# Lesson 11

Use Factors to Find Equivalent Fractions

## Warm-up: Which One Doesn’t Belong: Four Representations (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare representations of fractions. To make comparisons, students need to draw on their knowledge about fractional parts, the size of fractions, and equivalent fractions.

### Launch

• Groups of 2
• Display the image.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

Which one doesn't belong?

1. $$\frac{1}{4}$$

### Activity Synthesis

• “Let’s find at least one reason why each one doesn’t belong.”

## Activity 1: The Other Way Around (20 minutes)

### Narrative

In this activity, students see that they can find equivalent fractions by dividing the numerator and denominator by a common factor. They connect this strategy to the process of grouping unit fractions on a number line into larger equal-size parts. The result is a fewer number of parts, and smaller numbers for the numerator and denominator of equivalent fractions.

MLR8 Discussion Supports. Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain the strategies used in the given examples. Invite groups to rehearse what they will say when they share with the whole class.

• Groups of 2

### Activity

• “Be prepared to explain how you think Andre’s and Kiran’s strategies are related.”
• 7–8 minutes: partner work time
• “Take a few minutes to answer the last problem independently.”
• 2–3 minutes: independent work time for the last problem

### Student Facing

1. Andre drew a number line and marked a point on it. Label the point with the fraction it represents.

2. To find other fractions that the point represents, Andre made copies of the number line. He drew darker marks on some of the existing tick marks.

Label the darker tick marks Andre made on each number line.

3. Kiran wrote the same fractions for the points but used a different strategy, as shown. Analyze his reasoning.

$$\frac{8 \ \div \ 4}{12 \ \div \ 4}=\frac{2}{3}$$

$$\frac{8 \ \div \ 2}{12 \ \div \ 2}=\frac{4}{6}$$

How do you think Andre’s and Kiran’s strategies are related?

4. Try using Kiran’s strategy to find one or more fractions that are equivalent to $$\frac{10}{12}$$ and $$\frac{18}{12}$$.

### Activity Synthesis

• “What did Andre do with the number lines? How would it help him find equivalent fractions?” (Andre grouped the 12 parts into equal groups of different sizes—2s, 4s—to make bigger parts. Then, he counted the number of those new parts.)
• “What did Kiran do? How is his strategy related to Andre’s?” (Kiran divided 12 by 4 and then by 2, similar to how Andre put 12 parts into groups of 4 and then of 2.)
• “Notice that the equivalent fractions $$\frac{2}{3}$$ and $$\frac{4}{6}$$ both have smaller numbers for the numerator and denominator than the original fraction. Can you use Andre’s number line to show why this might be?” (The size of the parts are bigger, so there are fewer parts in 1 whole.)

• “How many equivalent fractions can you find for $$\frac{10}{12}$$ using Kiran’s way?” (One) “How many can you find for $$\frac{18}{12}$$?” (Three)
• “What might be a reason that you could find more equivalent fractions for $$\frac{18}{12}$$ than for $$\frac{10}{12}$$?” (18 and 12 have more factors in common than 10 and 12.)
• “How would you show $$\frac{9 \ \div \ 3}{12 \ \div \ 3}=\frac{3}{4}$$ on the number line?” (Put the original 12 parts into groups of 3 to get 4 parts, each being a fourth. Mark 3 of those 4 parts to show $$\frac{3}{4}$$.)

## Activity 2: How Would You Find Them? (15 minutes)

### Narrative

In this activity, students generate equivalent fractions by applying the numerical strategies they learned. (Students might opt to use other strategies, but most of the given fractions have numbers that would make visual representation and reasoning inconvenient.) Depending on the given fractions, students need to decide whether it makes sense to multiply or divide the numerator and denominator by a common number.

Engagement: Provide Access by Recruiting Interest. Synthesis: Leverage choice around perceived challenge. Invite students to select at least 3 of the 5 given fractions for this activity.
Supports accessibility for: Organization, Attention, Social-Emotional Functioning

• Groups of 2

### Activity

• “Work on the activity independently. Then, share your responses with your partner and check each other’s work.”
• 8–10 minutes: independent work time
• 3–5 minutes: partner discussion

### Student Facing

Find at least two fractions that are equivalent to each fraction. Show your reasoning.

1. $$\frac{16}{8}$$
2. $$\frac{40}{10}$$
3. $$\frac{7}{6}$$
4. $$\frac{90}{100}$$
5. $$\frac{5}{4}$$

### Student Response

If students attempt to partition the number line into 30, 60, or 90 parts, consider asking: “How can we use the patterns from the previous activity to help us here?”

### Activity Synthesis

• See lesson synthesis.

## Activity 3: Card Sort: Fractions Galore [OPTIONAL] (15 minutes)

### Narrative

This activity is optional because it provides an opportunity for students to apply concepts from previous activities that not all classes may need. It allows students to practice using numerical strategies to find equivalent fractions by sorting a set of 36 cards. Students are not expected to find all equivalent fractions in the set. When students look for equivalent fractions they use their understanding of multiples and the meaning of fractions (MP7).

### Required Materials

Materials to Copy

• Fractions Galore

### Required Preparation

• Create a set of Fraction Galore cards from the blackline for each group of 3.

### Launch

• Groups of 3–4
• Give each group one set of cards created from the blackline master.

### Activity

• “Work with your group to sort the cards by equivalence. Find as many sets of equivalent fractions as you can. Some fractions have no equivalent fractions.”
• 8–10 minutes: small group work time

### Student Facing

Your teacher will give you a set of cards. Find as many sets of equivalent fractions as you can. Be prepared to explain or show your reasoning.

Record the sets of equivalent fractions here.

Record fractions that do not have an equivalent fraction here.

### Activity Synthesis

• “What strategy did your group use to find equivalent fractions? How well did the strategy work? How efficient was it?” (We looked at the numerators and denominators to see if they were multiples or factors we recognized.)
• “Did you notice any new patterns in the fractions that are equivalent?”

## Lesson Synthesis

### Lesson Synthesis

“Today we looked at another way to find equivalent fractions. We divided the numerator and denominator of a fraction by a factor they have in common.”

“How did you decide whether to use multiplication or division to write an equivalent fraction?” (Sample response: It depends on the numbers in the fraction. When the numbers are large to start with and both have a factor in common, we’d divide by that factor. When the numbers are small and have no shared factors, we’d multiply.)

## Student Section Summary

### Student Facing

In this section, we learned to identify and write equivalent fractions. We placed fractions on number lines and saw that two fractions that occupy the same spot on a number line are equivalent.

We also looked at strategies for finding equivalent fractions and learned that multiplying or dividing the numerator and denominator by the same number will result in an equivalent fraction. Here are some examples:

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

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

$$\frac{1}{5}$$ is equivalent to $$\frac{2}{10}$$ and $$\frac{4}{20}$$.

$$\frac{8 \ \div \ 2}{12 \ \div \ 2} = \frac{4}{6}$$

$$\frac{8 \ \div \ 4}{12 \ \div \ 4} = \frac{2}{3}$$

$$\frac{8}{12}$$ is equivalent to $$\frac{4}{6}$$ and $$\frac{2}{3}$$.