# Lesson 12

Ways to Compare Fractions

## Warm-up: Estimation Exploration: What’s That Point? (10 minutes)

### Narrative

The purpose of this warm-up is for students to practice estimating a reasonable fractional value on a number line. The reasoning here prepares students to use these benchmarks as a way to compare fractions later in the lesson.

The warm-up gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3).

### Launch

• Groups of 2
• Display the number line.
• “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 value represented by the point on the number line?

Make 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

• “How did you decide what fraction would be ‘about right’?” (The point is a little to the left of the middle point, so the fraction must be a little less than $$\frac{1}{2}$$.)
• “Would writing the label ‘1’ as ‘$$\frac{10}{10}$$’ or as ‘$$\frac{100}{100}$$’ help us make better estimates? Why or why not?” (Sample response: It could, because it would help us mentally partition the number line into 10 or 100 parts, which makes it possible to estimate more precisely.)

## Activity 1: The Greatest of Them All (15 minutes)

### Narrative

In earlier lessons, students compared two fractions that share the same denominator or the same numerator. In this activity, students use that understanding to compare a large set of fractions that are arranged into rows and columns. The fractions in each row share the same numerator and those in each column share the same denominator.

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Invite students to look at column A first, then column B, then row 1. Provide access to pre-made fraction strips for thirds and fifths to help them get started. Check in with students to provide feedback and encouragement after each chunk, particularly in terms of looking for and making use of structure.
Supports accessibility for: Conceptual Processing, Organization, Social-Emotional Functioning

### Launch

• Groups of 2
• Display the table of fractions.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: partner discussion

### Activity

• “Take a few quiet minutes to complete the problems. Afterward, discuss your responses with your partner.”
• 6–7 minutes: independent work time
• “When discussing with your partner, explain how you know which fraction is the greatest in each row, each column, and the entire table.”
• 3–4 minutes: partner discussion

### Student Facing

Here are 25 fractions in a table.

A B C D E
1 $$\frac{2}{3}$$ $$\frac{2}{5}$$ $$\frac{2}{10}$$ $$\frac{2}{12}$$ $$\frac{2}{100}$$
2 $$\frac{4}{3}$$ $$\frac{4}{5}$$ $$\frac{4}{10}$$ $$\frac{4}{12}$$ $$\frac{4}{100}$$
3 $$\frac{7}{3}$$ $$\frac{7}{5}$$ $$\frac{7}{10}$$ $$\frac{7}{12}$$ $$\frac{7}{100}$$
4 $$\frac{11}{3}$$ $$\frac{11}{5}$$ $$\frac{11}{10}$$ $$\frac{11}{12}$$ $$\frac{11}{100}$$
5 $$\frac{26}{3}$$ $$\frac{26}{5}$$ $$\frac{26}{10}$$ $$\frac{26}{12}$$ $$\frac{26}{100}$$

For each question, be prepared to explain your reasoning.

1. Identify the greatest fraction in each column (A, B, C, D, and E).
2. Identify the greatest fraction in each row (1, 2, 3, 4, and 5).
3. Which fraction is the greatest fraction in the entire table?

### Student Response

Students may decide $$\frac{26}{3}$$ and $$\frac{26}{5}$$ are less than $$\frac{26}{12}$$ and $$\frac{26}{100}$$ because the former two involve smaller numbers than the latter two. Suggest that students compare fractions with the same numerator, but one that is more familiar (such as those in row 1). Consider asking: “Which is greater, $$\frac{2}{5}$$ or $$\frac{2}{10}$$? $$\frac{8}{5}$$ or $$\frac{8}{10}$$?” Refer them to the diagram of fraction strips to make a similar comparison, if helpful.

### Activity Synthesis

• Invite students to share their responses and reasoning. Highlight responses that clarify that:
• In each column, the fraction in row 5 is the greatest because it has the greatest numerator of all fractions with the same denominator (with fractional parts of the same size).
• In each row, the fraction in column A is greater than others to its right because it has the greatest fractional part of all fractions with the same numerator ($$\frac{1}{3}$$ is greater than $$\frac{1}{5}$$, $$\frac{1}{10}$$, $$\frac{1}{12}$$, and $$\frac{1}{100}$$).
• “How did you know that $$\frac{26}{3}$$ is the greatest fraction in the entire table?” (Sample responses:
• It is the greatest fraction in row 5 and in column A.
• It is more than 8 wholes. All the other fractions are less than that.)

