Park City Mathematics Institute 2004 Course Notes Number Theory - Bowen Kerins, Ben Sinwell, and Ryota Matsuura

Stories that Count: The Art and Craft of Combinatorial Proofs: Combinatorics, sometimes called, "the art of counting without counting" has made some inroads into the middle and high school curriculum. This course will look at how combinatorics itself can fit into the 5-12 program, but it will also look at how combinatorics and combinatorial thinking can be used to illuminate ideas from more mainstream courses like algebra, arithmetic, and geometry. Look at any row of Pascal's triangle. What is the sum of the entries? What is the sum of the squares of the entries? Combinatorial proofs allow you to answer questions like these by "telling a story" - establishing a formula by showing that each side of the formula represents a different way to count the same thing. Starting from scratch, we'll learn in this course how to use combinatorial proofs to create and establish - in very simple ways - many beautiful results that will delight your students.