Why is a negative multiplied by a negative positive?
There are numerous ways one can attempt to answer this question but a simple practical application sometimes makes the most sense.
So here is one that you can use with your students to help them answer that question.
To begin with give the student a handful of green and red unit blocks. Tell them that for each green block in their hand you owe them one candy and for each red block they owe you one candy. Have the student count their green and red blocks to see how many candies they will receive, give away, and keep.
Here’s an example of the full application.
I gave a student a handful of blocks that contained 13 green blocks and 9 red blocks.
He counted the green and red blocks in his hand and figured out that he would receive 13 candies and give away 9 candies leaving him with a total of 4 candies to keep.
I then told him that I had given him too many red blocks and was going to take away one red block one time (-1)(-1).
He recounted his blocks to see how many candies he would now get to keep. He quickly realized that taking away one red block one time meant a gain of +1 candies giving him a total of 5 candies to keep.
I then told him he still had too many red blocks and I was going to take away two red blocks three times (-3)(-2).
He again recounted his blocks to see how many candies he would now get to keep. He quickly realized that taking away two red blocks three times meant a gain of +6 candies giving him a total of 11 candies to keep.
Being the kid that he was he took all eleven of his candies home to share with his family
Another time we used pennies instead of candies. He now understood that whenever he multiplied a negative by a negative he was taking away some of what was owed which meant a gain of the same amount. This gave meaning to the concept which meant he no longer had to memorize rules when multiplying negatives by negatives.
Note: the green and red unit blocks we used are from the Math U See lot.