# Course Descriptions

*Am I ready?*

**Level 1:** We do not require any formal background. As a measure of maturity, we ask that students have some experience with algebra and manipulating equations that have variables. (The advanced middle school or early high school level should be enough.) We do not have age restrictions for students.

**Level 2:** We ask that you have taken a Level 1 class, or have a similar experience in your school or other math activities. Contact us (paul@puremathacademy.com) if you are not sure!

**[Level 1] Games.** In this class we will play games! At the same time we will learn about strategies and logic, with a strong emphasis on clear and complete explanations. We will work through a series of take-away games, as well as balance scale puzzles and truth teller puzzles. We end the class with a tournament of the game SET.

*This class was the inspiration for Pure **Math Academy. Over many years, Paul developed this material as a way to introduce rigorous mathematical reasoning to students with no formal training.*

**[Level 1] Counting.** In this class, we will see many methods of counting: the multiplication rule, the addition rule, careful overcounting, counting by negation. If time permits, we will see how these methods are used in the development of probability, and develop the formula for counting combinations. Activities include Pascal's triangle, nontransitive dice, probability bingo, designing geometric objects, and no-bluff poker.

**[Level 1] Numbers.** This course introduces several number concepts: parity, divisibility, greatest common divisor, binary numbers, modular arithmetic, and the Fibonacci sequence. We explore these topics through several engaging activities, for example, Sona, Bulgarian Solitaire, and the Four Numbers Game.

**[Level 1] Graphs.** We will start with some problems that naturally show the power of graphs: the Königsberg problem, the traveling salesperson problem, and minimal cost spanning trees. We will also explore basic ideas about graphs including isomorphism and degree sequence. If time permits, we will explore other topics, such as trees, planarity, matching, and/or coloring.

**[Level 2]**** ****Groups****.** Groups begin with permutations. Then come multiplication tables and basic properties like Lagrange's Theorem. Then we explore applications like the Futurama Theorem, the 15-puzzle, and the 100 Prisoners Problem.