# Lesson 12

Equivalent Fractions on a Number Line

## Warm-up: Notice and Wonder: Running on a Trail (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the idea that fractions can be used to describe lengths. While students may notice and wonder many things about this statement, the idea that Han and Tyler could have run the same distance or different distances are the important discussion points.

### Launch

• Groups of 2
• Display the statement.
• “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?

Tyler ran part of the length of a trail.
Han ran part of the length of the same trail.

### Activity Synthesis

• “How could fractions give us more information about how far Tyler and Han ran?” (Tyler and Han ran $$\frac{1}{2}$$ of the field. Tyler ran $$\frac{3}{4}$$ of the field and Han ran $$\frac{7}{8}$$ of the field.)
• “What questions could we ask about the situation?” (Did they run the same distance? Who ran farthest? How much farther did one student run than the other?)

## Activity 1: Running Part of a Trail (10 minutes)

### Narrative

The purpose of this activity is for students to explain equivalence using a number line. Students are given situations in a measurement context and have to determine whether the distance is the same. Students are encouraged to use a number line to provide an opportunity to explain fraction equivalence as fractions that are at the same location. They may choose to use two number lines for each question (one for each fraction). Choosing to use one number line or two will be discussed in the synthesis of the next activity.

When they identify whether or not two fractions of the same trail represent the same distance, students reason abstractly and quantitatively (MP2).

MLR8 Discussion Supports. Display sentence frames to support whole group discussion. “First, I _____ because . . .”, “I noticed _____ so I . . . .”

• Groups of 2

### Activity

• “Work with your partner to decide whether each pair of students ran the same distance or not. You can use number lines to explain your reasoning if they’re helpful to you.”
• 5–7 minutes: partner work time
• Monitor for students who use the number lines to explain that the students ran the same distance if the fractions are at the same location on the number line.

### Student Facing

Some students are running on a trail at a park. Decide if each pair of students ran the same distance.

You can use number lines if they are helpful to you.

1. Elena ran $$\frac{3}{6}$$ of the trail.

Han ran $$\frac{1}{2}$$ of the trail.

2. Jada ran $$\frac{1}{4}$$ of the trail.

Kiran ran $$\frac{2}{8}$$ of the trail.

3. Lin ran $$\frac{2}{3}$$ of the trail.

Mai ran $$\frac{5}{6}$$ of the trail.

### Activity Synthesis

• Display a student-created number line that shows $$\frac{1}{4}$$ and $$\frac{2}{8}$$ at the same location.
• “How does this show that Jada and Kiran ran the same distance?” (The points that represent them are at the same location between 0 and 1 on the number line.)
• “We’ve learned that two fractions are equivalent if they are the same size. Now we also know that two numbers are equivalent if they are at the same location on a number line. Because $$\frac{1}{4}$$ and $$\frac{2}{8}$$ are at the same location, we can say they are equivalent.”
• “How could we use the equal sign to record fractions that are equivalent?” ($$\frac{3}{6} = \frac{1}{2}$$, $$\frac{1}{4} = \frac{2}{8}$$)
• Share and record responses.

## Activity 2: Locate and Pair (10 minutes)

### Narrative

The purpose of this activity is for students to locate fractions on the number line, and find pairs of fractions that are equivalent. Students can use a separate number line for each denominator, but they can also place fractions with different denominators on the same number line to show equivalence. Focus explanations about why fractions are equivalent on the fact that they share the same location. In the synthesis, discuss how one number line or two can be used to compare fractions.

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each chunk.
Supports accessibility for: Attention, Organization

• Groups of 2

### Activity

• “Work independently to locate these numbers on the number line. Then, find 4 pairs of fractions that are equivalent. Be prepared to explain your reasoning.”
• 3–5 minutes: independent work time
• “Now, share the pairs of fractions you wrote with your partner and explain how you know they are equivalent.”
• 2–3 minutes: partner discussion
• Monitor for students who compare fractions on a single number line and those who compare fractions on separate number lines.

### Student Facing

1. Locate and label the following numbers on a number line. You can use more than one number line if you wish.

$$\frac{1}{2}$$, $$\frac{1}{3}$$, $$\frac{1}{4}$$, $$\frac{2}{3}$$, $$\frac{2}{6}$$, $$\frac{3}{8}$$, $$\frac{3}{4}$$, $$\frac{4}{6}$$, $$\frac{4}{8}$$, $$\frac{6}{8}$$, $$\frac{7}{8}$$

2. Find 4 pairs of fractions that are equivalent. Write equations to represent them.

$$\underline{\hspace{1 cm}} = \underline{\hspace{1 cm}}$$

$$\underline{\hspace{1 cm}} = \underline{\hspace{1 cm}}$$

$$\underline{\hspace{1 cm}} = \underline{\hspace{1 cm}}$$

$$\underline{\hspace{1 cm}} = \underline{\hspace{1 cm}}$$

If you have time: Use the number lines to generate as many equivalent fractions as you can.

