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Cycle sat in a classroom, completely alone. The clock's ticking echoed across the room like a bomb ticking towards a death sentence. Atleast, that's what the fourteen year-old saw Mathematics as; Complete psychological torture. The blackboard stood in front of him, still bearing the remnants of other lessons' chalk marks. Cycle slumped backwards into his chair, his lips pressing into a thin line. Solar had told him to wait for a bit while she prepared the required materials for their little session.
Soon enough, Solar did enter the classroom, her heels clacking against the floor tiles. The sound travels through the room immediately, echoing in the sparse space. "Hey, kiddo. Honestly-.. I'm surprised you haven't ransacked the entire classroom bouncing around by now." Solar was surprised and amused. And even if he did, she could never get mad at her little brother. "Eager?" Curiosity was very much visible in her voice. "NO." Cycle responded immediately, fearing Solar thought he was actually... INTERESTED... In MATHEMATICS.... But she only chuckles in response. "It's fun, I promise." She smirks. "You can't decide that for me!!" Cycle retaliates, and Solar has to hold back a laugh due to his small voice.
She walks over to the whiteboard and writes "Algebra" on it with chalk, in big bold letters. Cycle winces at the mere word itself, feeling like he was about to faint. It was like he was looking at a Cease-and-Desist letter. She then wrote the word "Polynomials" under it.
"So, polynomials are basically just terms smooshed together." She explains softly, writing 3ab^6 on the board. "This, is a term." She pointed at the term, and Cycle nodded slowly. His dramatic demeanor seemed to rub off a bit as he started to focus. "Polynomials have different names based on the number of terms. This here, is a monomial. The name is derived from the word 'mono', which basically means one." She goes to label the monomial with ease. Then, she writes the same term (3ab^6) on the board, accompanied by a plus sign and another term; 7ab. "Now, these are two terms. You call this a binomial, deriving from the word 'bi' which means two."
"But I thought that was the sexuality." Cycle squints at the word. Solar just laughs. "There's a lot of words that start with 'bi'. Bilingual, bicycle, bipolar..."
"Like Hypen?" Cycle asked, genuinely curious. "What- CYCLE!" Solar didn't even know how he came to that conclusion. No anger was shown in her voice though, just complete shock.
Solar takes a deep breath, then continues, copying the same two terms and adding a third term; - 4. Yep, just - 4. "Okay, so now there are three terms, 3ab^6 + 7ab - 4. You call this a trinomial, which the word is self-explanatory." She labeled those terms too. "Every other polynomial from four and above? Those are called polynomial or multinomial, meant only for algebraic expressions with four or more terms."
"Ohh, okay, I understand it now!" Cycle exclaimed, happiness booming from his voice. This earned a smile from Solar.
"Okay, okay, good. Next, let's try evaluating polynomials." This earned a groan from Cycle, but all Solar did was be amused. She started writing on the other side of the board; the same expression (3ab^6 + 7ab - 4.) Then, she started writing the values of each variable on the board. "Okay, let's say that (a) is equal to 2 and (b) is equal to 1. We substitute the variables with their values, making the expression 3(2)(1)^6 + 7(2)(1) - 4." Cycle writes the same expression on a piece of paper.
"What's three times two?" Solar asks. "Six!" Cycle's response is eager and quick.
Solar smiles. "Okay, now one raised to the power of six, what is it?" At the question, Cycle starts counting on his fingers. As he realizes the difficulty (or lack thereof) of the question, he immediately lowers his hands in embarrassment. "Uh- One. Yeah, one, I knew that. Hah. Yeah." His response was rushed, tipped with clear nervousness. Solar raises an eyebrow, smirking. "Oh really?" "Yes, DEFINITELY!" He says, pouting. She lets out a laugh. "Six times one?" Solar turns to face Cycle. "Six!" He responds just as he's writing the answer down. "Okay, now let's solve the second term. Seven times two plus one?" She waits, staring at him. Cycle was focused on the paper, muttering to himself: "Seven times two? Fourteen.. Like my age! Times one... well, that's still fourteen."
"Fourteen!" He finally yells out, and Solar writes the answer down with a smile. He follows quickly.
