I saw two neat ideas about parametric equations in Desmos during the last week. First from Mr S. on twitter:

And then later from Patrick Honner:

So, I modified the Desmos program that Mr. S. shared to show velocity and acceleration and asked the kids what they saw in the animation. Here we are looking at the parametric curve defined by the equations

(When you watch the videos, keep in mind that my older son has been studying parametric equations in calculus but my younger son has essentially never seen them)

Next I asked my older son to pick a new set of equations and we looked at what the velocity and acceleration vectors looked like now:

Now my younger son picked some new equations – this time there was a lot of wiggling around!

Finally, I wrapped up by showing them a fun little surprise – what velocity and acceleration look like for an ellipse. This example shows what’s going on with planetary motion.