This document is an introduction to our first expedition, and you are encouraged to think about the questions before our first meeting. It also contains some additional expeditions that we may undertake throughout the year.

This seminar investigates topics that are not on the "traditional" path, but are sometimes introduced in math clubs. It draws on material that is often taught as an upper-level math class at the university level in the first few weeks of an abstract algebra class. The unique part of this seminar is the way it is organized and tailored to reinforce the important questions and techniques learned in Algebra I.

By wrapping the number line around a circle, participants will need to go back to things they mastered in elementary school and see if they still work. For example, negative numbers and fractions will need some serious revision. By reinterrogating these foundational notions, participants will not only explore new material but develop a deeper facility with skills they need for future success in mathematics.

The material for this seminary is imported from established topics like modular arithmetic and finite cyclic groups \(\mathbb{Z}/n\mathbb{Z}\). This material is repackaged to ask the same questions as an Algebra I course, but now we study Diophantine Equations and Quadratic Reciprocity. Chaos will also be discovered by looking at the Logistic Map as a quadratic function \(f:\mathbb{R}/\mathbb{Z}\rightarrow\mathbb{R}/\mathbb{Z}\).