Oh, I thought. Oh that's difference of two squares isn't it. Oh dear. I have no idea how to do that. Algebra wasn't my forte, especially not quadratic equations, but the atmosphere wasn't helping.
Three different pieces of music, if they could be called that, played too loud through three different sets of headphones, so that rather than the music being heard only by the person wanting to listen, the songs charged at each other headlong and crashed into each other, producing a clash of harmonics which were never meant to be heard together. Lipstick and nail varnish lay scattered on the desks and in the corner Lucy, the girl who's party I hadn't been invited to, was industriously scrubbing at a desk with nail varnish remover, attempting to clean off a stain before anyone realised it was there.
A half eaten apple crunched under the leg of my chair as I rocked backwards, trying to catch the thin shaft of light filtering through the blinds. Just as I had found it, Jessica came running in past me, catching my arm with her bag and knocking me backwards into the desk lid of the girl behind, which promptly fell forwards onto her head as she rummaged for books.
"Watch it Olivia. That could have taken my head clean off." It really couldn't have, was one thing I could have said, that or it was Jessica's fault. Both were equally true.
"Sorry." I muttered, turning back to my work. Difference of two squares. I couldn't even remember when we had done that. Not recently. Possibly last year.
It was too loud. Next time, I promised myself, I wouldn't try and do homework in the classroom. I'd go to the library and detention for missing registration would be a small price to pay. Actually, they made you do homework in detention and no one could talk in there. Perfect.
Difference of two squares. That meant factorising, didn't it? That was usually what you had to do to quadratic equations. Or had I already done that? I might have done. Then again, possibly not.
"Do you want me to do that for you? It's been a while, but I used to be quite good at equations." I took a deep breath. It couldn't be her, could it? She'd never been here before. The first time I had ever seen her had been two days ago by the river and again in the same place yesterday. Besides, someone like her would be sent to a special school, wouldn't they? Yet it was her voice. I turned slowly, praying I was mistaken.
She sat on the desk of the girl who's neck I had apparently so nearly severed, with arms folded and a small smile playing across her lips. She wasn't wearing uniform. If anything, she looked more scruffy than when I had seen her before. There was a large hole in the left knee of her faded blue jeans, which looked more like it had been made through too much wear than put in by the makers as part of a stylistic choice, like the holes in the fashionably shabby clothes you saw others wearing outside of school. There was a dark stain on her bright red t-shirt, the laces of her trainers were undone and dragged on the floor behind her and there was a twig sticking out from behind one ear, accompanied by some leaves which had become ensnared in her knotted hair.
"What are you doing here?" I asked. "I've never seen you here before. Have you come to tell me that I made you up again? Because if you have, don't expect me to listen." First I had liked the girl, then I had felt sorry for her, now her persistence and apparent pursuit of me to the school was starting to grate. It might have been a coincidence she was here, but surely not.
"Well, I could do a lot of different things, it really depends on you." Said Araera. "For a start, I can do that algebra question for you. X is equal to five or seven, it's quite simple really." I raised my eyebrows, but she just shrugged. I turned back to the paper. X equals five or seven.
"You wouldn't like to do the rest of it for me, would you?" I offered her the pen, but she put her hand over mine and forced it back onto the desk.
"I can give you the answers to the questions, but I can't write them down. I'll tell you what to write, you write it." I wouldn't ask, I just wouldn't ask.
It took her all of five minutes to complete what would have taken me hours. It was possible she had bluffed her way through it and got everything wrong, but her answers generally seemed to make sense. She even told me the working to put down, so that it wouldn't look like I had just copied the answers. I wondered why she hadn't wanted to write anything down herself, unless it was to disguise handwriting. That would make sense.
"No, though that's a good idea. I suppose if everything was written in my handwriting and you could see it all it would make it a bit obvious the work wasn't yours, but that wasn't why I made you do the writing. " That mind reading trick again. One day, I would have to work out how she did that.
"Now, I've done you a favour. Could you possibly give me a favour in return?" So that was why she had helped me, she wanted something. That was probably why she had come here in the first place and why she had talked to me by the river. Did she need money for something? Money for what? Perhaps she had run away from home, perhaps that was why I had never seen her before. I slipped my hand into my pocket to look for loose change. Maybe if I just gave her something she would go away. I felt the crinkle of paper and withdrew a twenty pound note from my pocket.
"Not that kind of favour." She assured. "I don't want your money, I don't need any. All I want you to do is let me prove I'm not real." This again.
"If you don't manage to convince me by the end of the day, you leave me alone, is that a deal?" Araera smiled and clapped her hands.
