Lesson 2

More Multiplication

Warm-up: Estimation Exploration: Large Product (10 minutes)

Narrative

The purpose of an Estimation Exploration is for students to practice the skill of estimating a reasonable answer based on experience and known information. 

Launch

  • Groups of 2
  • Display the expression.
  • “What is an estimate that’s too high? Too low? About right?”

Activity

  • 1 minute: quiet think time
  • 1 minute: partner discussion
  • Record responses.
  • Monitor for students who estimate by using \(10,\!000 \times 900\).

Student Facing

\(9,\!999  \times 896\)

Record an estimate that is:

too low about right too high
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Student Response

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Activity Synthesis

  • Invite students to share estimates.
  • “Why is \(10,\!000 \times 900\) a good estimate for the product?” (10,000 is just 1 more than 9,999 and 896 is close to 900.)
  • “What is the value of \(10,\!000 \times 900\)? How do you know?” (9,000,000 because the two numbers have 6 factors of 10 combined.)

Activity 1: Kiran’s Work (20 minutes)

Narrative

The purpose of this activity is for students to consider possible mistakes when multiplying large numbers. Monitor for students who:
  • revise their answer after examining Kiran’s mistake.
  • recognize that \(650 \times 10 = 6,\!500\) so \(650 \times 27\) has to be much greater than 5,850.
  • can explain why Kiran should be multiplying \(650 \times 2 \times 10\).
  • recognize that \(20 \times 50 = 1,\!000\) so there should be three zeros in the second partial product.

When students determine Kiran's error and make sense of his work, they interpret and critique the work of others (MP3).

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “what parts of Kiran’s work do you agree and disagree with?” Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.
Advances: Writing, Speaking, Listening

Launch

  • Display or write for all to see.

    \(650 \times 27\)

  • Display each number in a different corner of the room:
    14,000
    18,000
    13,000
    19,000

  • “When I say go, stand in the corner with the number that you think is the most reasonable estimate for \(650 \times 27\). Be prepared to explain your reasoning.”
  • 1 minute: quiet think time
  • Ask a representative from each corner to explain their reasoning.
  • “Does anyone want to switch corners?”
  • Ask a student who switched corners to explain their reasoning.
  • “Now you are going to find this product and analyze some work.”

Activity

  • Groups of 2
  • 5–7 minutes: partner work time

Student Facing

  1. Find the value of the product.

    multiply, 6 hundred fifty, times, 27, equals.

  2. Below is Kiran’s work finding the value of the product \(650 \times 27\). Is his answer reasonable? Explain your reasoning.

    multiply. 6 hundred fifty times 27. 
  3. What parts of the work do you agree with? Be prepared to explain your reasoning.
  4. What parts of the work do you disagree with? Be prepared to explain your reasoning.

  5. Look at your solution to problem 1. Is there anything you want to revise? Be prepared to explain.

Student Response

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Activity Synthesis

  • Ask previously identified students to share their thinking.
  • Display Kiran's work.
  • “Why doesn’t 5,850 make sense?” (\(650 \times 10 = 6,\!500 \) so \(650 \times 27\) should be a lot larger than 6,500.)
  • “What makes sense about Kiran’s work?” (\(650 \times 2 =1,\!300\), but he needs to multiply \(650\times2\times10\).)
  • Display a student's solution or the image from the student solution.
  • “How do we know that 17,550 is a reasonable value for the product?” (Because \(600 \times 30 =18,\!000\).)

Activity 2: Zero the Hero (15 minutes)

Narrative

The purpose of this activity is for students to practice multiplying multi-digit numbers that have one or more digits of 0 at the end. Monitor for students who:

  • use the standard algorithm to evaluate \(6,\!700 \times 89\).
  • multiply the product \(67 \times 89\) by 10 to find the value of the product \(670 \times 89\).
  • multiply the product of \(670 \times 89\) by 10 to find the value of the product \(6,\!700 \times 89\).

Students who observe that \(670 = 10 \times 67\) and \(6,\!700 = 10 \times 670\) and use these relationships to find the values of the products are observing regularity in repeated reasoning and using their knowledge of how to multiply a whole number by 10 (MP7, MP8).

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were important to find products of multi-digit numbers that have zero digits. Display the sentence frame, “The next time I multiply a number that contains zero digits, I will pay attention to . . . .“
Supports accessibility for: Conceptual Processing, Memory

Launch

  • Groups of 2

Activity

  • 5–7 minutes: independent work time
  • 5–7 minutes: partner discussion

Student Facing

Find the value of each product.

1.multiply. 2 hundred sixty, times, 35, equals.

2.Multiply. 2 thousand, 6 hundred, times, 35 equals.

3.multiply, 6 hundred seventy, times, 89, equals.

4.Multiply. 6 thousand, 7 hundred, times, eighty, 9, equals. 

Student Response

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Activity Synthesis

  • Display the product \(6,\!700 \times 89\):
  • Ask previously identified students to share their solutions.
  • Display student work or the image from the sample response:
    multiplication algorithm
  • “What is the relationship between \(670\times89\) and \(6,\!700\times89\)?” (The product \(6,\!700\times89\) is ten times larger because one of the factors is ten times greater.)
  • “What is the relationship between \(67 \times 89\) and \(6,\!700 \times 89\)?” (The product \(6,\!700 \times 89\) is 100 times as large as \(67 \times 89\)& since 6,700 is \(100 \times 67\).)

Lesson Synthesis

Lesson Synthesis

“Today we multiplied multi-digit numbers using the standard algorithm. What was challenging about the problems we solved today?” (It was hard to keep track of the numbers as we multiplied. I wasn’t sure how many zeroes to write in the second partial product.)

“What is important to remember when using a standard algorithm to multiply \(350 \times 74\)?” (Estimate first so you know if your answer is reasonable. Pay attention to which place each digit is in.)

Display or write the product for all to see.

multiplication algorithm

“What is a reasonable estimate for \(350 \times 74\)?” (Sample responses: 21,000, 24,000, 28,000.)

Ask students to describe to a partner how they would use the standard algorithm to find the value of the product. 

Cool-down: What is Important? (5 minutes)

Cool-Down

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