# Lesson 10

Using Algorithms with Partial Products: 2 Two-digit Numbers

## Warm-up: Number Talk: Products (10 minutes)

### Narrative

This Number Talk encourages students to think about the strategies they can use to multiply 2 two-digit numbers. Students can decompose factors by place value to multiply by multiples of ten, or they can use a doubling and halving strategy to create an equivalent expression. The strategies elicited here will be helpful later in developing a flexible sense of numbers and using this sense to make decisions when multiplying mentally.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$30 \times 7$$
• $$15 \times 14$$
• $$50 \times 8$$
• $$25 \times 16$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How can $$30 \times 7$$ help you solve $$15 \times 14$$?” ($$30 \times 7$$ is twice as many groups with half as much in each group, which results in the same product with easier factors to multiply mentally.)

## Activity 1: Partial Products, Recorded (25 minutes)

### Narrative

In this activity, students analyze multiplication involving 2 two-digit factors. Students make sense of work where the partial products are recorded, but there is no indication of where each product came from. After making sense of work, students use the same strategy of finding and keeping track of partial products to evaluate $$31 \times 15$$.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing

Representation: Internalize Comprehension. Provide students with a partially completed version of the itemized steps as described in the Activity Synthesis. For example, offer students a page that shows four copies of $$64 \times 87$$. Each equation could have one of the partial products written in its corresponding color underneath. Invite students to use a green, yellow, red, and purple colored pencil to highlight the digits involved in each step and to write the corresponding multiplication expression next to each partial product.
Supports accessibility for: Visual Spatial Processing, Organization, Attention

### Launch

• Groups of 2
• “Find the value of $$64\times87$$ and share your reasoning with a partner.”

### Activity

• “Take a few minutes to make sense of Tyler’s calculation. Be prepared to explain your thinking.”
• 3–4 minutes: independent work time on the first problem

MLR1 Stronger and Clearer Each Time

• “Share your analysis of Tyler’s calculation with a partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 2 minutes: structured partner discussion
• Consider displaying these prompts to support students’ conversations:
• “Can you explain how multiplying _____ and _____ gives _____?”
• “Can you use the phrase ‘partial products’ in your explanation?”
• Repeat with 1–2 different partners.
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time
• “Now try using Tyler’s method to complete the last problem and use a diagram to check your work.”
• 5 minutes: independent work time on the last problem

### Student Facing

1. Tyler used an algorithm to find the value of $$64 \times 87$$.

How do you think he arrived at the last five numbers? Record your thinking. Be prepared to share it with a partner.

2. Use Tyler’s method to find the value of $$31 \times 15$$. Then, draw a diagram to check your answer.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• To clarify the steps in Tyler’s notation, consider itemizing each step and using color coding, as shown here:
• Also consider displaying a corresponding diagram for $$64 \times 87$$ and inviting students to make connections to an algorithm.

## Activity 2: Han’s Multiplication Mishap (10 minutes)

### Narrative

When using an algorithm that uses partial products, students may be inclined to pay attention only to single digits in each number and pay little attention to the value of the digits. For example, $$32 \times 19$$ might sound like “9 times 2 is 18 and 9 times 3 is 27.” In this activity, students analyze this error (MP3) and also look at the commutativity of multiplication when finding partial products.

To help students see that the place value of the digits impacts each partial product, consider displaying a diagram that shows partial products alongside the vertical notation.

MLR8 Discussion Supports. Synthesis: Display sentence frames to support whole-class discussion: “_____ and _____ are the same/alike because . . .” and “Why did you . . .?”

• Groups of 2

### Activity

• “Work with your partner on the first problem. Each partner should find the value of one product.”
• 3–4 minutes: partner work time on the first problem
• Pause for a discussion. Select a group whose calculations yield the same product to display their work.
• “Should the calculations show the same result? Why or why not?” (Yes, because the same two numbers are being multiplied.)
• “How are the two calculations different?” (The partial products are written in different orders.)
• “Does it matter which number is listed first and which is listed second?” (No)
• “Complete the last problem independently.”
• 3–4 minutes: independent work time

### Student Facing

1. Decide with your partner who will find each product. Show your reasoning.

2. Here is Han’s computation of $$51 \times 47$$.

1. What error or errors did Han make?
2. Show the correct computation for finding the value of $$51 \times 47$$.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we used partial products in an algorithm to multiply pairs of two-digit numbers. We wrote expressions and equations to help us find and keep track of the partial products as we were multiplying.”

Select 1–2 examples of student work for $$31\times15$$ for the focus of class discussion.

“Let’s look at the strategies used by one student from our class to solve $$31\times15$$.”

“Before we analyze this sample closely, can you tell whether his answer is correct or seems reasonable? Why or why not?”

Select students to explain each step in the computation and any questions the sample raises.

Display a different calculation for $$31\times15$$. Consider using a diagram to support students in making sense of the products of the decomposed factors in the vertical calculation.

Reiterate the importance of paying attention to the place value of each digit being multiplied as we find and record the partial products.

## Cool-down: Choose Your Own Strategy (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.