You can do a jazz version of Happy Birthday, for example. And you can also do jazz versions of classical tunes. Here's a link to a cool Bach-based jazz impro: . On the other hand, once you learn the techniques you can create impros of pop tunes in a classical style. Here's a wicked cool Bach-ish Beatles tune:
Also, listen to the jazz piano "gods" like Art Tatum, Thelonius Monk, Oscar Peterson, Bill Evans---et. al. Pick one of their tunes and seriously impress your teacher!
I would say be warned about listening to Art Tatum and Oscar Peterson especially when just beginning to explore jazz, they'll scare the shite out of you
They are seriously mindblowing and are two of my all time favourite musicians, but they will scare the shite out of you.
I'm more the Bill Evans type--or--let me say -- I aspire to be...