180 Days of Ideas for Discussion in Math Class.
(as of 9July2014, we're in overtime!)
I *hate* the terms "cross-add," "cross-cancel," and "cross-multiply" (though I tend to slip up and use "cross-multiply" sometimes). Kids always seem to mix these tricks up. They cross-multiply when they don't have an equation, or they cross-cancel when they are trying to add things, etc.The kids I get in my regular Calculus class tend to have weak fraction/algebra skills, so when we get to things like limits involving rational functions, or second derivatives of implicitly-defined relations where the first derivative is a rational expression, I try to emphasize why they use each of the little tricks that they use. They go, "Oh, that's the first time anyone has ever explained it to me!"
I wouldn't have included them if they didn't aggravate me, too. That question "why" is the only antidote to the confusion.