# Lesson 2

Representations of Fractions (Part 2)

## Warm-up: Which One Doesn’t Belong: All Cut Up (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare the features of four partitioned shapes. It allows the teacher to hear the terminologies students use to talk about fractions and fractional parts. In making comparisons, students have a reason to use language precisely (MP6).

### Launch

• Groups of 2
• Display the image.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

Which one doesn’t belong?

### Activity Synthesis

• “What does the shaded part in D represent?” ($$\frac{1}{3}$$ or one-third of the shape).
• Shade one part of B and C.
• “Is each shaded part one-third of the shape as well?” (Yes for B, no for C.)
• “Why is the shaded part not one-third of the square in C?” (The parts aren’t equal in size.)
• Shade one part of A. “Is it a third of the square?” (No, it is $$\frac{1}{4}$$ or one-fourth.)

## Activity 1: A Diagram for Each Fraction (20 minutes)

### Narrative

The purpose of this activity is to activate what students know about the meaning and size of non-unit fractions. Students match a set of fractions with diagrams that represent them. There are a 3 sets of equivalent fractions to prompt students to share what they know about equivalent fractions.

To add movement to the activity, students can check their matches with other groups in the room before the synthesis.

Representation: Internalize Comprehension. Use visual details such as color or arrows to illustrate connections between representations. For example, use the same color for the numerator and the shaded portion of the corresponding diagram.
Supports accessibility for: Visual-Spatial Processing

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give each student a straightedge
• Record and display the fraction $$\frac{1}{4}$$
• “Describe to your partner what the diagram would look like for this fraction.”
• 30 seconds: partner discussion
• Record and display the fraction $$\frac{2}{4}$$.
• “Describe what the diagram would look like for this fraction.”
• 30 seconds: partner discussion
• Share responses.
• “In an earlier lesson, we looked at fractions with 1 for the numerator. Now let’s look at fractions with other numbers for the numerator.”
• As a class, read aloud the word name of each fraction in the task.

### Activity

• “Take a minute to think quietly about how you might match each fraction with a diagram that represents it.”
• 1 minute: quiet think time
• “Work with a partner to match each fraction with a diagram. Two of the fractions have no matching diagrams. Use the blank diagrams to create representations for them.”
• 10 minutes: group work time

### Student Facing

Each full diagram represents 1. Match each fraction to a diagram whose shaded parts represents it.

Two of the fractions are not represented. Create a representation for each of them.

$$\frac{2}{3}: \underline{\hspace{0.5in}} \qquad \frac{3}{8}: \underline{\hspace{0.5in}} \qquad \frac{4}{10}: \underline{\hspace{0.5in}} \qquad \frac{4}{6}: \underline{\hspace{0.5in}} \qquad \frac{6}{6}: \underline{\hspace{0.5in}}$$

$$\frac{3}{5}: \underline{\hspace{0.5in}} \qquad \frac{4}{8}: \underline{\hspace{0.5in}} \qquad \frac{6}{12}: \underline{\hspace{0.5in}} \qquad \frac{6}{10}: \underline{\hspace{0.5in}} \qquad \frac{3}{4}: \underline{\hspace{0.5in}}$$

$$\frac{5}{6}: \underline{\hspace{0.5in}} \qquad \frac{2}{5}: \underline{\hspace{0.5in}} \qquad \frac{5}{12}: \underline{\hspace{0.5in}} \qquad \frac{7}{10}: \underline{\hspace{0.5in}} \qquad \frac{7}{8}: \underline{\hspace{0.5in}}$$

### Activity Synthesis

• Invite students to share how they went about making the matches.
• Highlight explanations that emphasize the meaning of numerator and denominator in a fraction.
• Ask students if they noticed that some diagrams have the same amount shaded but the fractions they represent have different numbers. “Which diagrams show this?” (A and L, B and H, C and E, I and K)
• “What does it mean that the diagrams representing those fractions are the same?” (The fractions are the same size. The term “equivalent” may or may not come up at this point.)

