# Expressing Ratios

## Overview

Students work with a set of cards showing different ways of expressing ratios, including both part-part statements and part-whole statements. They group the cards that show the same ratio of boys to girls, but without the explicit use of the term *equivalent*.

# Key Concepts

Ratios can be represented in *a*:*b* form, as fractions, as decimals, as factors, and in words; they can be expressed in part-part statements or in part-whole statements.

# Goals and Learning Objectives

- Group cards showing ratios that are equivalent but expressed in different forms.

# Ratio of Boys and Girls

# Lesson Guide

Read the prompt about boys and girls at the middle school aloud. Then have students work together to complete the given statements using each given ratio exactly once.

When the class is done, have students share how they knew which values went with which statements.

SWD: When students think aloud, it provides them with opportunities to practice the thinking process involved in solving a set of problems. Listen to the students’ thought processes, correct misconceptions, and fill in incomplete processes.

ELL: Read-aloud routines and procedures are important for ELLs because teachers explicitly model strategies and behaviors used by effective readers. These valuable strategies can be used to access texts, retrieve information, build comprehension, and organize information.

# Mathematics

Note that some of the given ratio values represent part-whole relationships. Students will work with part-whole relationships in Lesson 13, when percents are introduced.

Point out that some of the statements describe the relationship between a “part” and “the whole” and some of the statements describe the relationship between one “part” and the other “part.”

# Answers

- The ratio of boys to girls is 2:1.
- Of all the students, $\frac{1}{3}$ are girls.
- $\frac{2}{3}$ of all the students are boys.
- One out of every 3 students is a girl.
- For every 2 boys, there is 1 girl.
- There are half as many girls as boys.
- There are 2 times as many boys as girls.

## Opening

# Ratio of Boys and Girls

There are 200 boys and 100 girls at Thurgood Marshall Middle School.

Complete the following sentences using each of the given quantities shown below only once.

- The ratio of boys to girls is
**_____**. - Of all the students,
**_____**are girls. **_____**of all the students are boys.- One out of every
**_____**students is a girl. - For every 2 boys, there is
**_____**girl. - There are
**_____**as many girls as boys. - There are
**_____**times as many boys as girls.

2:1 1 $\frac{2}{3}$ 3 $\frac{1}{3}$ half 2

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will represent ratios in different ways.

## Opening

Represent ratios in different ways.

# Four Schools Card Sort

# Lesson Guide

Have students work in pairs on the Card Sort Interactive. Have them take turns sorting the cards into the school with the same ratio of girls and boys. Encourage students to discuss their reasoning with their partner about their card placement

SWD: Students with disabilities may have difficulty with operations involving more complex numbers (decimals). Supporting students to practice skills related to ratios more independently may include:

- Allowing students access to a calculator
- Providing partially scaffolded answers

ELL: Encourage students to verbalize their explanations. Allow students to speak in small groups to gain confidence.

# Mathematical Practices

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

Listen for students who are challenging one another’s grouping choices and who are developing clearer, more coherent arguments to justify their choices.

# Interventions

**Student doesn’t understand the task.**

- Each of the cards shows a comparison of boys and girls; your job is to group them so all of the cards that express the same ratio are in a group together.

**Student relates cards without reference to the data from each school.**

- How can you tell this card should be grouped with these cards? Justify your choice to your partner. Can you think of another way to justify the choice?
- For which cards do you need to know the actual number of boys and girls at the school?
- Explain how you knew which group this card belonged to without having to check it against the number of boys and girls at the school.

**Student has difficulty setting up appropriate calculations.**

- Talk to your partner about why it is difficult to set up the calculation.
- Some of the statements are part-part statements and some are part-whole statements. What is the whole—the number of all students—at each school?

# Answers

## Work Time

# Four Schools Card Sort

Four high schools (A, B, C, and D) have different numbers of students and different ratios of boys to girls.

- Working with a partner, take turns matching cards that represent the same school.
- Explain to your partner how you know the cards match.
- Your partner should either agree with your explanation or challenge it if your explanation is not correct, clear, or complete.

To match the cards, find a strategy that will help you narrow down the choices. For example, you might start by choosing a school in which the ratio of boys to girls is easy for you to see. Then find all the cards that match that school.

# Prepare a Presentation

# Preparing for Ways of Thinking

Listen for the following student thinking to highlight during the Ways of Thinking discussion:

- Students who talk about which statements are about “part-part” relationships and which are about “part-whole” relationships, in those terms or in other terms
- Students who debate about the equivalence of the ratios on the cards in a group
- Students who have difficulty with calculations

# Challenge Problem

## Possible Answers

- Answers will vary. Check students’ work. They should create one group of cards or statements, each with a different representation: ratio, decimal, fraction, factor, and part-part statement in words.

## Work Time

# Prepare a Presentation

Select one of your card matches and explain why those cards represent the same school.

# Challenge Problem

- Make a set of cards for another school using a ratio of girls to boys that is different from any of the ratios in the card sort.

Prepare a Presentation—HANDOUT: Four Schools Card Sort: Plan a Presentation

Challenge—HANDOUT: Four Schools Card Sort: Create Your Own School

# Make Connections

# Mathematics

Have students share their presentations.

During the discussion, make explicit the distinction between statements about part-whole relationships and statements about part-part relationships.

Ask students to share their thinking for the most difficult calculations, and ask if the difficulty came from the computations themselves or from translating between types of ratio comparisons (part-part vs. part-whole).

Then ask students to share their thinking for the easiest calculations, and ask if the calcuations were easy based on the computations themselves or on translating between the part-part and part-whole comparisons.

# Mathematical Practices

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

As students share their work, have them explain any disagreements they had and how they resolved them. Have them justify their grouping choice with a clear and coherent argument, clarifying and elaborating with help from the class as needed.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about the different approaches your classmates used to match the cards.

As your classmates present, ask questions such as:

- How do you know that those two cards describe the same school?
- What was the most difficult calculation that you had to make?
- What was difficult about it? What were the easiest calculations?
- What do you think made them easy?

# Different Ways to Represent Ratios

# Lesson Guide

Have pairs quietly discuss the definition of a ratio. As student pairs work together, make a note to clarify any misunderstandings in the class discussion. After a few minutes, discuss the Summary as a class.

## Formative Assessment

# Summary of the Math: Different Ways to Represent Ratios

**Read and Discuss**

- You learned the definition of a 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 21:3 is 7, which we find by computing 21 ÷ 3 = 7.

- A
- A ratio is a
*relative*comparison. It tells us about one quantity in terms of another quantity. In contrast, a comparison by subtraction is an*absolute*comparison. It tells us the difference between two quantities. - There are different ways to represent ratios, including:
- Using colon notation (2:5)
- As a decimal (0.4)
- As a fraction ($\frac{2}{5}$)

Can you:

- Define a ratio?
- Represent a ratio in different ways?
- Explain the difference between comparisons that use ratios and comparisons that use subtraction?

# 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 connections students see between ratios, fractions, and decimals.

## 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.

**A connection I see between ratios and fractions and decimals is …**