Welcome to Precalculus

[Adam Deulane]

Second period, first day back from winter break. Adam's first day as a teacher would be starting in nine and a half minutes. He had written his name on the board and erased it twice because it didn't look authoritative enough. Mr. Deulane. He finally settled with a heavy, properly illegible scrawl for a math teacher. As he fiddled with his notes and waited for the students to show up, he found himself thinking wistfully of his old job at the research lab. Why had he ever left? Well, he hadn't, they had fired him after he shouted an intern into tears for using a permanent marker on his whiteboard.

The final bell rang, noise rose in the halls, and students began to filter in. Adam thought he recognized one or two that his sister had pointed out to him, but he remained standing grimly at the front of the class with his arms folded across his chest. His nerves grew with every student that slouched through the door.

From his own brief stint in high school, Adam remembered how crucial first impressions were. Most of the time, Adam's first impressions were about what could be expected from a cornered wolverine. But most of the time, he was not teaching math. He clenched his teeth against the mounting stage fright. He'd be damned if the little monsters didn't learn something by the time he finished with them.

The bell rang again, and shuffling silence fell on the room. Adam, unsmiling, spoke for the first time. “My name is Mr. Deulane. Welcome to precalculus.”

He picked up a marker from the front of the room and began rolling it between his fingers. “In Tibet, there's a monk who lives at the bottom of a mountain. At sunrise, he leaves his house and walks up the path to the top of the mountain, where he arrives at sunset. The next morning, he leaves the mountaintop again at sunrise, and arrives back at his house at sunset again.

“The monk can walk at whatever speed he wants, speed up, slow down, even turn around and go backwards. But he never leaves the path, and he always arrives at exactly sunrise and sunset.” Adam placed the marker on the desk next to him. “During the second trip, is the monk ever in exactly the same place on the path at exactly the same time as he was the day before? If so, can you prove it?” Adam looked around at his students, adjusted the marker again, and waited.

[Adam Deulane]

((We went over this problem at the beginning of my calculus class. If anyone wants to know more about it for a mathy character or just out of curiosity, send me a message! ))

[Deleted]

Why me? Dawn repeatedly asked them not to put her in a math class, but of course you have to have math or you fail. Dawn sighed and walked toward the class room with the impression of being neutral. She didn't even mind the teacher's expression as she walked in. His impression showed her that he was strict and not to try to sleep.

She sat down in the front since nobody else seemed to be there and listened to the teacher. Just because she had a nonchalant attitude didn't mean she didn't at least pay attention. She wondered why the guy always walked up the hill. Was he bored? It was smart to do it in the day time, that way he didn't get killed by wolves or something.

The question confused her. If he said that he didn't walk the same speed and everything? Sure he arrives at the sunrise and sundown, but what about daylight savings time and leap years? She didn't see how someone could answer this question. She looked around and didn't see any brave people...man up! She sighed and shrugged her shoulders. "Uh..I don't see how we could tell. Since he does not walk the same way. I understand that he comes out on sunrise and comes back on sundown. Daylight savings time and leap year make me question on how to answer this question correctly.." She was shocked to hear herself speak. Maybe it was because this guy meant business and he seemed to be blunt like her.

[Adam Deulane]

Adam had been hoping there wouldn't be a temporal elitist in the class, but, as usual, Murphy's law rang true. He resisted rolling his eyes. He didn't want to shoot the girl down or no one else would try and answer, but she was completely missing the point of the problem. "That's true, the timing is important," he conceded. "If the monk doesn't leave and arrive at exactly the same time each day, the problem becomes too ambiguous to solve. But for the sake of clarity, we assume that this mountain in Tibet has no leap years, the length of each day is exactly the same, and there are no earthquakes or continental drift or anything else to alter the length of the path."

To try and help, maybe the class had some of those visual math people he always heard about, Adam turned around, took the whiteboard marker, and drew a large upside-down V for a mountain, a small house next to it, and a long wiggly line connecting the house and the mountain top. As he drew, he said, "Don't get caught up in the details of how long the day is or if the monk is on the left or right side of the path. This is a reasoning problem, not a riddle, there are no tricks and you have all the information you need." He capped the marker and turned back around. "You don't need to have the answer to speak up, just ideas is fine."

