## Division Facts and Patterns

Multiplication of whole numbers can be defined as repeated addition. The statement 5 × 3 = 15 is based on the fact that 3 + 3 + 3 + 3 + 3 = 15. The multiplication fact can be represented on the number line in terms of skip counting from 0 to 15 in five steps of 3 each.

Because division is the inverse operation of multiplication, division facts can be represented in terms of repeated subtraction. The division fact 15 ÷ 3 = 5 corresponds to repeated subtractions of 3 from 15. That it takes five such subtractions to reduce 15 to 0 can be represented as follows.

While skip counting provides a useful representation of multiplication and division, it is important for students to be able to arrive at 5 × 3 = 15 and 15 ÷ 3 = 5 without making use of such strategies. In the case of multiplication, students are expected to memorize the 1-digit multiplication facts contained in a 10 × 10 multiplication table.

Having committed such facts to memory, students should now learn to use the multiplication table in the other direction. To find 15 ÷ 3, students find the row representing the divisor, 3. They move across the row to find the column that shows the dividend, 15. Then they follow the column to the top to find the quotient, 5. So 15 ÷ 3 = 5.

One way of helping students understand the relationship between multiplication and division is to ask them to write down the fact families for the entries in the multiplication table. For the product 42, this calls for listing four equivalent statements.

6 × 7 = 42
7 × 6 = 42
42 ÷ 6 = 7
42 ÷ 7 = 6

Given this fact family, the problem 42 ÷ 7 = ? can be rewritten as 7 × ? = 42. In this way a mastery of the multiplication facts becomes a tool for solving division problems as well. If a, b, and c are nonzero numbers satisfying a × b = c, the corresponding four-member fact family is

a × b = c
b × a = c
c ÷ a = b
c ÷ b = a

If a or b is allowed to be 0, the resulting fact family consists of only three equations. For example, 5 × 0 = 0 has the fact family

5 × 0 = 0
0 × 5 = 0
0 ÷ 5 = 0

Teaching Model 11.1: Divide Using a Multiplication Table