Relating      Multiplication      and Division   Introducing the Concept   Developing the Concept

## Relating Multiplication and Division

An arrangement of objects, pictures, or numbers in columns and rows is called an array. Your students used arrays in Chapters 5 and 6 to multiply. See Using Arrays to Show Multiplication Concepts. In this chapter your students will learn how to use arrays to show the relationship between multiplication and division.

Students will learn that division can be thought of in two ways, partitioning and measurement. Although at this level students may not use these names, you can convey the meaning of both kinds of division so that they can have a better understanding of the division process. When you divide to find the number of objects in each group, the division is called fair sharing or partitioning. For example:

A farmer is filling baskets of apples. The farmer has 24 apples and 4 baskets. If she divides them equally, how many apples will she put in each basket?

When you divide to find the number of groups, the division is called measuring or repeated subtraction. It is easy to see that you can keep subtracting 4 from 24 until you reach zero. Each 4 you subtract is a group or basket.

A farmer has 24 apples. She wants to sell them at 4 apples for \$1.00. How many baskets of 4 can she fill?

Manipulatives and visual aids are important when teaching multiplication and division. Students have used arrays to illustrate the multiplication process. Arrays can also illustrate division.

Since division is the inverse, or opposite, of multiplication, you can use arrays to help students understand how multiplication and division are related. If in multiplication we find the product of two factors, in division we find the missing factor if the other factor and the product are known.

In the multiplication model below, you multiply to find the number of counters in all. In the division model you divide to find the number of counters in each group. The same three numbers are used. The model shows that division "undoes" multiplication and multiplication "undoes" division. So when multiplying or dividing, students can use a fact from the inverse operation. For example, since you know that 4 5 = 20, you also know the related division fact 20 ÷ 4 = 5 or 20 ÷ 5 = 4. Students can also check their work by using the inverse operation.

Notice that the numbers in multiplication and division sentences have special names. In multiplication the numbers you multiply are called factors; the answer is called the product. In division the number being divided is the dividend, the number that divides it is the divisor, and the answer is the quotient. Discuss the relationship of these numbers as you explain how multiplication and division are related.

There are other models your students can use to explore the relationship between multiplication and division. Expose your students to the different models and let each student choose which model is most helpful to him or her. Here is an example using counters to multiply and divide.

 factor4number ofgroups factor3counters ineach group = product12total number ofcounters dividend12total numberof counters ÷ divisor4number ofgroups = quotient3counters ineach group

Here is an example using a number line.
 factor4 factor5 = product20 dividend20 ÷ divisor5 = quotient4

Another strategy your students may find helpful is using a related multiplication fact to divide. The lesson Relating Multiplication and Division focuses on this strategy. Here is an example.

18 ÷ 6 = ?

Think: 6 ? = 18 Six times what number is 18?
6 3 = 18,
so 18 ÷ 6 = 3.

When students understand the concept of division, they can proceed to explore the rules for dividing with 0 and 1. Lead students to discover the rules themselves by having them use counters to model the division. A few examples follow.

 Divide 4 counters into 4 groups. 4 ÷ 4 = 1 Divide 2 counters into 2 groups. 2 ÷ 2 = 1

When any number except 0 is divided by itself, the quotient is 1.

 Put 3 counters in 1 group. Put 5 counters in 1 group. 3 ÷ 1 = 3 5 ÷ 1 = 5

When any number is divided by 1, the quotient is that number.

 Divide 0 counters into 2 groups. 0 ÷ 2 = 0 Divide 0 counters into 4 groups. 0 ÷ 4 = 0

When 0 is divided by any number except 0, the quotient is 0.

 Divide 6 counters into 0 groups. Divide 1 counter into 0 groups. You cannot divide a number by 0.

Encourage students to think about the relationship between multiplication and division when they solve problems. For example, they can use a related multiplication fact to find the unit cost of an item—for example the cost of one baseball cap priced at 3 for \$18.

\$18 ÷ 3 = \$6
Think: 3 ? = \$18
3 \$6 = \$18
So \$18 ÷ 3 = ?

The cost is \$6 for one baseball cap.