What I usually did was create a link to the full video and then during the lesson/presentation I would drag the play-head to the exact time I wanted to start showing. I usually ended up waiting forever for the video to load taking up tons of time that I didn't have, and it never working out as I had planned.
Now TubeChop makes this task simple.
1. Before your lesson or presentation copy the YouTube video URL and then go to TubeChop.
2. Insert the URL in the text field at the top of the page and click search.
Hopefully this easy to use tool will help you to have more effective and efficient lessons and presentations. Here is the clip that I chopped: