This paper develops a mathematical description of the NTP and walks through the construction of linear solutions, highlighting unique features with theorems and diagrams. While it was motivated as an exercise based on a fun problem, we found that the results were interesting in their own right and had connections to several areas of mathematics, including number theory, combinatorics and a particular type of group permutations called orthomorphisms, which in turn have applications in various other areas of mathematics. Section 1.1 offers a preliminary, qualitative description of the mathematics involved, detailing some of the conceptual difficulties that accompanied its formulation. Section 1.2 proceeds to work out the corresponding mathematical notation of the description from section 1.1. Once supported with a mathematical theory, section 2 constructs linear solutions to the NTP and highlights their number-theoretic properties with examples and diagrams. Section 3 is dedicated to counting linear solutions to the NTP for any given n. Finally, section 4 offers a brief introduction to the definition and applications of orthomorphic functions, establishes the equivalence of the orthomorphisms of ℤ/*n*ℤ and solutions to the NTP, and provides resources for further reading.

These lessons are laid out as individual lessons that could be taught at any given point of a class that was dealing with the topic of the lesson at the time. These lessons are snapshots of what would be happening in a classroom and the idea is that lessons and teaching happen in-between each of the individual lessons and ideas presented here. Each chapter will begin with a summary of the main concepts and big ideas to be addressed in the chapter. I then offer the general structure of the lesson and how it could be taught. This includes what the teacher would say in the lesson and student misconceptions and questions. My hope is that this document would act as a teaching resource for teachers looking for individual lesson plans to be implemented in their own classroom during moments that they feel are appropriate. A lesson in this paper should take one class period to teach, which I have timed out at an hour. Being that most class periods are about 45 to 50 minutes this can be shortened or it could be spread out over several days as needed and appropriate. I appreciate your reading of this document and wish you a lovely day, Joseph Willert

]]>In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite. In truth, they are not! The set containing all the natural numbers {1, 2, 3...} is an infinite set. Our main goal for this paper will be the discussion of Axiom of Choice (AC) and its equivalents.

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