Math Background

Lesson: Relating Multiplication and Division
Developing the Concept

In this lesson, students use arrays to explore the relationship between multiplication and division.

Materials: 24 counters for each student or pair of students

Preparation: Distribute counters to students. Write 3 x ? = 15 on the board and label the factors and product.

factor factor product
3 x ? = 15

Prerequisite Skills and Background: Students should know how to model multiplication by using arrays.

  • Say: We can use an array to find a missing factor in a multiplication sentence. Use 15 counters to make an array with 3 equal rows.
  • Ask: How many counters are in each row? (5) What is the missing factor in the multiplication sentence? (5)
    Replace the question mark in 3 x ? = 15 with 5.
  • Say: Finding a missing factor in multiplication is the same as dividing.
  • Ask: How many counters are in the array? (15)
    Write15 below the 3 in 3 x 5 = 15.
  • Ask: How many rows of counters are in the array? (3)
    Write ÷3 below the x 5 in 3 x 5 = 15.
  • Ask: How many counters are in each row? (5)
    Write5 below the 15 in 3 x 5 =15.
  • Say: Read the division sentence aloud.
    (Fifteen divided by three is equal to five.)
  • Say: The product in a multiplication sentence is the dividend in a division sentence. The factors are the divisor and quotient.
    Label the dividend, divisor, and quotient.
    dividend divisor quotient
    15 ÷ 3 = 5
  • Say: Let's find a missing factor in another multiplication sentence.
    Replace 3 x 5 = 15 with? x 9 = 18 on the board. Erase 15 ÷ 3 = 5.
  • Say: Use 18 counters to make an array with 9 counters in each row.
  • Ask: How many equal rows are in the array? (2) What is the missing factor in the multiplication sentence? (2)
    Replace the question mark with 2.
  • Ask: What division sentence describes the array?
    Have a volunteer write 18 ÷ 9 = 2 on the board.
  • Ask: Which number is the dividend? (18) the divisor? (9) the quotient? (2)
  • Ask: Can you use this array to show another multiplication sentence?
    Discuss how the array can be “turned” so that there are 9 rows of 2.
  • Ask: What multiplication sentence does this array model? What division sentence does it model?
    (2 x 9 = 18; 18 ÷ 2 = 9)

    Repeat the activity with other arrays until students understand the inverse relationship of multiplication and division.

Wrap-Up and Assessment Hints
Have students make an array with 24 counters and write two multiplication sentences and two division sentences that describe the array. Check their arrays and have them read their number sentences aloud.


Houghton Mifflin Math Grade 3