<human>: Can you condense the text into a few key points?

There's this girl who's been in my English class since september, and from about 5 months ago I've somehow developed this insane crush on her without even properly knowing her. We have exchanged words on occasion, just casual stuff about the lesson and once about a sandwich I was gonna eat during a mock exam, but never really had a full on conversation. It's weirding me out.
And now we come to the big reveal: the reason I bring this to you fine folk is that the 27th of May was the last of our English lessons together. There's gonna be an English exam on the 24th of June which we both have and a study period on the 21st with my English teacher which both of us are probably going to go to (I assume she's going to be there, but might be wrong.) These are probably the last two times I will see her if I don't make a move and I don't know how I'd deal with that fact.
I'm an exceptionally shy guy, which is probably the reason I thus far haven't done anything remotely like asking her out. I was thinking about asking her to the college prom (happened on the 3rd June) but the window in which to buy tickets was only a couple of days and that was a few months back, when this thing was in its adolescence. 
Further, she did go to the prom with a few of her friends, and I saw a photo on facebook with her and a guy captioned 'Prom Date <3' but that's the only photo of the two together out of about 50 in her Prom album and no relationship status updates have occurred since so I'm not sure what's going on there.
So my question; on either the 21st of 24th, IF I do see her, how do I approach the situation? I think I should say here that I've never had a girlfriend before, so what lies in the future is unchartered territory for myself. I think I could deal with a no but I don't know whether not asking would be worse for me than a rejection.
<bot>: I wanna ask this girl out I don't really know that well, and may only have two opportunities to see or speak to her.