I've noticed that a lot of my tutoring students have been taught to cross multiply when dividing fractions, whereas I was taught to multiply by the reciprocal. OK, you may think, they both wind up getting you the right answer, so what? I guess it doesn't matter if your only goal is to pass the next quiz. What you are forgetting is the power of the commutative and associative properties of multiplication. By converting all of your divisions into multiplication by reciprocals, you are no longer forced to follow the left to right order of operation, but get to rearrange your problem to suit your convenience. This lets you group your terms for easy elimination. Basically, it turns the drudgery of a long series of calculations into a game with numbers. The same goes for adding and subtracting. By turning all of your subtractions into additions of negatives, you again get to rearrange and group your calculations to your convenience. Of course to play these games, you have to stop looking at numbers for what they are and look at them in terms of what they are made of. A 10 is not just a 10, its 2x5, or 1+9, or 14-4, or... As teachers, we have to put more emphasis on questions like "How do you make 10?" and using fact families when teaching operations. When you're teaching your grade 2 class the 2 time table, you're really just starting the lesson on algebra, so do it right.
