# In Fundamentals of Geometry, students have found the slope of a steep San Francisco hill, the length of the labyrinth at Grace Cathedral, and the height of a lamppost from its shadow.

As a mathematician whose area of expertise is “probabilistic phenomena in high-dimensional convex geometry," I approached Fundamentals of Geometry with some trepidation. Would I be able to make angles and polygons and parallel lines interesting to my 6th and 7th and 8th graders? Would they struggle to understand what I’ve taken for granted for so long that I can’t remember learning it?

I needn’t have worried: put a rich enough puzzle in front of these kids, and they will make the discoveries all on their own. The hardest part for me is to stay out of the way! In this spirit, we’ve used paper strips and hinges to explore what makes a shape rigid or flexible; tiled tabletops with “pattern blocks” to find out what combinations of angles would work; and worked out trick billiards shots in theory before trying them in practice. (One of the younger members of the class turned out to be quite the latent pool shark.)

In one of the most popular activities, we walked around the Proof School neighborhood and solved problems as we went—finding the slope of a steep San Francisco hill, the length of the labyrinth at Grace Cathedral, the height of a lamppost from its shadow, and so forth. It has been delightful to rediscover all the “simple” geometry in our immediate environment right alongside my students.

Yet, for all my attempts to make geometry relevant, I’m even more pleased to see my students getting satisfaction from the conjectures they make and the theories they are building. When the class found out that side-side-angle wasn’t quite enough information to decide that two triangles are congruent—except when it is!—they competed to figure out the exact circumstances under which they could rescue this “invalid” congruence test. When we worked out the exact height that billboards would have to stand above the road to line up from a special vantage point, I held out the payoff of actually getting to build a model at the end from chopsticks and clothespins. The model turned out not to be very satisfying (it seems my choice of materials did not support precision), but no one seemed to mind, since the class felt intrinsic pleasure at getting the math right.

Along with making theorems (and even the occasional proof), we’ve also made up our own problems. The students deconstructed some textbook algebra problems (How many cows and chickens have X heads and Y legs?), figured out what makes them work as problems, and then wrote some in their own inimitable styles: “Some hydras get 2 new heads when they lose an old one; others get 3. Hercules cuts off all their heads twice more, then runs away at the sight of 213 heads…” “There are two types of dinosaurs: naustidactyls, which have 1-inch noses, and pteranodons, which have 20-inch noses…”

While the class sometimes has to buckle down to master basic skills, they’re also looking ahead with eagerness to bigger challenges ahead. Students have made observations that anticipate trigonometry, integer sequences and combinatorics. In trying to persuade others of their assertions about geometric figures, they are learning how to distinguish what looks true to the eye from what they know to be true. Even at its most fundamental level, geometry turns out to be about much more than angles and polygons and parallel lines.

--Austin Shapiro