Improvisation can be a fun part of piano lessons, and it should be introduced early to show the student that they don’t always have to reproduce music, but they can create it themselves! My favourite way to do this is with young children and adult beginners is to play a 12 bar blues chord progression in F# major (Gb major) and let them make up whatever they want on any black keys. They can play the same note the whole time, or play 20 notes at a time, and it will all sound good. Try it out below!
If my students get comfortable with the improv in F#, I like them to try improv in C major. It’s a little more complicated because E and B don’t sound good all the time, so I either accept that, or tell them to play Eb and Bb instead. If I haven’t already introduced some easy licks to incorporate, I do that now. They can mix those in with their own ideas. Here’s the one in C major.