# Lesson 13

Use Equivalent Fractions to Compare

## Warm-up: Notice and Wonder: Pairs of Numbers (10 minutes)

### Narrative

The purpose of this warm-up is to draw students’ attention to inequality statements. It reminds them of the meaning of inequality symbols and how to read the statements, which will be useful when students compare fractions later in the lesson. The warm-up also elicits observations that an equation or inequality can be true or false. While students may notice and wonder many things, highlight observations about comparison and about the meaning of the symbols and statements.

### Launch

• Groups of 2
• Display the four statements.
• “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?

$$5 < 8$$

$$\frac{9}{2} >4\frac{1}{2}$$

$$4 = \frac{3}{2}$$

$$\frac{1}{3} <\frac{1}{2}$$

### Activity Synthesis

• “What does each statement say?”
• “Which of these statements are true? Which ones are not?” (The first and last are true. The second and third are false.)
• “Why are they false?” ($$\frac{9}{2}$$ is equal to, not greater than, 4 wholes and $$\frac{1}{2}$$. Four is greater than $$\frac{3}{2}$$.)

## Activity 1: Pairs to Compare (20 minutes)

### Narrative

Previously, students classified fractions based on their relationship to $$\frac{1}{2}$$ and 1 (whether they are less than or more than these benchmarks). They used these classifications to compare fractions. In this activity, students are presented with fractions that are in the same group (for example, both less than $$\frac{1}{2}$$, or both greater than $$\frac{1}{2}$$ but less than 1), so they need to reason in other ways to make comparisons.

Students can reason in a number of ways—by thinking about size and number of parts, drawing a diagram or number line, or reasoning numerically, but in most cases, they need to also rely on the idea of equivalence.

MLR7 Compare and Connect. Synthesis: After each strategy has been presented, lead a whole-class discussion comparing, contrasting, and connecting the different approaches. Ask, “Did anyone solve the problem the same way, but would explain it differently?” and “Why did the different approaches lead to the same outcome?”
Representation: Internalize Comprehension. Activate background knowledge. Invite students to review the strategies they know for comparing fractions (reasoning about denominators or numerators, comparing to a benchmark, and writing equivalent fractions). Record students’ strategies on a visible display, including details (words or pictures) that will help them remember how to use the strategy.
Supports accessibility for: Conceptual Processing, Memory, Attention

### Launch

• Groups of 2
• “Here are some fractions you’ve sorted in an earlier lesson. We compared them to $$\frac{1}{2}$$ and 1.”
• “What do the fractions in group 3 have in common? Why might they be in the same group?” (They are all greater than 1.)
• “How are the fractions in group 1 different than those in group 2?” (Those in group 1 are less than $$\frac{1}{2}$$, and those in group 2 greater than $$\frac{1}{2}$$ but less than 1.)
• “We can tell that the fractions in group 2 are greater than those in group 1, and the fractions in group 3 are greater than those in the other groups.”
• “Now compare the fractions in each group.”

### Activity

• “Take a few quiet minutes to work on the problems. Afterward, share your responses with your partner.”
• 7–8 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who:
• reason by drawing number lines or tape diagrams
• reason about the distance of each fraction from 0, $$\frac{1}{2}$$, or 1
• reason about equivalent fractions (even if they don’t write out the multiplication numerically)
• reason about equivalent fractions numerically by writing out the multiplication

### Student Facing

Here are some pairs of fractions sorted into three groups. Circle the greater fraction in each pair. Explain or show your reasoning.

1. Group 1:

1. $$\frac{2}{10}$$  or  $$\frac{26}{100}$$

2. $$\frac{2}{5}$$  or  $$\frac{11}{100}$$

2. Group 2:

1. $$\frac{2}{3}$$  or  $$\frac{7}{12}$$

2. $$\frac{4}{5}$$  or  $$\frac{7}{10}$$

3. Group 3:

1. $$\frac{11}{5}$$  or  $$\frac{26}{10}$$

2. $$\frac{11}{3}$$  or  $$\frac{26}{12}$$

### Activity Synthesis

• Select students who used different strategies to share their responses.
• “How do we compare two fractions that are in the same group—say, both less than $$\frac{1}{2}$$ or both greater than 1?” (We can think about how close or far away from $$\frac{1}{2}$$ each fraction is.)
• Highlight how equivalent fractions came into play in each strategy. For example, ask, “When comparing $$\frac{2}{3}$$ and $$\frac{7}{12}$$, why was it helpful to think of the $$\frac{2}{3}$$ as $$\frac{8}{12}$$?” Or, “When comparing $$\frac{7}{10}$$ and $$\frac{4}{5}$$, why did you think of $$\frac{4}{5}$$ as $$\frac{8}{10}$$?”
• If no students mention that it is often easier to compare two fractions when they have the same denominator, ask them about it.

## Activity 2: New Pairs to Compare (15 minutes)

### Narrative

The purpose of this activity is for students to compare pairs of fractions by writing one or more equivalent fractions. In all pairs of fractions given here, one denominator is a factor or a multiple of the other, which encourages students to convert one into an equivalent fraction with the same denominator as the other fraction. On repeated reasoning, students see that writing an equivalent fraction can facilitate the comparison (though in some cases, students may still find it efficient to reason in other ways).

This is the first time in grade 4 that students use the symbols $$<$$ and $$>$$ to express comparison, so some supports for reading aloud inequality statements are suggested in the launch.

### Launch

• Groups of 2
• Read together the four statements in the first question.
• Consider writing out in words the meaning of the symbols $$<$$ and $$>$$ (“is greater than” and “is less than”) and display them for students’ reference.

### Activity

• 7–8 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who make comparisons by:
• using the relationship and distance to benchmark numbers
• writing an equivalent fraction either by dividing or multiplying the numerator and denominator by a number

### Student Facing

1. Decide whether each statement is true or false. Be prepared to show how you know.

1. $$\frac{5}{12}=\frac{2}{6}$$
2. $$\frac{10}{3}<\frac{44}{12}$$
3. $$\frac{1}{4}>\frac{25}{100}$$
4. $$\frac{8}{15}<\frac{3}{5}$$
2. Compare each pair of fractions. Use the symbols $$<$$, $$=$$, and $$>$$ to make each statement true.

1. $$\frac{6}{12} \ \underline{\hspace{1.05cm}} \ \frac{4}{6}$$
2. $$\frac{4}{3} \ \underline{\hspace{1.05cm}} \ \frac{7}{6}$$
3. $$\frac{8}{5} \ \underline{\hspace{1.05cm}} \ \frac{400}{100}$$
4. $$\frac{12}{10} \ \underline{\hspace{1.05cm}} \ \frac{35}{5}$$
5. $$\frac{11}{4} \ \underline{\hspace{1.05cm}} \ \frac{17}{8}$$
6. $$\frac{7}{12} \ \underline{\hspace{1.05cm}} \ \frac{4}{3}$$

### Activity Synthesis

• Select students to share their responses and how they reasoned about them.

## Lesson Synthesis

### Lesson Synthesis

“Today we compared fractions by writing equivalent fractions and by using some other ways.”

Ask students to find an example of a pair of fractions in today’s activity that it was helpful to compare by:

• reasoning about the denominators and numerators
• seeing where the fractions are in relation to $$\frac{1}{2}$$, 1, or another benchmark
• writing an equivalent fraction for one of the fractions