My friend, let me help you.
Firstly, the day of the dance I want you to wake up early that morning and go to your school gym / ymca / home gym... Whatever! Just find some weights!

Do a few of your favorite lifts for about 3 x 6 and then have a light jog for about .5 miles.
Your probably wondering why I'm telling you this, but trust me, the exercise in the morning will get your metabolism going, your brain sharper and not to mention lifting weights / cardio increases the testosterone in your system. Simply put, you'll be a well-oiled, testosterone filled freshmen stud for the date. Okay.
Now, about working with shy girls, I see eye-to-eye with you. Last summer I was at a dance with a this banging girl, who had some sort of Obsessive Compulsive Disorder and was the biggest introvert I'd ever met. You could hold a 30 minute conversation with her (or I would) and her responses would be short 1-line snippets. Just to let you know the severity....
So, I was at the dance and she would NOT even get from her chair and get on the dance floor. She had really big boobs, I think she was self-conscious about. Well, at first I tried to convince her to come by making a fool of myself on the dance floor. (Point 1) Always, always , always when dealing with a shy girl make yourself to be overly outgoing. This instills trust their trust in you and also now your connecting with one of the most basic human emotions: laughter. If your enjoying yourself out there, dancing like a fool with some of your guy friends and your all laughing and having a good time, she'll see how fun you can be, right?
Okay, so back to the story. I'm making a fool of myself, trying to do the robot and cripwalk, just letting loose and not giving a rat's ass about anyone elses opinions about how i should dance. (Honestly, at a freshmen high school dance, who's opinion really matters on what dance moves are 'hot' ? Better yet, what dance moves are 'hot' I'd sure like to know! Haha. ) I went back up to Ms. Shy and asked her again to dance.
She told me, 'No.'
That sucked, but it was okay, I still kept chatting with her, trying to see if she'd laugh. Then I finally asked the simplest question, 'What kind of dancing do you do then?'
Turns out she was big into Swing, so I had her go outside the club with me and teach me a few moves in seclusion from everyone else. It was some sweet alone time and I had lots of chances to mess up in the Swing dancing to touch her boobs. She just laughed and really opened up to me.
So, in your case. I think you don't have too much to worry about. Mine was an extreme case, yours doesn't seem as bad.
What you need to do after your amazing day (trust me, aftering working out in the morning you'll be feeling good all day) is go to the dance and make sure to make some eye contact with her from across the gym, whatever.
My advice, get in a group of your guy friends and while talking to them, try to catch her eye, hold the eye contact until she breaks it off then go back to your conversation. Once your done talking with your buds, go over to her and ask her how things are going. Chat her up about your work project-thingy. Talk about some guy / girl you two both don't like's goofy shirt/pants/blouse/shoes/hat, etc. Don't be overly harsh, just joke around, make her laugh.
Ask her why she isn't dancing, if she hasn't been. All this time your talking with her, make sure your touching her. It doesn't have to be something heavy, serious, just a light punch in the shoulder. A flick of her hair jokingly. Tap your on the side/top of the thigh with your hand when pointing out someone.
Then, after you've gotten her comfortable, say this:
"So apparently I heard your really awesome at grind dancing or somethin..."
Usually a shy girl at this point int he convo will laugh and say, 'Haha, who told you that?'
Of course she'll be thinking about some random time she was dancing crazily with some of her girlfriends, or family, or whatever.
Obviously, you wanna keep playing around with her and dance around the question with, 'Oh i dunno, I've just heard from a few people...'
Do this until she gives up and then just ask her nonchalantly, 'Hey do you wanna go dance then?'
Of course she'll say , 'yes.' I mean, if she doesn't then you NEXT that ***** and tell her how all my advice was in vain.
Let me know if you need anymore help, I've been there, man, and I wish I had these forums back then.