Top student: All my teachers are academic whizzes!

The math monthly exam in March is scheduled for the end of the month.

After taking up a physical education class taught by a "frail and sickly" physical education teacher, the third and fourth periods on Thursday afternoon were combined.

Carrying that familiar enamel mug, Cao Wen walked into the classroom with a stack of exam papers tucked under his arm, and began speaking in his familiar words.

"Your PE teacher is sick, so there will be a test over the next two periods!"

"Math class representative, come up here and hand out the exam papers. Don't even think about cheating; it's pointless for your lives!"

As soon as he finished speaking, a suppressed restlessness erupted in the classroom.

"Another exam."

"Why is our PE teacher always sick?"

"I've never even seen a physical education class!"

Cao Wen glanced around the classroom, then threw the test paper in his hand onto the podium, kicking up a cloud of chalk dust.

"What's all the noise about? If you don't want to do it, you can go to your PE class next period!"

The classroom immediately fell silent upon hearing this.

Math teachers can say that, but they wouldn't dare actually do it.

If we really have to go out to play after the next class, it'll be fine if we do well on the test, but if we don't, then we're completely doomed.

Sitting in the back row by the window, Han Chuan looked at the test papers handed out in front of him, took a deep breath, and thought it was time to test the results of his more than half a month of learning.

"If you don't even get a 90 on this exam... hmph..."

On the desk, Ge Jun's handwriting from the math textbook appeared.

Han Chuan nodded seriously and said, "If I don't pass the test, Teacher Ge, you can put me through the wringer!"

Although he has made rapid progress under Ge Jun's guidance in the past two weeks, his foundation is still too weak.

He wasn't entirely confident that he could pass the test in 90 minutes.

Ge Jun: "Very good, that's spirited!"

The classroom was quiet; the only sound was the scratching of pens on paper.

Despite all the complaints from the students in Class 09 of Grade 11, they still had to face the exams and the college entrance examination. This was their fate, and they couldn't escape it.

Cao Wen stood on the podium, holding an enamel mug, looking down at everything and taking in the scene in the classroom.

He had two expectations for this monthly exam: one was whether the top students in the class could surpass their previous scores.

The other one is Han Chuan, whom I've been paying attention to recently.

Thinking about it, Cao Wen turned his gaze to the middle and back row by the window, where the student he was looking at was currently engrossed in doing his homework.

The fact that Han Chuan was able to persist for more than half a month is enough to prove that he has really changed his ways and is studying seriously instead of just pretending to fool himself.

He was actually quite looking forward to seeing how many points Han Chuan would get in this exam after more than half a month.

......

When the exam paper was handed to Han Chuan, he didn't rush to start writing. Instead, he quickly flipped through the entire paper, following the method taught by Ge Jun.

There were eight multiple-choice questions, six fill-in-the-blank questions, and four essay questions.

Solid geometry made up half of the material, and functions made up the other half.

He looked at the multiple-choice questions first.

The first question tested set operations, the second was about the domain of a function, and the third question and so on involved the relationship between lines and planes in solid geometry...

Of the eight multiple-choice questions, he was only confident that he could answer four correctly.

Of the remaining four questions, two can eliminate some obviously wrong options, but the correct answer cannot be determined at a glance and requires further calculation.

I knew absolutely nothing about the last two questions.

As for multiple-choice and fill-in-the-blank questions, they were more difficult than single-choice questions, and he was unsure whether he could answer more than half of them.

However, Han Chuan was not in a hurry, nor did he give up on things he couldn't do, as he did in his previous life.

After quickly solving the problems he could, he began tackling the ones he was relatively confident about.

As time went on, the formulas and drawings on the drafts gradually increased, and he would quickly skip over the problems that he solved in one go.

This wasn't a problem-solving approach taught to him by Ge Jun, but rather a method that was already etched into his mind and hadn't faded with time.

However, while skipping over this problem, Han Chuan has developed a new habit compared to his previous life: writing down his analysis process, including the known conditions, the requirements, and where he got stuck.

This is a habit that Ge Jun has been teaching and forcing him to develop over this period of time: it's okay if you can't do it, but you need to figure out exactly where you're stuck.

When the bell rang to signal the end of the first class, Han Chuan had already finished all the problems he knew how to solve.

What remains are the long questions that I don't have much of a clue about, or the last few multiple-choice and fill-in-the-blank questions.

Rubbing his slightly sore temples, Han Chuan looked at the major questions in the problem-solving section.

