## Multiplication and Division

Multiplication is one of the four basic operations that form the foundation of arithmetic and is an essential part of the computation work in the elementary school grades. Multiplication can be thought of as repeated addition. For example, 4 × 2 can be thought of as 4 groups of 2.

2 + 2 + 2 + 2

The 4 × 2 can also be thought of as 2 × 4 because multiplication is commutative; that is, you can multiply in any order and the answer, or product, will be the same.

Using variables, multiplying n × b can be represented as

 n × b = b + b + … + b n addends

The general form of a multiplication sentence using variables is a × b = c. It is read “a times b equals c.” The numbers being multiplied (a and b) are factors and the result of the multiplication is the product. Besides being commutative, multiplication is also associative (changing the grouping of factors does not change the product).

The identity element for multiplication is 1. An identity element is a number that combines with other numbers, in any order, without changing the original number. With variables, a × 1 = a and 1 × a = a.

Multiplication can also be represented using arrays, the number line, or by an area model.

The arrangement of objects in equal rows is called an array. For example, the array below represents the product 6 × 3 because it consists of 6 rows of 3 in each row.

6 × 3 = 18

Multiplication can also be represented by the area of a rectangle. For example, the rectangle below represents the product of 6 × 3.

A number line provides a graphic representation of counting by multiples to model multiplication. For example, consider the product 3 × 5. This can be viewed as skip counting by 5 three times (or, since multiplication is commutative, as skip counting by 3 five times).

The Concept of Division
Division is the inverse, or the opposite, of multiplication. If you multiply a number by some number and then divide by the same number you multiplied by, you end up with the original number. Because division is the inverse of multiplication, the models used to illustrate division are the inverse of those used to model multiplication. Division can be understood as repeated subtraction, separating a set into a number of equal groups.

For example, 15 ÷ 3 can be thought of as repeatedly subtracting 3 until the difference is 0. The number of times you subtract 3 is the answer, or quotient, as shown below.

 15 − 3 12 − 3   9 − 3   6 − 3   3 − 3    0

Therefore, 15 ÷ 3 = 5.

15 ÷ 3 can also be viewed as 15 separated into 3 equal groups

15 ÷ 3 = 5

or 15 separated into groups with 3 objects in each group.

15 ÷ 3 = 5

Since children can use skip counting to model multiplication and multiplication and division are inverse operations, they can skip count back on a number line to find a quotient. For example, to find 15 × 3, a child could start at 15 and skip count backward by three 5 times in order to reach 0.

15 ÷ 3 = 5

When division is formally thought of as the inverse of multiplication, it leads to the missing factor model of division, which involves using a related multiplication fact to find the answer.

For example, to find the answer to 20 ÷ 4, think 4 × ? = 20.

The general form of a division sentence using variables is a ÷ b = c where b > 0. It is read “a divided by b equals c.” The dividend is a, the divisor is b, and c is the quotient.

The equation a ÷ b = c can be rewritten as b × c = a. Substituting 6 for a and 0 for b, leads to the following equations.

6 ÷ 0 = c
0 × c = 6

Since any number times zero is zero, no number times zero can be 6. Therefore division by zero is undefined. It is also important to note that division is not commutative, not associative, and does not have an identity element.

Teaching Model 19.2: Multiply With 2 and 5