What if you could simplify a problem so that all which remains, and needs to be known, is a friendly rational number (often an integer)? As Dustin Hoffman might have heard it after graduation, “I want to say two words to you. Just two words: scaling exponents!” Scaling exponents, the heart of proportional reasoning, declutter our thinking even more than do twiddle analyses.1 A scaling analysis discards even constants of proportionality (including often dimensionless constants) until only the exponent remains.
The cost in accuracy is repaid in lower cognitive load and greater understanding and insight. Unfortunately, many students loathe and avoid scaling exponents and proportional reasoning. Try asking for the length of the horizontal line that bisects an equilateral triangle with side length L (Fig. 1).2 The common solution first finds the area Al of the small triangle. One fraught step is finding its height. That...
