The steps we teach in our algebra courses are not the same steps that skilled practitioner use. I have already alluded to the utility of “criss-cross applesauce” when adding and subtracting fractional expressions and how to solve proportion problems quickly. We teach hoping that students will remember a concept and thus the process that the concept motivates. In fact many concepts have depths that beginning students lack the sophistication to understand much less remember. My students respond best to pattern matching demonstrations. Here is an example.
As can be seen, we (I) demonstrated a pattern to justify the meaning of a negative exponent, thus getting the first rule. I then did a little “algebra” to get the second rule which is really the same as the first! This demonstration “justified” the rule which generalizes as stated in the first pink box. The generalization however would require taking an honors calculus course to understand. Anyway at this point the rule as a visual guide has been presented and I could have started my students practicing problems by referring to said rule and saying “Have at it”. But in practice I created the second pink box. (They seem to like pink.) This is the rule that fast calculators use as they push symbols around on paper – easy to remember and immensely satisfying to use!