# Lesson 2

Representations of Equal Groups of Fractions

## Warm-up: Number Talk: Three, Six, Nine, Twelve (10 minutes)

### Narrative

This Number Talk encourages students to use their knowledge of multiplication facts, properties of operations, and the structure of the given expressions to mentally solve problems. The reasoning elicited here will be helpful in upcoming lessons as students find products of whole numbers and non-unit fractions (such as $$3 \times \frac{6}{10}$$ or $$6 \times \frac{9}{4}$$).

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$3 \times 6$$
• $$3 \times 9$$
• $$6 \times 9$$
• $$12 \times 9$$

### Activity Synthesis

• “What did you notice about the factors in all of the expressions?” (They are all multiples of 3.)
• “Did noticing that all the factors are multiples of 3 help you find the values?” (Sample responses:
• Yes, I was able to think of “3 more groups of something.”
• Yes, it helped me see how the factors were related, which helped me reason about the products.
• No, it didn't, but it made me think that the values would be multiples of 3 as well.)

## Activity 1: Card Sort: Expressions and Diagrams (25 minutes)

### Narrative

In this activity, students interpret multiplication expressions and diagrams as the number of groups and amount in each group and match representations of the same quantity. They then use their insight from the matching activity to generate diagrams for expressions without a match and to find their values (MP2).

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____ , so I matched . . .” Encourage students to challenge each other when they disagree.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Organization, Conceptual Processing

### Required Materials

Materials to Copy

• Expressions and Diagrams

### Required Preparation

• Create a set of cards from the blackline master for each group of 2.

### Launch

• Groups of 2
• Give each group a set of cards from the blackline master.

### Activity

• “Work with your partner to match each expression to a diagram that represents the same equal-group situation and the same amount.”
• “Be prepared to explain how you know the two representations belong together.”
• 5 minutes: partner work time
• Monitor for students who reason about the number of groups and amount in each group as they match.
• Pause for a discussion. Select students to share their matches and reasoning.
• Highlight reasoning that clearly connects one factor in the expression to the number of groups and the other factor to the size of each group.
• “Now you will complete an unfinished diagram for $$7 \times \frac{1}{8}$$ and then draw a new diagram for an expression without a match.”
• 5 minutes: independent work time

### Student Facing

Your teacher will give you a set of cards with expressions and diagrams.

1. Match each expression with a diagram that represents the same quantity.
2. Record each expression without a match.
3. Han started drawing a diagram to represent $$7 \times \frac{1}{8}$$ and did not finish. Complete his diagram. Be prepared to explain your reasoning.

4. Choose one expression that you recorded earlier that didn't have a match.

Draw a diagram that can be represented by the expression. What value do the shaded parts of your diagram represent?

### Student Response

If students are not yet matching expressions to appropriate diagrams, consider asking them to compare the diagrams for $$5 \times 3$$ and $$5 \times \frac{1}{3}$$ and reason about the number of groups and the size of each group. Consider asking: “How are these alike? How are they different?”

### Activity Synthesis

• “What was missing from Han’s diagram? How do you know?” (4 more groups of $$\frac{1}{8}$$ were missing, because $$7 \times \frac{1}{8}$$ means 7 groups of $$\frac{1}{8}$$and there are only 3 in Han’s diagram.)
• “If the expression was for 7 groups of $$\frac{1}{3}$$ instead of $$\frac{1}{8}$$ how would Han’s diagram change?” (Each rectangle representing 1 would have 3 equal parts with 1 shaded.)
• Select students to share the diagrams they drew for the expressions without a match. Ask them to point out the number of groups and size of each group in each diagram.

## Activity 2: Different Representations (10 minutes)

### Narrative

This activity prompts students to use their earlier observations to generate a diagram or expression that represents equal groups of unit fractions when one or the other is given. In one of the problems, only the total quantity ($$\frac{7}{2}$$) is given, so students need to reason in about the number of groups and the size of each group that could lead to this value. Finally, they analyze two different ways of representing $$4 \times\frac{1}{3}$$ with a diagram, which further illustrates that the value of the expression is $$\frac{4}{3}$$.

### Launch

• Groups of 2
• “Turn to a partner and explain what needs to be done to complete the first problem.”

### Activity

• Complete the first problem independently. Afterwards, pause for a class discussion.
• 5 minutes: independent work time
• Pause to discuss the fraction $$\frac{7}{2}$$ in the first problem.
• “How did you know what diagram and expression would have the value $$\frac{7}{2}$$?” (Sample response:
• For the diagram, the numerator, 7, is the number of groups, and the denominator, 2, shows how many parts are in 1 whole.
• For the expression, I multiplied a whole number and a fraction. The whole number was the same as the number in the numerator of $$\frac{7}{2}$$ and the fraction has the same number as the denominator of $$\frac{7}{2}$$.)
• Work on the last problem with your partner.
• 5 minutes: partner work time

### Student Facing

1. Write a multiplication expression that represents the shaded parts of the diagram. Then, find the value of the expression.

Diagram:

Expression:

Value:

2. Draw a diagram that the expression $$6 \times \frac{1}{3}$$ could represent. Then, find the value of the expression.

Diagram:

Expression: $$6 \times \frac{1}{3}$$

Value:

3. Draw a diagram and write an expression that gives the value $$\frac{7}{2}$$.

Diagram:

Expression:

Value: $$\frac{7}{2}$$

1. To represent $$4 \times \frac{1}{3}$$, Diego drew this diagram:

Elena drew this diagram:

Are they representing the same expression and value? Explain or show how you know.

### Student Response

Students may be unsure about how to begin writing expressions for fractions. Remind students that the fraction will be written as a whole number times a unit fraction. Consider asking: “How might this help to write the expression?”

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we analyzed expressions and diagrams that represent equal groups and created some of these representations.”

Display or sketch these diagrams:

“How do we know which diagram represents $$3 \times \frac{1}{5}$$? Where do we see each number in the diagram?” (B represents $$3 \times \frac{1}{5}$$ because it shows 3 groups of $$\frac{1}{5}$$.)

“What expression does the other diagram represent?” (A represents $$5 \times \frac{1}{3}$$, because it shows 5 groups with $$\frac{1}{3}$$ in each group.)

“What is the value of $$3 \times \frac{1}{5}$$? How do we know?” ($$\frac{3}{5}$$. We can count the number of shaded fifths and see that there are 3.)

“What is the value of $$5 \times \frac{1}{3}$$?” ($$\frac{5}{3}$$)