"Okay, now it's 6+14-4." She's already doing the math in her mind, just waiting for Cycle to answer.
6+14=20
20-4=16.
"The answer is sixteen!" Cycle exclaims, holding his paper up.
"Mhm, yeah, you're right lil' man." Solar smiles again, erasing the things on the board. Cycle frowns slightly, like the numbers and letters were somehow sentimental to him.
"Okay, now we're going to add and subtract polynomials. This is very different from evaluation, since there are no given values for the variables and you can only add liked terms." Solar started to explain again, writing things down on a board. "What are liked terms?" Cycle's head tilted in confusion as he asked the small question. "Liked terms are terms with the same variables and exponent. Like 5bc^2 and 7bc^2." She writes the terms down on the blackboard. "Liked terms like these can be combined by adding the numerical coefficients and copying the exponent and variables."
5bc^2 + 7bc^2 = 12bc^2.
That's what Solar writes down, "Like this." She moves out of the way so Cycle can see the equation clearly. "The only thing that changes is the numerical coefficient. The variable and the exponent stay the same since they are liked terms." She writes another expression below; (11ab^2 - 4^2 + 7) + (-10ab^2 + 6). Cycle groans. "This one looks hard..." Cycle complains, squinting his eyes at the board intensely. "Unless you understand it, I have to agree." Solar's response is one of mild amusement; an emotion that grows with every comment from Cycle. "What you need to do first is multiply the second group with positive one. Since the first term is negative, it'll be converted into positive. Six stays the same."
She rewrites the group below the expression, so it becomes 10ab^2 + 6. The parentheses are removed, and the plus sign turns into a subtraction sign.
11ab^2 - 4^2 + 7 - 10ab^2 + 6.
"Then, you evaluate the exponent. Four times four?" She taps the chalk on the teacher's desk, dust flying off. Cycle pauses, looking down at his paper. His handwriting wasn't the best, but it was still legible atleast. As the answer crosses his mind, he looks up. "Sixteen!"
Solar was observing his behavior, and he definitely seemed more enthusiastic than before. She smiled to herself.
He never got to finish school, and now he was experiencing the thrill of learning.
She was proud of him.
She was pulled back into reality by the sound of pencil against paper. "Good, good. Now the expression is 11ab^2 - 16 + 7 - 10ab^2 + 6." She was rewriting the expression already. "Negative sixteen plus seven?" She asked. You must always follow the sign before the number. "Hahah, that's so obviously negative nine!" Cycle was definitely enjoying this now. Solar rewrites the expression again; it becomes 11ab^2 - 9 - 10ab^2 + 6. (-16 + 7 = -9, the negative sign becomes a subtraction sign.)
11ab^2 - 9
- 10ab^2 + 6
11ab^2 - 10ab^2 - 9 + 6
"You follow the signs before when adding or subtracting polynomials, like 11ab^2 and -10ab^2. Same goes for -9 and +6." Solar encircles the signs. Cycle nods eagerly. "11ab^2 minus 10ab^2?" Solar asks, "1ab^2!" Cycle responds. "You can simplify 1ab^2 into ab^2." Solar writes the answer down. "Ohhh, okay." Cycle does the same. "ab^2 - 9 + 6." Cycle stares at the expression. -9+6. "Hmm... Negative three!" He said as he started writing the sum down. "So, after combining the liked terms, the expression becomes ab^2 - 3." Solar writes the final answer down with an obvious smile, "Good job, Cycle."
Solar granted Cycle a mandatory break, eating churros with him and talking about cats and dreams and whatnot.
After their break had finished, Solar got up from her seat beside Cycle, walking up to the blackboard and picking up the piece of chalk again. She erases everything on the blackboard and writes "Properties of Equality" on the upper part of the board. Cycle looks up at the sound of chalk hitting the surface, immediately writing down what Solar was writing. "You ready?" Solar asked, looking over her shoulder, at him. She sees him nodding eagerly, genuinely interested, and her smile widens.
