For a while now, I’ve been intrigued by the ways in which the study of geometric transformations can provide students with a very effective introduction to function concepts. Daniel and I have written a couple of articles about this topic, and we created a number of activities to take advantage of what can arguably be … Continue Reading ››

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 ››

Yesterday, I led a webinar that demonstrated how Sketchpad and Web Sketchpad can be a powerful tools for exploring Common Core algebra topics. My examples included solving for unknowns with a pan balance, exploring the slopes of lines, maximizing the area of a fixed-perimeter rectangle, and graphing trigonometric functions. I touched only briefly on each example during the … Continue Reading ››

At the 2013 Baltimore Regional NCTM Meeting, I gave a presentation on Picturing Functions and Functions of Pictures. In the description of my presentation I’d promised to show how to use Sketchpad to create special effects, so when it came time to prepare, I figured I needed to deliver on my promise. So I wrote a … Continue Reading ››

I’m excited to be making my first presentation of the 2013-14 school year next week in Baltimore. Daniel Scher and I are presenting Picturing Functions and Functions of Pictures. We'll be discussing the connections between pictures and functions. These connections are even richer than I realized when I first began to work with pictures in Sketchpad. In the … Continue Reading ››

After writing yesterday's post on the connections between polar and Cartesian graphs, I realized that I hadn't said anything about how easy it is to start from scratch and create a polar graph in Sketchpad, so I decided to write this post, and include an instructional video. Here are … Continue Reading ››

The May 2013 Mathematics Teacher has an excellent article by Jonathan F. Lawes ("Graphing Polar Curves") on the value of plotting the same function in both polar and rectangular coordinates. Doing so not only helps students understand how polar coordinates work, but also gives them a novel and revealing … Continue Reading ››

Last week was the fourth session of my spring Advanced Secondary Math Methods class at the University of Pennsylvania. Each year I assign a semester project in which groups of three students use lesson-study techniques—on a small scale—to create, test, refine, teach, evaluate, and document specific shared instructional products, composed of a (possibly multi-day) lesson … Continue Reading ››

Like other enthusiasts of mathematics, I’m captivated by the way that mathematical ideas can explain things in the physical world around me, and by the way that I can carry out mathematical thought experiments in my mind and then apply the results to control my external physical environment.