Okay so let me start off by saying I attend university, and there isn't really a greek scene / real frats like there are in the United States, and from what I've observed over the past couple years is that most people are NOT connected whatsoever, and most people just have a small social group, or are completely alone. I'm going to say about 95% of the school is this way.
So anyways, new semester = new classes = new opportunities to open girls in class before the class actually starts. Of course checking myself out in the mirror before going to class making sure i'm looking fresh.
I'll try to post the interaction as detailed as I can, so people can help tell me where things (rather where I) went wrong.

sit down beside HB8
me: hey uh this is the room for **** right?
HB8: uhh *smiles* yeah i hope so
me: haha me too
HB8: *giggles* you confused me for a second there
me: I'm just as confused as you are
*Body lang opens up slightly towards me*
--short period of not talking, we were both checking our phones
me: so have you had this guy before as a prof?
HB8: no i had a different guy
me: oh i had this guy last year and hes awesome he's even really funny
HB8: *little smile*
me: so did you take part 1 for this course (not a stupid question, because part 1 was not a prereq)
HB8: yeah i took it last term with some guy
me: oh i took it last year with this guy
--again no talking going on for ~5 minutes
me: do you know what course number this is?
HB8: umm im not sure i'll check
me: thanks my internet on my phone isnt working
HB8: the number is *****
me: *writes it down on top of my page*
--another ~5 minutes not talking, then we get the syllabuses handed out to us
me: uhoh i hate when a class has only tests and no assignments
HB8: yeah, last term i had a class where 5% was just participation
me: i would hate that class so much. usually i can get an A+ on an assignment but only like B's on tests
*prof starts the class going over the syllabus*
*body lang starts to close up
*prof talks about how he'll ask people who text to put their phone away*
*me noticing she was texting lots i whispered "dont worry he wont actually hes lying"
*she started to giggle, but it may have been from a text she got
*end of class she leaves, i leave.

Okay now i know one of the key things is to not get stuck in the trend of being the boring normal academic guy (which i was clearly displaying here) my biggest questions are what other topics could i have brought up to have a more interactive conversation. I would much prefer being told the words/how to transition into these topics also, over just the general ideas of topics to bring up.

At what point did you introduce yourself? Steer the convo by asking the things that she likes to do outside of class. Where she goes to have fun, music, food, etc. Then, once you find out, you can ask for her # to meet up to do the thing that you both liked and had in common. Start the flirting/teasing NOW so you don't end up the friend zone.