Starting to teach private lessons back in 2008 was an intimidating experience; I wasn't quite sure yet how to establish authority while teaching, but luckily I have been graced with willing, engaged students who don't require much in the way of classroom management.
One thing that helped me to feel more comfortable and in control while teaching was to figure out a routine for the lesson. The beauty of private lessons is that they're highly customizable to individual students' needs, but the basic layout for any lesson I teach is the same. It goes something like this:
1.) Guided warmup time
Figuring out a useful warmup routine is challenging, so I like to present different methods to my students so they'll have options over the course of the week.
2.) Technique study
Depending on ability level, this could mean scales, etudes, or experimentation with extended techniques such as multiphonics.
3.) Solo pieces
Once we've spent some focused time working on individual skills during the technique portion of the lesson, we combine those skills in the student's solo piece.
4.) Sight reading duets together
Most intermediate students can sight read duets, and I've found it to be a nice way to build both sight reading and chamber music skills. It's also a low-pressure way to end the lesson; I make sure my students know that sight reading will, by its very nature, be imperfect, and that it's absolutely fine to mess up as long as they recover and keep going.
This plan of attack isn't set in stone, but it seems to work for my students most of the time.
Do you have a routine that you use during your teaching? Share in the comments!