# Lesson 9

Recording Partial Products: One-digit and Three- or Four-digit Factors

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

### Narrative

This warm-up prompts students to carefully analyze and compare four equivalent expressions. In making comparisons, students have a reason to attend closely to the features and the value of the expressions. The activity also enables the teacher to gain insight into students’ understanding of properties of operations and how they talk about them.

### Launch

• Groups of 2
• Display the expressions.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

Which one doesn’t belong?

1. $$7 \times 50$$

2. $$(3 \times 50) + (4 \times 50)$$

3. $$(5 \times 10) \times 7$$

4. $$50 + 50 + 50 + 50 + 50 + 50 + 50$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How are the expressions alike?” (They all have a value of 350.)
• “Were there expressions that you knew right away were equivalent? Which ones? How did you know?”

## Activity 1: An Algorithm for Noah (20 minutes)

### Narrative

The purpose of this activity is to analyze an algorithm that uses partial products. Students are not required to use a specific notation, but analyzing each algorithm deepens their understanding of the structure of place value in multiplication.

When students interpret and make sense of Noah's work, they construct viable arguments and critique the reasoning of others (MP3).

MLR8 Discussion Supports. Synthesis: Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement: “I added all the numbers together to find the answer” as “I added all the partial products to calculate the final product.”

### Launch

• Groups of 2
• Display the diagram in the activity.
• “What do you notice? What do you wonder?”
• 1 minute: partner discussion
• Share and record responses.

### Activity

• 4 minutes: independent work time on the first problem about Noah’s diagram
• 4 minutes: partner discussion
• 5 minutes: group work time on the rest of the activity
• Monitor for students who include the place value of each digit in 124 in explaining what is happening in the algorithm.

### Student Facing

1. Noah drew a diagram and wrote expressions to show his thinking as he multiplied two numbers.

$$7 \times 124$$
$$7 \times (100 + 20 + 4)$$
$$(7 \times 100) + (7 \times 20) + (7 \times 4)$$
$$700 + 140 + 28$$

How does each expression represent Noah’s diagram? Be prepared to share your thinking with a partner.

2. Later, Noah learned another way to record the multiplication, as shown here.

Make sense of each step of the calculations and record your thoughts. Be prepared to explain Noah’s steps to a partner.

3. Complete the diagram to find the value of $$217 \times 8$$. Use Noah’s recording method to check your work.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Select students to share their interpretations of the steps in the second problem.
• Highlight these key ideas along the way:
• Multiply the 7 by the 4 ones in 124 and write the product below the line.
• Multiply the 7 by 2 tens in 124 and record this product below the first product.
• Multiply the 7 by the 1 hundred and record this product below the previous product.
• Add all the partial products to calculate the final product.
• If needed, review these steps for $$8 \times 217$$.
• Reiterate that we typically begin by multiplying the single-digit factor by the digit in the ones place, and then repeat with then the digit in the tens place, and so on.
• “The recording strategy that Noah learned is an algorithm that uses partial products. We used an algorithm to add and subtract large numbers in our last unit.”
• “How do we know if we have finished finding all the partial products?” (We keep track of all the partial products or smaller parts of the whole product as we go. We went in order from the smallest place value to the largest place value.)

## Activity 2: Try an Algorithm with Partial Products (15 minutes)

### Narrative

In this activity, students continue to analyze an algorithm that uses partial products and learn that there are different ways to write the partial products. While students are not required to use a specific algorithm for multiplication, analyzing variations in the partial products notation deepens their understanding of how to use base-ten structure to multiply efficiently (MP7).

Action and Expression: Internalize Executive Functions. Provide access to colored pencils and sticky notes. To support organization, invite students to color code each partial product as seen in the previous activity. To support working memory, invite students to use the sticky notes to record the expanded form of the multi-digit number they are multiplying by.
Supports accessibility for: Memory, Organization

### Launch

• Groups of 2
• “What strategy would you use to find the value of $$8 \times 3,\!419$$?”
• 1 minute: partner discussion
• Share responses.
• Display Noah’s and Mai’s computations.
• “Take a quiet moment to make sense of Noah and Mai’s work. How do you think they arrived at the last four numbers?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• “Which representation is more like how you thought about it?”

### Activity

• “Work with your partner on the first two questions.”
• 5 minutes: partner work time on the first two problems
• Pause for a brief discussion.
• “How are Mai’s and Noah’s notation the same and how are they different?” (Both of these methods calculate partial products, but Noah’s goes in order from multiplying the ones, tens, hundreds, then the thousands. Mai’s goes in order the other way from the thousands, hundreds, tens, and then the ones.)
• “Are both methods correct? How do you know?” (Yes, because they both result in the same product when you calculate it.)
• “Work on the last set of problems independently before discussing with your partner.”
• 5 minutes: independent work time on the last set of problems

### Student Facing

Noah and Mai want to find the value of $$8 \times 3,\!419$$. They recorded their steps in different ways, as shown.

1. How are Mai’s and Noah’s notation alike? How are they different?
2. Use a diagram to show what each of the partial products 72, 80, 3,200 and 24,000 represent. Then, find the value of $$8 \times 3,\!419$$.
3. Find the value of each expression. For at least one expression, use the algorithm that Noah used. Show your reasoning.

1. $$4 \times 5,\!342$$
2. $$7 \times 983$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we learned different ways of recording partial products to multiply four-digit by one-digit numbers. We made connections between a diagram and using algorithms that use partial products.”

Display the expression $$4 \times 5,\!342$$ and corresponding diagram and computation.

Invite students to take turns making connections between each part of the diagram to the algorithm.

Highlighting the following:

• Each number in the smaller rectangles in the diagram is a partial product—a result of multiplying a part of one factor by the second factor.
• When we use a diagram, we can find the partial products for different smaller rectangles and add the pieces together in any order. The product stays the same no matter how we decompose the diagram.
• When we use an algorithm that uses partial products, we multiply the ones, tens, hundreds, then thousands, and record the partial products vertically. Changing the order of multiplying doesn’t change the final product.

## Cool-down: Partial Products (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.