pjbooks Chapter 10: Is this what it feels like to be a top student?
Chapter 10: Is this what it feels like to be a top student?
"Is this what it feels like to be a top student?"
In the classroom, listening to the praise and exclamations from his classmates, Han Chuan felt a little giddy for a moment.
That's actually quite normal, after all, it's a feeling he's never experienced before!
In his past life, he was a poor student and had never experienced what it felt like to have good grades.
Forget about being praised by everyone; even when the teacher calls on him to answer questions, eight out of ten times it's because he fell asleep in class.
The only time he was noticed by the whole class was when he only scored 43 points on the midterm math exam in the second semester of his senior year. The homeroom teacher called him to the front of the class and criticized him, making him a laughing stock among the other students.
But it's different now.
The people surrounding him now aren't there because they want to laugh at him.
How to describe this feeling? He could only say, "Is this what it feels like to be a top student? It's truly wonderful!"
......
"A bet's a bet!"
After a brief period of noise in the classroom, Li Hao looked at the ten math problems in his exercise book, gritted his teeth, and stuffed the CD into Han Chuan's arms.
"This latest anime episode featuring Akiho Yoshizawa is yours starting today!"
"Besides that, I can't use my phone or read novels for the next three days. Everyone can keep an eye on me!"
As soon as these words were spoken, the classroom immediately became lively again.
"Holy crap, that's awesome! Brother Chuan, you actually won Li Hao's treasured collection!"
"Let me see it! Let me see it! I haven't seen it before!"
"Me too!"
"Awesome! Brother Chuan, you can't keep this all to yourself! This is our class's cultural heritage!"
Han Chuan took the disc in his hand and looked at it.
These days, pirated DVDs are all the rage, but the latest anime featuring Yoshizawa Akiho in Li Hao's hand should be an original. I wonder where this kid got it from.
He's lost interest in these kinds of anime. In his previous life, his Xunlei cloud drive was filled with several gigabytes of torrents, and he had a super membership that was paid for annually.
In his eyes, this thing was just like the street food and barbecue outside of school when he was in high school. He couldn't sleep because he craved it when he was in school. Later, when he started working, he ordered takeout every day and ate whatever he wanted. So he realized it was just so-so.
Waving the disc in his hand, Han Chuan smiled at Li Hao and said, "Haozi, wanna continue the bet?"
"You won, so here's your disc back. Want it or not?"
Upon hearing this, Li Hao paused for a moment, then gritted his teeth and said, "Come on! What are we betting on?!"
Han Chuan glanced at the training book on the table and said with a smile, "Let's make a bet like today. I'll give you a week to explain these ten questions you've chosen to me."
"Once you truly understand and comprehend it, I'll return this disc to you."
"If you can't do it..."
At this point, Han Chuan suddenly looked up at the other students in the class and said with a smile, "Then I'll have to share your treasure with the other students."
"Li Hao, you wouldn't want your precious item to be shared by others, would you?"
Although he had already roughly explained the problem-solving ideas and methods for the ten problems he had just solved, Han Chuan knew that Li Hao definitely hadn't fully understood them.
It was quite challenging for Li Hao to understand these questions and the underlying concepts within a week.
Across from him, Li Hao's face turned pale. Although the concept of "small days" hadn't yet emerged in this era, Han Chuan's "Japanese-style speech" still made him grit his teeth in anger.
This disc is a treasured possession of his; if it were shared by countless people...
Taking a deep breath, Li Hao gritted his teeth and nodded, saying, "You wait, even if I don't eat or drink this week, I'll get it back!"
Han Chuan smiled and said, "Okay, I'll be waiting!"
.....
The commotion surrounding the bet gradually subsided as the bell rang for the second evening self-study session.
Han Chuan stuffed the CD into the innermost part of his backpack, zipped it up, and prepared to take it back to his dormitory.
This thing didn't appeal to him much, but it was surprisingly useful as a bargaining chip to continue competing with Li Hao.
Stimulated, Li Hao was flipping through his textbooks on the table, muttering to himself, and marking the ten math problems he had bet on with his pen over and over again.
"You've really started reading?" Han Chuan asked with a smile.
"nonsense!"
Li Hao rolled his eyes and said, "I have a week, ten questions. I have to figure them out even if I don't eat or drink."
He paused, then added, "Nobody is allowed to touch that disc."
Han Chuan smiled, said nothing more, opened his notebook, and prepared to continue looking at the word problems of systems of two linear equations in two variables.
