# Lesson 8

Apply Fraction Multiplication

## Warm-up: Number Talk: Fraction Multiplication (10 minutes)

### Narrative

The purpose of this Number Talk is to for students to demonstrate strategies and understandings they have for multiplying fractions. These understandings help students develop fluency and will be helpful later in this lesson when students solve problems involving fraction multiplication.

### 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.

• $$\frac {1}{3} \times \frac {3}{5}$$
• $$\frac {2}{3} \times \frac {3}{5}$$
• $$\frac {5}{3} \times \frac {3}{5}$$
• $$\frac {2}{3} \times \frac {13}{5}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Did you use the same strategy to find the product of the first and last expressions? Why or why not?” (For the first one, it was easy to think about what $$\frac{1}{3}$$ of $$\frac{3}{5}$$ was. For the last one, I needed to multiply the numerators and then the denominators because it wasn’t easy for me to picture in my head.)

## Activity 1: Flags (20 minutes)

### Narrative

The purpose of this activity is for students to calculate areas in context. Students are not offered an area diagram but rather an image of a flag. Students may label the image of the flag with measurements or make their own area diagram. The lengths are not always presented in fraction form so students may rewrite them as fractions before calculating areas or they may use the distributive property of multiplication and multiply the whole number and fractional parts separately before adding them. Students reason abstractly and quantitatively when they interpret the given information about the flags and make calculations to solve problems (MP2).

MLR7 Compare and Connect. Invite students to prepare a visual display that shows the strategy they used to calculate the area of different parts of the flag. Encourage students to include details that will help others interpret their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams, or drawings. Give students time to investigate each other’s work. During the whole-class discussion, ask students, “What did the approaches have in common?”, “How were they different?”, and “Did anyone solve the problem the same way, but would explain it differently?”

### Launch

• Display image of flags from student workbook.
• “What do you notice? What do you wonder?” (There are lots of different flags, some of the colors are the same, there are a lot of stripes, these flags are from 1968, have any of the flags changed since then?)
• “These are flags from different countries. We are going to solve some problems about flags.”
• Groups of 2

### Activity

• 2 minutes: quiet think time
• 8–10 minutes: partner work time
• Monitor for students who:
• notice that the blue stripe is twice as wide as the red stripe and use the area of the red stripe to find the area of the blue stripe
• find the area of the full flag by converting $$7 \frac{1}{2}$$ to a fraction
• use the distributive property

### Student Facing

Jada has a small replica of a flag of Thailand.

It is 5 inches wide and $$7\frac{1}{2}$$ inches long.

1. What is the area of the flag? Explain or show your reasoning.
2. Each red stripe is $$\frac{5}{6}$$ inches wide. What is the area of each red stripe? Explain or show your reasoning.
3. The blue stripe is $$\frac{10}{6}$$ inches wide. What is the area of the blue stripe? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite previously selected students to share their solutions.
• Display equation: $$5 \times 7\frac{1}{2} = (5\times 7) + (5 \times \frac{1}{2})$$
• “How does the equation represent the area of the flag?” (I can divide the flag into two pieces, one of them 7 inches long, and the other $$\frac{1}{2}$$ inch long, and then add them to get the area of the flag.)
• Invite students to share responses for the area of the red stripe.
• “How can I use the area of the red stripe to find the area of the blue stripe?” (The blue stripe is twice as wide so its area is twice as much.)

## Activity 2: More Flags (15 minutes)

### Narrative

The goal of this activity is to examine calculations with measurements of a flag and try to figure out what question the calculations answer. The answers include units and this can serve as a guide to students. Since the first calculation has an answer in inches, the question it answers must ask for a length. Since the second calculation has an answer in square inches, the question it answers must ask for an area. This is an important step in solving the problems as students can then look at the diagram and the measurements and decide what the question could be.

One important part of the modeling cycle (MP4) is interpreting information. That information may be presented in words or graphs or with mathematical symbols. In this case, students interpret equations in light of given numerical relationships and diagrams.

Representation: Access for Perception. Read the statement aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Conceptual Processing, Language, Attention

### Launch

• Display image of the flag of Colombia from student workbook.
• “This is the flag of Colombia. It represents independence from Spain on July 20, 1810.”
• “What do you notice about the size of each stripe?” (The yellow stripe is about twice the size of each of the blue and red stripes.)
• Display the flag and information about the flag from the activity.
• “A student was answering a question about this flag and wrote $$\frac{1}{2} \times 3 \frac{1}{2} = 1 \frac{3}{4}$$.”
• “What question do you think the student is answering?” (What is the width of the yellow rectangle?)
• “You are going to solve more problems like this one where you are given the answer and you have to write the question.”

### Activity

• 4–5 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for students who use the units of the answers to help guide them to finding an appropriate question.

### Student Facing

Han has a replica of the flag of Colombia.

It is $$3 \frac{1}{2}$$ inches wide and $$5\frac{1}{4}$$ inches long. The yellow stripe is $$\frac{1}{2}$$ of the width of the flag and the blue and red stripes are each $$\frac{1}{4}$$ of the width.

1. $$\frac{1}{4} \times 3 \frac{1}{2} = \frac{7}{8}$$. The answer is $$\frac{7}{8}$$ inch. What is the question?
2. $$\frac{1}{2} \times 3\frac{1}{2} = \frac{7}{4}$$ and $$\frac{7}{4} \times \frac{21}{4} = \frac{147}{16}$$. The answer is $$\frac{147}{16}$$ square inches. What is the question?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share their responses for the first question.
• “How did you know the question was about length?” (The answer is $$\frac{7}{8}$$ inch, which is the measurement of length. So the question had to be about length.)
• “How did you decide which length?” (The calculation took a quarter of the flag width. Since the red and blue stripes are each $$\frac{1}{4}$$ of the width, the calculation could be for their width.)
• Invite students to share their responses for the second question.
• “How did you know the question was about area?” (The answer is a measurement in square inches so that’s the area of something.)

## Lesson Synthesis

### Lesson Synthesis

“In this section, we have multiplied fractions using area diagrams. What are you most proud of from your work in this section? What questions about fraction multiplication do you still have?”

## Cool-down: The Flag of Chad (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Section Summary

### Student Facing

In this unit, we learned to multiply fractions. First we learned to multiply unit fractions. For example, we learned that $$\frac{2}{5} \times \frac {1}{3} = \frac {2}{15}$$

In diagram A, we can see that $$\frac{2}{5}$$ of $$\frac{1}{3}$$ of a square is the same size as $$\frac {2}{15}$$ of the whole square. Next, we learned how to multiply any fraction by a fraction.
In diagram B, we can see that $$\frac {4}{6} \times \frac {5}{7} = \frac {20}{42}$$. We can multiply the numerators, $$4 \times 5$$ to find the numerator in the product. We can multiply the denominators, $$6 \times 7$$, to find the denominator in the product. We can represent this relationship with the equation: $$\frac {(4\times5)}{(6\times7)} = \frac {20}{42}$$. Diagram B shows $$4 \times 5$$ or 20 pieces with $$6 \times 7$$ or 42 pieces in the whole square.