# Lesson 9

Relate Area to Multiplication

## Warm-up: Which One Doesn’t Belong: Area (10 minutes)

### Narrative

### 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

- “Discuss your thinking with your partner.”
- 2–3 minutes: partner discussion
- Share and record responses.

### Student Facing

Which one doesn’t belong?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

- “Let’s find at least one reason why each one doesn’t belong.”

## Activity 1: Find the Area (15 minutes)

### Narrative

The purpose of this activity is for students to find the area of rectangles with one fractional side length and one whole number side length. Students begin by considering a rectangle with whole number side lengths and then look at a series of rectangles with unit fraction side length. All of the rectangles have the same whole number width to help students see how the area changes when the fractional width changes. Students should use a strategy that makes sense to them. These strategies might include counting the individual shaded parts in the diagram or thinking about moving them to fill in unit squares. Some students may use multiplication or division. These ideas will be brought out in future lessons. During discussion, connect the different strategies students use to calculate the areas. As they choose a strategy, they have an opportunity to use appropriate tools, whether it be expressions that represent the shaded area or physical manipulations of the diagrams, strategically (MP5).

*MLR8 Discussion Supports.*Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.

*Advances: Reading, Representing*

*Action and Expression: Internalize Executive Functions*. Invite students to plan a strategy, including the tools they will use, for finding the area of a rectangle that has one side length that is a fraction. If time allows, invite students to share their plan with a partner before they begin.

*Supports accessibility for: Conceptual Processing, Memory, Language*

### Launch

- Groups of 2
- Display the images of the shaded rectangles.
- “What is the same about all of the rectangles? What is different?” (They are all shaded. They have different amounts shaded. They have different widths.)
- “We are going to figure out how much of each rectangle is shaded. We call this finding the area of the shaded region. What are some strategies we could use to find the area of each of the shaded regions?” (Move the pieces around to make full squares, count the number of blue pieces and multiply the number of pieces by their size.)

### Activity

- 5–7 minutes: partner work time
- Encourage students to find the area in a way that makes sense to them.
- Monitor for students who:
- count the number of shaded parts and multiply the total number of parts by their fractional area.
- visualize moving the shaded parts to fill whole unit squares.

### Student Facing

Find the area of the shaded region. Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

If students do not find the area of the shaded region, ask “How can you use the rectangle that has 6 unit squares shaded in to help you find the area of the other shaded regions?”

### Activity Synthesis

- Ask previously selected students to share their reasoning.
- “What is the same about the strategies? What is different?” (They all counted the number of shaded parts, but they counted them in different ways. Some people multiplied and some people moved the parts to make whole unit squares.)
- Display image from final problem.
- “How does the expression \(6 \times \frac{1}{4}\) represent the shaded area in square units?” (There are 6 shaded parts and each one has an area of \(\frac{1}{4}\) square unit.)
- “How does the expression \(\frac{1}{4} \times 6\) represent the shaded area in square units?” (There is a rectangle whose area is 6 square units and \(\frac{1}{4}\) of the rectangle is shaded.)

## Activity 2: Draw Rectangles (20 minutes)

### Narrative

The purpose of this activity is for students draw and shade rectangles with a unit fraction side length and a whole number side length. Then they find the areas of the shaded regions. The tactile experience of drawing and shading encourages students to count the number of shaded parts and then either reason about their size or think about moving them to make full unit squares. They also consider a diagram where not all of the unit squares are shown. Students estimate how many of the unit squares are hidden. This helps to highlight that finding the total area can be done with multiplication where one factor is the area of each shaded part and the other factor is the total number of shaded parts (MP7).

### Required Materials

Materials to Copy

- Grid Paper 5

### Launch

- Groups of 2
- Give students grid paper.

### Activity

- 5–7 minutes: independent work time
- 5 minutes: partner work time
- Monitor for students who:
- use the grid structure on the paper to draw their rectangles
- count the number of unit squares that might be hidden under the yellow rectangle

### Student Facing

- Represent each rectangle on grid paper:
- \(\frac{1}{2}\) unit by 1 unit
- \(\frac{1}{2}\) unit by 2 units
- \(\frac{1}{2}\) unit by 3 units
- \(\frac{1}{2}\) unit by 4 units

- Find the area of each rectangle that you drew.
- What information do you need to find the area of the shaded region?
- What might the area of the shaded region be? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Advancing Student Thinking

If students do not draw the rectangles correctly, show them the shaded region from the previous activity with side lengths 6 units and \(\frac{1}{2}\) unit and ask “How could you adapt this diagram to show a rectangle that is 4 units by \(\frac{1}{2}\) unit?”

### Activity Synthesis

- Display a student generated image of the \(\frac{1}{2}\) unit by 4 unit rectangle.
- Ask previously selected students to describe how they drew they 4 by \(\frac{1}{2}\) rectangle.
- Display image from student workbook of the rectangle that is partly covered.
- “What do you need to know to determine the area of the shaded region?” (We need to know how many unit squares are under the yellow rectangle.)
- “How many unit squares do you think make up the rectangle?” (6 or 7)
- “How did you use the number of these unit squares to make an estimate for the shaded region?” (I know that each shaded blue rectangle is \(\frac{1}{2}\) of a square so if there are 6 or 7 of those, that would be \(\frac{6}{2}\) or \(\frac{7}{2}\) square units.)

## Lesson Synthesis

### Lesson Synthesis

Display the shaded rectangle that has an area of 6 whole units:

“What strategies do we use to find the area of rectangles with 2 whole number side lengths?” (We can count the number of squares. We can multiply the side lengths.)

“What strategies did we use today to find the area of rectangles with a whole number side length and a unit fraction side length?” (We counted the number of unit squares and multiplied by the size of the shaded region in each unit square.)

“How are the strategies we used to find the area of rectangles with whole number side lengths the same as and different from the strategies we use to find the area of rectangles with a whole number side length and a fractional side length?” (We can use the same strategies, but we count area that is less than one unit square. We are still multiplying, but one of the numbers is a fraction.)

## Cool-down: Fractional Pieces (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.