I have switched schools, and am now at Rosedale Middle School with Sam Martin. Last Thursday was my first week interacting with the students, and I think it went fairly well.
I had them draw a planar graph, and count the number of vertices (V), edges (E), and regions (F). We then computed the quantity V-E+F, and everyone (who didn’t make counting errors or arithmetic errors) got 2. After the first class, I switched to writing V+F-E because of the number of students making errors in computation, and from then on everyone’s arithmetic was error-free.
I talked them through a “proof” that they should always get 2, and we had a nice discussion about how it is sometimes better to consider a simplification of the problem and then work up to the whole problem. The proof I led them toward was one in which we “built” the graph, starting with a single vertex and added edges and vertices. At each step V+F-E is unchanged.
In class they were talking about three dimensional shapes, so in some groups, if there was time, I also mentioned that the same formula holds for all (convex) polyhedra. Luckily there were some see-through plastic polyhedra in the class. I showed them that if you put your eye up close to a face of a polyhedron, then you can see it’s vertices and edges, and they form a planar graph (this is called a Schlegel diagram if you want to read more about it).