What it's like to have a boring gym partner

Today I was contemplating a certain situation that has been brought to my attention.

There is a new girl at school (as am I) and she seems to be very lonely. I only got to know her during gym class on thursday afternoons. My other supposedly 'friends' seem to enjoy leaving me with her while they have a laugh among themselves.
I have always been small minded in a way. I take on board other peoples' judgements. For instance, I am embarrassed to be seen with an ugly person as I fear it may harm the way people percieve me. I like to make a good impression all the time. But this time I know I am being small minded but I don't seem to want to change that fact because it's just so embarrassing to be seen with her.

Anyway, so in gym we have to get into pairs so I went with her as there was on one else available. While we were crossing the road she asked me how my week had been. I responded saying it had been better for me socially as I had found it difficult to make friends in the beginning. I looked at her and I could see the tiniest bit of sadness in her face but quickly, she slipped her arm around my shoulders and I instantly froze under her. She said that was good then took her arm off me. I was like, what just happened there??

So the rest of the while when we ran I constantly tried to keep up with the other 2 girls but I could see they didn't really care about me, even now. The new girl could see what I was doing so she constantly kept moving closer, closer, and closer. I had to gradually keep moving further and further away.
Then I got stuck with her while we were doing strength training. At that time I was muttering under my breath, why do the others keep ignoring me? Why do they always go with each other and not me? How come they are so close to each other and I am not? I better not be stuck with this girl every week.
The new girl is constantly talking about stuff that just doesn't particularly interest me, like what subjects I have the next day (I hadn't even looked that far ahead yet), or how her sister and her fight a lot, or how heavy her sister's bag was. I was bored out of my mind!! And the others couldn't care in the slightest.

But now I'm thinking, this has to mean something. Why is she in my life?
I believed that the girl had come into my life to teach me what it is like to have no friends. But I know what it is like, I had to eat my recess on the toilet. But now I have friends. Even though I sometimes got the feeling that some people thought I was boring, I perservered and now I have around 20 friends in the school (and counting).

But I have ffigured that she's tring to teach me what it is like to be different. She is trying to teach me not to feel like I'm stuck with her when I'm not, to stop judging her through my eyes, to understand that I have my life and she has her life and that she may not want to have friends yet.
There is no right way to make friends, no body makes a better effort than anybody. We are all experiencing the same thing but it dfifferent ways. I have to learn to not take her problems on board and to not feel guilty about things do not actually concern me.

I have to learn to think about myself but respect others.