Taryn has shared a great question:

*So I am teaching Algebra II and will be doing a review session on multiplication and division and solving equations using multiplication and division. As I was preparing through my lesson a couple questions came up. I loved how we represented most of our problems graphically in the online course using manipulatives. I am stuck trying to figure out how to represent -4 x-4= 16. I realized that I truly do not understand why two negatives equal a positive even
*

*though I’ve always been told that it was the rule. I have the same problem demonstrating how to divide 15/-3=-5. I was wondering if you could help me understand the underlying mathematics here.*

*Thanks!
Taryn*

Hi Taryn -

Thanks for the great question. I hope others will weigh in and share their thoughts and experiences, but for starters, Dr. Math from the Math Forum has a couple nice explanations:

Algebraic Proof That Positive Times Negative Equals Negative: http://mathforum.org/library/drmath/view/65129.html

Prove That -(-a) = a: http://mathforum.org/library/drmath/view/55712.html

Why is a negative times a negative a positive?: http://mathforum.org/dr.math/faq/faq.negxneg.html

Do any of those help or appeal to you?

Best,

Valerie

Taryn,

I liked the link about debt and often use that with my students.

Now, I have to be honest and say, sometimes making meaning does not support the students as much as I would like it to and this may be one of those places. So, there are times when I try mathematical reasons instead.

Here’s the language I often use when I am talking about lots of negatives. I begin to speak as though they are functions, “the function that ‘negates’ or ‘opposites’. I know that’s not really a verb, but I often use it. Symbolically that looks like f(x) = -x.

So for (-3)(-4) I might say “3 times 4 is 12, then opposite that and get -12, then opposite that and get 12.

OR I might say, think of it as -(3)(-4). That’s -4, three times, (-4, -8, -12–just counting up -4 three times), then opposite that or opposite -12 and get 12.

Let us know if any of this stuff works for you….

Or, ask more questions.

Ellen

Taryn,

I had another thought about teaching (-3)(-4).

If you use the language 3 tens * 4 tens is 12 hundreds “because 3 * 4 is 12 and tens * tens is 10^2 or a hundred.”

And if you use the language that 3x * 4x is 12 x^2 ” because 3 * 4 is 12 and x’s * x’s give x squares.”

Then how about “3 negatives * 4 negatives is 12 (negative squares)” Then you can use the language of a negative squared meaning ‘opposite and opposite again and you get your original number?”

If you had to explain this to kids, it might help to compare it to English language rules. A double negative in the English language would also result in a positive. For example, if you “Don’t not drive,” then it would imply that you do, in fact, drive.