"Okay, so the properties of equality express that both sides of an equation are equal." Her back is turned to Cycle as she writes an equation on the board; 5b+10=25. Cycle follows suit. "One of the properties of equality is the subtraction property of equality, where you subtract the number with the opposite side from it, in this case, a plus sign, and do it on the other side. So.. 5b+10-10=25-10. You have to remove the constant before the number with the coefficient when finding the value of the variable." As she wrote, she heard Cycle's pencil scratching against the paper, and she knew he was learning quickly. "You cancel out 10, and subtract 10 from 25, which makes it 15." She writes the answers below the equation on the board. "Well... What do you do with 5b?" Cycle's small voice echoed across the room.
"You use the division property of equality, where you divide both sides of the equation."
Solar writes 5b as 5b/5, a fraction that suggests division. She does the same for 15, making it 15/5. "You cancel out 5 from 5b, and divide 15 by 3. The result will be b=3. So, (b)'s value is 3."
"To check if our answer is correct, we substitute the variable with it's given value."
5b+10=25
=5(3)+10=25
15+10=25.
"That makes our answer correct. (b)'s value is 3."
She erases everything on the board as Cycle claps enthusiastically like he was at a talent show. Well... Mathematics can be considered a talent, right? She thought about it for a bit as she wrote "Laws of Exponent" casually on the board. "An exponent has laws?" Cycle asked, definitely thinking about political laws. Solar chuckles. "Not the political kind, if that answers your question?" The amusement was clear as day in her voice. "Oh, hm... Okay then!" Cycle tried to maintain a tone of understanding; he didn't know what she was talking about.
"Okay, okay, so... The laws of exponent are rules you have to follow when evaluating a number containing an exponent, like 5^0." She writes the number on the board. "This number uses the Zero Exponent rule, wherein any number raised to the power of zero is equal to one." She writes "=1" beside the number. "So, take [7^23(5b^10-4)]^0. It's all equal to one, because of the exponent's value outside of the parenthesis; zero." As she wrote, the sound of Cycle's voice saying "Oh.." reached her ears, making her snicker.
She writes another example; a^3/a^2. "With this, you use the Quotient rule. It's where you subtract the exponents and copy the base."
=a^3-2.
"So, the answer is a, because any number raised to one is just the number itself." She swore she could see the gears in Cycle's head turning. She paused for a moment, waiting for him to finish so she could make sure he fully understood everything. "Done yet?"
"Yeah!" Cycle's voice was happy. The type of happiness that came from understanding a difficult topic. "You seem happy now. I swore you were groaning and wincing a while ago.." Solar said teasingly. "Hey! That was the old me, I'm a changed person now!" Cycle responded dramatically, making her snicker.
"Okay, okay, fine.. I get it! Anyway, the next rule is..." Solar wrote down another expression; (4x^2)^2. "The power rule. There are two rules related to this, actually. Power of a product and power of a quotient. It's where you distribute the exponent outside the parenthesis into all the digits without an exponent. So for this expression, we distribute the exponent outside the parenthesis. It becomes 4^2(x^2)^2."
"Using the power rule, we multiply the exponents in the parenthesis by the exponent. (x) raised to two becomes (x) raised to four."
4x4=16
16x^4
"The answer will be 16x^4."
She looks over her shoulder at Cycle, who was quietly taking notes and looking up occasionally. Then, she continued with the product rule, wherein the exponents were to be added if they had the same base number.
(a^5)(a^4)=a^9
She also covered three properties of Equality she had missed a while ago. The reflexive property, wherein a number is equal to itself (a=a), the symmetric property, wherein a number is equal to another number (a=b), and the transitive property, wherein if a number and another number are equal to a third number, the first two numbers are equal to each other (a=c, for a=b and c=b.)
After the oddly thrilling Mathematics session, Cycle and Solar were sitting together in their shared bedroom, reviewing the notes Cycle had wrote. His part of the room was painted in hues chosen to replicate the night and decorated with stars and moon-themed appliances and toys. Solar's was the opposite, warm, sunny hues, and sun-themed items everywhere. To be fair, they were both elementals of the moon and sun respectively.
"Do you still hate math, Cy?" Solar asked, amused.
Cycle looked up at her with big innocent eyes.
"When can we start the next lesson?"
Solar just... started laughing. Hard. Cycle didn't know what was so funny, but he started laughing because she was laughing. Maybe Cycle COULD continue his studies, after all. And he seemed pretty enthusiastic about it too...