Math 8 – Today the kids practiced adding and subtracting integers by working on suduko type puzzles together. Next we took at look at multiplying integers. We reasoned that a negative multiplied by a positive must be a negative, by modeling expressions like 4 x (-2), which corresponds to 4 groups of negative two. Things get tricky when you consider (-3) x 4, because exactly what does it mean to have negative three groups?
There are ways to model negative groups but I find them completely unsatisfactory. I believe that models in science and math are meant to represent a feature of the world such that it is easier to understand. Some of the modeling in mathematics clearly does not do this. Adding a bunch of matched pairs of positive and negatives such that we can subtract a negative where one didn’t used to exist is the opposite of easy. And certainly very few people would come up with this model or use.
Instead we used the communitive property of multiplication to show that (-3) x 4 = 4 x (-3) and then easily modeled the latter. At this point I asked the classes to come up with a rule for when a negative is multiplied by a positive. With examples on the boards and having discussed it for a bit, I was surprised that many kids couldn’t pick up the pattern – that the product is always negative. I think we were dealing with mental fatigue at this point.
We also took at the pattern generated from a table: