Lesson 1

Find the Largest Product

Warm-up: Notice and Wonder: Digits (10 minutes)

Narrative

The purpose of this warm-up is for students to discuss the location of digits in the products, which will be useful when students try to find the greatest product in a later activity. While students may notice and wonder many things, the location of the digits 6, 4, 1, and 8 is the important discussion point.

Launch

  • Groups of 2
  • Display the image.

Activity

  • “What do you notice? What do you wonder?”
  • 1 minute: quiet think time
  • 1 minute: partner discussion
  • Share and record responses.

Student Facing

What do you notice? What do you wonder?

multiply. 8 hundred 41 times 6 equals.
multiply. 6 hundred 41 times 8 equals.

Student Response

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Activity Synthesis

  • “Without finding the values, which product do you think will be greater? Explain your reasoning.” (I think \(841 \times 6\) will be greater because 841 is a lot more than 641. I think \(641 \times 8\) will be larger because there is 4,800 in each and the second product has more 41s.)
  • “We are going to revisit these problems in the lesson synthesis.”

Activity 1: Talk About it (15 minutes)

Narrative

The purpose of this activity is for students to practice using the standard algorithm and explain how the placement of the digits in factors impacts the value of the product when multiplying a two-digit number by a one-digit number. Students multiply different factors which use the same 3 digits and determine which combination yields the greatest product. While the problems were intentionally structured to encourage students to use an efficient strategy, such as the algorithm, students should use whatever strategy makes sense to them when solving these problems.

Students critically analyze a claim about the largest product that can be made with 3 digits and discuss their reasoning with several partners (MP3). 

MLR8 Discussion Supports. During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner.
Advances: Listening, Speaking
Action and Expression: Internalize Executive Functions. Before they begin, invite students to verbalize a strategy they can use to determine whether they agree or disagree with each statement. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language

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

Activity

  • 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 Facing

  1. Consider the statement below. Decide whether you agree, disagree, or are unsure. Be prepared to explain your reasoning.
    agree disagree unsure
    Given the digits 7, 5, and 2, the largest product you can make is \(75\times2\) because 75 is the largest number you can make.
    After round 1: Given the digits 7, 5, and 2, the largest product you can make is \(75\times2\) because 75 is the largest number you can make.

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

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

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Display:

    \(72\times 5 =(70\times 5)+(2\times5)\)

    \(52\times 7 =(50\times 7)+(2\times 7)\)

  • “Why is \(52\times 7\) greater than \(72\times5\)?” (The product of ones and tens is the same in these products but \(52 \times 7\) 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?”

Activity 2: More Digits (20 minutes)

Narrative

The purpose of this activity is for students to practice using the standard algorithm and to use place value reasoning to explain how the placement of the digits in factors impacts the value of the product when multiplying (MP7). 

Launch

  • Groups of 2
  • Display:

    7, 3, 2, 5

  • “Using only these digits, what multiplication expressions could we write?” (\(723 \times 5\), \(32 \times 57\), \(7 \times 3 \times2 \times 5\), \(73 \times 5 \times 2\).)
  • 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.”

Activity

  • 5–7 minutes: partner work time

Student Facing

  1. Use the digits 7, 3, 2, and 5 to make the greatest product.
  2. Explain or show how you know you have made the greatest product.

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students need help getting started, refer to the products generated during the launch and ask students to evaluate them. Ask them to order the products from least to greatest and explain what they notice. Next, ask them to generate other products using the same digits. Continue to check in with them and ask them to explain any patterns they notice.

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:
    multiplication algorithm
    multiplication algorithm
  • “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:
    multiplication algorithm
    multiplication algorithm
  • “Why does switching the placement of the digits 2 and 3 increase the value of the product?” (Both products have partial products \(50 \times 70\) and \(2 \times 3\), but when we switch the digits 2 and 3, the partial products in \(72 \times 53\) are greater.)

Lesson Synthesis

Lesson Synthesis

Display or write these products for all to see.

multiply. 8 hundred 41 times 6 equals.
multiply. 6 hundred 41 times 8 equals.

“Here are the problems from the warm-up. Does anyone want to revise their thinking about which one is the greater product?” (\(641 \times 8\) because both products will have 4,800, but there will be two more groups of 41 in \(641 \times 8\).)

“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?” (I never realized how many different problems you could create with the same digits. I was surprised by some of the largest products. I thought \(841 \times 6\) would be larger than \(641 \times 8\).)

Cool-down: Multiply 2 Digits by 2 Digits (5 minutes)

Cool-Down

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