[Deleted]

She nodded her head and listened. She didn't know that it didn't matter but she was glad there were no tricks. She watched him turn around and draw. She was glad he was because she was all visual. She took out a pencil and paper and wrote down what he told her and then the picture. She wasn't upset that she had it wrong, nobody else seemed to be answering.

She looked back up to listen to him once again. She nodded and waited for someone to answer him. Someone had to have an answer for this. She looked at her notes and thought some more about the problem. She felt so dumb, she wondered if she was thinking too hard, she did that a lot. She sighed and just looked at him for a bit. She didn't think he would be in a different spot. She knew a smart answer would be just to do math, but she wasn't like that. She shrugged her shoulders and folded her arms as she looked at the question again. Then she looked around while rolling her eyes. Someone had to know the answer to this easy question.

[Terrance Kellam]

First day back... and he had precalculus. The gods were just TRYING to torment him. TK had this bright idea to drop out of school almost two years ago after failing his junior year for the second time, only to want to come back and try to finish his junior year in the middle of the school year.

Great plan, genius.

But he didn't have much of a choice. Life wasn't too great and if he didn't want his dreams to continue to torment him, he was going to have to change. The Air Force was in his future; he wanted to become a pilot. And according to the internet, you had to have a full bachelor's degree. Meaning, TK had no choice but to go back and finish high school.

When he was 17, he always sat in the back, always cracked jokes, never payed attention. What for? His powers were the gateway to his future. He didn't need to know anything about World War II, or what the diameter of the Earth was, or how volcanos worked. Granted it was all still true, at 19 he knew he would have to think about things a bit differently. Depending on his power for his life meant that if it was ever taken away or became irrelevant, then he'd become nothing.

So, T.K. chose a seat in the third row -- the front was a bit too intimidating for him. His eyes scanned the board, looking over the monk and the mountain problem. None of it made sense. How do you spend your entire day walking from the top to the bottom and vice-versa? What was so important that he had to make the trip? WHEN WAS THIS GOING TO BE USEFUL OUTSIDE OF THIS CLASSROOM?

No, he thought to himself, you can't think like that. Try to answer the question... He sighed, listening to the teacher and the female student. All he wanted to do was talk to people, go out and have drinks, celebrate nothing. But instead, he was trying to get an education. "This is ridiculous," he accidentally said out loud. The old him was popping up. He had to keep going to get it out of his system. "Maybe it's just a little hill and he took forever to walk like two feet," he laughed. That felt muuuuch better.

[Adam Deulane]

Adam's nerves and disbelief swelled with the silence in the classroom. Had he given them a problem that was too difficult, or were they all idiots? This was one of the highest math classes offered here, and no one had so much as a whisper of an idea? And here he was, standing at the front of a class like a moron, barely older than most of them...

That was when an older boy from the third row spoke up. Any hope Adam had for some traction on the problem was quickly dashed, and his dark eyebrows sank dangerously low over his forehead. "Shut up and learn something, Mr. Kellam, or leave." He pointed to the door with his whiteboard marker. "By all means, I'm sure Ms. Rivera would love you have you back in beginner's algebra."

[Deleted]

When all hope felt drained away, that was when a moron blurted something stupid out of his mouth. Dawn's opened mouth of almost answering the question turned into a straight line and she turned sharply toward the moron. "Oh really? Then I guess you can answer it..." She glared at the guy, not even caring that the teacher was speaking now.

She felt dumb already because of how hard the question was. She smirked at what the teacher said and stared at the guy. At least there was some excitement. She then sighed, ignoring the moron, and turned toward the teacher. "Okay, back to what we were doing before a moron interrupted. So the monk can walk anyway he wants as long as he comes back by sundown and he starts walkin' out at sunrise." She nodded her head at what she understood and kept thinking. Bad enough that she was trying to think, know there was a trouble maker. "Um...I feel so dumb..suck at math." With that she slouched and thought about at least giving an opinion. " I don't think he can be at the same place since he can go wherever he likes. I believe we can prove it...but I have no idea how to prove it." She nodded her head and let a smile cross her lips. She wasn't being a show off, she felt that this poor teacher needed someone to try to answer it. "What are the steps to prove it?..." Yep, a blunt Dawn was a good Dawn.

[Terrance Kellam]

Well, he should have expected such a reaction. He did from the teacher, at least. T.K. thought he was just the slightest bit meaner than a normal teacher. What teacher tells high schoolers to shut up?! Oh, wait... I'm almost twenty years old... Nevermind... Being a "child" was something he couldn't claim anymore.

The comments from Dawn were completely unexpected, but it wouldn't be the first time he had encountered someone like her in his classes. Being the class clown wasn't an easy job, and he was bound to piss a few students like her off. "La chouchoute du professeur..." T.K. said in her direction. His power came in handy when he wanted to call someone a name in public and out loud and have only select people understand. And if the intended target understood? Then it was even funnier.

But aside from his remark, he knew he had to at least give this ridiculous problem a try. Rolling his eyes at the two who didn't seem to find his antics amusing, he turned his attention back on the board. What was even better than annoying the teacher's pet and having the teacher think you were a complete idiot? Getting the answer right.

His "joke" wasn't entirely that; he noticed he could have been on the right track after looking things over again. "Well, why can't the mountain be a hill? And why can't the actual trip only take two minutes? You say that he can stop and do whatever he wants on his journey, correct? So why can't he walk for a minute, sit on a bench for the rest of the day, and then get up when it's time to leave?" He leaned back in his chair and crossed his arms. To him, this could be the only answer. And if the real answer required the use of formulas and graphs and mathy-stuff, he was most likely going to be visiting Ms. Rivera soon.

[Adam Deulane]

The lights in the back of Adam's head clicked on. Finally! He gave Dawn a small nod. "Yes, it does seem that way at first. Don't get caught up on thinking about this as math. This is a logic problem that we can express mathematically, as I'll show you, but no numbers at all are required for the solution."

Adam turned back to the white board. "Your questions are related. Mr. Kellam, you're asking why the trip couldn't take two minutes. I'm going to make the problem a little more concrete. The road from the bottom to the top of the mountain is 12 miles. According to the problem, yes, the monk could spend only two minutes traveling." He looked over his shoulder and raised his eyebrows at Terrance. "He would just have to go 12 miles in two minutes while waiting around for the rest of the day." He hesitated. A joke...? "Not an easy trip, but he's a very fit monk."

Adam chewed on his lip for a minute, crossed his arms, and leaned against his desk. "Let's take a step back and make this simpler. Let's say that instead of walking however he wants along the path, the monk has to walk at a constant speed. That is, he can't speed up or slow down. How about now? Will the monk ever be in the same place at the same time?

"Ms. Ramona, you asked about problem solving methods." Despite the students not having introduced themselves, he seemed to know everyone's names. He was the sort of teacher who sat down and memorized the enrollment list just to try and impress his students. "Graphs are a good place to start. They're a way to visualize patterns that can be difficult to see conceptually or in equations." He drew a large L on the whiteboard and labeled the bottom of the L 'time' and the stem of the L 'distance.' "Units don't matter here, since we're just using the graph as a tool to think about the problem. Here," he marked the bottom corner of the L, "Is where the monk starts on the first day, at the bottom of the mountain at dawn. And here," he marked the upper right corner, "Is where he ends, the top of the mountain at sunset. Can anyone tell me what the line of the monk's path might look like if he traveled at a constant speed?"

[Deleted]

He nodded at me? Dawn had to blink a couple of times to wrap that thought in her head. Not many teachers liked her bluntness, guess he was different. Dawn was good at logic problems, so she thought. This was confusing her, but maybe it was because it was math. She nodded back at him anyway and listened.

She chuckled at his joke. It sounded like a joke too Dawn anyway. Nobody could take two minutes in twelve miles unless they were superman or someone with a gift that had to do with speed. She made sure to ignore the student who was a clown since that would probably get him bored. Back to what matters She wrote down what he drew and looked at repeatedly as he talked. She was more visual than anything else. Thank God, it's a constant speed. Much easier. She nodded her head while trying to remember not to think too hard. She was about to say 'unless there was a storm or something', but she stopped herself.