## Activity 2: Relative to $\frac{1}{2}$ and 1 (20 minutes)

### Narrative

In this activity, students apply previous reasoning about the size of fractions and their knowledge about fractions that are equivalent to $$\frac{1}{2}$$ to classify and compare fractions. Along the way, students have opportunities to make new observations about the structure of fractions that are less than $$\frac{1}{2}$$, greater than $$\frac{1}{2}$$ but less than 1, and greater than 1 (MP7).

The activity calls for the use of colors as a way to code fractions in different groups. If colored pencils are not available, students can code the fractions by putting circles, triangles, and squares around the fractions. In either case, a key or legend should be created.

MLR8 Discussion Supports. Encourage students to begin partner discussions by reading their written responses aloud. If time allows, invite students to revise or add to their responses based on the conversation that follows.

### Required Materials

Materials to Gather

### Required Preparation

• Each group of 2 needs 3 colored pencils (3 different colors).

### Launch

• Groups of 2
• Give each group 3 colored pencils.

### Activity

• “Take a few quiet minutes to answer the first 4 questions. Afterward, discuss your responses with your partner.”
• “You will need to code the fractions by color or by shape.”
• 7–8 minutes: individual work time
• 5 minutes: partner discussion
• Pause for a whole-class discussion before proceeding to the last question.
• Invite students to share how they knew if a fraction is less than $$\frac{1}{2}$$, or if it is greater than $$\frac{1}{2}$$ but less than 1. Record their responses.
• “How did you describe the last group of fractions that don’t fall into the first two groups?” (fractions greater than 1)
• “Now compare some fractions in the last problem.”
• 5 minutes: individual work time

### Student Facing

Here is the same table you saw earlier.

A B C D E
1 $$\frac{2}{3}$$ $$\frac{2}{5}$$ $$\frac{2}{10}$$ $$\frac{2}{12}$$ $$\frac{2}{100}$$
2 $$\frac{4}{3}$$ $$\frac{4}{5}$$ $$\frac{4}{10}$$ $$\frac{4}{12}$$ $$\frac{4}{100}$$
3 $$\frac{7}{3}$$ $$\frac{7}{5}$$ $$\frac{7}{10}$$ $$\frac{7}{12}$$ $$\frac{7}{100}$$
4 $$\frac{11}{3}$$ $$\frac{11}{5}$$ $$\frac{11}{10}$$ $$\frac{11}{12}$$ $$\frac{11}{100}$$
5 $$\frac{26}{3}$$ $$\frac{26}{5}$$ $$\frac{26}{10}$$ $$\frac{26}{12}$$ $$\frac{26}{100}$$
1. Which fractions are less than $$\frac{1}{2}$$? Circle each one of them. Then, complete this sentence:

I know a fraction is less than $$\frac{1}{2}$$ when . . .

2. Which are greater than $$\frac{1}{2}$$ but less than 1? Circle each of them with a pencil of a different color (or draw a triangle around each one). Then, complete this sentence:

I know a fraction is greater than $$\frac{1}{2}$$ but less than 1 when . . .

3. Circle the remaining fractions with a pencil of a third color (or draw a square around each one). How would you describe the size of these fractions?
4. Next to the table, create a legend or key to show what each color (or each shape) represents.
5. Here are some pairs of fractions from the table. In each pair, which fraction is greater?

1. $$\frac{2}{5}$$ or $$\frac{7}{10}$$
2. $$\frac{4}{10}$$ or $$\frac{7}{12}$$
3. $$\frac{11}{100}$$ or $$\frac{4}{3}$$
4. $$\frac{26}{10}$$ or $$\frac{11}{12}$$

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

Invite students to share their strategies for comparing fractions in the last question of the last activity.

“How did you compare fractions in which neither the numerator nor the denominator are the same?” (Sample response: We compared them to $$\frac{1}{2}$$ or 1.)

“How did the color coding (or comparison to $$\frac{1}{2}$$ or 1) help?” (Sample responses:

• Knowing whether a fraction is more or less than 1, or more or less than $$\frac{1}{2}$$, can help us tell which one is greater.
• All the fractions circled in yellow [less than $$\frac{1}{2}$$] are less than all the numbers in green [greater than $$\frac{1}{2}$$ but less than 1]. All fractions in green are less than all fractions in purple [greater than 1].)