# Lesson 8

Multiply 2 Two-digit Numbers

## Warm-up: Number Talk: Extra Groups (10 minutes)

### Narrative

This Number Talk encourages students to use multiples of 10 to mentally multiply two-digit numbers that are close to multiples of 10. Students can use place value reasoning and the distributive property of multiplication over addition or subtraction to find the value of the products. The work here prompts students to think flexibly about how numbers can be decomposed strategically when multiplying.

### 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.

• $$20 \times 60$$
• $$21 \times 60$$
• $$20 \times 62$$
• $$19 \times 60$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How can $$20 \times 60$$ help us find the value of $$19 \times 60$$?” ($$19 \times 60$$ is one group of 60 less than $$20 \times 60$$, so we can subtract 60 from $$20 \times 60$$.)

## Activity 1: Two by Two (20 minutes)

### Narrative

In this activity, students use rectangular diagrams and similar reasoning as in earlier activities to represent the multiplication of 2 two-digit numbers. They analyze a progression of diagrams, starting with those that represent multiplication of two-digit and one-digit numbers (18 and 6), a two-digit number and a ten (18 and 10), and then 2 two-digit numbers (18 and 16).

Students may decompose factors in different ways. For example, those who are familiar with multiples of 25 may find it intuitive to decompose $$25 \times 46$$ as $$25 \times 40$$ and $$25 \times 6$$, rather than decomposing 25 into $$20 + 5$$.

MLR8 Discussion Supports. Pair gestures with verbal directions to clarify the meaning of any unfamiliar terms such as partial product.
Action and Expression: Develop Expression and Communication. Give students access to base-ten blocks. If students use the blocks for the last question, ask them to also draw a diagram that represents their work with the blocks.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Launch

• Display the three diagrams in the activity.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: partner discussion

### Activity

• “Take a few quiet minutes to work on the first problem. Then, share your thinking with your partner.”
• 3 minutes: independent work time
• 3 minutes: partner discussion
• Invite students to share their responses: “What multiplication expression can be represented by each diagram?”
• “Complete the rest of the activity.”
• 3 minutes: independent or partner work time
• Monitor for students who decompose both factors into tens and ones and those who choose to keep one of the factors intact.

### Student Facing

1. For each diagram, write a multiplication expression that the diagram can represent. Then, find the value of the expression. Use equations to show or explain your reasoning.

2. How are the diagrams alike? How are they different? Discuss with your partner.
3. Use a diagram to find each product.

1. $$13 \times 21$$
2. $$25 \times 46$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Select two students to share their diagrams, solutions, and reasoning for the last problem.
• “How did you decompose the factors and the diagram?” (I decomposed the 13 into 10 and 3 and the 21 into 20 and 1.)
• “What expression could we write to show the partial product represented by each part of the diagram?”
• “How do these partial products help us find the value of $$13 \times 21$$?” (Adding them gives us the value of $$13 \times 21$$: $$200 + 10 + 3 + 60 = 273$$)

## Activity 2: Decompose by Place Value (15 minutes)

### Narrative

In this activity, students analyze two ways of decomposing a factor: by place value and not by place value. As they write the corresponding partial products, they see more clearly why it is helpful to decompose each of the factors by place value (MP7). Students may notice that when the factors are decomposed by place value, they end up finding multiples of 10 and multiplying a number by single-digit factors—both of which they can do with some degree of fluency.

• Groups of 2

### Activity

MLR5 Co-Craft Questions
• Display only the two diagrams, without revealing the opening sentence or the question(s).
• 2 minutes: independent work time
• 2 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “What do these questions have in common? How are they different?”
• “Take two quiet minutes to think about the first question. Then, share your thinking with your partner.”
• 2 minutes: independent work time on the first problem
• 1 minute: partner discussion
• “Now complete the rest of the activity.”
• Monitor for students who:
• decompose the side lengths of their diagrams by place value
• write expressions to show the sum of the partial products (This is not required, but it is helpful for the synthesis.)
• If extra time is available, add more two-digit by two-digit expressions to the last problem:
• $$83 \times 39$$
• $$64 \times 92$$

### Student Facing

These diagrams could be used to find the value of $$49 \times 57$$.

1. Which diagram is more helpful when finding the value of $$49\times57$$?  Why?

2. Use a diagram to find each product.

1. $$49 \times 57$$
2. $$29 \times 55$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How can both diagrams be used to find the value of $$49 \times 57$$?” (We can partition each side of a rectangle in many ways without changing the total side lengths.)
• “Why was the partitioning in diagram A more helpful than in diagram B?” ($$40 \times 50$$ is easier than $$37 \times 50$$ or $$12 \times 50$$ to find mentally, because we are multiplying multiples of 10.)

## Lesson Synthesis

### Lesson Synthesis

“Today we learned how to represent the multiplication of 2 two-digit numbers using a rectangular diagram. We learned that we can decompose each factor by place value and show the tens and ones on each side of the rectangle, and that doing this can help us to multiply efficiently.”

Select students with different strategies to share their reasoning for finding the value of $$29 \times 55$$ (the last problem of the last activity).

Display the following diagram as an example of how decomposing can result in facts that are not helpful when multiplying to support using place value to decompose.

“Why might it be more helpful to decompose 55 into $$50 + 5$$ than into $$42 + 13$$?” (Multiplying by multiples of 10 and by single-digit numbers is easier than multiplying numbers like 42 and 9.)

## Cool-down: What’s the Product? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.