Teaching Students to Multiply and Divide Positive and Negative Numbers

The set of whole numbers and their opposites are 'integers' (...-3, -2, -1, 0, 1, 2, 3...). They are also called positive and negative numbers or signed numbers. Students generally learn to add, subtract, multiply and divide integers beginning in fifth or sixth grade. It's vitally important for a student to develop a strong base in integers before embarking on an algebra class. Schools are nudging boys and girls into algebra classes sooner and sooner-no more waiting until ninth grade. A student will experience much more success if he is confident in the integer operations.

When I teach students how to multiply integers, I always begin by telling them how pleased they are going to be. They've already mastered adding and subtracting integers, and multiplying positive and negative numbers is much easier. A sigh of relief generally ensues.

As always, be sure to provide concrete examples at the onset of any math lesson. Experiment with algebra tiles and number lines in regard to the multiplication of integers. The students should be familiar with "product" as the name of the answer in multiplication.

Understand the Look
At this age, students generally believe that multiplication problems always contain the 'x' sign. They need to realize that multiplication problems take on different 'looks'.

"a times b" can be written in several ways:
a x b
a * b
a(b)
(a)b
ab
a • b

Make Sense of It
Positive x Positive
I always tell the students that they've been multiplying positive integers since third grade-it's just that the positive signs have been omitted. Emphasize that positive numbers don't require + signs. Just be sure they realize that ⁺2 x ⁺3 is the same as 2 x 3. Never take anything for granted!
(+) x (+) = +

Positive x Negative
It's logical that 3 x -2 = -6 because if a team loses 2 yards on each of 3 football plays, they'll lose a total of 6 yards.
Show students a pattern for multiplying a positive number times a negative number.
3 x 2 = 6
3 x 1 = 3
3 x 0 = 0
3 x -1 = -3
3 x -2 = -6
(+) x (-) = -

Negative x Negative
This pattern will reinforce to students that when two negative factors are multiplied, the product is always positive.
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
-1 x -3 = 3
-2 x -3 = 6
(-) x (-) = +

Everyone Loves a Shortcut
First multiply the absolute values of the numbers.¹
Then decide what the answer's sign should be.
Positive x Positive or Negative x Negative = Positive Answer
Positive x Negative or Negative x Positive = Negative Answer

In other words...
If the signs are the same, the product is positive.
If the signs are different, the product is negative.
It's really that simple.

Rules for dividing integers are the same as multiplying.
If the signs are the same, the quotient is positive.
If the signs are different, the quotient is negative.

Click here for online practice.

¹ absolute value is always positive

See also
An Interactive Approach to Adding Positive and Negative Numbers
Teaching Math: Subtracting Integers