## Tuesday, November 1, 2016

### Why Do Signs Work This Way?

Most of the students in my afternoon Algebra I class are having trouble understanding why a negative times a negative is a positive visually.  These students are very ELL and struggle with mathematics every day.

I've been able to create illustrations for adding two negative numbers, adding one negative and one positive, multiplying a negative by a positive, and dividing a negative by a positive but I have not managed to create drawings for multiplying a negative times a negative or dividing a negative by a negative.

I can find all sorts of examples showing the usual but most of the information I find is with the if it works this way, it has to work that way but after a lot of searching I finally found an analogy explaining it.

" If you film a man running forwards (+) and then play the film forward (+) he is still running forward (+). If you play the film backward () he appears to be running backwards () so the result of multiplying a positive and a negative is negative. Same goes for if you film a man running backwards () and play it normally (+) he appears to be still running backwards (). Now, if you film a man running backwards () and play it backwards () he appears to be running forward (+). The level to which you speed up the rewind doesn't matter (3x or 4x) these results hold true.
backward×backward=forward
negative×negative=positive""

I got the above from the Stack Exchange.  It was really one of the only explanations I found that my students might be able to relate to.

Dr Math at the Math forum uses the idea of a mortgage payment to illustrate this particular operation.  If you pay $700 per month for your house payment each month you'll spend$8400 every year which is subtracted from the money you have in your bank account.  So a total of -$8400 or 12 times -$700 illustrating a positive times a negative is a negative.

But what if your employer decides to pay the 12 months for you instead so you are not paying the 12 months which is minus 12 months of -$700 or the payment so its -12 x -$700 or a positive \$8400 because you have that much more in your pocket at the end of the year.

So for subtracting a negative from a negative could possibly be viewed as you are going forward, someone calls your name so you turn to face backwards.  You don't see anyone so turn to face forward and run forward so its a positive.

I'd love to hear your thoughts on this topic?  Do you have other ways to show it other than using the usual mathematical methods.