## Multiplication Concepts

Multiplication can be defined in terms of repeated addition. For example, 3 × 6 can be viewed as 6 + 6 + 6. More generally, for any positive integer n, n × b can be represented as n × b = b + b + … + b
where the sum on the right consists of n addends.

A rectangular array provides a visual model for multiplication. For example, the product 3 × 6 can be represented as

By displaying 18 dots as 3 rows with 6 dots in each row, this array provides a visual representation of 3 × 6 as 6 + 6 + 6.

An equivalent area model can be made in which the dots of the array are replaced by unit squares.

Besides representing 3 × 6 as an array of 18 unit squares, this model also shows that the area of a rectangle with a height of 3 units and a base of 6 units is 3 × 6 square units, or 18 square units.

Multiplication is a binary operation that operates on a pair of numbers to produce another number. Given a pair of numbers a and b called factors, multiplication assigns them a value a × b = c, called their product.

Multiplication has certain fundamental properties that are of great importance in arithmetic. The Commutative Property of Multiplication states that changing the order in which two numbers are multiplied does not change the product. That is, for all numbers a and b, a × b = b × a.

The array model can be used to make this plausible. For example, because 3 × 6 = 6 × 3, an array with 3 rows and 6 dots in each row has the same number of dots as an array with 6 rows and 3 dots in each row.

Another important property of multiplication is the Identity Property of Multiplication. It states that the product of any number and 1 is that number. That is, for all numbers a, a × 1 = 1 × a = a.

The Zero Property of Multiplication states that when a number is multiplied by zero, the product is zero. That is, for all numbers a, a × 0 = 0 × a = 0.

Teaching Model 8.1: Model Multiplication as Repeated Addition