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Find the Largest Product

Grade 5: Unit 8: Lesson 1

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Lesson Purpose

The purpose of this lesson is for students to recognize and explain place value patterns as they find products using the standard algorithm they learned in Unit 4.

Lesson Narrative

In previous units, students learned the standard algorithm for multiplying whole numbers. In this lesson, students compare and contrast different products that can be made with the same set of 3 or 4 digits. They look for patterns in the arrangement of the digits and explain those patterns in terms of place value. For example, students might notice that the product of 2 two-digit numbers results in a greater product than that of a three-digit number and 1 one-digit number. If students need additional support with the concepts in this lesson, refer back to Unit 4, Section A in the curriculum materials.

Learning Goals

•Fluently multiply multi-digit whole numbers using the standard algorithm.

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Let’s look for patterns when we multiply multi-digit numbers.

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Notice and Wonder: Digits

Warm-Up

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What do you notice? What do you wonder?

Launch

•Groups of 2

•Display the image.

Student Work Time

•“What do you notice? What do you wonder?”

•1 minute: quiet think time

•1 minute: partner discussion

•Share and record responses.

Student Responses

Students may notice:

• Both products have the same digits, but some of the digits are in different places.

• Both products have factors with the number 41 in them.

• The 8 and the 6 switch places.

Students may wonder:

• What are the values of the products?

• Are they equal?

• Which one is larger?

Activity Synthesis

“Without finding the values, which product do you think will be greater? Explain your reasoning.”

I think 841 x 6 will be greater because 841 is a lot more than 641. I think will 641 x 8 be

larger because there is 4,800 in each and the second product has more 41s.

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Talk About It

Activity 1

MLR8: Discussion Supports

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Consider the statement below. Decide whether you agree, disagree, or are unsure. Be prepared to explain your reasoning.

Launch

•Groups of 2

•“You and your partner will discuss a talking point in 2 rounds. In the first round, read the statement and take turns explaining whether you agree, disagree, or are unsure. Then we will switch groups and complete Round 2.”

•Partner work time: 2–3 minutes

Student Work Time

•After partner work time, rearrange students into groups of 4. New groups of 4 should be formed where partners are not in the same group.

•“Now, you will complete Round 2 in your new group. Each person in the group will be given time to restate their reasoning.”

•1–2 minutes: group discussion

•“Decide if you want to revise your thinking and be prepared to explain why you changed your thinking. Each person in the group will say whether or not they changed their answer and explain why or why not.”

•1–2 minutes: small-group discussion

•“Think about something new that you learned from your group or something that you are still wondering about.”

•1–2 minutes: independent work time

•1–2 minutes: partner discussion

•“Now, use what you learned to complete the second problem.”

•1–2 minutes: independent work time

Student Responses

a.

Sample response: 52 x 7 is the largest product you can make with the digits 5, 2, and 7. I learned that 52 x 7 is 4 greater than 72 x 5 because both products have 350, but making 7 the second factor gives a larger partial product than making 5 the second factor.

B.31 x 6 is the largest product you can make with the digits 6, 3, and 1 because 61 x 3 =183, but 31 x 6 = 186. Putting the 1 in the tens place or multiplying by 1 gives a much smaller product.

Activity Synthesis

Display:

72 x 5 = (70 x 5) + (2 x 5)

52 x 7= (50 x 7) + (2 x 7)

Why is 52 x 7 greater than 72 x 5?

The product of ones and tens is the same in these products but 52 x 7 has a larger product of ones and ones.

“What did you learn about placement of digits when multiplying a two-digit number by a one-digit number?”

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Write about something new that you learned from your group or something you still wonder about.

Launch

•Groups of 2

•“You and your partner will discuss a talking point in 2 rounds. In the first round, read the statement and take turns explaining whether you agree, disagree, or are unsure. Then we will switch groups and complete Round 2.”

•Partner work time: 2–3 minutes

Student Work Time

•After partner work time, rearrange students into groups of 4. New groups of 4 should be formed where partners are not in the same group.

•“Now, you will complete Round 2 in your new group. Each person in the group will be given time to restate their reasoning.”

•1–2 minutes: group discussion

•“Decide if you want to revise your thinking and be prepared to explain why you changed your thinking. Each person in the group will say whether or not they changed their answer and explain why or why not.”

•1–2 minutes: small-group discussion

•“Think about something new that you learned from your group or something that you are still wondering about.”

•1–2 minutes: independent work time

•1–2 minutes: partner discussion

•“Now, use what you learned to complete the second problem.”

•1–2 minutes: independent work time

Student Responses

a.

