Estimating Temperatures and Where it Gets You

One of the joys of creating a mathematical model for a real-world situation is that you can never be sure what mathematics you are going to get into. When the first author of the present essay created the Estimating Temperatures module for the Teachers College/COMAP Mathematical Modeling Handbook, it looked like an interesting exercise in interpolation. When it came time to teach this module, however, there emerged an unusual proof of a lesser-known theorem in Euclidean geometry. Who would have expected it?

The modeling situation in the module is to estimate the temperature at a location for which a direct reading is not available. We idealize the problem by assuming that the nature of the terrain permits one to assume that linear interpolation from nearby temperature readings is a reasonable way to proceed. In other words, if you had a reading of 71∞ at one location, and a reading of 75∞ at another, we assume that at the place halfway between them, a guess of 73∞ would be acceptable.