# Lesson 17

Compare Fractions

## Warm-up: Estimation Exploration: Ladybug Length (10 minutes)

### Narrative

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. In this warm-up, students apply what they know about fractions to estimate the length of an insect that is less than 1 inch.

### Launch

• Groups of 2
• Display the image.
• “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Record responses.

### Student Facing

What is the length of this ladybug?

Record an estimate that is:

too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Activity Synthesis

• “Is anyone’s estimate less than _____? Is anyone’s estimate greater than _____?”
• “Based on this discussion, does anyone want to revise their estimate?”

## Activity 1: Comparison Problems (15 minutes)

### Narrative

The purpose of this activity is for students to compare two numbers in context, to explain or show their reasoning, and record the results of the comparisons with the symbols >, =, or < (MP2). The numbers may be fractions with the same numerator or the same denominator, or a fraction and a whole number.

Students are likely to generate different comparison statements for the same situation. For example, they may write $$\frac{5}{8} > \frac{3}{8}$$ or $$\frac{3}{8}<\frac{5}{8}$$ to represent $$\frac{5}{8}$$ being the greater fraction. During synthesis, discuss how both statements capture the comparison and are valid.

MLR8 Discussion Supports. Synthesis: As students share the similarities and differences between the representations and comparison statements, use gestures to emphasize what is being described. For example, show with your fingers the partitions such as fourths or eighths that are the same in the representations that are being compared.
Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share a connection between activity content and their own lives. Ask “How can I use this in my own life?”
Supports accessibility for: Conceptual Processing

### Launch

• Groups of 2
• “Let’s use what we learned about comparing fractions and recognizing equivalent fractions to solve problems about lengths.”

### Activity

• “Work independently to solve the problems. For each one, be sure to show your thinking and to write a comparison statement.”
• 6–8 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for:
• different representations or reasoning strategies used for the same problem, such as diagrams, fraction strips, number lines, or explanations in words
• different statements written for the same problem, such as $$4=\frac{12}{3}$$ and $$\frac{12}{3}=4$$, or $$\frac{2}{3} > \frac{2}{8}$$ and $$\frac{2}{8} < \frac{2}{3}$$

### Student Facing

For each problem:

• Represent your answer with a statement that uses the symbols >, <, or =.
1. A beetle crawled $$\frac{2}{8}$$ of the length of a log. A caterpillar crawled $$\frac{2}{3}$$ of the length of the same log. Which insect crawled farther?

2. A grasshopper is 4 centimeters long. A caterpillar is $$\frac{12}{3}$$ centimeters long. Which insect is longer?

3. A ladybug crawled $$\frac{3}{8}$$ the length of a branch. An ant crawled $$\frac{5}{8}$$ the length of the same branch. Which insect crawled farther?

4. A grasshopper jumped $$\frac{5}{8}$$ the width of the sidewalk. A frog jumped $$\frac{5}{6}$$ the width of the same sidewalk. Which jumped a longer distance?

### Activity Synthesis

• Select 2–3 students to share their reasoning strategies or representations for at least one of the situations.
• “How did you use what you’ve learned in earlier lessons to compare fractions?” (If the fractions had the same numerator, I thought about the size of the denominators. If they had the same denominator, I compared the numerators.)
• Select students who wrote different but equally valid comparison statements (for instance, $$\frac{2}{8}<\frac{2}{3}$$ and $$\frac{2}{3} > \frac{2}{8}$$) to share.
• Discuss how to read each statement and ask students whether both accurately represent the comparison.
• Emphasize that we can write comparison statements in more than one way, but we need to check that the statements make sense given the numbers we write and the symbols we use.

## Activity 2: What Fraction Makes Sense? (15 minutes)

### Narrative

The purpose of this activity is for students to generalize what they have learned about comparing fractions to complete comparison statements and to generate new ones, using the symbols <, >, or =. Students first consider all numbers that could make an incomplete comparison statement true. Then, they find a fraction less than, greater than, and equivalent to a given fraction and write statements to record the comparisons. As in the previous activity, students see that there are different ways to record the same comparison of two numbers.

### Launch

• Groups of 2
• “Now that we have practiced comparing fractions, let's come up with fractions that are greater than, less than, or equivalent to a given fraction.”