【Given the function f(x) = x³ - 3x + 1, x ∈ 0, 2. (1) Find the monotonic intervals of the function within the interval [0, 2]. (2) Prove that for any x₁, x₂ ∈ 0, 2, |f(x₁) - f(x₂)| ≤ 4.】

Without a doubt, the core of this problem is finding the derivative.

However, he was taught the concept of derivatives in his second year of high school, which was just before his rebirth, and he has almost no memory of it now.

The junior high school math lessons taught under Ge Jun's guidance did not cover concepts related to derivatives.

If it were him before his rebirth, this amount of time would have been enough for him to write "Solution" on the answer sheet and leave it blank.

But things are different now.

Even knowing he couldn't do it, he still thought of ways to solve it.

If you don't know how to use the derivative, are there any alternative solutions?

The first question asks for the monotonic intervals, which means determining when the function increases and when it decreases.

After staring at the problem for a while, Han Chuan wrote down f(x)=x³-3x+1 on the draft paper.

He doesn't know how to differentiate, but he learned about quadratic functions in junior high school.

How do you determine the monotonicity of a quadratic function?

Just look at the direction of the opening and the axis of symmetry.

A cubic function doesn't have an axis of symmetry to observe, but it can observe the changes in the function value.

If we take two points in the interval, that is, x₁

Thinking of this, Han Chuan's eyes lit up.

This method seems a bit clumsy, but it seems like it might work?

Thinking it over, he quickly scribbled a few dots on the manuscript paper with his pen.

x=0时，f(0)=1；x=0.5时，f(0.5)=0.125-1.5+1=-0.375；x=1时，f(1)=1-3+1=-1；x=2时，f(2)=8-6+1=3.....

"There it is!"

Looking at the calculations on the draft paper, Han Chuan grinned, copied the calculations onto the test paper, and wrote down the answers.

"The monotonic intervals are: monotonically decreasing on 0 and 1, and monotonically increasing on 1 and 2."

......

The bell rang in the classroom, signaling the end of the second period.

On the podium, math teacher Cao Wen stood up: "Stop writing, pass the papers from back to front."

As Han Chuan passed the exam papers forward, Li Hao came over and glanced at the exam paper in his hand. The dense handwriting caught his eye.

"Holy crap, Brother Chuan, you've finished everything?"

"No," Han Chuan replied, shaking his head.

"But I see you've filled it all up?" Li Hao asked in surprise.

Han Chuan: "I only did what I knew how to do, and for what I didn't know how to do, I just wrote down a little bit of the process."

"oh."

Li Hao gave a long reply, then slumped onto the table, looking rather sullen.

"What's wrong? Missing your DVDs?"

Seeing him like this, Han Chuan asked curiously.

"no."

Li Hao shook his head, not looking at Han Chuan, and suddenly sighed, asking, "Brother Chuan, do you think I'm really not suited for studying?"

Why did you suddenly think of this?

Li Hao sighed again and said, "I just did a test and realized that I didn't know how to answer many of the questions; I didn't know how to answer most of them."

"I've been studying diligently for almost a week now, after all."

Although there was a bet involved, he had indeed been studying diligently over the past week, without playing on his phone or reading novels.

After all, seeing his deskmate soaring to new heights and far surpassing him in grades, it's impossible not to have some thoughts about it.

Both being academic underachievers and students at the same high school, Li Hao didn't think he was any worse than Han Chuan.

Moreover, his score in the last math monthly exam was several points higher.

But now he realizes that he seems to be getting further and further away from his deskmate.

Han Chuan insisted that if he studied diligently for ten days, he could bet that he could solve ten problems that he couldn't solve before.

He studied diligently for a week, but when he took the exam, he found that it seemed no different from before.

This is very frustrating.

Upon hearing this, Han Chuan put down his pen, turned his head to look at the dejected Li Hao, and spoke in a calm tone without any lecturing airs.

"You've only been studying seriously for a week, what's the rush?"

Li Hao hung his head and said sullenly, "But I feel like I haven't made any progress at all. I can't understand most of the test papers. It's exactly the same as when I was just wasting my time."

Han Chuan laughed and said, "Learning isn't like having a cheat code. How can you make rapid progress in just one week? I'm the same. I still couldn't figure out most of the difficult questions in this monthly exam."

Li Hao looked up, his eyes filled with confusion: "Is it even worth it for me to keep pushing myself like this every day?"

"certainly!"

Han Chuan looked at him and said seriously, "You used to never touch a book, but now at least you can sit still and listen attentively. That's progress."

