Math Skills - Kids Math

Tweet

A common question when teaching students how to multiply positive and negative numbers is “Why is a Negative times a Negative a Positive?”

It is easy to simply say because that is the way it works; however, it is much better to teach students why this happens as opposed to rote memorization.

The following provides examples of explanations for students.

Why is a negative times a negative positive?

A real-world approach modeling the product of two negatives requires a situation involving two quantities, both of which have two directions. Consider delivering mail by a postal deliverer who makes many mistakes. We can receive (+) a check (+) or a bill (-), and if we receive a check or a bill by mistake, then we may have to return (-) a check (+) or a bill (-).

If we receive (+) checks (+), then we have more (+) money.

If we receive (+) bills (-), then we have less (-) money.

If we were erroneously sent checks then we have to return (-) checks (+), and we have less (-) money.

If we were erroneously sent bills, then we have to return (-) bills (-), and we have more (+)money.

A Pattern Approach

Consider the following table showing the products of a positive number multiplied by a second number that is decreasing in value:

3 x 3 = 9

3 x 2 = 6

3 x 1 = 3

Notice that as the second factor decreases by one, the product decreases by three. It is moving down the number line in steps of three. Continuing this pattern gives us:

3 x 0 = 0

3 x -1 = -3

3 x -2 = -6

3 x -3 = -9

Because your brain detects patterns so well, this is a very reasonable way to see that when positive and negative numbers are multiplied, a negative number results.

Now lets take the last multiplication fact and begin decreasing the lead factor by one.

3 x -3 = -9

2 x -3 = -6

1 x -3 = -3

Now the product is advancing up the number line in increments of three. The product is increasing. If we continue this pattern the following table results:

0 x -3 = 0

-1 x -3 = 3

-2 x -3 = 6

-3 x -3 = 9

Since our brain trusts patterns, the student is confident in the resulting conclusion that a negative multiplied by a negative yields a positive product.

A Number Sense Approach

Let’s consider the problem 2 x 3 which we know to be equal to 6. One way to think about negative numbers is as inverses of positive numbers. In fact, every non-zero number has an inverse, and the sum of a number and it’s inverse is zero. Thus 5 + -5 = 0.

That means that we could read the following problem as, “What is the inverse of the product of two and three?”

-2 x 3 = ?

Since 2 x 3 is six, then the inverse of this must be -6. Thus:

-2 x 3 = -6

Now we have demonstrated by example that when a negative is multiplied by a positive, the product is negative. Since the Commutative Property applies to multiplication, we can rewrite the problem this way:

3 x -2 = -6

Next let’s ask ourselves, “What is the inverse of 3 x -2 = -6?” Clearly it would be:

-3 x -2 = ?

Since we know that the answer to the previous problem was -6, the answer to the inverse problem should be the inverse of -6, which is +6. Thus we have a demonstration that a negative multiplied by a negative is positive.

Source

Adapted from “Answering Your Students Why Questions in Mathematics” by Teacher to Teacher Press