What should I do about a bad TA as another TA for the same professor? I have been a math TA since my freshman year, and this year as a junior I was made head department TA. One of the TAs under me is what I deem to be a poor fit. 

He is incredibly sexist(at a liberal arts school, so a huge no-no, and he is especially so to me), he talks down to students instead of trying to correct them or nudge them in the right direction, and the smallest mistakes set him off. When we TA the same class, he acts as if I am always wrong, belittles me, then teaches students the incorrect way(i.e. , the students are just beginning to learn limits, and instead of giving them the basics, he's trying to teach them derivatives). 

The biggest, in my opinion, is that this student is a known cheater, and didn't get the grades needed to TA the honest way, and I am worried he will encourage our shared students to follow in his footsteps(he was discovered cheating in every class he attended last semester, and a couple of them the semester prior to that). 

Unfortunately, he is the department head's favorite so I am at a loss of what to do. I don't have control over who is hired or fired, I am just head TA in the way that I have been doing it the longest, and I answer the questions the others can't. 

Should I bring my complaints to the department head or my adviser? Do I risk my position to help my students? I can't bring myself to work with him any more, and don't want to lose the job I love because of a sexist, cheating TA.
Write down everything that happened and when.
Tell your advisor about it.  It's their job to help you.
There's no excuse for sexist teaching.  It cannot be tolerated.  I suggest you limit your complaints to this main issue.  The other issues just complicate the situation unnecessarily.  


It's unclear who supervises TAs in your situation, but your advisor should be able to figure that out.