Tutoring [Tammy]

Ugh...

If there was one class that Zhang Linhua did not like, it was math. She remembered how the boys at her old school always made cracks about how girls were supposed to be super smart at math and it always annoyed the crap right out of her. Truth was that not only did Linhua hate math, but she really wasn't doing well in it. Not that she'd managed to completely bomb and actually flunk the class. But at a constant C- or D+ always bordering each other, she definitely wasn't doing so hot.

The young woman stared at the sheet of paper before her and the textbook right beside it. All that was listed on her paper was the numbers of problems. Not a single solution or even a view of mathematical progress. Just... blank. Much like her mind felt right now. By no means was Linhua a stupid girl, but the numbers just seemed to completely wipe out whatever thought process was going on in her mind.

After a moment, the silence of her mind was interrupted by the ringing of a bell to dismiss the class. Linhua took a deep breathe and sighed in a bit of relief. Now she didn't need to worry about it until later. Much later. Like other students, she began to pack her things away, stuffing the paper in between the pages of textbook before placing the book itself into her bag. Slinging it over a shoulder, she began to depart the class. Albeit trailing somewhat behind most of the other girls who were already way outside of the door and into the hall beyond.

Tammy was one of the teachers in the mathematics department of Fukui; now while this was an academic institution meant to cultivate the artistic talents of its fine young women, there was still some degree of importance given to core subjects. She was really into the whole maths gig, and she had accomplished a lot in the field of mathematics. Regardless, in this class, Tammy had distributed some problem sheets for the girls to work on. But time had passed by really quickly, since the teacher got carried away trying to teach some tips and tricks of the trade with various difficult concepts in trigonometry. Well, Tammy knew most of the kids had a difficulty with the identities, so she was teaching some ways to help memorize them. Yes they would be on the hypothetical test.

The bell rang, and the students began to mill about putting away their things and packing up to leave class. Tammy looked and saw a particular student trail behind most of the other girls, Tammy quickly took a look at her attendance sheet, matched a name to the face, and realized she was one of the students who was not doing as well as the others. With a look of concern on her face, Tammy spoke up and called out to Linhua's attention. "Miss Zhang! Please wait for a moment, I need to have a word with you." she said with a strong voice, but not one of anger or disappointment.

And just as Linhua was about to get out the door, there was the voice of the teacher calling her to wait. The silver haired girl looked over at her teacher, a redheaded woman. Tammy Mu as she introduced herself, or just Ms. Mu. It was easy for Linhua to see the same Chinese ethnicity in the other woman. Honestly, it made her want to ask more about the older woman. But why would she want to talk to a student outside of class? Eh. At any rate however, she stepped away from the door frame and towards Ms. Mu's desk.

"Yes, Ms. Mu? What do you need?" Her parents taught her to always be polite, even when agitated or otherwise. Though right now, she was just plain wondering what the older woman wanted to talk to the younger girl about.

Tammy smiled warmly, though it was small and not a very wide smile. She looked at the young student - up close, she did look rather good. But now was not the time for flirtatious thoughts. Tammy had to focus on the problem at hand - Zhang Linhua was not doing well in her maths and it seemed to show, both in her attitude towards the subject and her grades. Tammy gave Linhua a gentle tap on the shoulder. "I'd like to say you're not in trouble, but I've noticed that compared to your peers in class your mathematics grades are not doing too hot," she said. "I'd like to offer some extra tutoring so I can help you out. I understand that one of the top reasons for poor performance in maths is a lack of understanding, but a lack of understanding is usually derived either from being slow in logical thinking, or just having a sheer lack of appreciation for the subject as a whole." Tammy said.

"Therefore, I'd like to see if I can work together with you to try and improve your grades and hopefully help you to like the subject more. Would that be alright? If you like, we can have a consultation and tutoring session in my office."

Linhua winced slightly as the redheaded teacher pointed out what was surely a glaringly obvious thing to her. At the offer of tutoring from Ms. Mu herself though, Linhua's expression brightened. She didn't know about liking the subject anymore than she did now, but having a teacher tutor her would definitely help her understanding of trigonometry at least. "Yes, Ms. Mu! I'd love to have you tutor me!" She gave an excited smile, her face lighting up much more from the tired look she possessed just a moment ago.

And since it was the end of classes for the day, that meant she had all afternoon to devote to learning from Ms. Mu's tutoring session. And well, it didn't hurt to be able to spend that time with an attractive older wo- Okay, she needed to stop thinking along that line before she did something stupid like embarrass herself in front of said attractive redheaded teacher.

The student's expression seemed to light up upon being given the offer to be personally tutored by Tammy. It would almost seem like it was too sudden of a change of heart, but perhaps Linhua was just rather eager to try and understand the subject matter, which gave Tammy the idea that perhaps she was more open to the idea of being able to appreciate maths; she just needed some help and guidance on how to better understand these concepts. And on top of this, Tammy certainly wouldn't object to having some bonding time with one of the prettier students in her class. Hehe. But again, that could come at a later time. She didn't want to scare Linhua after all. Tammy smiled and nodded, gently giving the student a pat on the back. "Alright, excellent! I'd be happy to try my best to help you out. Now, please follow me to my office. If you want, you can talk to me about anything you like as we go there." Tammy smiled, giving Linhua the chance to talk about whatever she wished, Tammy wanted to get to know her better as well.

It would be some time though before Tammy led Linhua to her office, it was of course the plain standard-issue office that each faculty member had, but there was a distinctly Chinese flair to it - a lot of scrolls with Chinese poems painted on it with calligraphy. There were also more potted plants than usual. Of course, there was a desk but there were two chairs next to each other instead of chairs facing each other. They were both smaller plush chairs. "Well, have a seat next to me here and take out your things so we can chat about where you're having problems, then we can work together to figure out how we can help you understand everything better." Tammy spoke with a more gentle tone compared to her teaching voice which was more assertive and more authoritative.

Linhua walked alongside Tammy, enjoying the presence of the attractive woman. Ah, how she did wish she could find someone as attractive as her. But again, it was about time to get off that train of thought. Especially as some thoughts came to mind during the walk towards the office. "Ms. Mu, do you come from China? My mother and I do, but she only really talks about culture and not much about what it was like over there. I'm curious to know more." She hoped it wasn't too intrusive.

And after actually arriving at Tammy's office, she could definitely see all the Chinese influences around it. As if there was any doubt at this point. Though the way the seats were arranged did make her slightly... nervous. Nonetheless, she took one of the seats, sitting upon it carefully before retrieving the paper and textbook, as well as a pencil. She flipped open to the appropriate page, a hand shifting towards the edge of the textbook while her other hand held the pencil with the actual worksheet beneath. "Well... I'm having some trouble with understanding that since, cosine, and tangent stuff we went over. How does that work again, exactly?"

Along the way, Linhua asked Tammy if she came from China before they arrived at the office. Nodding at that time, Tammy explained it as such. "Well, actually, my parents also came from mainland China, but I was born in Hong Kong. Learned a lot there" she said, smiling. Later on when Linhua was comfortably seated at the desk next to Tammy, the teacher gently and reassuringly wrapped an arm around the student's shoulder, pointing out some of the problems on the sheet that had been given to her. "Ah, so you are having some problems with how the whole trigonometric identity jazz actually works," Tammy said, nodding confidently. "I got it. Yes, some people do have better understanding of some concepts when they're shown how they logically work. So, let's go through this together." Tammy said, getting a piece of paper from the side and placing it in between them. Tammy drew a right-angled triangle on the paper.

"Let's start from the beginning. The identities I showed to the class earlier today are separate from the ones I'm going to start with here, which was from a previous lesson - triangle identities, which include both angles and the side lengths of a triangle like this one," Tammy said. "But I feel we should make sure you know the easier stuff first before we move on to this one."

Tammy proceeded to slowly explain the SOH-CAH-TOA mnemonic, which basically showed how to use 'sine,' 'cosine' and 'tangent' to be able to calculate the angles and side lengths of any given right angle triangle. She pointed out that the sides had special names which corresponded to which length and function you needed to use. "Yes, remember that these sides are named the way they are due to their position relative to the right angle. So, this side is the opposite since it's opposite the right angle, and this one's adjacent because it's right under the right angle, and so on..."

The teacher tried her best to explain the logic behind the basic triangle identity as a basis to give her some idea on how the real trigonometric identities operated. They were basically ratios, and so trigonometric identities served to help simplify really complicated expressions involving these functions. With this in mind, Tammy moved on to the coverage of today's lesson after she was sure Linhua was at least alright with the content reviewed so far.

"Any further questions before I move on to today's coverage?"

Linhua felt Tammy's arm around her shoulders, a slight flush coloring her cheeks. After what she was thinking earlier, it was hard not to get a bit embarrassed or excited or something when that happened. The girl nodded as her teacher pointed out the problems, trying not to act awkward around her. Afterwords the latter retrieved another sheet of paper and drew out a triangle before beginning to slowly and carefully explain the details of the various formulas to Linhua. And admittedly, it did start to make some more sense than before. There was still some parts that seemed odd to her, but that was more to do with the stuff that the math teacher had lectured about today.

"I think... I do get it better now." She repeated what Tammy had said before, even doodling out a triangle of her own and writing the basic formulas nearby it. "Like that, right?" She asked quickly for confirmation before nodding her assent.