She was shocked he knew her name without her even saying so, was that his power? Did he even have one? Listen!! She blinked before listening again. She nodded her head already knowing that, but she liked that he was explaining things more clearer. She drew the L thing while listening to him and drew everything else. She wasn't a teacher's pet, but she wanted to be a clinical psychologist and for that she needed to pass. Hahaha! 'at dawn'...pay attention. She sighed and looked at her paper and back at him. Oookayyy...A constant speed means a straight line right? Since if it went up or down it would be him slowing down or speeding up..okay. With that thought she just drew the line on her paper and waited. It was just a straight, horizontal line that went from the bottom corner of the L and then just straight across as if she was marking something off. "There..." She wasn't sure if she had it right and she didn't feel like talking so she just turned her paper toward the teacher and looked at her desk. I suck, this is horrible. Probably got the whole thing wrong.

[Terrance Kellam]

". . ."

T.K. stared as Mr. Deulane continued. Sadly for him, he was done. He was hoping the problem was more of a puzzle, something they could just look and figure out. Seeing that it now involved 'mathy-stuff,' the stuff he could barely grasp, he knew it was over. His attempt at impressing his teacher and pissing off the teacher's pet had failed.

He didn't fully give up though. After all, he had to pass this class so he could move on. So he could graduate. So he could go to college. So he could join the Air Force. So he could become a pilot. Some stupid monk's adventure up a mountain was not going to stop him. T.K. scribbled the notes onto a sheet of paper, trying his best to grasp the concepts that were being explained.

The only way he made it through his other Math classes through a combination of understanding small things that his power could grasp (small patterns and such) and cheating. He didn't want to have to do the latter, but it was becoming apparent it might be his only way out.

"The path would look... like... a path?" He wasn't understanding how all this was supposed to make sense.

[Adam Deulane]

A girl in the back put her hand in the air as if she was dipping her fingers into shark-infested water. Adam nodded at her. "It would be a straight line, right?" she said, echoing Dawn's thoughts. "Going diagonally from the bottom to the top of the graph?"

"Yes! And in a way, Mr. Kellam, that does represent the monk's path, though I don't think that's what you meant. Or rather, it represents the monk's journey as a whole. Where he goes, and how long he takes to get there." Adam drew a line splitting the graph in two.

"In this graph, the x-axis represents time, and the y-axis represents the monk's position on the path, with the top being the mountain and the bottom being his house. According to the problem, we know he starts at his house at 6AM and ends up at the mountain at 6PM, so that's where the line has to begin and end. The slope of the graph," he ran his pen over the line again, "represents his speed. In this case, his speed never changes, so the line is a straight line.

Adam glanced over his shoulder at the class. A few of them were starting to sit up in their chairs. The sick knot of nerves in his stomach loosened a bit. Maybe he could really do this. "This is what his trip to the mountain looks like on the first day. Can anyone tell me what his trip on the second day would look like, when he starts on top of the mountain and ends up at his house?"

[Deleted]

She looked at what he was doing and redrew her graph. She was getting it finally. This man was what they needed, not one of those 'do whatever' teachers. She smiled and nodded her head as she wrote down a few notes of her own. Ask him to be your tutor.

She looked up and sighed. "It would be from the top to the bottom?" She shrugged her shoulders and looked down. Why didn't she let someone else answer?

{sorry for the shortness!!}

[Terrance Kellam]

T.K. saw his teacher begin drawing things on the board and sighed. He was officially tapping out. There was no way he was going to be able to figure out this problem now. He leaned back in his chair fully now, even drooping a bit. Maybe someone behind him could answer it now that he had moved out of the way.

He couldn't help but feel the entire answer, or at least part of it, was glaringly obvious. So, from bottom of the mountain is diagonal...Maybe... Another sigh. This was when Dawn answered. He looked from her to the board, cocking his head to the side as he examined it a bit. I mean... it SEEMS right...

T.K. was going to seriously have to join a study group or get personal tutoring now. This year wasn't going to just fall in his lap just because he wanted to try and learn like he had hoped.