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I am teaching a class in the Spring for abstract algebra for high school teachers. The goals of this class is to give future high school teachers a deeper understanding of many of the patterns and concepts in geometry, introductory algebra, and trigonometry. What concepts would you consider necessary for students to learn? Also, what topics in high school/middle school math could be explained using abstract algebra? I have a general course outline in place listed below.

First couple of weeks- Review sets, functions, relations, binary operations, and common proof techniques

Next 6-8 weeks - Focus on groups. We will talk about the definition of groups, subgroups, cyclic groups, Cayley diagrams, multiplication tables, Direct Products, quotient groups, LaGrange's Theorem, cosets isomorphisms, homohorphisms, and plenty of examples of groups in geometry, trig, and algebra (such as rotations of a circle, plane isometries, symmetries of regular polygons, symmetry groups of the conics, modular arithmetic, even and odd functions, ect.) These concepts are not listed in the order they will be presented.

The last 4-6 weeks - Rings and fields. We will talk about the definitions of rings, fields, subrings, modules, ideals, Fermat's little theorem, and focus a fair bit on rings of polynomials. We will also talk about the differences between various number systems and compare them to less common number systems (naturals, integers, rationals, reals, complex, dual, and split numbers, and Gaussian integers). If we have time, we will wrap up with splitting fields and field extensions.

Are there any important concepts that are missing?

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Your curriculum looks really good. In Romania we study abstract algebra in our senior year in high school and this is more or less what is covered here, so I believe that your outline is pretty solid.

However, I think that since you will be talking about groups you should also consider covering group actions. I suggest that you do this because Burnside's Lemma from group theory has several applications in counting problems and these certainly do pop up a lot in high school/middle school maths. You can find plenty of examples of such problems online and you may also consider looking at this paper which briefly discusses the Polya Enumeration Theorem (a generalisation of Burnside's Lemma) and presents applications of this result in chemistry and music theory (which is something pretty cool and useful for a high school maths teacher to know if you ask me).

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