Gaokao Mathematics — Key Knowledge Points, Formulas, and Past Papers

When it comes to Gaokao Mathematics, the first priority isn't memorizing the entire textbook again. Instead, it's about figuring out which modules are tested most frequently, when to apply specific formulas, and how to effectively practice with past papers. For most students, the key to raising scores isn't "knowing more," but rather being able to recognize problem types faster based on the given conditions and making fewer careless mistakes.

If you are short on time, prioritize these areas: Functions & Derivatives, Sequences, Trigonometric Functions, Analytic Geometry, Probability & Statistics, and Solid Geometry & Vectors. These appear frequently and are often combined into a single comprehensive problem.

What is Gaokao Math Actually Testing?

On the surface, Gaokao Math tests knowledge points; however, from a problem-solving perspective, it's more about three things:

• Can you first determine which category of problem this is?
• Can you select the correct formula once the conditions are met?
• Can you write out the process consistently and accurately?

This is why many students perform well on specific topics but get stuck on comprehensive questions. Once a problem combines functions, equations, geometric meaning, and calculations, the real gap in performance isn't caused by memory, but by the ability to judge the correct method.

Which Key Knowledge Points Give the Best Return?

If you rank them by efficiency in raising scores, it usually looks like this:

• Functions & Derivatives: Common in multiple-choice, fill-in-the-blank, and final "challenge" problems. Focus on monotonicity, extrema, maximum/minimum values, and parameter discussion.
• Sequences: Focus on general terms, summation, and recursive transformations. These are often paired with inequalities or functions.
• Trigonometric Functions & Solving Triangles: Focus on identity transformations, graph properties, and the Law of Sines and Cosines.
• Analytic Geometry: Focus on the positional relationship between lines and conic sections, ranges, extrema, and converting geometric relationships into algebraic expressions.
• Probability & Statistics: Focus on counting, classical probability models, conditional probability, and basic applications of distribution and expectation.
• Solid Geometry & Vectors: Focus on line-plane relationships, angles, distances, and handling problems via coordinates or vectors.

Breaking it down this way is more practical than following textbook chapters because it corresponds directly to how you identify problem types during the exam.

Memorize Formulas Along with Their Conditions

Formulas are important, but the conditions for their application are even more critical. The following sets of formulas are high-frequency and easily misused.

• Derivatives & Monotonicity: If a function is differentiable on a certain interval and $f'(x) > 0$ for any $x$ in that interval, then $f(x)$ is monotonically increasing on that interval; if $f'(x) < 0$, then $f(x)$ is monotonically decreasing on that interval.
• Arithmetic Sequences: $a_n = a_1 + (n-1)d$, $S_n = \frac{n(a_1 + a_n)}{2}$. Suitable for problems where the first term, common difference, and the relationship between terms are known.
• Geometric Sequences: $a_n = a_1 q^{n-1}$. If $q \ne 1$, then $S_n = a_1 \frac{1-q^n}{1-q}$. Before using this, confirm that the sequence indeed satisfies a geometric relationship.
• Law of Cosines: For any triangle, $a^2 = b^2 + c^2 - 2bc\cos A$. This is very useful when the problem provides two sides and the included angle, or when you need to relate sides and angles.
• Classical Probability Model: When all basic outcomes are equally likely, it can be written as $P(A)=\frac{|A|}{|\Omega|}$. If the basic outcomes are not equally likely, this formula cannot be applied directly.

Many students only memorize the conclusion and ignore the conditions, which leads to confusion in comprehensive problems. A truly useful formula sheet should be written as "Formula + Condition + Common Traps."

A Common Type in Past Papers: Using Derivatives to Determine Monotonic Intervals and Extrema

The following problem is typical because it doesn't just test your ability to differentiate; it tests whether you can transition from the sign of the derivative to monotonicity and extrema.

Given:

$f(x)=x^3-3x+1$

Find the monotonic intervals and the extrema of the function.