The pen had just touched the paper when a shadow suddenly fell from in front of me, covering most of the tabletop.
He looked up and saw Jiang Wei standing by the table.
The top student sitting in the front row by the window wore a pair of black glasses, held a folded piece of draft paper in his hand, and had a flat expression on his face, revealing no emotion.
However, his very act of standing there had already drawn the attention of several "underachievers" around him.
"Han Chuan".
Jiang Wei unfolded the training book she had brought and laid it out in front of Han Chuan, saying, "This question seems a bit similar to the one we just made a bet on, but I still don't quite understand it. Could you explain it to me?"
"Let me see."
Han Chuan took the training book and glanced at the questions on it.
Given the function f(x) = x² + ax + 1, if the inequality f(x) ≥ 0 always holds for any x ∈ [1, 2], find the range of values for the real number a.
This is a problem combining functions and geometry. The parameter 'a' is nested within a quadratic function, and the inequality must always hold true on a specified interval. Find the range of values for 'a'.
"That's easy."
After glancing at the title, Han Chuan reached for the pen on the table, grabbed a piece of paper, and began to write.
"ax≥−x²−1, then divide both sides by x, x∈[1,2], always positive, the direction of the inequality sign remains unchanged."
"Construct g(x) = −(x + 1/x), x ∈ [1, 2]... "A checkmark function, monotonically increasing on [1, 2]"
"....."
Han Chuan quickly wrote several lines of calculations on the manuscript paper with his ballpoint pen, then stopped writing and continued.
"First, transform the inequality, separate the parameters, construct a new function, and then use the positional relationship between the graphs of the two functions to deduce the constant condition to get the answer."
Beside her, Jiang Wei frowned as she looked at the solution on the manuscript paper. After thinking for a while, she asked in confusion.
"Your problem-solving method seems to be different from what Teacher Cao taught?"
In his understanding of learning, solving this kind of problem should not involve first rearranging terms, and then classifying and discussing the discriminant, the position of the axis of symmetry, the endpoints of the interval, and the sign of the function value.
However, Han Chuan's solution seemed to have nothing to do with these things.
Li Hao, who was working on his ten problems, leaned over and glanced at the circled checkmark function on his draft paper, his face full of confusion.
"What's this 'checkmark function'? The name sounds like an insult."
Han Chuan smiled and said, "It's a function of the form y=x+a/x, and its graph looks like a checkmark."
As he spoke, he casually sketched a rough diagram on the blank space of the manuscript paper and continued, "For example, in the first quadrant, the minimum value of 2 is reached when x=1, and it increases to the left and right."
"The interval in this problem is exactly on the monotonically increasing segment, so the maximum value is at the right endpoint of the interval."
Li Hao stared at the sketch for a few seconds, his face full of confusion; he didn't understand it.
Just then, a steady voice came from behind the three of them.
"I don't think I taught you this problem-solving approach in class. How did you come up with it?"
The three of them were startled and turned around at the same time. Their math teacher, Cao Wen, was standing behind them without them noticing.
He was holding an enamel mug, the steam from its rim had stopped rising, indicating he'd been standing there for a while. No one knew when he'd arrived, or how much he'd overheard.
"Teacher Cao."
Han Chuan and the other two greeted each other at the same time.
Cao Wen nodded, which was taken as a response.
His gaze fell on the manuscript paper that Han Chuan had spread out—the one with the parameter separation, the one with the construction of the checkmark function, and the one with the sketch of the coordinate axes.
He looked at it for a few seconds, then reached out, picked up the manuscript from the table, and looked at it again.
My thumb unconsciously rubbed slowly along the edge of the paper.
This is his unique reading habit: while solving the problem, he also reflects on and studies the problem solver's personal thought process.
Clearly, Han Chuan's research method is novel even for him.
The standard solution to this problem is to solve it by case analysis.
After looking at it for a while, Cao Wen put down the manuscript and asked Han Chuan with interest, "First, move the terms, then classify them in three rounds according to the discriminant, the position of the axis of symmetry, and the sign of the function value at the endpoints of the interval."
"I've explained this method twice in class, and there was a similar question on last week's practice test. Most of your class uses this method."
"But your solution is different. You directly separate the parameters, construct a new function, and then use the monotonicity of the new function to directly determine the position of the maximum value."
"You didn't use any of the three rounds of classification and discussion I mentioned before."
How did you come up with this method to solve the problem?
Cao Wen's tone was flat, making it impossible to tell whether it was praise or criticism.
