Polynomials and the Reverse Galley Method (Essay!)
The Common Core State Standard A-APR-1 asks students to understand that “polynomials form a system analagous to the integers.” The whole number arithmetic one learns in grade school, in particular, the area model for multiplication (and backwards, division) translates directly to polynomial algbebra.
Here is chapter 7 from my book THINKING MATHEMATICS! Vol 1: Arithmetic = Gateway to All (available at lulu.com ) which shows the how to do this arithmetic of polynomials. For early work on the arithmetic of numbers and the build up of the story to this chapter, see chapters 1 through 6.