How to Explain Why a Negative Times a Negative is a Positive

A good way to explain why a negative times a negative is a positive is by means of pattern recognition.

Start with a sequence of multiplications, gradually introducing negative numbers. 

    4 × 2 = 8
    4 × 1 = 4
    4 × 0 = 0
    4 × (−1) = − 4
    4 × (−2) = −8

Tell students to examine carefully the sequence of multiplication above for like 5 minutes. Then, tell students that each time the number we multiply 4 by is reduced by 1, subtract 4 from the product.

Make sure you point to the number 4 is multiplied by (The number is 2 in 4 × 2 = 8 )

Make sure you point to the product and show how it is 4 less each time. 

Then, do the same as above. However, this this time use -4 instead of 4

    -4 × 2 = -8
    -4 × 1 = -4
    -4 × 0 = 0
    -4 × (−1) = 4
    -4 × (−2) = 8

Tell students to examine carefully the sequence of multiplication above for like 5 minutes. Then, tell students that each time the number we multiply -4 by is reduced by 1, add 4 to the product.

In other words, point out that as the multiplier (2, 1, 0, -1 or -2) decreases, the product increases back into positive territory, showing that −4 × (−2) must equal 8.

Another way to explain why a negative times a negative is a positive is to use a number line

Start by using a number line to illustrate multiplication as repeated addition.

Then, show how multiplying a positive number by a negative number works.

For instance, 4 × (−3) can be seen as starting at 0 and moving left 3 units four times, ending at -12.

Ask students what will happen if we multiple -3 by -4? Will we move left or right?

Next, emphasize that multiplying -3 by a negative number means reversing direction. This means we have to move to the right this time.

Starting at 0 and moving right 3 units four times, we end at 12.

Use of double negatives

You could use analogy

Suppose happy means positive number

unhappy means negative number

Not unhappy is a double negative and it means happy. 

Just like in language, where saying "not unhappy" means "happy," multiplying two negatives cancels out the negativity, resulting in a positive.

Algebraic proof showing why a negative times a negative is a positive

Let x and y be any numbers.

0 = (−x) × 0

0 = (−x) × [y + (−y)]

Use the distributive property:

0 = (−x) × y + (−x) × (−y) 

Since (−x) × y = −xy, we have:

0 = −xy + (−x) × (−y) 

Add xy to both sides of the equation

0 + xy = −xy+ xy + (−x) × (−y)

xy = (−x) × (−y)