Can someone help me arguing a teacher about how to skip pre calculus to calculus without taking a test?
First of all you have to understand that the curriculum is designed specifically by educators, so you must remember that there's a reason why you have to take certain subjects, in a particular order. In that case, if you really need to take pre-calculus before calculus, then you can't do anything -- unless...
Some schools might allow you to take exams -- credit exams, or advanced placement exams, or some other name. This makes sense because the exam tests your ability to actually comprehend and understand calculus, and exhibit skills that would be developed if you were to take pre-calculus.
Hence, if I were you, (and here I'm assuming that you're good in math, and are confident that you can already take on calculus), I'd either 1) take pre-calculus anyway, and master the topics, and eventually get a really high mark, or 2) take a credit exam, if I really want to skip it. You won't lose anything in either track.
If you really want to skip pre-calculus and not take an exam, then that'd be a very hard route -- not because of the subject, but because of the rules of the school. If you're really confident, your teacher/administrator would probably have you take an exam (can't see another way) -- UNLESS you have already taken a similar subject and got good grades in that prior subject. This could be used as an argument that you've already covered the necessary skills to be taught in pre-calculus. It doesn't need to be a math subject -- for instance a physics or chemistry subject in some schools that involve hardcore mathematics would probably suffice; just as well, computer science subjects (numerical analysis) would also probably forward your cause.
In any case, make sure you think of your decision properly, and seek advice of educators in your school. They really know the path that you should take and I believe they're not going to choose anything that would be bad for your education and well-being.. (Personal opinion: take an exam, and prove that you can do it! Good luck)
I strongly agree with mvcdc about taking the test. There are plenty of students who are in classes that are too easy for them and suffer academically as a result, and you may be one of them. But, for every student I've had like that there have been about five other students who think they know a subject, get themselves into a class above their level, and then fail or have to change classes. They probably really believe they can do it, maybe as a result of their previous class being too easy and giving them a false sense of how much math they know.
Anyway, my point is that your teacher isn't being difficult by insisting on a test, just realistic. Just take the test!