An arrangement of objects, pictures, or numbers in columns and rows is called an array. In this article, you will learn how to use arrays to show the relationship between multiplication and division.

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Key standard: Use multiplication and division within 100 to solve word problems in situations involving arrays. (3.OA.A.3)

Students in Grades 3 and up 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?


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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 students choose which model they find most helpful. Here is an example using counters to multiply and divide.


Teaching Multiplication Division With Arrays Graphics 05
Another strategy your students may find helpful is using a related multiplication fact to divide. Here is an example.

18 ÷ 6 = ?

Think: 6 × ? = 18Six times what number is 18?

6 × 3 = 18,so 18 ÷ 6 = 3.

Dividing with 0 and 1

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.

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


Students may be curious what happens if they divide by 0. Explain that it is not an easy concept, and even professional mathematicians struggle to explain it! One strategy to show why it is not possible is to have students try to divide any number into groups of zero. No matter how many groups you make, it doesn"t work.


Division in the Real World

Encourage students to think about the relationship between multiplication and division when they solve real-world 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 = ?Think: 3 × ? = $18

3 × $6 = $18,so $18 ÷ 3 = $6.

The cost is $6 for one baseball cap.

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