"It's just that you and I are both lacking in knowledge. A week is simply not enough to digest it all and not enough to see any results. That's normal."

"If you give up after a week, you'll still be at the bottom."

The frustration and disappointment in Li Hao's heart dissipated considerably. He scratched his head and asked, "Can we really catch up slowly?"

Han Chuan smiled and said, "Don't compare your progress with mine, just compare it with your past self."

"If you don't believe me, try it for a month and see. It will definitely be different now. It's okay to do poorly on this test. Just treat it as filling in the gaps in your knowledge. It's much better than just drifting through life aimlessly."

He paused for a moment, then continued jokingly, "Besides, don't forget your treasures are still with me. If you just give up now, wouldn't that be like surrendering and admitting defeat?"

Upon hearing this, Li Hao immediately perked up.

"Damn it! You're not allowed to share my treasured collection with anyone else!"

Han Chuan laughed and said, "That depends on you."

......

It was during evening self-study in the Grade 11 office.

Cao Wen held an enamel mug in one hand and a pen in the other, flipping through the test papers collected that morning and grading them one by one.

Class 09 of Grade 11 is a regular class. The average level of math is around the passing mark. Most students leave the last few big questions blank or only write a sentence or two.

Many of them even have only one solution.

In this situation, Cao Wen didn't even look at it and directly drew a big red X on it.

The phrase I heard most often during my school days was probably, "Even if you don't know how to do it, you still have to write down the solution!"

The school teachers teach this because it meets the exam requirements, helps demonstrate the clarity of the problem-solving steps, and may earn some points.

However, this possibility of obtaining partial credit is based on having demonstrated part of the problem-solving process.

In the official college entrance examination, not writing the solution may result in a deduction of points, but only writing the solution will definitely result in no points.

He flipped through the test papers one by one, and soon, a test paper with the answer area filled with answers came into his view.

Cao Wen glanced at the name; this test paper belonged to Han Chuan, the little guy he had been focusing on for the past two weeks.

His gaze shifted downwards.

There were eight single-choice questions, and I got five right. There were three multiple-answer questions, and I got two right.

Not particularly high, but within Class 09, it would rank slightly above average.

There were six fill-in-the-blank questions, and I only answered three, but I got them all right.

What caught Cao Wen's attention were the problem-solving questions.

Han Chuan scored around 50-60% on the first three major questions, completing half of some and more than half of others.

Although Han Chuan did not get full marks on these three major questions, judging from the solution process and the final answer, he left complete traces of his thinking for each question.

What truly caught his attention was the solution to the last major problem.

Find the monotonic intervals of the function within the interval.

The standard solution he taught this class of students was to first find the derivative, set the derivative to zero to find the extreme points, and then determine the sign of the derivative.

However, Han Chuan did not provide this solution. Instead, he listed the function values ​​for five points on the answer sheet.

"x=0, 0.5, 1, 1.5, 2."

After calculating all five values, the final conclusion is: the function decreases in the interval [0, 1] and increases in the interval [1, 2]. By comparing the function values ​​at different points, we find that x=1 is the minimum point.

"That's interesting. They don't know derivatives, but they managed to derive the correct answer using the definition of increment."

Looking at the answers on the test paper, Cao Wen smiled.

This little guy can't identify extreme points, but he manually found the minimum value using five-point sampling.

Each step was neither the method he taught nor a standard problem-solving process, yet the result was obtained in the end.

After thinking for a while, Cao Wen put a ⍻ next to the answer to the last big question, which is a combination of √ and ×, and then gave the first sub-question full marks (6-1) points, which is 5 points.

In fact, if you want to be strict with the scoring, using this numerical trial-and-error method, that is, guessing the answer by substituting specific values, estimating or trying values, usually will not get you full marks, and may even get you zero marks.

Because the college entrance examination emphasizes "points awarded according to steps", problem-solving questions must show a clear, standardized and complete reasoning process or derivation steps.

If only the final answer is given, even if the answer is correct, only partial points will be awarded.

But Cao Wen knew very well that this answer deserved more points than those papers that got full marks by simply applying standard solutions.

To be honest, the (6-1) points he gave were not for the correct final answer, but for the "thinking" demonstrated by a "poor student".

Of course, 6 points is a reward, and -1 point is a necessary reminder.

Ultimately, this method is not applicable to the current college entrance examination system.

......

(Please read on, vote, and comment! I'm a newbie author, so I'm asking for everything~ Thank you~)