 # The Virtual Classroom: Further Math

Making the choice to study Further Math in high school can be challenging enough, but imagine how much more difficult the class could be when classrooms are all virtual? Perhaps not surprisingly, George School students reported their experience with Math teacher Hamilton Davis was engaging, thoughtful, and even fun.

In a recent Zoom session, Hamilton and his students reviewed different problems and equations related to graph theory. Hamilton held up a soccer ball, not necessarily the first image that comes to mind when you think of Further Math. However, this turns out to be a whimsical and unique visual aide for students considering a convex polygon. Hamilton counted the pentagons making up the surface of the soccer ball and then posed the questions: Why do there have to be twelve pentagons on a complex polyhedron made up of only hexagons and pentagons? How do we think that through?

Will Carson ’20 came to the conclusion that there must be three edges emanating from each vertex, while Anney Ye ’20 asked if they are dealing with regular hexagons and pentagons. The next challenge for the class was how to turn a polyhedron into a graph in two dimensions.

They used the following formulas to get there. Since the graph was planar, Hamilton provided the equation v-e+f=2, where v represents the number of vertices, e represents the number of edges, and f represents the number of faces. Hamilton asked the students what kinds of questions they needed to answer to start solving the problem. One student kicked things off with a method to prove that there need to be at least 12 heptagonal face. Another student built off of this with a proof that there need to be exactly 12 hexagonal faces.

Throughout the session, Hamilton used an interactive whiteboard on Zoom. This way he could write equations and also use drawing tools. Will said that this strategy helped to create a collaborative environment where Hamilton was “engaging us with small-group problem solving and even applicable games involving graph theory and other pertinent topics.” Will added that math is a subject “where visual and verbal engagement is critical for many students.” Hamilton’s creative use of the Zoom platform’s capabilities allowed him to implement both the visual and the verbal in the online setting.