### Activity

• “Noah was working with fractions when some juice spilled. Now he can’t tell what some numbers were. Help him figure out what was written before the juice was spilled.”
• 5 minutes: partner work time
• Pause for a discussion and invite students to share the numbers that they think make sense in the first statement ($$\frac{2}{8}< \frac{?}{8}$$).
• Display or write the comparison statements using students’ numbers.
• “Do all of these statements make sense? How do you know?”
• “Are there any more statements that we could write?”
• If time permits, repeat with the next two parts.
• “Now work independently on the last set of problems.”
• 5 minutes: independent work time

### Student Facing

1. Oh, no! Some juice spilled on Noah’s fractions. Help him figure out what was written before the juice was spilled.

Find as many numbers as you can to make each statement true. Explain or show your reasoning.

1.

2.

3.

2. For each fraction, find a fraction that is less, one that is greater, and one that is equivalent. Then, write a statement that uses the symbols >, <, or = to record each comparison.

1. Less than $$\frac{4}{6}$$: __________

Statement:

More than $$\frac{4}{6}$$: __________

Statement:

Equivalent to $$\frac{4}{6}$$: __________

Statement:

2. Less than $$\frac{3}{4}$$: __________

Statement:

More than $$\frac{3}{4}$$: __________

Statement:

Equivalent to $$\frac{3}{4}$$: __________

Statement:

### Student Response

If students don’t find more than one number that would make the statements in the first problem true, consider asking:

• “You found one number that made the statement true. How did you find that number and know that it made the statement true?”
• “How could you use a similar strategy to find another number that would make the statement true?”

### Activity Synthesis

• Invite students to share their responses to the last set of problems.
• “How did you find a fraction that was less than (or greater than or equivalent to) the given fraction?”

## Activity 3: Ultimate Locate and Label [OPTIONAL] (10 minutes)

### Narrative

The purpose of this activity is for students to use their knowledge of fractions to locate fractions with different denominators on the number line. Students may use a variety of reasoning to locate the fractions, including their knowledge of equivalence, strategies about the same numerator or denominator, or benchmark numbers they are familiar with. The synthesis focuses on the variety of strategies that make sense, and students should be encouraged to use different strategies for different fractions as needed.

Although students have represented fractions on number lines (including those with two different denominators, when reasoning about equivalence), this activity is optional because representing multiple fractions of different denominators on the same number line involves a deeper understanding than required by the standards.

• Groups of 2

### Activity

• “Work independently to start placing these fractions on the number line.”
• 3–5 minutes: independent work time
• “Share your strategies with your partner and place any fractions you have left together.”
• 5–7 minutes: partner work time
• Monitor for students who:
• place each fraction separately by partitioning for that single fraction
• compare the fraction they are placing to others they’ve already placed
• use equivalent fractions
• use strategies about same numerators or denominators
• use benchmarks like whole numbers or halves

### Student Facing

Locate and label each fraction on the number line. Be prepared to share your reasoning.

$$\frac{1}{2},\frac{3}{8},\frac{13}{8},\frac{2}{4},\frac{3}{4},\frac{9}{8},\frac{5}{4},\frac{12}{6},\frac{5}{2},\frac{9}{3},\frac{20}{8}$$

### Activity Synthesis

• Invite students to share a variety of strategies for placing fractions on the number line.
• “Which fractions were easier to place on the number line?”
• “Which fractions were more difficult?”
• “Did anyone have the same strategy but would explain it differently?”

## Lesson Synthesis

### Lesson Synthesis

“We have compared a lot of different fractions. Fractions with the same denominator, fractions with the same numerator, and in this lesson, we saw fractions that were equivalent again.”

“What do you think would be some of the most important things to tell a friend who wanted to learn about comparing two fractions?” (I would tell my friend to think about whether they can draw a representation, like a number line or a diagram to see which fraction is greater. I think they need to know whether the fractions have the same numerator or the same denominator. They can check to see if the fractions are the same size or are the same location because that means they are equivalent.)

Consider asking: “Does your strategy for comparing fractions change depending on the fractions?”

## Student Section Summary

### Student Facing

In this section, we compared fractions with the same numerator or denominator and used the symbols >, =, or < to record our results. We used diagrams and number lines to represent our thinking.

$$\frac{4}{6} < \frac{5}{6}$$
$$\frac{5}{6} > \frac{5}{8}$$