Math Background

Lesson: Multiplication by Two-Digit Numbers
Introducing the Concept

Your students will have multiplied by two-digit numbers before, but they may not have related the process to the Distributive Property. Knowing the connection between the Distributive Property and multiplying by two-digit numbers will benefit your students, especially when solving real-life problems. Of course, students will need to know the basic multiplication facts. Have your students use a multiplication fact chart as necessary.

Materials: a large model of a multiplication fact table, sheets of prepared large 11 x 11 grid paper for students to make their own fact charts (see below), and a large 7 x 26 rectangular grid

multiplication fact table and 11 by 11 grid paper
7 by 26 rectangular grid

Preparation: Construct a large multiplication fact chart and post it where the students can see it. Prepare sheets of 11 x 11 large grid paper, like the one shown above, for students to make their own fact tables. Construct a large 7 x 26 rectangular grid.

Prerequisite Skills and Concepts: Your students should be familiar with the basic multiplication facts and the properties of multiplication, and they should be able to tell you about the Distributive Property of Multiplication. They should also have had some experience in determining the factors and multiples of a given number. They may find it helpful to review multiplication by one-digit numbers.

Post the large 7 x 26 rectangular grid for your students to see.

7 by 26 rectangular grid
  • Ask: Does anyone know what multiplication problem this model shows?
    Someone will probably respond, “7 x 26.” If not, write the number 26 above the grid and 7 on the side of the grid, and ask the question again. On the board, write 7 x 26 in vertical form.

    Cut your large 7 x 26 rectangular grid into two grids, one 7 x 20 and the other 7 x 6.

    7 by 26 rectangular grid and 7 by 6 rectangular grid
  • Ask: Does anyone know what multiplication problems this model shows now?
    Someone will probably respond, “7 x 20 and 7 x 6.” If not, write the numbers 20 and 6 above the grid and 7 on the side of the grid, and ask the question again. On the board, write 7 x 26 = 7 x (20 + 6) = (7 x 20) + (7 x 6).
  • Ask: What property of multiplication says that 7 x 26 = 7 x (20 + 6) = (7 x 20) + (7 x 6)?
    Someone will probably respond, “the Distributive Property.” Write “Distributive Property” on the board.
  • Ask: Can anyone show us how to use the Distributive Property of Multiplication to solve 7 x 26?
    Direct your students in solving 7 x 26: 7 x 26 = 7 x (20 + 6) = (7 x 20) + (7 x 6) = 140 + 42 = 182. Write this on the board for your students. Explain how the Distributive Property distributes the factor 7 to each part of the expanded form of 26.
  • Ask: (pointing to the 20 + 6) Why did I break the model into 20 + 6 instead of 18 + 8, or 24 + 2?
    Students should be able to tell you that the expanded form for 26 is 20 + 6 and it makes the multiplication easier since it involves basic multiplication facts. Ask your students to tell you which is easier to solve—7 x 20, 7 x 18, or 7 x 24—and why. Help students see that 18 + 8 or 24 + 2 are not good choices because they do not permit the use of basic multiplication facts.

    Write 7 x 326 on the board.

  • Say: Tell me how I can use the Distributive Property to solve 7 x 326.
    Students should reply that 326 can be rewritten in expanded form (300 + 20 + 6). Write 7 x 326 = 7 x (300 + 20 + 6) on the board.
  • Ask: Can anyone show us how to use the Distributive Property of Multiplication to solve 326 x 7?
    Direct your students to solve 7 x 326 in this manner: 7 x 326 = 7 x (300 + 20 + 6) = (7 x 300) + (7 x 20) + (7 x 6) = 2,100 + 140 + 42 = 2,282. Write this on the board. Explain how the Distributive Property distributes the factor 7 to each part of the expanded form of 326.
  • Say: Now that you have a good idea of how to rewrite a multiplication problem into a form that allows you to use the basic multiplication facts, each of you will make a basic multiplication fact chart by using the form I give you.
    Ask students to write the correct answer to each multiplication fact on the chart neatly and correctly.

Houghton Mifflin Math Grade 5