 or 3:10. This comparison uses the operation of division. - The value of the ratio of triangles to stars is $\frac{3}{10}$, or 0.3.

# Ratio of Egginess

# Lesson Guide

Have a volunteer read the definition of a ratio aloud.

Review the definition of ratio: A ratio is a comparison of two numbers by division.

The value of a ratio is the quotient that results from dividing the two numbers. For example, the value of the ratio 35:7 is 5, which you find by computing 35 ÷ 7 = 5.

Have students review the video of the egginess problem from the previous lesson. Discuss the ratio in the egginess problem. Demonstrate how to write the ratio. Talk about the fact that the ratio could be flour to eggs or eggs to flour. The ratio of flour to eggs is 3:2; the ratio of eggs to flour is 2:3.

ELL: When showing the video, be sure that ELLs are following the explanations. 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. Remind students that they are finding the ratio in the egginess problem.

## Opening

# Ratio of Egginess

A *ratio* is a comparison of two numbers by division.

The *value* of a ratio is the quotient that results from dividing the two numbers. For example, the value of the ratio 35:7 is 5, which you find by computing 35 ÷ 7 = 5.

In the previous lesson, you looked at how to fix the egginess in a mixture. Watch the Egginess

Part 2 video.

- What is the ratio in the egginess problem?

VIDEO: Egginess Part 2

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will explain how ratios are used to compare quantities.

ELL: Discuss the concept of ratio. Sketch diagrams of the examples so students can make associations with comparison of quantities. Use manipulatives to demonstrate the comparison of two quantities using subtraction and division. Use illustrations or manipulatives to model effective learning to explain ratios. It is important to emphasize that you can describe ratios by saying “for every” or “per” since these terms will be used interchangeably throughout the unit. Underline the two quantities so that students know exactly which quantities you are comparing.

## Opening

Explain how ratios are used to compare quantities.

# Ms. Lee’s Class

# Lesson Guide

Have students work in pairs on the problems and the presentation.

SWD: Help students with disabilities build their mathematical vocabulary by continually modeling the use of new terms in the context of classroom work and activities.

# Mathematical Practices

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

Listen for students who use the problem situations to help them make sense of the values they are working with.

**Mathematical Practice 6: Attend to precision.**

Listen also for students who use the term *ratio* correctly or who discuss the correct usage of the term as they work together to solve the problems.

# Interventions

**Student thinks that you write a ratio as a subtraction.**

- Look at your definition of ratio. Is it a difference?

**Student reverses the boys and girls in the ratio.**

- What are the two things you are comparing? What is the order that you are comparing them in?

# Answers

- There are 2 more girls than boys.
- The ratio of boys to girls is 15:17.
- The ratio of girls to boys is 17:15.

## Work Time

# Ms. Lee's Class

There are 15 boys and 17 girls in Ms. Lee's math class.

- What is the difference between the number of girls and the number of boys in the class?
- What is the ratio of boys to girls?
- What is the ratio of girls to boys?

Ask yourself:

- When you need to find the difference between two numbers, what operation do you use?
- For the ratio of boys to girls, what should the first number be, ”the number of girls or the number of boys?
- For the ratio of girls to boys, what should the first number be, the number of girls or the number of boys

# A Tennis Game

# Mathematical Practices

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

Listen for students who use the problem situations to help them make sense of the values they are working with.

**Mathematical Practice 6: Attend to precision.**

Listen also for students who use the term *ratio* correctly or who discuss the correct usage of the term as they work together to solve the problems.

# Interventions

**Student thinks there are exactly 3 females and 2 males watching the tennis game.**

- The ratio is 3:2. Here are some possibilities that fit this ratio: 6 females to 4 males, 30 females to 20 males, 300 females to 200 males. How can you tell these all have a ratio of 3:2?

**Student thinks there is 1 more female than male watching the tennis game.**

- Look at the definition of a ratio at the start of the lesson. Does a ratio compare by subtraction or by division?
- You are thinking of the difference between 3 and 2, but the problem is about the ratio between them. There could be 6 females and 4 males or 30 females and 20 males.

**Student believes the number of females, males, or people watching the tennis game can be calculated with only the ratio.**

- If there were 20 males, how many females would there be? If there were 15 females, how many males would there be? If you don’t know one of these numbers, can you find the other?

# Possible Answers

- No, the ratio tells you that for every 2 males watching the game, there are 3 females watching the game, but it does not tell you the total number of females.
- No, the ratio of 3 females to 2 males does not tell you the total number of people watching the game. It tells you that for every 5 people watching, 3 are female and 2 are male.
- Yes, the ratio tells you there are 1.5 times as many females as males watching the game.
- No, without knowing the actual number of each quantity, you can’t find their difference.
- Yes, the ratio of the number of males to the number of females is 2:3.

