The moment the bell rang, cutting through the classroom, the Class A teacher sensibly put down the chalk. There were no extra reminders and no intention of keeping the class late. He left the podium with a spring in his step, practically making a "quick getaway."
Only a few "study addicts" in the back refused to be outdone. Clutching their workbooks, they hurried after him, and soon the hallway filled with the faint sound of questions. Clearly, they meant to live by the motto, "If studying won't kill you, study until it does."
"Uesugi Hara, how do you approach this problem?" Morishita Ran's voice came from up front. She turned around with an open notebook in her hand, her finger pressed to the prompt of a function problem.
Uesugi Hara glanced at it and explained the key idea in a few words. Then he picked out several similar problems from his workbook and handed them over. "Do these, then let me check them."
Morishita Ran took the problems, her eyes widening as if to say, "What kind of person are you?" She looked utterly disbelieving.
Uesugi Hara paid it no mind and turned to Kamuro Masumi beside him, the corner of his mouth curling up. "Kamuro, after 'swimming' in a sea of knowledge for so long, do you feel like you're about to turn into a 'freshwater fish'?" The teasing in his voice was impossible to miss.
"What new way of insulting me is this?" Kamuro Masumi lay sprawled across her desk, her voice carrying the exasperation of someone who was about to laugh in spite of herself.
She had still been mulling over what happened in class. Just a moment ago, the teacher's explanation had been like a clear map, and she could follow the reasoning. But in the blink of an eye—no, she was sure she hadn't even blinked—the next moment she'd suddenly wandered into a fog, unable to find her bearings. Where the hell had the path gone?!
Seeing her like that, Uesugi Hara stopped teasing her and pulled his notebook from his desk, handing it over. Its edges were a little worn, but the pages were neat and orderly. He'd marked the key points from the lesson in red, added small diagrams beside the important formulas, and even annotated the teacher's passing remarks in blue.
"Don't just sit there staring into space." He pointed to the section labeled "Function Monotonicity and Graph Transformations." "The teacher said, 'The sign of the derivative determines whether a function increases or decreases, just as the slope determines which way you climb.' Where did you get stuck?"
Kamuro Masumi stared at the analogy circled in red, frowning. "But then he suddenly jumped to 'the second derivative affects the curve's concavity.' I can't connect 'concavity' to the 'slope' from before."
Uesugi Hara picked up a pen and drew a little mountain on a scrap sheet. "Just think of it as climbing a mountain. The first derivative tells you how steep the path beneath your feet is. If it's positive, you're going uphill; if it's negative, you're going downhill. What about the second derivative?" He drew a little person on each side of the mountain. "On the left, the farther you climb, the gentler the slope gets, like you're nearing the summit. That's a 'convex function.' On the right, the farther you climb, the steeper the slope gets, like you've only just started. That's a 'concave function.' Basically, it's about whether the slope itself is getting steeper or gentler. It has nothing to do with whether you're going uphill or downhill."
Kamuro Masumi stared at the two little figures for a few seconds, then suddenly went, "Ah!" She tapped the notebook. "So in that example just now, when the second derivative is positive, the curve bulges upward? Like the slope on the right?"
"At least my drawing wasn't wasted." Uesugi Hara tossed the pen back into his case. "Now when you look at the teacher's example, does it feel like you're looking at a map?"
Kamuro Masumi hugged Uesugi Hara's notebook and flipped through a couple of pages. The reasoning that had seemed like a tangled mess suddenly fell into place, and even her expression relaxed. "I think... I really get it now. Your notes are a lot easier to understand than the teacher's boardwork."
Uesugi Hara raised an eyebrow, just about to say, "Of course they are," when Morishita Ran came over with her finished exercises, pointing her pen at the last problem. "Uesugi Hara, I can never get the auxiliary line right for this one. How do I find the 'point of symmetry' you mentioned?"
When he turned to look, Kamuro Masumi was already working through the example on her own, her pen scratching across the paper. Every so often she paused, then quickly started writing again. She no longer looked like she was about to fall apart.
"Here." Uesugi Hara took Morishita Ran's notebook and drew a small right-angle symbol beside the figure. "The axis of symmetry is the y-axis. Reflect this vertex across it, and you can calculate the coordinates of the new point directly. You don't need to guess." As he spoke, he wrote out the steps, then tapped the paper. "It's like folding a sheet of paper in half. The points that overlap are points of symmetry."
Morishita Ran gave a little "Oh!" Her eyes lit up as she quickly corrected her steps. Kamuro Masumi looked up just then, pen in hand, and pointed to Uesugi Hara's notebook. "That problem about the extrema—are the little 'mountain' and 'valley' you drew supposed to represent the maximum and minimum?"
"Why else would I draw them? For decoration?" Uesugi Hara shot her a look, but still explained. "The summit is the maximum, and the valley floor is the minimum. Remember: 'uphill, then downhill, means a summit; downhill, then uphill, means a valley.' It's easier than memorizing the definition."
Kamuro Masumi couldn't help laughing. "You sure know how to take shortcuts." But she had already started drawing little peaks in her own notebook.
Sunlight slanted through the window, falling just right across the three open notebooks. The "study addicts" in the hallway were still asking questions, but the classroom had, for once, taken on a lighter air. The scratch of pens across paper mingled with Morishita Ran's soft mutter of "So that's how it works" and Kamuro Masumi's finally relaxed breathing.
Seeing Morishita Ran and Kamuro Masumi bent over their work, solving problems without looking up to ask anything else, Uesugi Hara closed his workbook and got up to cross to the other side of the classroom. Yamamura Miki was staring blankly at her open textbook, absently picking at the corner of her notebook. She didn't even notice him come up to her desk.
"Yamamura." Uesugi Hara's voice softened.
Yamamura Miki jerked her head up, her cheeks instantly turning faintly red. She hurriedly pulled her fingers away from the notebook, a little pencil dust still on her fingertips. "U-Uesugi?"
"I saw you frowning at your textbook for a while." Uesugi Hara handed her his notebook. The points marked in red stood out clearly on the white page. "If there's anything you didn't understand, you can always ask me. You can borrow these notes for now. They have the approach to the example the teacher went over just now."
Yamamura Miki's fingers paused on the notebook cover before she gently took it from him. Her fingertips accidentally brushed his, and she pulled her hand back as if she'd been shocked. Her voice was barely audible. "Th-thank you, Uesugi..." She lowered her head and flipped through a couple of pages. Her eyes brightened a little at the notes accompanied by small diagrams, but she still didn't dare look up.
"You're welcome." Uesugi Hara didn't linger. As he turned away, he noticed the folded corner of a page in her textbook. It was the section where Kamuro Masumi had gotten stuck, so he added casually, "I drew a mountain-climbing diagram for the part on second derivatives. If it doesn't make sense, just ask me."
Yamamura Miki tightened her grip on the notebook and gave a soft "Mm." By the time she looked up, Uesugi Hara had already returned to his seat and was organizing the test papers they'd just been handed out. She stared at the simple little drawing of a person climbing a mountain in the notebook. The flush on her cheeks slowly spread to the tips of her ears. Her pen hovered above the scrap paper for a long time before she carefully wrote down the first formula.
The sound of Morishita Ran and Kamuro Masumi discussing problems came from the front of the classroom, mingling with the breeze outside. It brought a gentle, cozy hum to the classroom during the break.