Screencastify isn’t the best screencasting tool, but it might be the easiest one for in the classroom. And the numbers prove that: 22 million videos were recorded in Screencastify during the 2017-18 school year. There are good reasons for those numbers, too: it’s free, it’s easy to use and it syncs up with Google Drive.
If you happen to have access to Screencastify Premium, you have the ability to create GIFs, too! Just click save to disk and export as animated GIF after recording a screencast. That’s it! Since you’re using Premium, you can even do some editing prior to exporting the video. Check out the process in the animated GIF below (note: I made the GIF in Camtasia, not Screencastify):