Math Background

Distributive Property of Multiplication
Introducing the Concept

Introduce students to the Distributive Property by showing them how to use the property for mental math in multiplying a one-digit number by a two-digit number. Then advance to multiplication of 2 two-digit numbers.

Materials: none

Preparation: none

Prerequisite Skills and Background: Students should know the basic addition and multiplication facts, how to write a number in expanded form, and how to use mental math to multiply with multiples of 10.

  • Ask: Yesterday I bought 5 CDs at the music store. Each one cost $8. Today I bought 7 CDs at $8 each. How can I found the total cost of all CDs?
    Elicit two ways of solving the problem. Students should realize that they could find the cost of the CDs yesterday, then find the cost of the CDs today, and then add the products. Or they can find the total number of CDs first, then multiply by the cost of one CD.
  • Say: Let's write each method mathematically. We can show the cost of yesterday's CDs by finding the product of 5 and $8. We can show the cost of today's CDs by finding the product of 7 and $8.

    Write $5 x 8 and $7 x 8 on the board.

  • Ask: What is the product of $5 and 8? ($40) What is the product of $7 and 8? ($56)

    Write $40 below $5 x 8 and write $56 below $7 x 8.

  • Ask: How do we find the total cost? (Add the products.)

    Write a plus sign between the products on the board: $5 x 8 + $7 x 8 and $40 + $56.

  • Ask: What is the total cost of the CDs? ($96)

    Write $96 below the addends.

  • Say: Notice that the factor 8 is repeated. We can use the Distributive Property to find the answer in another way. Our second method of solving the problem involves finding the total number of CDs first, then multiplying by the cost of one CD.
  • Ask: How can we find the total number of CDs? (Add 5 and 7.)

    Write (5 + 7) on the board.

  • Ask: How can we find the total cost of the CDs? (Multiply by 8.)\

    Write 8 x in front of (5 + 7) on the board.

  • Ask: How can we find the total cost? (Add 5 and 7, then multiply by 8.) What is the total cost? (The total cost is $96.)

    Write 8 x 12 under 8 x (5 + 7). Write $96 below the factors.

  • Ask: What do you notice about both answers? (They are the same.)
  • Say: Both answers are the same because of the Distributive Property. The $8 had been distributed to yesterday's CDs and to today's CDs. Let's try another problem. Suppose I buy 5 CDs tomorrow and 7 CDs today, but this time they cost $18 each. How can we find the total cost?
    Elicit two ways to solve the problem. Multiply the total number of CDs by 18 or find separate products and then add.
  • Say: Let's find 18 x 12.

    Write 18 x 12 in vertical form on the board.

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  • Ask: How is this problem similar to the last problem? (There we multiplied 8 x 12, now we are multiplying 18 x 12.)
  • Ask: How can we use what we already know to solve the problem? Students should know 8 x 12 = 96, so use 96 as part of the product.
  • Say: We can use 96 as part of our answer by writing it below the line. That means we multiplied 8 x 12.
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  • Ask: What else needs to be multiplied?
    Guide students to realize that 18 is 10 + 8, so 10 x 12 must be done next.
  • Say: The expanded form of 18 is 10 plus 8. We already found 8 times 12. Now we need to find 10 times 12.
  • Ask: What is 10 times 12? (120)

    Write 120 in the appropriate place on the board. Show that it is 10 x 12.

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  • Ask: How can we find the final product?
    Guide students in realizing that adding 96 and 120 will result in the final product because it represents (8 x 12) + (10 x 12).

    Record the product on the board.

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  • Ask: What is the total cost of 12 CDs at $18 each? ($216)
  • Ask: How did the Distributive Property help us do this problem?
    Lead students to understand that they used the Distributive Property to rename 18 as the sum of 10 and 8, and then to multiply each addend by 12.

    Continue with additional examples that emphasize the connection between multiplication and the Distributive Property.


Houghton Mifflin Math Grade 4