Do You Think She Likes Me?

ok, so i'm a freshman. I came from the smallest of the main feeder schools. I know many kids from other schools that came into my high school, but not everyone. Since like day one, I've had a feeling that this one girl may kinda like me. She was a bit too nice, and she smiled at me a few times. But I thought she was just being friendly. I didn't really react. Then, i noticed that she really is kinda shy whenever I'm around. We're both in honors algebra and health, and i've seen her talking to her friends very openly and happily. But whenever I'm around, whenever that happens, that's usually when we just pass each other in the halls, she would kinda smile and walk past. Before, she would kinda blush and look down and pass by. Back then, I didn't really notice her much because I didn't know her. Then, after a while, I realized that I kinda like her because she's really nice, smart, well-mannered, and a good person all-around. I'm not meaning to brag, but I'm kinda the same. However, after a few months, I started to realize that she either doesn't like me anymore or she never did, because I started to look at her across the class more often, and she rarely ever made an look at me or anything. We didn't really ever pass each other in the halls, and we never had an honest conversation. The only time we ever talked before was when somehow one of those backpack buckles or ropes wutever u call it got stuck to my desk when she passed by.. It 's kinda weird to describe how it happened. it just sounds stupid describing it as some buckle. So I assumed she didn't like me. However, at our school's open house, I was there as part of the school newspaper interviewing clubs and students. She was part of the swim team. I started talking to a female friend on the swim team, and the girl that i like started to giggle to one of her friends, and then rushed over to get into the conversation. She seemed a little fastidious in responding but then kinda eased off. You know how when you get nervous about something but you keep your cool by trying to talk all suave? Well, that's kinda how it was. We spoke for about 5 minutes. I should've stayed longer, but I left to go interview the baseball team. The next day, we didn't talk, but I think she was trying to get me to start a conversation because we had gym that day (health and gym rotate every day) and when we were out of the locker rooms waiting in this little commence area for the bell, she was talking to a friend and i was talking to some friends, and i noticed she was kinda looking over at us. She even put her jacket beside where I put my hoodie, probably so when i get it, we would be beside each other. Honestly, I might be thinking that she doesn't like me. Maybe she's just really nice and well-mannered. But whatever it is, I think there's potential that she might like me. I'm not really that hot nor am I really a GENIUS, so I don't know why she would like me. But if she does, I have no idea how i can talk to her more. She doesn't have a facebook nor do I know her number. I don't know her friends very well. And the ones I do are more of my friends than hers, so they don't know her contacts either. We're not in any of the same classes. And we have assigned seats in math and health. And gym kinda seems awkward to just random approach her and start talking to her. All I know is she's in swimming and she plays violin in orchestra. I'm a music guy, but I'm a band geek, so we're naturally enemies. I play piano classically but I don't know how I could possibly just approach her and start a conversation about that. Please can anybody tell me how I should get into talking with her, far enough to ask her out?