"Yeah, um... Would you mind talking about that again too? I think it'll make more sense now at least." Linhua admitted with a bit of shyness in her voice. It felt pretty humbling to have Tammy doing this for her.

Tammy caught the slight flush coloring Linhua's cheeks, a sight that made the teacher smile (though it was a genuine one). The student seemed to try her best too, she was repeating what Tammy was saying as they went along, she even drew a triangle herself and tried to write the basic identities nearby it. Tammy smiled and nodded; it seemed she at least was able to get down the triangle identities fairly well, though she wanted some confirmation as to how it actually worked. Tammy smiled again. "Of course I don't mind going through it again," Tammy said. "Let's review it again, and make sure that you've gotten it." Tammy pointed out how each functions were ratios of the side lengths of the triangles. The function of a given angle was basically dependent on which sides you divided: sin of an angle was determined by dividing the side opposite the angle over the longest side called the 'hypotenuse,' and such.

Tammy went on to explain that this basic logic was how we could come up with trigonometric identities. For today's coverage, Tammy discussed Pythagorean identities, which was grounded on the basic relationship that sine squared of an angle plus cosine squared of an angle added together gave you one. It was called as such because it is a variation of the quintessential Pythagorean theorem. "Today's coverage was with these identities, and how we could write literally any trigonometric function in terms of the other functions. I can explain how we can reach the base theory, if you'd like."

Linhua began to realize something. Her teacher's smile was pretty. Really pretty.

The younger girl nodded as Tammy spoke, trying to put her mind back into the work. The redhead went about explaining the terms in a bit more detail, as well as explaining how one found the numerical value of certain ones. She also mentioned some kind of theorem and explained how it was pretty essential. All in all, it was quite a bit to memorize, but at least now Linhua was starting to figure it out. And there was that really pretty smile again.

"I-I think I'm starting to get it, but how do we reach the base theory again?" She responded to the question after a moment more.

Tammy nodded as Linhua asked how everything pretty much began. "This is actually described in the textbook, but it seems pretty intimidating," Tammy said, before drawing a pair of graph axes and then using the axes to draw a circle, marking it with a radius of 1. "This is a unit circle, which has a radius of 1. If, coordinate (x,y) is a point on the unit circle's circumference... Or this outline here," Tammy said, putting a dot on the circle itself. "Then the positive values of x and y are the lengths of the legs of a right triangle whose longest side has a length of 1. That's how we get the formula x squared plus y squared equals 1..."

Tammy proceeded to explain, using the most simple terms she could muster, how using this knowledge of Pythagorean theorem on the unit circle leads to the Pythagorean identity Tammy mentioned earlier. "And that's the basic foundation for the coverage in this class. Normally, teachers would just force students to memorize the related identities, but I try to make an effort in making sure every person gets them clearly, so I start with just the basic ones today. I do not cover the more advanced reciprocal functions cosecant, secant and cotangent because we are still learning, so I just make sure everyone memorizes the basic nine identities."

"The legs of... the right... triangle..." Linhua repeated as she began to copy the work. But rather than the legs of the triangle, she couldn't help but let her eyes wander. Slowly, deliberately, she peeked down at Tammy's legs from the corners of her eyes. After a moment she noticed Tammy actually looking at her and not the paper once more, which caused the younger girl to quickly leap back in with the last thing she heard. "Ah, right. The x squared and y squared..."

Luckily the next part wasn't so distracting. Rather, they were kind of mind-numbing when Tammy said that there was actually even more and complicated parts to the theories. She was not looking forward to that. But nine basic? Ugh... Linhua winced slightly. Well, she was going to try and do her best anyways. Hopefully it'd get her somewhere.

As Tammy was explaining the theorem to Linhua, she wondered how the student was feeling about all this. However as she looked to Linhua, Tammy noticed the student was a bit startled; as if her attentions were directed elsewhere. Grinning, Tammy gave Linhua a gentle pat. "Hey, I understand that this must be all pretty overwhelming for you," she said. "But I promise you that it really isn't so bad once you set your mind to it. They all look pretty similar, you just have to replace bits and pieces of it with other bits and pieces." Tammy pointed out. "And writing the identities for functions in terms of themselves... Is pretty obvious, so that eliminates 3 of them - you only need to memorize 6 so far." she said.

"But before I move on, let me offer you a little something for being so patient and understanding once we finish our tutoring session today?" Tammy smiled, giving Linhua a small wink.

Tammy pat Linhua as she explained that there was only six more to go. Right... Only six more. That was three so far. Just six more to go. Yeah, she could remember six more as well as the other three. ...It was going to be so very annoying though. Linhua sighed slightly before Tammy spoke up once more, smiling and winking after she made an "offer" to the younger girl. A slight blush returned to her cheeks once more and... Well, she was definitely curious. What could Tammy be "offering" to her?

