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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.
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.
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.
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)
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