This is a tricky one. If your private teacher is a professor and they are your best recommender is it inappropriate to ask for letters of recommendations that you will use at schools other that theirs? I would think this is not that uncommon. Any advice?
Not a problem, unless your prof is unusually attached to their students. Mostly, they will have your best interests at heart and will write a single letter that can be used anywhere.
It shouldn’t be a problem, if the professor was your private teacher, in that role he would be expected to give letters of recommendation (as long, obviously, if he thought you were not that good…:). It would be unethical for him IMO to deny you letters to use with other schools, it would be in effect him trying to force you to go to his program by refusing to help with the admissions to others, it is a not so gentle form of coercion. Plus it is supposed to be what’s good for the student, if the professor teaches at a so-so program and you had the opportunity to go to one you liked better, that you felt offered more, it would not be in the student’s best interest to refuse to do the letters.