# Lesson 16

Compare Fractions with the Same Numerator

## Warm-up: True or False: Unit Fractions (10 minutes)

### Narrative

The purpose of this True or False is to elicit insights students have about comparing unit fractions. The reasoning students do helps to deepen their understanding of what the denominator of a fraction means. It will also be helpful later when students compare fractions with the same numerator.

In this activity, students have an opportunity to look for and make use of structure (MP7) because they notice that a larger denominator indicates that a whole is split into more parts. The more parts the whole is split into, the smaller those parts will be.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

Decide whether each statement is true or false. Be prepared to explain your reasoning.

• $$\frac{1}{2} > \frac{1}{4}$$
• $$\frac{1}{4} > \frac{1}{3}$$
• $$\frac{1}{6} > \frac{1}{8}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Who can restate _____’s reasoning in a different way?”
• “Does anyone want to add on to _____’s reasoning?”

## Activity 1: Five Parts of Something (20 minutes)

### Narrative

The purpose of this activity is for students to represent their thinking visually as they compare pairs of fractions with the same numerator. They also locate fractions with the same numerator on number lines and observe the relative locations of the points. Students see that fractions with larger denominator are smaller in size (or are closer to 0 on the number line). Their reasoning here reinforces the idea that the denominator of a fraction determines how many equal parts are in a whole, and that the more parts there are, the smaller each part is (MP7).

To compare $$\frac{5}{6}$$ and $$\frac{5}{8}$$, students are likely to draw one diagram or number line for sixths and a separate one for eighths. They may use a single diagram or number line, but find it more difficult to partition and represent both denominators.

### Launch

• Groups of 2
• “Priya and Tyler are comparing two fractions. Read their conclusions and decide who you agree with.”
• 2 minutes: independent work time

### Activity

• “Talk to your partner about who you agree with. Use diagrams or number lines to show your thinking.”
• 3–5 minutes: partner discussion
• As students work, consider asking:
• “How does your representation show which fraction is greater?”
• “How do you know that eighths are smaller than sixths?”
• Monitor for students who use diagrams and those who use number lines.
• Pause for a discussion.
• Select two students, one who uses each representation, to share. Display their work side-by-side for all to see.
• “How do these representations both show that $$\frac{5}{6}$$ is greater than $$\frac{5}{8}$$?” (Both show that we are looking at 5 parts in each fraction and that sixths are larger than eighths. That means that 5 sixths are larger than 5 eighths.)
• “Now complete the last two problems.”
• 5–7 minutes: independent or partner work time

### Student Facing

1. Priya says that $$\frac{5}{6}$$ is greater than $$\frac{5}{8}$$.

Tyler says that $$\frac{5}{8}$$ is greater than $$\frac{5}{6}$$.

Who do you agree with? Show your thinking using diagrams or number lines.

2. For each pair of fractions, which fraction do you think is greater?

1. $$\frac{5}{3}$$ or $$\frac{5}{4}$$

2. $$\frac{5}{8}$$ or $$\frac{5}{2}$$

3. $$\frac{5}{6}$$ or $$\frac{5}{4}$$

3. Locate and label each fraction on a number line: $$\frac{5}{2}$$, $$\frac{5}{3}$$, $$\frac{5}{4}$$, $$\frac{5}{6}$$, $$\frac{5}{8}$$.

What do you notice about the points? Make 1–2 observations.

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

If students are not sure how to get started, consider asking: “What do we know about these fractions?” and “How could we represent these fractions?”

### Activity Synthesis

• Display a set of number lines that a student completed.
• Invite the class to share their observations about the locations of the points.
• “How can the locations of the points help you decide which is greater, $$\frac{5}{3}$$ or $$\frac{5}{4}$$?” (I can see that $$\frac{5}{3}$$ is located to the right of $$\frac{5}{4}$$, so $$\frac{5}{3}$$ is greater.)
• Consider asking: “Why is the fraction with the smallest denominator, 2, the greatest fraction in the set?” (There are only 2 parts in 1 whole. With every other fraction the denominator is larger than 2 so there are more parts in the whole which makes the parts smaller.)

## Activity 2: Fractions with the Same Numerator (15 minutes)

### Narrative

The purpose of this activity is for students to compare two fractions with the same numerator. Students use any representation to reason about the number and size of the parts of each fraction. In the last problem, students may notice that more than one denominator can sometimes work, but do not need to generalize about all the denominators that make each statement true. There is an opportunity for that kind of generalization in a future lesson.

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “Which denominators did you choose and why do they make the statements true?” Invite listeners to ask questions, to press for details, and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.
Advances: Writing, Speaking, Listening
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, Social-Emotional Functioning

### Launch

• Groups of 2
• “We just saw several ways we could represent comparisons of fractions with the same numerator. As you compare fractions with the same numerator in this activity, you can use diagrams or number lines or write to explain your reasoning.”

### Activity

• 8–10 minutes: independent work time
• “Share your favorite way to represent your reasoning with your partner.”
• 3–5 minutes: partner discussion
• Monitor for students who choose different denominators in the last problem.

### Student Facing

1. For each pair of fractions, circle the fraction that is greater. Explain or show your reasoning.

1. $$\frac{1}{4}$$ and $$\frac{1}{3}$$
2. $$\frac{3}{4}$$ and $$\frac{3}{8}$$
3. $$\frac{5}{3}$$ and $$\frac{5}{6}$$
4. $$\frac{9}{8}$$ and $$\frac{9}{6}$$
2. Use the symbols > or < to make each statement true. Be prepared to explain your reasoning.

1. $$\frac{2}{2} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{2}{6}$$
2. $$\frac{4}{3} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{4}{8}$$
3. $$\frac{8}{8} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{8}{4}$$
4. $$\frac{5}{4} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{3}$$
3. Write in the missing denominator of the fraction to make each statement true. Be prepared to explain your reasoning.

1. $$\frac{1}{3} < \frac{1}{\phantom{0000}}$$
2. $$\frac{6}{4} > \frac{6}{\phantom{0000}}$$
3. $$\frac{4}{4} < \frac{4}{\phantom{0000}}$$
4. $$\frac{2}{6} < \frac{2}{\phantom{0000}}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

If students don’t determine which fraction is greater, consider asking: “What do we know about these fractions?” and “How could we represent these fractions?”

### Activity Synthesis

• Select 3–4 students to share the denominators they chose for one part in the last problem. Ask them to explain why their chosen denominator makes the statement true.

## Lesson Synthesis

### Lesson Synthesis

“Today we compared fractions with the same numerator.”

“How would you describe to a friend how to compare fractions with the same numerator?” (We have to think about how big the parts are since the denominators are different. We have the same number of parts, but we need to know which parts are bigger or smaller. If the denominator is larger, there are more parts in the whole, so the parts are smaller than those in a fraction with a smaller denominator.)

## Cool-down: Same Numerator (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.