Building a new site while old one is still active

I am trying to update my old site, with a brand new one. I want to build the new one totally then use a redirect from the old to the new... I know that I have to put the new one in a subdirectory and add wordpress to it. My site is http://www.example.org and I want the new one to be http://www.example.org/FCM ...What I don't know is where to put the subdirectory, does it go in the root so it will look like /FCM or does it go after public_html/FCM.... I have searched and searched and don't know the answer.