At some point in a student’s algebra career they need to lose the doctrinaire algorithms and start to feel the need for speed. Since my college algebra class seemed to accept the change depicted in this figure, I decided to teach them the fast way to work proportion problems. I wrote letters on small blocks of cardboard, placed them as in this picture,
and started moving them around without violating the equality. I felt an odd sense of pleasurable relief in the room. I think if students have worked enough algebra problems, they have a natural need to skip steps and I had just validated this need and confirmed the pattern that they had intuited. The class probably has more kinesthetic thinkers than usual so that my demonstration of moving numbers around appealed to them. Here is part of a handout that I gave them.
Two notes: First the idea of moving numbers and changing their signs, also works with exponents when simplifying exponential expressions. Secondly, I don’t think Tom Cruise in the referenced movie clip from Top Gun had a “need for speed.” He actually was “achin’ for acceleration.” At the elevations of most dogfights the visual clues of speed are not existent. That’s why the writers scripted low level chases. Tom really had the need for tight turns and sudden changes in velocity.