"A-Alright. I-I'll try my best." Crap, she was starting to grow even more self-conscious of how close she was to the beautiful teacher.

"Right, so as I said, the ones we've eliminated are sine of an angle, cosine of an angle, and tangent of an angle," Tammy explained. "Writing cosine in terms of sine and sine in terms of cosine is actually pretty easy to remember. Do you recall that when you use the Pythagorean theorem, to get the length of either a or b, you get the square root of the difference between side c and either side a or b, depending on what you were looking for? It's the same concept here, except with angles." Tammy said. On the paper, she wrote the following:

sine theta = ±square root of 1 minus cosine squared theta

cosine theta = ±square root of 1 minus sine squared theta

"This goes back to the Pythagorean identity of sine squared theta plus cosine squared theta being equal to one. You just copy the Pythagorean theorem's idea, but with angles. The other four are slightly different. When you want to write tangent theta in terms of sine, you have to divide sine theta by the square root of 1 minus sine squared theta. And for tangent theta in terms of cosine, you actually reverse the positions and replace sine by cosine, so you would be dividing the square root of 1 minus cosine squared theta by cosine theta," Tammy said.

She went on to elaborate that the square root of 1 minus sine squared theta was actually equivalent to cosine theta, and sine theta divided by cosine theta was the same as cosine. The square root of 1 minus cosine squared theta was the same as sine, and tangent theta is equal to sine theta divided by cosine theta.

Then she explained that when writing both sine and cosine theta in terms of tangent theta, both identities used the same divisor - the square root of 1 plus tangent squared theta. The only difference was the dividend. For sine theta, the dividend was tangent theta, and for cosine theta, the dividend was 1.

"Am I going too fast?"

...Half of it sounded like words and the other half sounded like nonsense.

Linhua scratched at her head with the end of her pencil, trying to make sense of what she'd been told. She got about halfway through what Tammy had said before confusion set in. "Ah... Could you repeat that one more time?" As Tammy did so, she tried to work with what she understood out of it already, eventually managing to catch on where she had failed to before. The repeat elaborations helped her understanding, but if this stuff wasn't starting to give her a headache, she didn't know what would.

"Hmm. I think I've got most of it now. There's still some parts that are confusing me, but I'm pretty sure I've got it from here." She faced Tammy and smiled, both proud of herself for understanding and happy to have the pretty teacher's help in doing so. "Would you mind tutoring me more after class? This is really helping me a lot."

Linhua seemed to get some of it but not all of it, yet it seemed to be easier for her to get a grasp on things once Tammy repeated herself and tried to get down the eccentricities of the whole thing. Mostly that substitution was a really cool thing. However, it seemed like Linhua could finally figure things out on her own, at least that's what she said, despite the fact some parts still confused her slightly. She then asked if Tammy could help tutor her more after class. "Oh, you think you've gotten it? Well, that's great!" Tammy smiled, giving Linhua's back a gentle rub. "I'd be more than happy to tutor you more after class especially after seeing that it's been a good help, but you do deserve a good break." Tammy grinned, before giving Linhua a gentle kiss on her cheek.

"I'd like to help you relax for a while. It's after class hours, after all. What would you like to do?" Tammy asked.

Linhua's cheeks absolutely flushed into a blossoming crimson red after Tammy's brief kiss upon her cheek. Even the gentle rub of her back felt absolutely divine at this instant. "M-Ms. Mu..." The words escaped her lips in a small voice, her eyes locked upon Tammy's beautiful face. "R-Relax? Ah, um..." She stumbled over her words, more than a little disoriented after the kiss. "W-well I um... I like to watch sew clothes, or watch kung fu movies, or..." The words fell out in a rapid succession and Linhua's eyes kind of fell slightly to look down at the floor, or rather attempted to before they were drawn towards Tammy's legs again. After the torrent of words ceased though, she managed to look back up to Tammy's eyes again.

"O-Or um... Whatever... You want to do... Ms. Mu."

Tammy smiled awkwardly as she saw Linhua's cheeks flush and she could barely even answer properly! Linhua mentioned she liked to sew clothes or watch Kung Fu movies or... Something else, but then eventually she couldn't speak straight and said she'd do whatever Tammy wanted to do. The teacher found it absolutely adorable and served only to make her like Linhua even more than she already did. "Oh my," Tammy said, now freely exposing her flirtatious side, and releasing her inhibitions. "Did anyone tell you how adorable and beautiful you look all flushed like that? Would you like to go out for some ice cream, then? Or perhaps you would just want to watch a film instead~?" Tammy suggested, trying to pull Linhua a bit closer to her as much as the chairs would permit.

"Or, if you're into it, we could get close and cuddly right in here if you want, I saw the way you look at me...~"