# Lesson 5

Represent Products as Areas

## Warm-up: How Many Do You See: One More (10 minutes)

### Narrative

The purpose of this How Many Do You See is for students to subitize or use grouping strategies to describe the images they see. The arrangement of the groups of dots encourages students to see 5 groups of dots in the first image and then 6 groups of dots in the next image. When students use equal groups and a known quantity to find an unknown quantity, they are looking for and making use of structure. (MP7).

### Launch

• Groups of 2
• “How many do you see? How do you see them?”
• Flash the image.
• 30 seconds: quiet think time

### Activity

• Display the image.
• 1 minute: partner discussion
• Record responses.
• Repeat for each image.

### Student Facing

How many do you see? How do you see them?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How did the first image help you find the number of dots in the second image?” (I know that 5 groups of 3 is 15, and one more group of 3 would be 18.)
• “How did the first and second images help you find the number of dots in the third image?” (I figured out 5 groups of 4 pretty quickly, then added another group of 4.)

## Activity 1: Match Expressions and Areas (15 minutes)

### Narrative

The purpose of this activity is for students to directly connect multiplication expressions to equal groups they see within rectangular areas. Students may decompose the rectangular areas in various ways to see equal groups, but they should relate the rows and columns to the factors of a multiplication expression. This will be helpful in future activities when students multiply side lengths to find the area.

### Required Materials

Materials to Copy

• Match Expressions and Areas

### Launch

• Groups of 3–4
• Sketch a 5-by-3 gridded rectangle, as shown.

• “What is one way you could describe this rectangle?” (It has 3 rows of 5 squares. There are 5 groups of 3. Its area is 15 square units. There are 15 squares.)
• Share and record responses. Save responses for discussion after the next activity.
• Display rectangles from the blackline master around the room.

### Activity

• “Match each expression to one of the rectangles posted around the room. Be ready to explain your reasoning.”
• 5–7 minutes: group work time

### Student Facing

Your teacher has posted images of rectangles around the room. Match each expression with a rectangle that can represent it. Be prepared to explain your reasoning.

1. $$9 \times 5$$
2. $$8 \times 2$$
3. $$7 \times 10$$
4. $$3 \times 3$$
5. $$2 \times 6$$
6. $$8 \times 4$$
7. $$5 \times 7$$

### Student Response

For access, consult one of our IM Certified Partners.

If students don’t mention the groups in the rows and columns of squares, consider asking:

• “How did you decide which rectangle matched each expression?”
• “Where do we see equal groups in the rectangles?”

### Activity Synthesis

• “How do you see each factor in the rectangle?” (I can see one factor in the number of squares in a row. I can see the other factor as the number of rows. I see one factor as the number of squares in a column. The other factor is the number of columns. It’s like I see the factors in an array, only it’s squares, not dots.)
• “How do you see the product in the rectangles?” (If we count the squares in each rectangle, it gives us the same number as the product of the factors. The product is the same as the total number of squares in each rectangle.)
• “Why does multiplication give the same number as counting one by one?”

## Activity 2: Create from Expressions (20 minutes)

### Narrative

The purpose of this activity is for students to represent multiplication expressions as rectangular areas. Students use a grid to draw the rectangular area that represents a multiplication expression. In the synthesis, students explain how they interpret the multiplication expression, specifically how they see the equal groups in the rows and columns of the rectangular area. Give students access to square tiles if needed. When students draw and relate area diagrams to multiplication expressions they are reasoning abstractly and quantitatively (MP2).

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of the 5 problems to complete.
Supports accessibility for: Organization, Attention, Social-emotional skills

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “Now you’re going to draw rectangles that match some multiplication expressions.”
• 1 minute: quiet think time

### Activity

• 7–10 minutes: partner work time
• Monitor for a rectangle that two students oriented differently.

### Student Facing

1. The numbers in each expression represent the number of rows (or columns) in a rectangle and how many squares are in each row (or column).

On the grid, draw each rectangle, label it with the numbers, and find its area.

1. $$3 \times 4$$
2. $$4 \times 6$$
3. $$6 \times 3$$
4. $$7 \times 4$$
5. $$3 \times 2$$
2. Explain why multiplying the numbers in each expression gives us the area of the rectangle.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Have 2–3 students share a rectangle for each expression.
• For each student sample ask:
• “How does the area of this rectangle match the expression?” (I see 4 equal groups of 6 because each row has 6 squares. I see 4 equal groups because each column has 6 squares.)
• “Did anyone draw a different rectangle for this expression?”
• Display a rectangle that two students oriented differently.
• “How can both of these rectangles match the same expression?” (They have the same number of squares. They have the same side lengths, just switched. I see the groups in the rows in the first rectangle and in the columns in the second rectangle.)
• Display the 3-by-5 rectangle and descriptions from the launch in the first activity.
• “Which way of describing a rectangle was the most helpful to you as you drew rectangles in this activity?” (It was helpful to describe how many rows were in the rectangle and how many squares were in each row. It was helpful to think about one factor as the number of columns and the other factor as the number of squares in each column.)

## Lesson Synthesis

### Lesson Synthesis

Display or sketch a 2-by-7 gridded rectangle with the side lengths labeled 2 and 7, as shown:

“How could you figure out the total number of squares in this rectangle?” (Count by one. Count by 2. Count by 7. Multiply $$2 \times 7$$. Multiply $$7 \times 2$$.)

“How are rectangular areas similar to other ways we’ve shown multiplication?” (We can see rows and columns like arrays. We can see equal groups in the rows. We can see equal groups in the columns.)

“How are rectangular areas different from other ways we’ve shown multiplication?” (We’re counting spaces instead of objects.)

## Cool-down: Create a Rectangular Area (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.