Directed numbers.
Students find calculating consistently with negative numbers hard to get used to. The ‘rules’ for multiplying and dividing allow you to state the sign of the answer from the signs of the numbers in the calculation. When adding and subtracting, the sign of the answer depends on the extent to which the total of the negative numbers is larger or smaller than the total of the positive numbers.
I tell ‘stories’ about bank accounts (cancelling a debit covers—4 and a spending spree provides a model of -150 – 120 -400). We draw number lines. I find that students find -15 / -5 easier to visualise than the multiplication. I stress common principles like – tends to switch the sign following. I usually manage to convince students that there isn’t in fact any such thing as subtraction; you are just adding negative quantities. Then things like 6 +- 5 and 6 -+5 become easier to deal with.
A few months later comes algebra basics, and we are ‘simplifying’ expressions like
4(3x + 2) – 5(2x – 8 )These questions put the ‘multiplication’ and ‘addition’ rule sets for directed numbers in close proximity and need the student to switch frame. They manage it but it takes a few examples and some one to one work. I have found it useful to explicitly mention the rule shift between multiplying out the terms and collecting them together. The result (as one might expect) is a better model of negative numbers.
The more I look at the wrong answers in homework and tests, the more I think this idea of ‘rule switch’ might help. I’m beginning to ask students to write the rules they use for each stage.