## Exploring Discrete Mathematics Summary

### Monday - Friday, July 11 - 15, 2011

This week we concentrated on creating teaching materials in our individual subgroups. Each group worked together separately and divided parts of the lesson to individual group members.

Graph Theory (Dan, Olimpia, Timon, and Gabe):
The overarching idea for these materials will be Euler paths and circuits. The materials will consist of 6 tasks of increasing difficulty. The initial task deals with how to traverse New York City by crossing each of the seven toll bridges exactly once. Later tasks add nodes/edges to the graph introducing new concepts. The fifth task deals with making conjectures and testing them using intentionally simplified graphs. The final task will have them apply their knowledge to a map of Lima, Peru. The teaching materials will include the tasks, teacher guide, and graphics/maps.

The graph theory group has three objectives with this unit: (1) develop problem solving strategies, (2) notice patterns and test conjectures, and (3) justify and critique ideas. The unit will be designed for middle school and older students but could easily be applied to any high school course.

Mastermind (Trevor, Marcelle, Todd, Mahen):
The Mastermind group has divided the unit into four parts: (1) Introductory Activity, (2) Combinations/Permutations, (3) Probability, (4) and Logic/Strategy. The modules can be done in sequence or separately. This allows teachers to pick and choose which activities they need for their class without needing to do the un-needed pieces.

The modules will be problem based allowing students to apply their own understanding and/or strategies to solve them. Each major question will be accompanied by scaffolding questions and extension problems to allow for differentiated instruction. The questions and teacher resources will be merged to a single document on a question by question basis. The completed unit will include an overview/description, suggested path differentiated by level, and tasks with teacher resources.

The Mastermind group asked the other PCMI participants for suggested strategies (not necessarily optimal). They are still waiting for responses which they can integrate into the unit. In addition, they have been pulling resources from the book "Mastermind Mathematics: Logic, Strategies, and Proofs," Mathew Mitchel printed by Key Curriculum Press.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.