Geometric Systems is a class that covers Euclidean and non-Euclidean geometry. In this class, I presented the students with the following assignment:
Prepare a lesson on an extra topic of your choice (see “other geometries”). It should include a 15–25 minute introduction to the topic, a handout with useful information/examples, and a in-class “homework assignment” that the others should be able to complete based on your presentation.
The students had some fun ideas. There were presentations on knot theory, topical geometry, and projective space.
Students completed the worksheets for each topic with assistance from each presenter. Before the end of class, they turned in a peer review which explained what they liked, what could be improved, and how well they learned the material.
In addition, I had students spend the last week of the class on a group project on graph theory where they formed conjectures, discovered (known) theorems, and wrote proofs for these theorems. We ended the week learning about graph theory on surfaces (for example, the chromatic number of a planar graph in projective space or on the torus).