## Work Time

# A Tennis Game

The ratio of the number of females watching a tennis game to the number of males watching the tennis game is 3 to 2. You can write that as $\frac{3}{2}$, or 3:2.

- Can you tell from this ratio how many females are watching the tennis game? Explain.
- Can you tell from this ratio how many people are watching the tennis game? Explain.
- Can you tell from this ratio whether more males or more females are watching the game? Explain.
- Can you tell from this ratio the difference between the number of males and the number of females watching the game? Explain.
- Could you use this ratio to write the ratio of the number of males watching the tennis game to the number of females watching the tennis game? Explain.

Ask yourself:

- Sketch a diagram showing several possible numbers of females and males watching the game. Can you tell which numbers are correct based on the ratio?
- In looking at a ratio, how can you tell which number represents the larger amount?
- What would you need to know to find the difference between the number of males and females watching the game?
- In determining whether the ratio can be rewritten to represent males to females, think about the definition of a ratio.

# 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 explicitly discuss subtraction and division as ways of comparing numbers
- Students who discuss the limits of what a ratio tells you (e.g., “It doesn’t tell us how many …”)
- Students who divide 3 by 2 to get 1.5 and then discuss the meaning of 1.5 with reference to the problem situation

# Challenge Problem

## Possible Answer

- The statement is always true. A number divided by itself is equal to 1, so if two quantities get closer to each other, their ratio’s value is closer to a number divided by itself, or 1.

## Work Time

# Prepare a Presentation

Explain what types of conclusions you can and cannot make based on the tennis game ratio.

In your own words, explain what a ratio is.

# Challenge Problem

As two quantities get closer to each other, the value of the ratio of the quantities approaches 1.

- Is the above statement
*always true*,*sometimes true*, or*never true*? Explain.

# Make Connections

# Mathematics

Have students share their presentations. If there are any misunderstandings, facilitate a class discussion that focuses on the meaning of a ratio and what it does and does not tell us.

Ask:

- What are some examples of numbers of females and males that could be watching the tennis game?

Compare these examples to the situation about the number of boys and girls in Ms. Lee’s class, in which the actual numbers of students are given rather than a ratio between numbers. Highlight correct use of the term *ratio*, or provide opportunities for students to revise their use of the term.

Conclude the discussion with a focus on whether the ratio tells you whether more males are watching the game or more females are watching the game. If there is a student who divided 3 by 2 to get 1.5, ask them to share their strategy. Ask the class what 1.5 means in this situation:

- Does it make sense for a ratio of numbers of people to have a value of 1.5?

Check that all students understand that the order of values in a ratio matters. The ratio of males to females (2:3) is different from the ratio of females to males (3:2).

# Mathematical Practices

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

Ask students to talk about how they can use the context of a problem situation to help them make sense of the values they are working with.

**Mathematical Practice 6: Attend to precision.**

Call attention to correct uses of the term *ratio*, or ask for clarifications about its use and meaning as needed as students present their work and ask questions of presenters.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates' explanations of the conclusions that can and cannot be made based on the tennis game ratio, and their explanations of what a ratio is.

As your classmates present, ask questions such as:

- What possible numbers of females and males watching the tennis game did you find?
- How did you determine these numbers?
- How did you decide what types of conclusions you can and cannot make based on the tennis game ratio?
- Is there anything you can add to your explanation to make it more specific or precise?

# What I Know about Ratios

# A Possible Summary

Ratios allow you to compare quantities, but by themselves, they do not tell you the actual values of the quantities. Using a ratio to compare quantities is different from using subtraction to find the difference between quantities because a ratio tells you the value of one quantity for a given value of the other quantity. For example, for a ratio of 3:2, you know that if the first quantity has a value of 6, the second quantity has a value of 4.

SWD: Create a resource for some students that includes this explanation in simpler language (perhaps with visual illustrations). Annotate illustrations that show ratios and what they represent.

# Additional Discussion Points

If there is time, discuss the following:

- A summary of different ways to compare numbers
- A definition of a ratio
- A description of what a ratio tells you and what it doesn’t tell you

## Formative Assessment

# Summary of the Math: What I Know about Ratios

Write a summary of what you learned about ratios.

Check your summary.

- Do you explain what a ratio is?
- Do you discuss what types of conclusions can and cannot be made based on a ratio?
- Do you explain how using ratios to compare two numbers is different from using subtraction to compare two numbers?

# 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 students wonder about 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.

**Something I wonder about ratios is …**