"I knew you would let me show you. Alright, let's start with these other people around me. How much notice have they taken of me?"
"What have they said to me?"
"Nothing." I hissed, "but nothing is precisely what that proves. I've been in this class three years and they've barely said a word to me in that time."
"They haven't said a nice word." Araera reasoned. "They've told you to get out of the way and to be quiet and that you're annoying. Lots of times." I laughed bitterly.
"Araera, there's this thing called tact, where you avoid saying things which might upset other people, have you ever heard of it?"
"Oh that." She waved a hand dismissively. "I've never really understood what that's for. Not mentioning a thing doesn't make it any less true, or any less upsetting, so if it's an effort not to mention it, then just talk about it and get it over with. Anyway, so the lack of attention payed to me by your classmates does nothing to convince you of my non-existence?" I just shook my head.
"Nothing whatsoever." Had she hoped that the lack of attention would be enough to allow me to share her fantasy? If so, she would be disappointed.
The door rattled in its frame as Mr Green, our form tutor, kicked it open. He strolled in with his eyes on the screen of his phone, lost in texting whoever it was he messaged constantly. He was only twenty one, I asked him once. Still fairly young, that was probably why he still behaved like a teenager. Including texting in class, instead of teaching us. That suited everyone else fine. Just not me.
"Right kids." He said languidly, dropping into his seat and sticking his feet up on the desk. I'm not a kid.
"That just makes you sound like a kid." Said Araera, who had produced a chair from somewhere and sat down next to me. I elbowed her in the ribs. Ah, the register. How did She plan to navigate her way through this one? She would be noticed. Mr Green was rattling off names.
When he reached the end of the list of names he turned back to his phone, but Araera stood and cleared her throat. "Excuse me? You haven't called my name. Well, of course you haven't, because I'm not supposed to be here, but I mean, you haven't noticed me, though I do look a bit out of place. Well, of course you haven't noticed me, because I don't exist, but my friend, who is the only one who can actually see me, doesn't believe that, so if you would kindly explain everything to her. No, of course, you can't. She's the only one who can see me, so you have no idea I'm here."
He didn't look up. He really didn't look up. None of them did. There wasn't a laugh, a smile or a glance in the direction of the mad girl. Araera stepped up onto my desk and started drumming a rhythm on the ceiling. Beethoven's fifth symphony. Still no reaction from the class. She started singing, though yelling to the hint of a tune might be a more accurate description. She was doing it deliberately, to annoy me. She probably had quite a nice voice, she spoke well. Her feet were doing a tap dance on the table top to replace her hands on the ceiling as she let them fall to her sides.
Still no reaction from anyone. Nothing. They talked as normal and ignored her antics.
So she had convinced the class to help her. She was probably the friend or cousin of one of them. They had brought her in to have a laugh at my expense, hoping against hope that I would believe her lies, because that would be so funny. It was all a practical joke and on no account was I going to fall for it.
Araera just looked at me, then stepped down from the desk again and moved to the window. Daintily, she perched on the windowsill. It took me a few seconds, but then I realised what she was going to do.
Her back bent out of the window and she flung herself out in a graceful backflip.
"Araera." I whispered, struggling to breath. Then breath came to me in a rush and I screamed. "Araera!"
There was salt in my mouth as a tear ran into it. She was dead. I couldn't cry properly, she wasn't my friend, but she had just jumped out of a window. Why? If this was a prank, why?
I had everyone's attention now. They had all heard my scream.
"Olivia?" Mr Green asked. "What's happened?"
"What's happened?" I shouted at him, my face ruby red. "What's happened? A girl just killed herself, you can't have missed it." He looked puzzled, genuinely puzzled. They all looked like they had no idea what I was talking about.
I ran to the window, bracing myself for blood and a broken body. Bracing myself to stare at death.
Below me, Araera lay on the ground, unmoving. For a second. Then she stood up, dusted herself down, looked up at me and winked. Just winked, casually. The classroom was on the fourth floor. How had she done it. She had jumped, but she was there. Unharmed. Mr Green had joined me at the window and gently placed a hand on my shoulder.
"Olivia? Olivia, there's no one there?" He was serious this time and worried. He wasn't joking. I should have left it there, but I had to check. Araera was still there, waving now. Waving, how could she still be alive, let alone standing there waving? I turned to Mr Green.
"You don't see anything? Dead or alive?" He stared at me and I could see concern in his eyes. He didn't look like a teenager anymore.
"I don't see anything. Dead or alive."