Discrete Dynamical Systems

This thesis is made up of two parts, which are connected by a common subject, Discrete Dynamical Systems. The first part treats the topic on a collegiate level, proving theorems about fixed points and their classification, such as the first and second derivative tests.\ud The main theorem that we work towards in this paper is Sharkovski's\ud Theorem, which takes all of the terms and classifications to help us determine the long term behavior of various periodic orbits in a discrete dynamical system.\ud The second section, Discrete Dynamical Systems in High School:\ud Lesson Plan and Rationale, adapts some aspects of the sophisticated mathematics from the first part and makes it accessible for high school students