Teaching Models

Division Concepts

Division is the inverse of multiplication. Given that b is not equal to zero(b ≠ 0), the statement

a ÷ b = c

is true if and only if a = b × c. This relationship provides a basis for checking the correctness of answers to division problems. The fact that 56 ÷ 7 = 8 can be confirmed by the observation that 56 = 7 × 8. The statement a ÷ b = c is read a divided by b equals c, where a is called the dividend, b the divisor, and c the quotient.

At this grade level, it is important to help students create mental models of division. The fact that the multiplication problem 3 × 5 = 15 can be represented by a 3 × 5 array of objects provides a basis for using arrays to represent the division problems 15 ÷ 3 = 5 and 15 ÷ 5 = 3.

15 divided by 3 15 divided by 3
15 ÷ 3 = 5 15 ÷ 5 = 3

In the first case, a group of 15 objects is separated into three groups of 5, while in the second case a group of 15 objects is separated into 5 groups of 3. Arrays can help students use known multiplication facts to solve certain division problems. For example, to solve “20 ÷ 4 = ?,” the student should learn to think “4 × ? = 20.”

The Identity Property of Multiplication states that for any number n, 1 × n = n. This property carries over to division in the following form.

For any number n, n ÷ 1 = n.
For any number n ≠ 0, n ÷ n = 1.

Because multiplication and division are inverse operations, the Zero Property of Multiplication states that for any number b, 0 × b = 0. This implies that:

For any number b ≠ 0, 0 ÷ b = 0.

However, the condition “b ≠ 0” is essential in the above statement. Neither a ÷ 0 nor 0 ÷ 0 are defined, and recreational mathematics is filled with paradoxical “proofs” of statements such as “1 = 2” that are based on a disguised form of division by zero.


Teaching Model 10.2: Model Division as Repeated Subtraction


Houghton Mifflin Math Grade 3