# Lesson 5

Multiplication of Multi-digit Numbers

## Warm-up: Estimation Exploration: A Silly Riddle (10 minutes)

### Narrative

In this warm-up, students practice estimating a reasonable answer using known information, rounding, and multiplicative reasoning strategies. For example, students may create a diagram and arrive at $$7 \times 7 \times 7 \times 7$$ as an estimate of the number of fish. Then, they may approximate $$(7 \times 7) \times (7 \times 7)$$ with $$50 \times 50$$, or estimate $$(7 \times 7 \times 7) \times 7$$ with $$(50 \times 7) \times 7$$ or $$350 \times 7$$. Some students may notice that the question is vague and asks, “How many are going to the park?” rather than “How many people are going to the park?” and account only for the number of children and teachers in their estimation. If students ask for clarification, ask them to make their own assumptions and explain why they made those assumptions when they share their estimates.

### Launch

• Groups of 2
• Display the description.

### Activity

• “What is an estimate that’s too high? Too low? About right?”
• 2 minutes: quiet think time
• 1 minute: partner discussion
• Record responses.

### Student Facing

• Seven teachers are going to the park.
• Each teacher is taking 7 students.
• Each student is bringing 7 fishbowls.
• Each fishbowl has 7 fish.

How many are going to the park?

Record an estimate that is:

too low about right too high
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### Activity Synthesis

• “Is anyone’s estimate less than 500? Greater than 5,000?”
• “Did anyone include the number of teachers and students in their estimate?”
• Invite students to share their estimation strategies. After each explanation, ask if others reasoned the same way.
• Consider revealing the actual value of 2,457 teachers, students, and fish.

## Activity 1: Two Methods Revisited (20 minutes)

### Narrative

In this activity, students revisit two algorithms for multiplying numbers. They recall that, in the standard algorithm, the digit in one factor is multiplied by each digit in the other factor, but the partial products are not recorded on separate lines. Rather, the standard algorithm condenses multiple partial products into a single product.

MLR8 Discussion Supports. Synthesis: For each strategy that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language.
Representation: Develop Language and Symbols. Activate or supply background knowledge. Display $$416 \times 2$$ without the solutions. Ask students to tell you what each digit in 416 represents (for example, the 1 represents 10).
Supports accessibility for: Conceptual Processing, Memory, Language

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students access to grid paper, if needed for aligning the digits in a multiplication algorithm.

### Activity

• 2 minutes: independent work time
• Pause to discuss the first set of questions. Display the two algorithms in the first question. Ask students to share responses.
• “How are the two algorithms alike? How are they different?”
• Highlight student responses to emphasize:
• In method A, each partial product is listed separately before being added at the end.
• In method B, only one digit is recorded at a time. The values for any place value unit are added and only one digit is recorded. Any new units are recorded in the next highest place.
• 6–10 minutes: independent work time
• 2–4 minutes: partner discussion

### Student Facing

1. Earlier in the course, we used these two ways to multiply numbers:

1. In method A, where do the 12, 20, and 800 come from?
2. In method B, where does the 1 above 416 come from?
2. Diego used both methods to find the value of $$215 \times 3$$ but ended up with very different results.

1. Without calculating anything, can you tell which method shows the correct product? How do you know the other one is not correct?
2. For the incorrect result, explain what was correct and what was incorrect in his steps. Then, show the correct calculation using method B.
3. Use either way to find the value of each product. Show your reasoning.

1. $$521 \times 3$$
2. $$6,\!121 \times 4$$
3. $$305 \times 9$$

### Activity Synthesis

• Select students to share their responses to the second set of questions.
• If not mentioned in students’ explanations, highlight that:
• The result of $$3 \times 5$$ is 15, a two-digit number, so the 1 ten should be carried over to the tens place and added to the 3 tens that result from $$3 \times 10$$.
• One ten and 3 tens make 4 tens.
• Poll the class on whether their preferred method is A, B, or is dependent on the problem. Select a student from each camp to explain their reason.

## Activity 2: Two by Two (15 minutes)

### Narrative

Earlier, students compared and made connections between two algorithms for multiplying a multi-digit number and a single-digit number. In this activity, students compare an algorithm that uses partial products with the standard algorithm for multiplying 2 two-digit numbers. As students analyze and critique each method, they practice looking for and making use of base-ten structure of whole numbers (MP7).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Access to grid paper, in case needed to align digits when multiplying

### Activity

• 3–4 minutes: independent work time on the first two problems
• 1–2 minutes: partner discussion
• Monitor for students who can explain the numbers in the standard algorithm.
• 5 minutes: independent work time on the remaining questions

### Student Facing

Here are two ways to find the value of $$34 \times 21$$.

1. In method A, where do the 4, 30, 80, and 600 come from?
2. In method B, which two numbers are multiplied to get:

1. 34?
2. 680?
3. Use the two methods to show that each equation is true.

### Activity Synthesis

• Invite students to share how they used each method to show that $$44 \times 12 = 528$$​ and $$63 \times 21 = 1,\!323$$

## Lesson Synthesis

### Lesson Synthesis

“Today we looked at several methods for multiplying a multi-digit number by a single-digit number and also multiplying 2 two-digit numbers.“

“Here are some reasoning or calculation strategies we have seen for multiplying 2 two-digit numbers.”

$$33 \times 2 = 66$$

$$33 \times 10 = 330$$

$$66 + 330 = 396$$

“What connections do you see among these strategies? Point out as many as you can.”

“Which of these strategies makes the most sense or is clearest to you?”