Category Archives: Advanced Mathematics

WCYDWT: A Ball, a Trash Can, and Web Sketchpad

Dan Meyer has posted a number of "What Can You Do With This?" activities on his blog. (Activities is probably too prescriptive a word; they're more in the nature of prompts for student thinking, noticing, and wondering.) One of the first was the image below, which he made by superimposing frames from a … Continue Reading ››

Pentaflake Chaos

Dan Anderson commented on my Pentaflake post to observe that the pentaflake can also be created by a random process, sometimes called the Chaos Game. In this game you start with an arbitrary point and dilate it toward a target point that's randomly chosen from some set … Continue Reading ››

The Brouwer Fixed Point Theorem

According to Wikipedia, the Brouwer Fixed Point Theorem, named after mathematician and philosopher Luitzen Brouwer, states that "for any continuous function f mapping a compact convex set into itself, there is a point x0 such that f(x0) = x0. This is a deep theorem,  but one aspect of it is lovely, surprising, and entirely approachable by high-school geometry … Continue Reading ››

Soccer Challenges: Angling for a Shot on Goal

With the World Cup in our hemisphere, and the US squad having started out with a win over Ghana, my thoughts turned to the mathematics of soccer. My friend Henri Picciotto has a nice page about the shooting angle, the angle within which a shot is on goal, so I thought of using … Continue Reading ››

Create Parametric Curves Graphically and Kinesthetically

In this guest post, Nate Burchell describes a sketch he uses with his students to explore parametric functions. In this process students work entirely in a graphical world, manipulating graphs directly rather than by way of equations. (Nate teaches in Seoul, Korea, where I enjoyed his family's hospitality when I attended ICME in … Continue Reading ››