Extended Analyses: Finding Deep Structure in Standard High School Mathematics

This paper describes an approach for pre-service or in-service mathematics education that teaches sophisticated mathematics using only the tools of high school mathematics. The idea is to start with a standard problem from high school mathematics and let the solution to this problem serve as a platform for asking good mathematical questions and searching for deeper mathematical structure beyond the obvious "answer" to the question. Surprisingly, it turns out that high school mathematics is remarkably open to this sort of analysis, and the results are more interesting mathematically than one might initially think.