How to initiate conversation with a girl you've never spoken to before.

Evening Reddit, cliché question coming up. There's a girl in the same school year as me (year 13). I've noticed that we've been exchanging a fair few looks recently, well, a lot of extended looks. However, I'm very nervous about initiating a conversation with her. I don't really have any excuse to talk to her as we've never ever talked before or done... Anything together at all. What in your infinite wisdom would be the best way to go about iniating such a conversation be it online or face to face? My ex-girlfriend and I were in the same situation around a year and a half ago, she initiated a conversation with me on Facebook (typical teenager, I know) by saying that she 'didn't like randomly adding people' and liked to 'get to know the people on my [her] friend list'. It worked for me back then, do you think it would work in my current situation?