A brief post this morning. We are winding down in our AP Calculus BC class and the last topic of the year is a short unit on vectors and parametric equations. Many of my students buy a (slightly) different version of our text book so some do not have the vector chapter. I use a curriculum module from the AP site as the spine for our work through these ideas. I have to admit that I do not have a great amount of enthusiasm for this topic, at least at the level that we work with it. But on Wednesday we had a fun breakthrough in class. We were working on a fairly typical example of a parametrically defined function on the Cartesian plane and found its derivative. The kiddos asked for a picture so we graphed both the position and velocity vectors on Desmos. One of my students expressed disappointment that we did not see the order of the graphs so it was hard to move our eyes from one graph to the other to see how they related. I am more comfortable making GeoGebra jump through hoops so I moved on to GeoGebra and graphed both with sliders and leaving a trace on. The kids seemed to perk up a bit liking this visual better. Then one student asked me to change the velocity vector. Instead of having it rooted at the origin, he asked me to redefine it so that it was attached to the point, so that it would be a tangent vector. I made this adjustment (you can find my GeoGebra of it here) and the kids seemed so much more engaged immediately. The power of seeing the trace points move apart from each other combined with the direction and length of the velocity vector changing along really caught their attention. I want to tweak it a bit still, there was a request for adding the acceleration vector as well. At a time of year when energy is running low, it was a fun blast of energy and engagement here.
That’s all for now, just wanted to share something fun.