Math Background

Lesson: Distributive Property of Multiplication
Developing the Concept

In this lesson, students use the Distributive Property to multiply three-digit numbers by two-digit numbers.

Materials: none

Preparation: none

Prerequisite Skills and Background: Students should know basic multiplication and addition facts, how to expand numbers, and how to multiply three-digit numbers by one-digit numbers.

Write 45 x 123 in vertical form on the board.

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  • Say: Today we will learn to multiply three-digit numbers by two-digit numbers.
  • Ask: What is 45 in expanded form? (4 tens + 5 ones or 40 + 5)
  • Say: Let's find 40 x 123 and 5 x 123.

    Write 40 x 123 and 5 x 123 on the board. Have volunteers go to the board and complete those exercises.

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  • Say: Now look at the original exercise. Where would we write those products in that exercise?
    Students should realize that 615 is the first partial product, and 4,920 is the second. Record the partial products.
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  • Ask: How did we use the Distributive Property?
    Lead students to understand that 123 was distributed when we found 40 x 123 and 5 x 123.
  • Ask: Two operations are used in the Distributive Property. What operation is needed to find the final product? (addition)

    Draw a line under the two products. Ask a volunteer to find the sum.

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  • Ask: Where else can we write 5,535?
    Allow students to discover another place for the sum.

    Write a plus sign between the products of 40 x 123 and 5 x 123. Write 5,535 after the products.

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  • Ask: What is the sum of the products 40 x 123 and 5 x 123? (5,535) What is the product of 45 x 123? (5,535)
  • Say: Let's multiply with a different factor.

    Write 45 x 103 in vertical form on the board.

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  • Ask: What is 45 in expanded form? (40 + 5)

    Write 40 x 103 and 5 x 103 on the board. Have volunteers explain how to find the partial products.

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  • Ask: Notice that 103 is multiplied by 40 and 103 is also multiplied by 5. What property is used? (Distributive Property)
  • Ask: How can we use the two products to find 45 x 103?
    Students should be able to identify the correct locations for the partial products 515 and 4,120. Record their answers in the original exercise.
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  • Ask: How can we find the final product? (Add 515 and 4,120.) Allow a volunteer to complete the exercise.
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    Write 4,635 as the sum of the products for 40 x 103 and 5 x 103.

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  • Say: We used the Distributive Property to find 45 x 103 by first finding 5 x 103 and then finding 40 x 103. Now let's use the property to find 45 x 120.

    Write 45 x 120 in vertical form on the board.

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  • Ask: How can we use the Distributive Property to find the product?
    Students should be able to identify 40 x 120 and 5 x 120.

    Write 40 x 120 and 5 x 120 on the board. Invite volunteers to orally explain how to find the products. Review multiplying by a multiple of 10 and multiplying by two multiples of 10.

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  • Ask: How will the products 4,800 and 600 help us find 45 x 120? (They are the partial products for 45 x 120.)
  • Ask: How can we find the final product? Invite a volunteer to complete the exercise.
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Write a plus sign, an equals sign, and 5,400 to show the connection between the partial products and the final product.

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On the board, write several multiplication exercises that have three-digit factors and two-digit factors. Invite volunteers to find the products. Have them explain how the Distributive Property was used.

Wrap-Up and Assessment Hints
Students require a lot of practice in multiplying three-digit numbers by two-digit numbers. Offer grid paper or use lined paper turned sideways to keep place values aligned correctly. Frequently revisit the Commutative, Associative, and Distributive Properties of Multiplication in order to reinforce their use in paper-and-pencil multiplication and in mental math. Display the properties, their names, and numerical examples around the room to aid students in their work.


Houghton Mifflin Math Grade 4