Sample response: 52 x 7 is the largest product you can make with the digits 5, 2, and 7. I learned that 52 x 7 is 4 greater than 72 x 5 because both products have 350, but making 7 the second factor gives a larger partial product than making 5 the second factor.

B.31 x 6 is the largest product you can make with the digits 6, 3, and 1 because 61 x 3 =183, but 31 x 6 = 186. Putting the 1 in the tens place or multiplying by 1 gives a much smaller product.

Activity Synthesis

Display:

72 x 5 = (70 x 5) + (2 x 5)

52 x 7= (50 x 7) + (2 x 7)

Why is 52 x 7 greater than 72 x 5?

The product of ones and tens is the same in these products but 52 x 7 has a larger product of ones and ones.

“What did you learn about placement of digits when multiplying a two-digit number by a one-digit number?”

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Use the digits 6, 3, and 1 to make the largest possible product. Be prepared to explain your reasoning.

Launch

•Groups of 2

•“You and your partner will discuss a talking point in 2 rounds. In the first round, read the statement and take turns explaining whether you agree, disagree, or are unsure. Then we will switch groups and complete Round 2.”

•Partner work time: 2–3 minutes

Student Work Time

•After partner work time, rearrange students into groups of 4. New groups of 4 should be formed where partners are not in the same group.

•“Now, you will complete Round 2 in your new group. Each person in the group will be given time to restate their reasoning.”

•1–2 minutes: group discussion

•“Decide if you want to revise your thinking and be prepared to explain why you changed your thinking. Each person in the group will say whether or not they changed their answer and explain why or why not.”

•1–2 minutes: small-group discussion

•“Think about something new that you learned from your group or something that you are still wondering about.”

•1–2 minutes: independent work time

•1–2 minutes: partner discussion

•“Now, use what you learned to complete the second problem.”

•1–2 minutes: independent work time

Student Responses

a.

Sample response: 52 x 7 is the largest product you can make with the digits 5, 2, and 7. I learned that 52 x 7 is 4 greater than 72 x 5 because both products have 350, but making 7 the second factor gives a larger partial product than making 5 the second factor.

B.31 x 6 is the largest product you can make with the digits 6, 3, and 1 because 61 x 3 =183, but 31 x 6 = 186. Putting the 1 in the tens place or multiplying by 1 gives a much smaller product.

Activity Synthesis

Display:

72 x 5 = (70 x 5) + (2 x 5)

52 x 7= (50 x 7) + (2 x 7)

Why is 52 x 7 greater than 72 x 5?

The product of ones and tens is the same in these products but 52 x 7 has a larger product of ones and ones.

“What did you learn about placement of digits when multiplying a two-digit number by a one-digit number?”

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More Digits

Activity 2

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Use the digits 7, 3, 2, and 5 to make the greatest product.

Explain or show how you know you have made the greatest product.

Launch

•Groups of 2

•Display:

7, 3, 2, 5

•“Using only these digits, what multiplication expressions could we write?”

•1 minute: quiet think time

•Record answers for all to see.

•“Which of these expressions do you think would make the greatest product? Be prepared to explain your reasoning.” (I think the three-digit by one-digit expression would make the greatest product because you can put the 7 in the hundreds place.)

•“Use the digits 7, 3, 2, and 5 to make the greatest product.”

Student Work Time

•5–7 minutes: partner work time

Student Responses

72 x 53
I figured out that the two-digit by two-digit multiplication problems made greater products and then I tried a few different problems until I found the one that made the greatest product.

Activity Synthesis

•“Which multiplication expression will have the greatest product?”

•Poll the class.

•Record responses for all to see.

•Display or write these expressions for all to see:

•“Why does switching the placement of the 5 and the 7 increase the value of the product?”

Both products have 35 hundreds, but when 7 is the second factor, the product has 7 groups of 32 instead of 5 groups.

•Display or write these expressions for all to see:

•“Why does switching the placement of the digits 2 and 3 increase the value of the product?”

Both products have partial products 50 x 70 and 2 x 3, but when we switch the digits 2 and 3, the partial products in 72 x 53 are greater.

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Lesson Synthesis

Here are the problems from the warm-up. Does anyone want to revise their thinking about which

one is the greater product?

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Lesson Synthesis

Today we explored ways to arrange digits to make the greatest product. We had to solve a lot of

multiplication problems. What is something new that you learned about multiplication today?

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Multiply 2 Digits by 2 Digits

Cool-Down

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Cool-Down: Multiply 2 Digits by 2 Digits

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Image Citations

Photo by Jeremy Thomas on Unsplash

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