Lesson 6

Multiply Fractions

Warm-up: Which One Doesn’t Belong: More Pieces (10 minutes)

Narrative

The purpose of this warm-up is for students to compare different shaded regions in order to introduce the new type of region that will be considered in this lesson, namely regions where neither side length is a unit fraction. The focus of the discussion is on diagram A where neither side length is a unit fraction.

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?

Student Response

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Activity Synthesis

• “Why doesn’t image A belong?" (It’s the only one where neither side length is a unit fraction.)
• “What is the area of the shaded region in image A? How do you know?” ($$\frac{6}{12}$$ because there are 6 shaded pieces and there are 12 pieces in the whole square.)

Activity 1: Many Expressions (15 minutes)

Narrative

The purpose of this activity is for students to relate the structure in an expression to an area diagram (MP7). As students work with their partners, make sure both partners have an opportunity to verbally explain how the diagram represents each expression.

• Groups of 2

Activity

• 2–3 minutes: independent think time
• 5–8 minutes: partner work time
• Monitor for students who can explain how each expression is represented in the diagram.

Student Facing

Explain or show how each expression can represent the area of the shaded region in square units. Be prepared to share your thinking.

1. $$\frac {8}{30}$$
2. $$2 \times 4 \times (\frac {1}{5} \times \frac {1}{6})$$
3. $$\frac {2}{6} \times \frac {4}{5}$$

Student Response

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If students do not explain how each expression represents the area of the shaded region, ask: “How would you describe the area of the shaded region?” Connect students' explanations to the given expressions.

Activity Synthesis

• Ask previously selected students to share their thinking.
• “How does $$\frac {8}{30}$$ represent the diagram?” (There are 8 pieces shaded and each piece is $$\frac {1}{30}$$ of the square.)
• “How does the expression $$2 \times 4 \times (\frac {1}{5} \times \frac {1}{6})$$ represent the diagram?” (The shaded region is a 2 by 4 array and each of the pieces in the array is $$\frac {1}{5}$$ of $$\frac {1}{6}$$ of the whole square.)

Activity 2: More Patterns (20 minutes)

Narrative

The purpose of this activity is for students to observe and use the structure of diagrams to find areas of shaded regions with non-unit fraction side lengths. Students build on what they learned in the previous activity, solidifying their understanding of why the numerator of a product of two fractions is the product of the numerators and the denominator of a product of fractions is the product of the denominators.

Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for writing multiplication expressions to represent the area of the shaded rectangle in each figure before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language

• Groups of 2

Activity

• “Start working on completing the table independently. After a couple minutes, you'll work with your partner to complete the table and answer the rest of the questions.”
• 1–2 minutes: independent work time
• 5-8 minutes: partner work time
• Monitor for students who:
• notice the area of the shaded regions is always twentieths
• write the expression $$\frac{6}{5}\times\frac{4}{5}$$ to represent the shaded region of the last diagram in the table
• explain that the expression $$\frac{6\times4}{5\times4}$$ represents the shaded part of the last diagram in the table because $$6\times4$$ represents the number of pieces that are shaded and $$4 \times 5$$ represents the number of those pieces in the unit square

Student Facing

1. Complete the table.
diagram multiplication
expression
(square units)
diagram multiplication
expression
(square units)
2. What patterns do you notice in the table?
3. Explain or show how the expression $$\frac{6\times4}{5\times4}$$ represents the last diagram in the table.

Student Response

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If students do not explain that the product of the numerators represents the number of pieces in the shaded region and the product of the denominators represents the number of pieces in the whole, consider asking: “What is the same and what is different about diagrams A and B?”

Activity Synthesis

• Ask previously selected students to share their reasoning.
• “How do the expressions in the table represent the number of pieces shaded in and the size of the pieces shaded in?” (If we multiply the numerators, we get the number of pieces that are shaded in. If we multiply the denominators, we get the size of the pieces.)
• Refer to diagrams and draw on each diagram to show how the multiplication of the numerators and denominators represents the number of shaded pieces and the size of the shaded pieces (that is, the number of those pieces in the whole).

Lesson Synthesis

Lesson Synthesis

Display diagram A from the last activity.

Display expression: $$\frac{2}{4} \times \frac{3}{5}$$

“We can multiply the numerators to find the numerator in the product. How does the diagram represent $$2\times3$$?” (The shaded pieces are a 2 by 3 array and there are 6 of them.)

“We can multiply the denominators to find the denominator in the product. How does the diagram represent $$4\times5$$?” (The unit square is a 4 by 5 array so there are 20 pieces in the whole unit square.)

“How does the diagram represent $$\frac{6}{20}$$?” (There are 6 pieces shaded in they are each $$\frac{1}{20}$$ of the unit square.)

Cool-down: What is the Area? (5 minutes)

Cool-Down

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