# Lesson 10

Use Multiples to Find Equivalent Fractions

## Warm-up: Notice and Wonder: Four Equations (10 minutes)

### Narrative

The purpose of this warm-up is to draw students’ attention to the multiplicative relationships between the numerators and denominators of two equivalent fractions. These observations will be helpful later as students use the idea of multiples to generate equivalent fractions.

While students may notice and wonder many things about these equations, highlight observations about a factor relating the numbers in the two sides of each equation.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

• $$\frac{1}{3} = \frac{2}{6}$$

• $$\frac{2}{3} = \frac{4}{6}$$

• $$\frac{3}{3} = \frac{6}{6}$$

• $$\frac{4}{3} = \frac{8}{6}$$

### Activity Synthesis

• “How are the numbers on the right side of each equal sign related to the numbers on the left?” (Each number on the right is twice the number on the left.)
• “Are the fractions on the right twice the size of the fractions on the left?” (No, they are the same size.)

## Activity 1: Elena’s Way (20 minutes)

### Narrative

In an earlier lesson, students used visual representations to generate equivalent fractions. They did so by partitioning each increment on a number line into smaller equal-size parts. In this activity, they connect that action to a numerical process—one that involves multiplying both the numerator and denominator by the same factor. When students notice that they can multiply the numerator and denominator of a fraction by any whole number to get an equivalent fraction they observe regularity in repeated reasoning (MP8).

MLR2 Collect and Display. Collect the language students use to reason about how to find equivalent fractions. Display words and phrases such as: equivalent fraction, equation, number line, numerator, denominator, multiply, multiples, etc. During the activity, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.

### Launch

• Groups of 2
• “Take a look at Andre’s number lines you worked with in a previous lesson.”

### Activity

• “Think quietly for a couple of minutes about what Elena did and how it relates to Andre’s number lines.”
• 1–2 minutes: quiet think time for the first problem
• 3–4 minutes: partner discussion on the first problem
• Pause for a brief whole-class discussion. Invite students to share their ideas about Elena’s work and how it is related to Andre’s number lines.
• 4–5 minutes: independent work time for the last problem
• Monitor for students who find equivalent fractions for $$\frac{1}{8}$$ by multiplying by a factor other than 2, 3 or 4.

### Student Facing

Elena thought of another way to find equivalent fractions. She wrote:

$$\frac{1 \ \times \ 2}{5 \ \times \ 2} = \frac{2}{10}$$

$$\frac{1 \ \times \ 3}{5 \ \times \ 3} = \frac{3}{15}$$

$$\frac{1 \ \times \ 4}{5 \ \times \ 4} = \frac{4}{20}$$

$$\frac{1 \ \times \ 5}{5 \ \times \ 5} = \frac{5}{25}$$

$$\frac{1 \ \times \ 10}{5 \ \times \ 10} = \frac{10}{50}$$

1. Analyze Elena’s work. Then, discuss with a partner:

1. How are Elena’s equations related to Andre’s number lines?
2. How might Elena find other fractions that are equivalent to $$\frac{1}{5}$$? Show a couple of examples.
2. Use Elena’s strategy to find five fractions that are equivalent to $$\frac{1}{8}$$. Use number lines to check your thinking, if they help.

### Activity Synthesis

• Select 1–2 students to share their equivalent fractions for $$\frac{1}{8}$$ and their reasoning. Display their equivalent fractions as equations (for example, $$\frac{1}{8} = \frac{3}{24}$$, $$\frac{1}{8} = \frac{4}{32}$$, $$\frac{1}{8} = \frac{7}{56}$$, and $$\frac{1}{8} = \frac{8}{64}$$).
• “Do these equations show the same patterns as the equations in the warm-up? How are they alike or different?” (Alike: In all of the equations the numerators and denominators are multiplied by the same amount. Different: In the warm up each numerator and denominator was multiplied by 2. In these problems each the numerator and denominator of the fraction is multiplied by different numbers each time.)

## Activity 2: Equivalence Hunting (15 minutes)

### Narrative

In this activity, students identify equivalent fractions. In the first problem, they use the numerical strategy they learned earlier to determine if two fractions are equivalent. In the second problem, they can use any strategy in their toolkit—which now includes a numerical method—to identify equivalent fractions.

Students encounter some fractions with unfamiliar denominators such as 9, 16, 32, 40, and 80, but they will not be assessed on such fractions. These denominators are multiples of familiar denominators such as 2, 3, 4, 5, 8, or 10, and are included to give students opportunities to generalize their reasoning about equivalence.

Action and Expression: Internalize Executive Functions. Synthesis: Check for understanding by inviting students to rephrase directions in their own words. Keep a display of directions visible throughout the activity.
Supports accessibility for: Memory, Organization

### Launch

• Groups of 2
• “Now you are going to see whether you can use Elena’s method to see if fractions are equivalent.”

### Activity

• 3–4 minutes: independent time to work on the first problem
• Pause for a brief whole-class discussion.
• “How did you know what number to multiply to the numerator and denominator to check equivalence?” (Sample responses:
• See if there’s a whole number that can be multiplied by 5 to get 10, multiplied by 2 to get 8, and so on.
• Divide 10 by 5, or 8 by 2, and so on, and see if what the result is and whether it’s a whole number.)
• “Work with your partner to identify all fractions on the list that are equivalent to $$\frac{3}{4}$$. Be prepared to show how you know.”
• 6–7 minutes: group work time for the second problem

### Student Facing

Look at Elena’s strategy from an earlier activity.

1. Could her strategy help us know whether two fractions are equivalent? Try using it to check the equivalence of these fractions:

1. $$\frac{5}{2}$$ and $$\frac{10}{8}$$
2. $$\frac{2}{6}$$ and $$\frac{4}{12}$$

For any two fractions that are equivalent, write an equation.

2. Find all fractions in the list that are equivalent to $$\frac{3}{4}$$. Be prepared to explain or show how you know.

$$\frac{2}{10}$$

$$\frac{6}{8}$$

$$\frac{12}{15}$$

$$\frac{30}{40}$$

$$\frac{8}{9}$$

$$\frac{12}{20}$$

$$\frac{12}{16}$$

$$\frac{15}{20}$$

$$\frac{8}{10}$$

$$\frac{24}{32}$$

$$\frac{75}{100}$$

$$\frac{60}{80}$$

### Activity Synthesis

• “Check your list of equivalent fractions with another group.”
• “Discuss any disagreement about a fraction until both groups agree whether or not it is equivalent to $$\frac{3}{4}$$.”
• 3 minutes: Check list with another group.

## Lesson Synthesis

### Lesson Synthesis

“Today we used a numerical strategy for finding equivalent fractions and for checking if fractions are equivalent.”

“Suppose a classmate was absent today. They later saw some examples of how to find equivalent fractions for $$\frac{1}{3}$$ using this strategy, but they don’t fully follow the examples.”

Display: $$\frac{1\times4}{3\times4} = \frac{4}{12}$$ and $$\frac{1\times6}{3\times6} = \frac{6}{18}$$

“What would you say to help your classmate understand what is happening in the equations? How would you explain the multiplication by 4 or by 6?”