### Activity Synthesis

• Select previously identified students to display how a single number line or separate number lines can be used to show equivalent fractions.
• “When might it make sense to use a single number line and when might it be helpful to use two number lines?“ (One number line might make sense if one fraction can be partitioned to get to another, like halves to fourths. If a number line is too crowded or has fractions that could be hard to partition together, like halves and thirds, it might be helpful to use two number lines.)

## Activity 3: Rolling for Equivalent Fractions (15 minutes)

### Narrative

The purpose of this activity is for students to practice generating equivalent fractions. The goal of each round is to use the numbers on the number cubes to complete a statement that shows that two fractions are equivalent. Students roll 6 number cubes and try to use 4 of the numbers to create a statement that shows two equivalent fractions. If students roll a 5 (or a blank), they may choose any number to use. Students may choose to re-roll any of their number cubes up to 2 times. Students get a point for every true statement they make. Students may choose to use fraction strips, diagrams, or number lines to prove that their fractions are equivalent. If students choose to use diagrams, monitor to make sure they are drawing equal-sized wholes.

### Required Materials

Materials to Gather

### Required Preparation

• Each group of 2 needs 6 number cubes.

### Launch

• Groups of 2
• Give each group 6 number cubes.
• “We’re going to play a game called Rolling for Equivalent Fractions. Let’s read through the directions and play 1 round together.”
• Read through the directions with the class and play a round against the class, displaying the fractions from the cards, drawing tape diagrams, and thinking through decisions aloud.

### Activity

• “Now, play the game with your partner. See if you can get 5 points.”
• 8–10 minutes: partner work time
• Monitor for students to highlight during the synthesis that:
• create a diagram of their fraction and generate an equivalent fraction with larger parts, such as picturing $$\frac{2}{8}$$ as $$\frac{1}{4}$$
• create a diagram of the fraction they draw and further partition their fraction to make smaller pieces, such as further partitioning $$\frac{1}{2}$$ to make $$\frac{2}{4}$$
• use a pattern to generate equivalent fractions, such as knowing that there are two sixths in each third, so $$\frac{2}{3}$$ is equivalent to $$\frac{4}{6}$$

### Student Facing

1. Roll 6 number cubes. If you roll any fives, they count as a wild card and can be any number you’d like.
2. Can you put the numbers you rolled in the boxes to make a statement that shows equivalent fractions? Work with your partner to find out.
3. If you cannot, re-roll as many number cubes as you’d like. You can re-roll your number cubes twice.
4. If you can make equivalent fractions, record your statement and show or explain how you know the fractions are equivalent. You get 1 point for each pair of equivalent fractions you write.

Round 1:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Show or explain how your fractions are equivalent.

Round 2:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Show or explain how your fractions are equivalent.

Round 3:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Show or explain how your fractions are equivalent.

Round 4:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Show or explain how your fractions are equivalent.

Round 5:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Show or explain how your fractions are equivalent.

Round 6:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Show or explain how your fractions are equivalent.

Round 7:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Show or explain how your fractions are equivalent.

Round 8:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Show or explain how your fractions are equivalent.

### Student Response

If students say they aren’t sure what fractions they can make that would be equivalent, consider asking:

• “What fractions could you make with what you rolled?”
• “How could you use your fraction strips to decide if any of the fractions are equivalent?”

### Activity Synthesis

• Display number cubes showing 1, 1, 4, 2, 2, 5
• “If you got these numbers on your last roll, what equivalent fractions could you make?” ($$\frac{2}{4} = \frac{1}{2}$$ or $$\frac{1}{4} = \frac{2}{8}$$)
• If needed, ask, “What number should I use for my wild card?”

## Lesson Synthesis

### Lesson Synthesis

Display a number line that shows two fractions that are at the same location, such as $$\frac {3}{2}$$ and $$\frac{6}{4}$$.

“Earlier in the unit, we used fraction strips to see and find equivalent fractions. Here, we use number lines to find equivalent fractions.”

“How are the two ways of showing equivalent fractions alike?” (They both involve partitioning a whole and identifying two or more fractions.)

“How are they different?” (Instead of looking for parts that are the same size, we are looking for the same point or location on the number line.)

“Today, we saw that it can be helpful to use one or two number lines to show that fractions are equivalent. Keep that in mind during the cool-down.”