Han Chuan thought for a moment and said, "It's easy to miss cases when using case-by-case discussion. There was a question on the last monthly exam paper that I had to do three times using case-by-case discussion to find all the cases."
"After I finished, I was thinking, could we separate the parameters directly? That way, we could at least reduce one round of discussion."
Upon hearing this, Cao Wen looked at him with some surprise and said, "So you came up with this method yourself?"
Hearing this, Han Chuan blushed slightly but still nodded.
In fact, this method was taught to him by Ge Jun. It is simpler than classification and discussion, but it requires a more comprehensive foundation and logical thinking ability.
But he couldn't say that Ge Jun taught him, since the real Ge Jun was in Jiangsu.
Cao Wen nodded, put the manuscript back on the table, picked up his enamel mug, took a sip of tea, and then suddenly looked at Jiang Wei beside him and asked.
"Jiang Wei, what do you think of his method?"
Jiang Wei was called on and paused for a moment before adjusting her glasses. After thinking for a moment, she said, "This seems simpler than using category discussion?"
Cao Wen smiled and said, "The premise of parameter separation is that the direction of the inequality sign does not change when transforming the inequality. This problem can be solved by x."
"But if the interval spans both positive and negative, we have to discuss the sign of x separately. In that case, separating the parameters is not as direct as discussing them separately."
He paused for a moment, then looked at Han Chuan and asked, "What if the interval for today's question is [0.5, 2]?"
Startled by the question, Han Chuan paused for a moment before replying, "If the interval crosses an extreme point, it needs to be split into two segments for separate discussion."
"right!"
Cao Wen nodded and smiled, "The fact that you realize this shows that you've reflected on the methods you used, and that you're not just done with the task. This habit is more valuable than getting a question right."
He paused, then reached out and tapped a spot on the other side of the manuscript.
"But here, separating the parameters does not mean not classifying. In fact, your step has changed the object of classification from parameter a to the monotonicity of function g(x) - originally we discussed a by category, now we discuss the behavior of g(x) in different intervals by category."
Upon hearing this, Han Chuan's eyes lit up, as if a lightbulb had suddenly flashed in his mind. He blurted out, "The classification itself hasn't disappeared; it's just been transferred!"
When he used the parameter separation method before, he always thought that he was discussing this problem by bypassing 'classification'.
But Cao Wen's words clearly told him that the classification had not actually disappeared, but had simply been transferred.
This also made him realize that he was actually just doing the same thing from a different perspective.
The original method directly discusses the parameter 'a', while his method discusses the behavior of the new function g(x). Both types of discussions are essentially different expressions of the same logic.
Cao Wen picked up the enamel mug, his expression revealing undisguised satisfaction. He nodded and smiled, saying, "Good that you understand."
He paused for a moment, then looked at Jiang Wei and added, "However, this kind of problem-solving approach is not needed in the normal college entrance examination. Don't worry too much about it. Just learn how to classify and discuss cases."
Jiang Wei nodded and said, "Okay, Teacher Cao."
Cao Wen smiled, picked up his enamel mug, and walked up to the podium to oversee the evening self-study session.
After scanning the classroom, his gaze returned to Han Chuan.
What he didn't say earlier was that although this method is not applicable to the college entrance examination, it is an essential problem-solving approach for students participating in competitions.
If Han Chuan's math scores were on par with Jiang Huai's, he would really like to have him try participating in the International Mathematical Olympiad.
After all, with such a unique ability to understand, he might be able to bring back a provincial award for No. 1 Middle School by participating in the Olympiad.
As for the higher-level national awards, he hadn't even considered them.
That's a field that a student who scores 130 or 140 in math can no longer access.
It's no exaggeration to say that any student who makes it to the national Olympiad has already achieved near-perfect scores in math since middle school. Even if they lose points, it's only a point or two due to handwriting or minor details.
Pity.
Thinking of this, Cao Wen gently shook his head and sighed, thus abandoning his idea.
If only Han Chuan had awakened to his own potential sooner.
For several years now, our school hasn't had a student reach the national finals of the Mathematical Olympiad. Last year, the best result was only a provincial first prize.
It's a pity that this little guy "awakened" a bit late. He's already in the second semester of his junior year, and his foundation is still so weak. It's too late for him to participate in competitions.
If I were in the first year of high school, I might have had a chance for special training, and who knows, I might have been able to compete in the national competition.
As for now, it would be great if I could catch up in my grades and get into a good university.
.....
(Please read on, vote, and comment! I'm a newbie author, so I'm asking for everything~ Thank you~)