It may depend on the system. When I taught in public schools, I got Yom Tovim off for free if I recall correctly. It’s probably covered by the union contract. If you don’t like the answer HR gives you, check with your union rep.
As mik5 points out, there aren’t a lot of days since most districts give off RH, YK, and Pesach. This year it’s probably 6: Sukkos, Shmini Atzeres/Simchas Torah, and Shavuos. You’ll probably have to work during Chol HaMoed. Some years you’ll luck out and have YK fall on Columbus Day or Shavuos fall on Memorial Day. If necessary, take Purim as a personal day (this year it’s on a Sunday). It’s a good idea to save sick days for when you’re sick.