We've mentioned before ways to display YouTube videos without the side content being shown with tools such as ViewPure. You can see the original post here.
Now fast forward to the many ways you can rid yourself of those annoying and sometimes inappropriate distractions.
Some of the best choices to use are Quietube, SafeShare.tv, and of course ViewPure. Each is easy to use and effective.
If you use Google Classroom (and really why wouldn't you?), you can post the YouTube link, so when your students open it they will only see the video. Side note - if your students are blocked from using YouTube in your school filter, this step will only work when they log into Google Classroom away from school.