# Lesson 10

Explore Multiplication Strategies with Rectangles

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

### Narrative

The purpose of this How Many Do You See is for students to use grouping strategies to describe the quantities they see.

### 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.
• Keep the images displayed for the launch of the next activity.

### Student Facing

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

### Activity Synthesis

• “How can we use amounts that we can see quickly to find the total number of squares?” (We can look for repetition of the number of squares that we can easily see. We can add to or multiply the number of squares we can easily see.)
• “Who can restate the way _____ saw the squares in different words?”
• “Did anyone see the squares the same way but would explain it differently?”
• “Does anyone want to add an observation to the way _____ saw the squares?”

## Activity 1: From Diagrams to Expressions (20 minutes)

### Narrative

The purpose of this activity is for students to analyze different ways of decomposing a gridded rectangle to find the total number of squares in a rectangle. For example, they see that the area of a rectangle that is 3 units by 6 units can be found adding $$3 \times 5$$ and $$3 \times 1$$ and relate that strategy to the expression $$(3 \times 5) + (3 \times 1)$$. The area can also be found by decomposing the rectangle into two halves or finding $$3 \times 3$$ twice, which is represented by $$2 \times (3 \times 3)$$.

The reasoning here allows students to visually make sense of strategies for multiplication that are based on the associative and distributive properties of multiplication. The focus is not on naming the properties, but rather on interpreting the expressions and relating them to the quantities in the diagrams (MP7).

### Launch

• Groups of 2
• Display the first problem.
• “Take a minute to make sense of how Andre and Elena found the area of a rectangle.”
• 1 minute: quiet think time

### Activity

• “Work with your partner to discuss how their strategies are alike and different, and how the numbers in each of their expressions relate to their diagrams.”
• 5–7 minutes: partner work time
• Share responses.
• “Work with your partner to complete the second problem.”
• 3–5 minutes: partner work time

MLR2 Collect and Display

• Circulate, listen for and collect ways that students decompose the rectangle and the language students use to describe the strategies they used. Listen for: decomposed, smaller parts, smaller rectangles, $$2 \times 2 \times 9$$, $$2 \times 18$$, $$4 \times 5 + 4 \times 4$$, and $$5 \times 4 + 4 \times 4$$.
• Record students’ diagrams, words, and phrases on a visual display and update it throughout the lesson.

### Student Facing

Andre and Elena are finding the area of this rectangle.

Andre writes $$6 \times 3$$.

He marks the rectangle like this:

He then writes:

$$2 \times (3 \times 3)$$
$$2 \times 9 = 18$$

Elena writes  $$3 \times 6$$

She marks the rectangle like this:

She then writes:

$$3 \times (5 + 1)$$
$$(3 \times 5) + (3 \times 1)$$
$$15 + 3$$
18

1. Discuss with a partner:

1. How are Andre and Elena’s strategies alike? How are they different?
2. How are the numbers in Andre’s expressions related to his diagram?
3. How are the numbers in Elena’s expressions related to her diagram?
2. Here is another rectangle.

Its area can be found by finding $$4 \times 9$$.

2. Write one or more expressions that can represent your work on the diagram and show how you find the area.

### Student Response

If students count one-by-one to find the area if the rectangle in the second problem, consider asking:
• “How did you find the area of the rectangle?”
• “How could you use a product you already know to find the area of the rectangle? How could you show your strategy on the rectangle?”

### Activity Synthesis

• “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.

## Activity 2: From Expressions to Diagrams (15 minutes)

### Narrative

In this activity, students are given expressions that represent strategies for finding the area of rectangles. The strategies are based on the distributive property and the associative property of multiplication. Students interpret the expressions by marking or shading area diagrams and connect each expression to the product of two factors (MP2). For instance, they see that to find the value of $$2 \times (2 \times 6)$$ is to find the value of $$4 \times 6$$ or $$6 \times 4$$.

MLR8 Discussion Supports: Synthesis: Create a visual display of the diagrams. As students share their strategies, annotate the display to illustrate connections. For example, trace the area showing 5 columns of 3, and write $$5 \times 3$$.
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 2 of the 3 problems to complete.
Supports accessibility for: Organization, Attention, Social-emotional skills

### Launch

• Groups of 2
• “Take a minute to read the directions of the activity. Then, talk to your partner about what you are asked to do.”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Answer any clarifying questions from students.

### Activity

• “Mark or shade each diagram to represent how each student found the area.”
• 3–5 minutes: independent work time
• “Share with your partner how you used the rectangles to show each expression.”
• 3–5 minutes: partner discussion

### Student Facing

Here are some rectangles and expressions that show how three students saw the area of the rectangles.

Noah

$$(5\times 3)+(2 \times 3)$$

Priya

$$2 \times (2 \times 6)$$

Tyler

$$(5 \times 8) + (3 \times 8)$$

For each rectangle:

1. Name the two factors that can be multiplied to find its area.
2. Mark or shade each rectangle to show how each student saw the area. Be prepared to explain your reasoning.

### Activity Synthesis

• “What are the two factors you can multiply to find the area of Noah’s rectangle?” (7 and 3)
• “How are those numbers related to the expression that he wrote: $$(5\times 3)+(2 \times 3)$$?”  ($$7 \times 3$$ is 21. Finding $$5\times3$$, which is 15, then adding $$2\times3$$, which is 6, also gives 21.)
• “Where do you see the two factors in his expression?” (The 7 is 5 and 2 combined. The 3 is in the $$5 \times 3$$ and $$2 \times 3$$.)
• Repeat the line of questioning with Priya’s rectangle and expression.

## Lesson Synthesis

### Lesson Synthesis

“Today, we used diagrams to find the area of rectangles with certain side lengths. We decomposed the rectangles in different ways and wrote different expressions.”

“What were some strategies for decomposing the rectangles to find their areas?” (Partition one side into smaller parts and find the area of smaller rectangles within the original one. Partitioning the rectangle into two halves and finding the area of each half and then doubling it.)

“How might these strategies help us multiply two numbers?” (They show that we can break apart or decompose one of the numbers and multiply smaller numbers and then combine the results. Using diagrams and writing expressions can help us see and record the parts.)