Step 1: Differentiate and Find Critical Points

$f'(x)=3x^2-3=3(x^2-1)=3(x-1)(x+1)$

When the derivative is zero:

$x=-1,\quad x=1$

These two points are the key boundaries for judging monotonicity because the sign of the derivative may change here.

Step 2: Check the Sign of the Derivative in Each Interval

Divide the number line into three intervals:

• When $x < -1$, $(x-1)(x+1) > 0$, so $f'(x) > 0$.
• When $-1 < x < 1$, $(x-1)(x+1) < 0$, so $f'(x) < 0$.
• When $x > 1$, $(x-1)(x+1) > 0$, so $f'(x) > 0$.

Therefore:

• $f(x)$ is monotonically increasing on $(-\infty,-1)$.
• $f(x)$ is monotonically decreasing on $(-1,1)$.
• $f(x)$ is monotonically increasing on $(1,+\infty)$.

Step 3: Use the Change in Derivative to Determine Extrema

First, calculate the function values:

$f(-1)=(-1)^3-3(-1)+1=3$

$f(1)=1-3+1=-1$

Since the derivative is positive to the left and negative to the right at $x=-1$, a local maximum of $3$ is reached at $x=-1$.

Since the derivative is negative to the left and positive to the right at $x=1$, a local minimum of $-1$ is reached at $x=1$.

What is This Problem Actually Testing?

This type of problem isn't just testing "can you differentiate," but rather the entire chain of logic:

• Can you factorize the derivative?
• Can you judge monotonicity based on the sign?
• Can you translate "change in derivative" into "conclusion about extrema"?

Many points are lost between Step 2 and Step 3. The derivative might be correct, but the monotonic interval is wrong, or the point of the maximum value is confused with the maximum value itself.

How to Practice Past Papers Effectively

The value of past papers isn't just seeing "what was tested," but seeing how the Gaokao combines knowledge points into a complete problem.

A more effective way to practice is:

1. Practice by module first: Specifically identify which type of problem causes you to get stuck.
2. Do full papers: Train your time allocation and your ability to switch between problem types.
3. Review by finding the "first point of error": Don't just write "I don't know this problem."

If you only check the answer, you only know you were wrong. If you find the first point where you went wrong, you know exactly what to change for next time.

Common Points of Point-Loss

Memorizing Conclusions, Not Conditions

For example, discussing monotonicity as soon as you see a derivative without first confirming that the function is differentiable on the corresponding interval. If the conditions aren't met, the conclusion cannot be applied directly.

Knowing the Formula, Not Recognizing the Problem Type

Sequence problems don't always tell you directly, "this is arithmetic" or "this is geometric." You need to judge the structure yourself from the recursive relationship or the change between terms.

Looking at Answers, Not Processes

Many marks in Gaokao Math are awarded for the process. Even if you only care about the result for multiple-choice and fill-in-the-blank questions, subjective questions rely heavily on the stability of your steps.

Underestimating Calculation Errors

Many lost points aren't due to a completely wrong approach, but due to instability in signs, intervals, completing the square, or substitution. In the late stages of senior year, controlling these "low-level" errors is a significant way to boost your score.

Which Area to Patch First?

If you often lose momentum before reaching the final challenge questions, first patch high-integration modules like Functions, Derivatives, and Analytic Geometry.

If you always "know how to do it but calculate it wrong," prioritize practicing sequences, trigonometric identity transformations, and algebraic operation control in analytic geometry.

If you don't know where to start when you see a problem, organize common problem types by "model" first, rather than continuing to blindly grind through new questions.

In other words, your review order shouldn't necessarily follow the textbook; it should follow the bottleneck that is currently limiting your score.

Practical Next Steps for Practice

Pick a module where you've recently been losing the most points and do only two things: write one page of "Formula + Condition + Common Traps," and then solve three corresponding past paper problems, noting the first point of error for each. This is often more effective than reviewing a long list of formulas.

If you want to keep practicing, try changing the function above to:

$g(x)=x^3-3x^2+2$

Then, determine its monotonic intervals and extrema on your own. Complete it independently first, then verify the process; the learning effect is usually much better.