Idea 1: Networks: A Mathematical Approach to Friendships

Networks or a digraph unit that would involve a student project of creating a personal friendship network connecting to people they know, etc, to see how far they can go in several steps, similar to the 6 degrees phenomenon of human relationships. In my math research class last semester, we used the book Random walks a=on electrical networks and I feel that this would be a valuable resource for this project. For the friendship network (which would be the final project) I have not yet found any research, but I see it more as an application to their lives. With the extreme use of Facebook, students might even be able to further develop these graphs than would otherwise be possible.

Primary References:

Doyle, Peter G. and J. Laurie Snell. Random Walks and Electric Networks. Version dated 5 July 2006. GNU FDL.

McBride, Michael. Position-specific information in social networks: Are you connected? Mathematical Social Sciences, Volume 56, Issue 2, September 2008, Pages 283-295.

Idea 2 : Voting: The Mathematically Impossible Fair Contest

An introduction to the mathematics of voting methods. I have seen a presentation on this and feel that it would be particularly beneficial for high school students, particularly those in an advanced course, such as pre-Calculus or Calculus. The mathematics behind this is necessarily more complex, so I feel that this is only suitable for students with the appropriate level of instruction. However, I would probably incorporate a few of the more simplistic systems into the curricula for a lower level class.

Primary Reference: Alon, Noga. Voting paradoxes and digraphs realizations. Advances in Applied Mathematics, Volume 29, Issue 1, July 2002, Pages 126-135.

Idea 3: How Many Colors Does It Take to Color a Map?

Graph coloring. After some basic instruction, students would be given the opportunity to construct a map of their own or color a map of the country, state, etc. Special attention would be given to the incorporation of vertex-edge graphs in this instruction. I have found some pre-existing curricula on this topic, so I would likely include a variety of alternative options for the use of graph coloring, or use it as a part of a unit on vertex-edge graphs.

Primary Reference:

Miao, Lianying and Shiyou Pang. On the size of edge-coloring critical graphs with maximum degree 4. Discrete Mathematics, Volume 308, Issue 23, 6 December 2008, Pages 5856-5859.

I selected these topics because I wish to write a teacher module and I feel that these topics would be both mathematically interesting as well as enjoyable for the students. The cross-disciplinary possibilities are also exciting for use in a high school.