### Algebraic Systems Notes

Introduction to Algebraic Systems is the required undergraduate course in theory of algebra for middle and secondary mathematics education majors. Disliking all the available textbooks, I taught the class this past fall without a book, replacing the textbook with daily lecture notes posted on the web. The notes for the first three lectures illustrate several different ways of making on-line notes more interactive.
Course Outline
Just a standard listing of course information posted on the web. No interactivity at all. Does include some introductory material about the definition of a field.
Modular Arithmetic
Basic rules of modular arithmetic. Here I use tables and pictures to make things look prettier, but it still isn't interactive.
Modular Inverses
Here the notes include an applet where students can see which elements have inverses for various moduli. It lists a few patterns found (by the students) during the lecture and invites them to look for other patterns.
The Euclidean Algorithm
Once students know when modular inverses exist, we go over the algorithm to find modular inverses (which completes the proof that the integers mod a prime form a field). At the end, an applet lets students choose their own examples. The applet not only produces the answers for the students, it also lists all the steps along the way, in the same fashion as I wrote out examples in class.