# Lesson 2

Partial Products in Diagrams

## Warm-up: Which One Doesn't Belong: Diagrams to Find Products (10 minutes)

### Narrative

The purpose of this warm-up is for students to compare and contrast different diagrams that can be used to represent and calculate products of two-digit numbers. Students used these partial products diagrams in Grade 4. They will extend them to represent the product of a three-digit number and a two-digit number later in the lesson.

These rectangular diagrams use the intuition and properties of area to support representing multiplication. But, a genuine area diagram would be difficult to read, so the individual pieces are not drawn to scale.

### 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

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### Activity Synthesis

- “How might Diagram C be helpful for calculating the product \(42 \times 33\)?” (I can add those numbers to get the value of \(42 \times 33\).)
- Highlight that this is the type of diagram that will be used throughout the next several lessons. The purpose of the diagram is to help see different ways to calculate products of numbers.

## Activity 1: Decompose in Many Ways (15 minutes)

### Narrative

The purpose of this activity is for students to use a diagram to help calculate the product of a three-digit number and a two-digit number. The diagram helps to organize the individual products that can be used to find the larger product. During the activity synthesis, students connect the diagram to the distributive property when they explain how the sum of the individual products gives the larger product (MP7).

*MLR8 Discussion Supports.*For each observation that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language.

*Advances: Listening, Speaking*

### Launch

- Groups of 2
- Display the image from the student workbook.
- “In these problems, write each product inside the part of the diagram that represents that product.”
- Demonstrate by writing 1,200 inside the rectangle with sides marked 30 and 40.

### Activity

- 1–2 minutes: quiet think time
- 6–8 minutes: partner work time
- Monitor for students who use their work for the first product to find the second product.

### Student Facing

- Write the value of each product inside the rectangles.
- Find the value of \(42 \times 33\).
- This diagram represents \(142 \times 33\). Write the value of each product inside the rectangles.
- Find the value of \(142 \times 33\).

### Student Response

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### Activity Synthesis

- Invite students to share their work for finding the product \(42 \times 33\).
- Display: \(42 \times 33 = (40 + 2) \times (30 + 3)\)
- “How does the diagram represent this equation?” (It shows 42 broken up into 40 and 2 and 33 broken up into 30 and 3.)
- Display: \((40 + 2) \times (30 + 3) =\) \((40 \times 30) + (2 \times 30) + (40 \times 3) + (2 \times 3)\)
- “How do you know this equation is true?” (The diagram shows \(42 \times 33\) broken up into those 4 partial products.)
- “How is finding the product \(142 \times 33\) related to finding the product \(42 \times 33\)?” (The products and partial products are the same, except that I also have \(100 \times 33\) in \(142 \times 33\).)

## Activity 2: Calculate in Many Ways (20 minutes)

### Narrative

The purpose of this activity is for students to write expressions to represent different ways to decompose a product. Then they choose one of the decompositions to find the product. Students consider how certain decompositions are more helpful than others, depending on the specific numbers in the problem. The diagrams used here relate to the partial products and standard algorithm methods which students will learn in future lessons.

*Engagement: Provide Access by Recruiting Interest.*Provide choice. Invite students to decide which problem to start with and how they want to write the expressions.

*Supports accessibility for: Attention*

### Launch

- Groups of 2
- Give students time to read the task statement.
- “This time, you will write an expression in each piece of the diagram, rather than a number.”

### Activity

- 1 minute: independent think time
- 7–8 minutes: partner work time
- Monitor for students who:
- use the first diagram to help calculate the values for the other two diagrams.
- choose different diagrams for their calculations.

### Student Facing

Here are some different diagrams that represent \(315 \times 24\). For each diagram, write a multiplication expression inside each rectangle to represent the product.

- Use one of the diagrams to find the value of \(315 \times 24\).
- Explain why you chose that diagram to find the product.

### Student Response

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### Advancing Student Thinking

If students do not write the correct partial product in the diagram, ask, “What is a reasonable estimate for the product of \(315 \times 24\)?”

### Activity Synthesis

- Display:

\(20 \times 300\) - “How does this expression relate to the product \(315 \times 24\)?” (It represents one of the products in the first diagram.)
- “Why isn’t this expression written in any of the other diagrams?” (Because the other diagrams are decomposed differently.)
- Invite students to share the diagram they chose to find the product and how it was helpful. As students share, record equations to represent each partial product.
- “What are the advantages or disadvantages of this way to calculate \(315 \times 24\)?” (For full partial products, each product is simple to calculate. I do have 6 different numbers to add up at the end. When I broke the full product into two products, the calculations I used to find each product were harder, but once I had them, there were only two things to add. When I broke the full product into 3 products, this was a good compromise. The products were not too hard to calculate and there were just 3 of them to add.)

## Lesson Synthesis

### Lesson Synthesis

“Today we multiplied numbers and thought about how diagrams could help.”

Display the first image from the last activity.

“How can the diagram help us find the value of the product \(315 \times 24\)?” (It helps me break up the product by place value. I take the hundreds, tens, and ones of one number and multiply them by the tens and ones of the other number.)

“Tomorrow we are going to work with partial products and organize them in a different way.”

## Cool-down: $222 \times 14$ (5 minutes)

### Cool-Down

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