## Activity 2: Diagrams for Some Other Fractions (15 minutes)

### Narrative

This activity extends students’ reasoning about the meaning of numerator and denominator and the size of non-unit fractions to include fractions greater than 1. Students see the size of 1 whole marked in a couple of diagrams and learn that the same size applies to all diagrams. They are prompted to both interpret diagrams and create them: they write a fraction to represent the shaded part of a diagram and partition a diagram to represent a given fraction.

Some students may benefit from having physical manipulatives to help them conceptualize fractions that are greater than 1. Consider using fraction strips to support them, for instance, by asking them to fold as many strips as needed to represent, say, 4 halves or 5 fourths.

MLR2 Collect and Display. Collect the language students use to reason about fractions greater than one. Display words and phrases such as: fraction, numerator, denominator, 1 whole, greater than, equal-size parts, etc. During the activity, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.

### Required Materials

Materials to Gather

### Required Preparation

• Each student needs access to their fraction strips from a previous lesson.

### Launch

• Groups of 2
• Give each student a straightedge and access to their fraction strips from a previous lesson.
• “How can you show $$\frac{3}{4}$$ with fraction strips?” (Find the strip showing fourths, highlight 3 parts of fourths.)
• “How can you show $$\frac{8}{4}$$?”
• 1 minute: partner discussion
• If students say that they don’t have enough strips to show 8 fourths, ask them to combine their strips with another group’s.
• Invite groups to share their representations of $$\frac{8}{4}$$. Students may use different fractional parts (fourths and halves, or fourths and eighths).
• Display two strips that show fourths side by side, as shown.
• Count the fourths: “$$\frac{1}{4}$$, $$\frac{2}{4}$$, . . . , $$\frac{8}{4}$$.”
• “How many fourths did we count?” (8) “How many wholes was that?” (2)

### Activity

• “Each bracket in the first diagram shows 1 whole. The size of 1 whole is the same in all the diagrams.”
• “Take a moment to work on the first question. Then, discuss your responses with your partner.”
• “Be prepared to explain how you know what fractions the diagrams represent.”
• 2–3 minutes: independent work time on the first question
• 2 minutes: partner discussion
• Pause for a discussion.
• “How did you determine what fractions the diagrams represent?” (Count the number of parts in 1, and then count the number of shaded parts.)
• Display students’ work, or display and annotate the tape diagrams as they explain.
• Consider labeling each part with the unit fraction and counting each shaded part aloud (“1 half, 2 halves, 3 halves,” or “1 third, 2 thirds, 3 thirds, 4 thirds”) before writing the represented fractions ($$\frac{3}{2}$$ or $$\frac{4}{3}$$).
• 5–7 minutes: partner work time

### Student Facing

1. What fraction do the shaded parts represent?

2. Here are four fractions and four blank diagrams. Partition each diagram and shade the parts to represent the fraction.

1. $$\frac{2}{2}$$
2. $$\frac{4}{2}$$
3. $$\frac{5}{4}$$
4. $$\frac{10}{8}$$

### Student Response

Students may not attend to the size of 1 whole as they partition the blank diagrams. Consider asking: “What does each part in your diagram represent? How do you know?” and “Where is 1 whole in your diagram?”

### Activity Synthesis

• Select students to share their completed diagrams.
• “How did you know how many parts to partition each diagram and how many parts to shade?“ (Cut each 1 whole portion into as many equal parts as the number in the denominator. Shade as many parts as the number in the numerator.)
• “How did you partition a diagram into 4 equal parts?” (Split each 1 whole into 2, and then split each half into 2 parts again.)
• “How did you partition a diagram into 8 equal parts?” (Split each fourth into 2 parts.)

## Lesson Synthesis

### Lesson Synthesis

“Today we made sense of and created diagrams that represent fractions, including fractions greater than 1.”

“Did you notice anything interesting about the last two diagrams you created and the fractions they represent?” (Students may or may not refer to equivalence. Sample responses:

• They are both greater than 1.
• The shaded parts are the same size. They have the same amount of shading.
• The numerator and denominator in one fraction are twice the numerator and denominator in the other.
• The fractions are equivalent.)