Had a great email conversation with a colleague today. She was asking about word problems that required multiplying a negative number by a negative number. I spent some time thinking about it (I’ve never introduced those ideas to students, so I’ve never had to dive too deep into their complexities).

My short answer was: no, I don’t have any great ideas for authentic word problems that involve multiplying two negatives.

My longer answer was this:

Multiplying two negatives, along with subtracting a negative, will always seem a bit forced when working with arithmetic. There’s almost always another way to phrase the question that both makes more sense and doesn’t require puzzling through negatives.

The classic example of losing debt is one that makes a bit of sense for subtracting negatives; I owe 15 dollars. Then I lose a debt of 5 dollars. How much do I owe? But that still seems forced to me. Why not just treat the debt as positive? There’s no reason not to, if you understand what debt means. Our society avoids negative numbers like the plague. We don’t have negative years; we have BC. We don’t have negative altitude, we have altitude below sea level. These situations are almost always forced.

For multiplying two negatives in arithmetic word problems, I have fewer good answers. I can’t think of anything that’s within the bounds of reality.

But these situations do come up — they come up in algebra. Example: A car is 500 feet away, traveling toward me at 25 feet per second. Great. It’s distance is 500 – 25x. That seems intuitive. We’re tracking distance away from me. It’s getting closer. Subtraction. But where was the car three seconds ago? Well now we have two negative numbers. The answer has to be positive (the car has to have been farther away than 500 feet 3 seconds ago). Must be 575.

I’m not arguing that this should be the only way multiplying two negatives is introduced. The logic of inverses builds on the structure of the rational numbers; considering the patterns in positives and negatives also creates a logical need for two negatives to multiply to a positive. My point is just this: I don’t think this double negative thing happens in natural arithmetic. I like that it falls in 7th grade in the Common Core — just as students are considering expressions and equations as a tool for looking at the world. But it seems like a lot to push some of the toughest parts of rational number operations off until expressions & equations. Are these ideas useful for introducing multiplying negatives, or should it still come first, and then be applied to expressions later? Not sure.

howardat58Draw a rectangle, sides 8, top and bottom 10, take a point on each of the sides 5 up, and points on the top and on the bottom 7 from the left side. Then the area of the little rectangle at top right is (8-5)x((10-7). Clearly this is equal to 9.

Now expand the brackets(UK speak) :

(8-5)x((10-7) = 8x(10-7) – 5x(10-7) = 80 – 56 – 50 – 5x(-7)

and the only way you are going to get 9 out of this is if (-5)x(-7)=35

In the positive/negative number system even multiplication of anything by a negative number is not particularly meaningful. Actions such as this (and dividing by a negative fraction) usually appear in some calculation. The interpretation does NOT have to have a real world meaning, it only has to preserve consistency on the formal side. After all, negative numbers do not exist. 250 years ago mathematicians were arguing about there being ANY useful interpretation.

Have a look at my post on this stuff:

http://howardat58.wordpress.com/2014/07/13/1-x-1-1-but-some-need-convincing/