I bought an IPEVO V4K document camera on he advice of a colleague. It costs about $100 but seems to have the features of a cameras from more prominent names like Elmo at the $200 price range. I started to experiment with it today.
My second video is the one I uploaded. I ran the software (IPEVO Visualizer) you can download from IPEVO to display your camera feed on screen. I captured that display using Screencast-O-Matic. In the video you will see I had some trouble staying in the frame I selected to record. (You will also see I had some trouble using the blade guard on my new hair clippers.) I had an earlier attempt at a different problem where I did not select my headphone mic and instead used the built in mic on the document camera so it caught all the background noise in my house. So those are two things to watch out for if you use this system to capture videos.
In post production I added the problem as an overlay and I added details about three parts of the problem.
I think if I practice this over the next couple weeks this would be a good way to answer students questions I receive via email or to create some examples of problems for students.
Thanks to Sam Bazzi for referring me to Screencast-O-Matic and to Linda Brandt for telling me about he IPEVO V4K.
Here is the video: