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The college entrance examination is only 100 days away. How can we improve our math scores?
Prepare for the science exam and enter sunny June.

-to the senior three students who are about to enter the examination room.

Every year, dozens of days before the college entrance examination, at least 2 million candidates in the country are in a state of hurry and confusion. If you don't get rid of this situation before entering the examination room, it will snow in your season.

This paper is willing to communicate with you frankly, discuss good strategies, find the way to win the college entrance examination, and help you step into sunny June and win brilliant June!

As we all know, the college entrance examination is a "national examination" and a serious state act. It must be based on the foundation, ability and psychological quality of this exam, and it is quite stable and rigorous. At the same time, after more than ten years of serious study, especially six years of middle school study, especially a round of review, I have experienced a lot of battles, and I have already seen, experienced and strengthened my strength. It can be said that up to now, most of our classmates have boiled the water to 99 degrees. Of course, it is this boiling point that makes us anxious, unable to eat or sleep well, but as long as we prepare for the exam scientifically, calmly deal with it, consolidate the double basics, make up for loopholes, improve our thinking ability and strengthen our psychological quality, we can walk into the examination room with confidence and achieve ideal results, and miracles always happen at the last minute.

Special reminder is that after the first round of review, grasping the double foundation is still your top priority.

Below, this article takes "eight changes" as the clue, paving a victory net for you to win the college entrance examination.

First, the concept is clear.

Basic concepts, especially definitions, are the growing points of all knowledge and must be clear and definite. We should not only know what things are, but also know what things are not, and we should understand things from both positive and negative aspects. For example—

Question 0 1, is y2=4x a function?

Question 02. Is y=4x2(x≥0) an even function?

Question 03. If x0 satisfies f' (x0) = 0, is f(x0) an extreme value?

Question 04: Is the locus of the moving point M(x, y) satisfying ︱ x- 1 ︱ x+ 1 = 2 an ellipse?

Question 05. Is it a (m+n) = is it?

Question 06. Is the inclination of the straight line y=tanθ x θ?

Question 07. Is it the intercept distance?

Question 08. Do you know what are the two geometric meanings of hyperbolic imaginary semi-axis B?

Question 09. Is the included angle between the normal vectors of two half planes the size of dihedral angle?

Question 10. Do you know how to define the spherical distance?

Question 1 1, (a b) c = a (b c), right? A > b, right? A-a=0, right?

Question 12. Do you know the geometric meaning of a b = | a || b | cos θ? What about the physical meaning?

……

Second, knowledge networking.

Knowledge is the building material for building a capacity building, but the random stacking of building materials is by no means equal to a building. Filling your brain with a lot of disorganized knowledge and information will only become a heavy burden and even make your thinking "collapse". A Russian scientist once said: Wisdom is just an orderly organization of knowledge. This shows that it is very important for us to take the time to sort out what we have learned and make it systematic and networked. We would rather do fewer topics than spend enough time on this job, because a clear head is more important than anything else.

How to do this work well?

First of all, learn to read the catalogue and establish a knowledge menu. First of all, read through all the catalogues of high school mathematics textbooks and see what you have learned from a big perspective; Secondly, read the contents of each chapter to see what knowledge this part has learned; Browse the contents of each chapter in chapter order to see what concepts, formulas, theorems and laws there are, and write them down while reading. Also look at the summary after each chapter to enrich what you have learned at ordinary times. A lot of knowledge should be framed, tabular, programmed, menu or formulated. This is a process from macro to micro. In a word, we should know everything we have learned, so that we can have a macro vision.

Second, learn to find connections and build knowledge modules. It is said that an excellent engineer has tens of thousands of knowledge modules in his mind, which makes him handy in solving problems. When we study mathematics, we should modularize our knowledge. For example, the content of trigonometric function is generally divided into three sections: preparatory knowledge section, function section and trigonometric solution section. Based on the definition of trigonometric function with arbitrary angle, the basic relationship of trigonometric function with the same angle is deduced, and then five groups of inductive formulas are deduced. Combined with the distance formula of two points, the sum and difference formula of two angles are derived, and the double angle formula is further derived, and finally the half angle formula is derived. This is basically a process from general to special, which is the first big plate. This paper studies the images and properties of trigonometric functions by using higher function theory (mainly "three properties and two fields"), in which periodicity and parity are newly learned function concepts with the help of trigonometric functions. In fact, as you know from analytic geometry, the so-called parity is only a special case of symmetry. This is the second largest plate, strictly speaking, this is a real trigonometric function; The third largest plate is the sine and cosine theorem. Can you successfully finish your homework on a blank sheet of paper in 30 minutes without reading the textbook?

Another example is the content of mean inequality, and there are many formulas related to it, which are also widely used. Memorizing them in isolation is not only time-consuming and laborious, but also prone to mistakes, not to mention flexible use. But if you take a little time to interpret them, you will find them very interesting. I suggest you infer that:

∫A A = A2≥0, a -b replaces A,

∴ (a -b)2≥0,

∴ a2+b2≥2ab, where a2 and b2 are replaced by A and B respectively.

Have a+b≥2 (a, b > 0)

∴ ≥ ,

Or ab ≤;

Similarly, add a2+b2 to both sides of a2+b2≥2ab to obtain

2 (a2+b2)≥(a+b)2,

∴︱a+b︱≤;

When a, b > 0, you can also get

≥2,a + ≥ 2b,b + ≥ 2a,…

And inequality string: ≤≤≤≤≤

Another example is analytic geometry, which is nothing more than studying the definitions and properties of five geometric curves (straight line, circle, ellipse, hyperbola and parabola) with algebraic tools, telling you how to write equations according to conditions and study the properties of curves with coordinate method. Please deduce this content yourself.

Third, skill proficiency.

Skilled skills are a prerequisite for accomplishing anything efficiently. If you stumble in the calculation process, your problem-solving efficiency and quality will be greatly reduced. Take the following questions as an example, please test yourself:

Question 0 1. Is it skillful and accurate to solve set problems with Webster's diagram or number axis?

Question 02: Is it skillful and accurate to match the quadratic function y=ax2+bx+c into "vertex" y=a(x-h)2+k or "zero" y=a(x-x 1)(x-x2) (if any)?

Question 03. Are you skilled and accurate in solving inequalities?

Question 04. Are you skilled and accurate in using the "five-point method" as the image of function y=Asin(ωx+φ)+k?

Question 05. Are you familiar with the "method of dividing angles and rounding" often used in trigonometric functions?

Question 06. Are you familiar with the skills of finding the general term formula from the recurrence formula that often appears in the sequence? Are you familiar with the common summation methods?

Question 07: Are you familiar with the skills of finding angles (three kinds) and distances (eight kinds) by geometric and vector methods in solid geometry?

Question 08. Can we get the results quickly and accurately without expanding the general formula for finding the coefficient of binomial specified term?

Question 09. Can you find the inverse function skillfully and accurately?

Question 10. Is it skillful to prove inequality by mathematical induction?

Question 1 1. When solving the problem about the positional relationship between straight lines and conic curves, have you mastered a series of deformation steps, especially the technology of integral deformation and integral substitution?

Question 12. Are you familiar with the solution of common permutation and combination problems (about 10)?

Question 13. Are the skills of finding the maximum value and monotone interval skillful and accurate?

Have you mastered the general steps to solve the problem 14, probability problem and expected variance problem?

……

Fourth, the question type is modeled.

The so-called question type is the general problem-solving procedures and practices (general methods) that are familiar with and summarize classic questions, which are relatively fixed and easy to use. In this way, when you encounter a problem, you will feel familiar and friendly, and of course it will be much smoother to solve it. At this point, it is closely related to the above-mentioned "skill familiarity", but the former focuses on local skills and technologies, while the latter focuses on the overall situation of the problem. The former is sharpening the knife, while the latter is chopping wood. This paper suggests that you carefully collect a certain number of classic questions, sort them out, and try to figure them out again and again. Very good!

5. Normalization of thinking.

The hardest thing to do in life is normal, because normal is not a simple realm. For example, the flight of an airplane does not need to be extraordinary. Being normal is a great blessing. If you don't believe me, you'll understand once you take the risk. For example, healthy, normal and healthy physique is the highest goal pursued by countless people. If you don't believe me, you can ask someone who has been seriously ill or physically disabled.

Without a normal strategy, no problem can be solved. If the strategy is right, it will be solved, and if the strategy is wrong, it will be difficult to move. Generally speaking, there are three points that must be paid great attention to-

First, we should form the habit of consciously using mathematical ideas to solve problems, which is really the biggest strategy. In addition to clarifying the ideas of functional equation, equivalent transformation, classified discussion and combination of numbers and shapes, the mathematical ideas to be tested in the college entrance examination are also very important, such as overall solution, elimination, reverse thinking and dialectical thinking. In the usual teaching, every math teacher has repeatedly stressed the need to use mathematical ideas to guide problem solving. Unfortunately, quite a few candidates have no feeling about the word "mathematical thought", probably because the theory is too abstract and abstruse. Maybe candidates always have natural resistance to so-called "ideas" at such an age, but they are not interested in some problem-solving skills. In fact, thought is the refinement of experience, and thought is a rainy cloud. Only when you have ideas can you have ideas, and only when you have correct ideas can you be in an invincible position. Mao Zedong defeated Chiang Kai-shek, who was much stronger than him. Besides Mao Zedong conforming to the historical trend, Mao Zedong's introduction of Marxism-Leninism is more brilliant than Chiang Kai-shek's introduction of American-made equipment: if I have advanced ideas, all your advanced weapons are mine-I wonder if you think so? In short, if you don't use mathematical thinking to solve problems, you will definitely be limited to luck, and you will take detours or even have no choice.

Here are a few examples, the topic is not difficult, intended to attract jade-

Question 1. It is known that the circumference of a sector is 4. What is the largest area of this area?

Question 2: ︱ x+1︱+x–2 ︱≥ 2a+1holds for X ∈ R, and find the range of a.

Question 3. At △ABC, Sina: sinb: sinc = 5: 7: 8, then the size of ∠B is (06 Beijing College Entrance Examination, 12).

Question 4. What is the solution set of inequality (x- 1)(x-a)≥0?

The above questions 1, 2, 3 and 4 correspond to four major ideas: the idea of examining functions, the idea of combining numbers with shapes, the idea of equivalent transformation, and the idea of classified discussion.

Although there is always not enough time to prepare for the exam, I hope candidates should pay attention to this problem and try their best to mature themselves. First of all, pay attention to what ideas the teacher used to solve the problem when listening to the class. You can also take some time to look at the topics about mathematical thought in the first round of materials. The second is to consciously use mathematical ideas to guide problem solving and strengthen "ideological consciousness".

Second, small problems and big problems are solved separately. In order to realize a comprehensive examination of candidates' knowledge, ability, quality and sustainable development space, the college entrance examination paper has set up small questions (multiple choice questions and fill-in-the-blank questions), most of which can and should be indirectly used (such as combination of numbers and shapes, substitution test of special values, logical exclusion, counterexample exclusion, trend judgment, intuitive judgment, estimation judgment, degradation judgment, on-site operation, extreme thinking, equivalent transformation, clever use of definitions, etc.). However, a considerable number of candidates are uneasy about solving problems indirectly, or even not liberated. They think this is "immoral" and don't understand that this is actually the true intention of the college entrance examination proposer.

Example 0 1, y=2sin(2x+ )- 3cos(2x+) has a period of (adapted from college entrance examination questions).

Example 02, point p (2cos 10, 2sin 10), q (2cos70, 2sin70), PQ=?

Example 03, ax2+2x+ 1=0 has at least one negative real root if and only if ().

A, 0 ~ a ≤ 1.1.c, a≤ 1 .d, 0 ~ A ≤1or a ~ 0. (Page 43 of Book 1 of the textbook)

Example 04: What is the maximum area (unit: cm) of a triangle composed of five thin sticks with lengths of 2, 3, 4, 5 and 6 (connection is allowed, but breaking is not allowed)?

A, 8 cm2 B, 6 cm2 C, 3 cm2 D, 20cm2 (National Volume I, 2006, 1 1)

Example 05. In the expansion of (x4+ 1/x) 10, the constant term is (Volume 2 of 2006, 13).

Example 06. The minimum value of the function f (x) =-x-n = is ().

A, 190 B, 17 1 C, 90 D, 45 (National Volume II, 2006, 12)

The wonderful solution is as follows:

Example 0 1: If it is expanded first and then simplified, it will take a lot of time; If we can directly understand that simplification must be in the form of y=Asin(2x+φ), then we will know the answer in an instant. (intuitive judgment)

Example 02: If you substitute the distance formula between two points, of course; If we can realize the geometric meaning of the question: p and q are two points on a circle with the origin as the center and the radius of 2, and ∠ POQ = 60, we will know that the answer is 2 at once. (number-shape combination)

Example 03: If the problem is solved directly, it may not take 10 minutes to solve it. If you can see from the options, 0 and 1 are two key numbers. If you substitute 0, you will get that x=- meets the requirements, excluding A and D; Then substitute 1, and x=- 1 meets the requirements, so choose C. (special value for inspection)

Example 04: For most candidates, this is a difficult multiple-choice question, because there may be many combinations when you arrange the length of each side, so the proposer uses this question to "check". In fact, we know from the title that the perimeter of this triangle is a triangle with constant 20 and constant perimeter. When its height or bottom tends to zero, it tends to a straight line and its area tends to zero. Therefore, it can be seen that the area is the largest only when the shape of the triangle tends to be the most "full", that is, when the shape is close to the regular triangle, the area is the largest, so the three sides should be 7, 7 and 6, so it is easy to know that the maximum area is 6 √ 65433. (trend judgment)

Example 05: To get the specified term, we usually use the general term formula Tr+ 1 to get r first, which is of course understandable. However, if we realize that the expansion term is determined by the degree x4 and both * * * are "divided" by 10, we immediately know that x4 should be divided by 2 by 8, that is, when r=2, we get the constant term, then the constant term is C 102=45. (intuitive judgment)

Example 06: This is also a "check" question. I believe many candidates have chosen to give up thinking and guess one. It's definitely not easy to do it in a direct way, but actually it forces you to do it in an indirect way. Here, n is a known quantity. Imagine that if you degenerate and take n=3, you will probably realize that f (x) = ︱ x–1︱ x–2 ︱+︱ x–3 only when x=2. If you have a good intuition, you should also realize that when X should take a special value, that is, the middle value of 10, f(x) will be the smallest; In addition, your method of combining numbers and shapes is also good: f(x) represents the sum of the distances from the moving point X to the fixed point 1, 2, 3, …, 19 on the number axis. From the geometric sense, when x= 10, f(x) takes the minimum value; You can even associate it with "variance", which is actually a deviation accumulation. F(x) is regarded as variance, and small variance makes random variables stable, but since random variables have been determined first, X should take their average value 10 (analogical judgment).

How to solve small problems quickly, please refer to related topics in detail.

There are the above strategies to solve small problems, but what are the strategies to solve big problems? Please look at the following five points-

First: give priority to the examination of questions and identify patterns. The examination room is like a battlefield, and solving problems is like fighting. Only by knowing ourselves and ourselves can we defeat the enemy. Examining questions is a process of detecting "the enemy's situation" and solving problems, and it is the premise of winning, so we must attach great importance to it. Reading a topic should be word for word, sentence by sentence, punctuated sentence by sentence, understanding the topic should adhere to the association while reading, grasping the topic should distinguish the known, unknown and implied knowledge, and finally achieve the situation of "grasping the key words and putting the topic in the chest".

Second: reverse thinking, aiming at leading the way. Have you ever played the game of walking the maze? If you really walk from the "entrance" to the "exit", there will be many roads and you will not be able to walk out for a long time; On the contrary, if you go from the "exit" to the "entrance", it will be like going downstream, which is a typical reverse thinking method. Goal guidance is actually a kind of reverse thinking, that is, starting from the goal to be achieved, step by step back until you find the conditions provided by the topic. You should think like this about a topic that cannot be seen through at a glance.

Third: difference analysis, two-way communication. This article is a supplement to the previous one. Some topics, conditions and goals are quite different, so it may be difficult to get through them in one fell swoop only by reverse thinking. It is necessary to change the conditions and objectives in two directions, constantly compare the differences between them, consciously narrow the differences between them, and finally achieve a breakthrough. It's like two construction teams digging the same tunnel at the same time, aiming at each other and walking in opposite directions.

Fourth: abstract problems and follow the law. Abstract problems can also be called regular problems, including abstract function problems, abstract inequality problems, abstract sequence problems and so on. In recent years, the so-called "new definition problem" is also a problem of regularity in essence. The forms of questions are large and small, and such questions are often annoying, which seriously affects the test mood. Why is this happening? Because the background of abstract questions is not specific, it seems that it is difficult for us to operate. In fact, abstract questions are cute paper tigers! Because one of the biggest advantages of abstract problems is that the rules are concrete. As long as the given rules are correctly used for deduction or assignment, or specific problems that meet the rules are found for exploration, it is not difficult to get what you want.

Fifth: key topics, break down one by one. This is a big article, and "content specialization" is the embodiment of systematization and modularization of knowledge. It can be said that every content of high school mathematics has a special topic, and students have spent a lot of effort in the first and second rounds of review, but for key topics, they need to make key breakthroughs. For example—

Title 1, given the recursive formula of series, how to find its general term formula?

Topic 2: How to find the sum of the first n terms of a given recursive formula or a general term formula of a sequence?

Topic 3: Problems related to eccentricity in analytic geometry, problems of focus triangle.

Topic 4: The relationship between straight lines and conic curves in analytic geometry.

Topic 5. Problems related to assembly in solid geometry.

Topic six, the topic of permanent institutions.

Topic 7, Application of Derivative.

Topic 8: Inference topic in function.

Topic 9, inequality problem with parameters.

Topic 10, permutation and combination to solve problems quickly and accurately.

……

For example, the function f(x) satisfies the condition f (x+2) for any real number x = if f (1) =–5, then f(f(5))= (06 Anhui College Entrance Examination, 15).

Analysis: According to the meaning of the question, f(x+2) = = f (x–2), so f (x) is a periodic function with a period of 4. ∴f(5)=f( 1)=-5, let x= 1, then f(3)= =-

∴f(f(5))= f(–5)= f(–5+8)= f(3)=-.

Sixth, the blind spots are cleared.

Just like Sun Tzu's Art of War said, we must be invincible before using troops, but we must wait for the enemy to win. He said, know yourself, know yourself. The college entrance examination, of course, should examine the rigor of candidates' ideas and thinking, so you must take this matter seriously, otherwise there will be many loopholes in the college entrance examination, and there will be regrets that it will be wrong or incomplete. What is your blind spot? May wish to refer to the following questions first-

Blind spot 0 1: ignore the denominator ≠0 and mistake ≥0 for (x-1) (x+1) ≥ 0;

Blind spot 02: missing solution. If sinx= is known, it is mistaken for x =+2 k, k ∈ z;

Blind spot 03: a stochastic model, such as finding the monotonic increasing interval of y=3sin(-x+ )+ 1, substituting -x+ into the monotonic increasing interval of y=sinx to solve the error;

Blind spot 04: Find the symmetrical center of y=tanx, misspelled as (k, 0), k ∈ z;

Blind spot 05: connect monotonous intervals with union symbols;

Blind spot 06: Ignore the domain of the new variable, such as substituting with sinx+cosx=t and forgetting to write t ∈ [-,];

Blind spot 07: Ignore self-limiting conditions, such as knowing 3sin2x+2sin2y-2sinx=0, and find the range of cos2x+cos2y;

Blind spot 08: Ignore implicit conditions, such as sinθ¢0, θ is considered to be in the third and fourth quadrants;

Blind spot 09: when using "equal intercept", forget to consider the case that the intercept is 0;

Blind spot 10: I forgot to define the priority domain, such as finding the monotonous interval of function y=lg(x2+2x);

Blind spot 1 1: I forgot to write the definition domain of the inverse function;

Blind spot 12: When using an= Sn- Sn- 1 to find the general term formula of series, the condition of n≥2 is forgotten;

Blind spot 20: Forgot to consider whether to take the interval endpoint? ;

Blind spot 13: Solve the related problems of ax2+bx+c, regardless of whether there is the possibility of a=0;

Blind spot 14: Is the common ratio of geometric series 1? ;

Blind spot 15: use trigonometric function value to find the angle, forget to discuss the range of angle value, and draw a conclusion;

Blind spot 16: to solve the distribution problem of roots of quadratic equation, the sign of discriminant is not preferred;

Blind spot 17: use the vector method to find the included angle of straight lines on different planes, forgetting to consider that what may be obtained is the complementary angle;

Blind spot 18: the concept of ball distance is unclear;

Blind spot 19: the extreme value is regarded as the maximum value;

……

I hope candidates can take the above-mentioned "typical blind spots" as an example, and collect your own knowledge blind spots, thinking blind spots and habit blind spots in the order of textbook content, and remove them one by one.

Seven. Abundance of conclusions

Singing, although some people sing "accurately", is not touching at all. Why? Because his timbre is not beautiful, the sound made by people with beautiful timbre is accompanied by a lot of homophones, just like there are several stars around the curved moon, which makes it more beautiful. The same is true of learning. In normal study, you must know that you have a lot of backbone knowledge, but if you only have backbone knowledge, your knowledge structure is dry. You must have come across many small conclusions. Although these small conclusions are lower than theorem formulas, they greatly enrich the original theorems and formulas and are very useful. Therefore, you should carefully collect and memorize more than 80 books in the order of the textbook catalogue. For example—

1 、( 1i)2 = 2i;

2、 1 sinθ=(sin cos)2≥0;

3、 1+cosθ=2cos2, 1-cosθ= 2 sin 2;

4. odd function+odd function = odd function, even function+even function = even function, …

5. increasing function+increasing function = increasing function, minus function minus function = increasing function, …

6. Monotonicity of compound function: same increase but different decrease;

7. The range of exponential function and logarithmic function: same big but different small;

8、x2 x+ 1 ≥> 0; | x | x≥0; If |f(x)|≥ f(x) is known, then f(x)

9. If f(x) is odd function and f(x) is significant at x=0, then f (0) = 0;

10, geometric progression+geometric progression = geometric progression, arithmetic progression+geometric progression = arithmetic progression, and arithmetic progression+geometric progression are generally neither arithmetic series nor proportional series.

In 1 1 and △ABC, Sina: sinB:sinC = a;; b:c;

In 12 and △ABC, if a, b and c become arithmetic progression, then B = 60.

13. In hyperbola (a, b > 0), the distance from focus F to asymptote is equal to b;

14, in the parabola y2 = 2px (P > 0), let the straight line l passing through the focus f intersect with the parabola at point A (x 1, y 1) and point B (x2, y2), and the projections of a, f and b on the directrix are,, and respectively. ; ; │AB │= =; ; The circle with diameter AB is tangent to the directrix, and the circle with diameter AB is tangent to the focus F; , o, b three-point * * * line; A, O, three-point * * * line; The tangent equation is: if this tangent is at point t, then FT=FP, ...

Eight, writing standardization

This is a cliche topic, and this article is nothing new. Let's give a few examples-

1、2、3、

By the way, if you are not a genius, if you don't want to lose more than 60 points in the college entrance examination, then start writing well and learn typesetting from reading this article.

What is the reason? Let me tell you a big secret-it is ordinary people, not aliens, who grade the college entrance examination papers. They all eat, drink and sleep; Everyone loves beauty (American); ..... They also like to doze off when doing repetitive work; When they read the papers, they all said, "The first day is tight, the second day is loose, and the third day is drums." . If your writing is not good, he will make you "speak out after eating a dark loss", and no one will hold the responsibility of the reviewers-the leaders who supervise them are also from the earth, so don't expect them to "get justice" for you. According to the rules, the college entrance examination paper only checks whether the total score is wrong, not whether it has been changed. That's the reason.

Therefore, if you want to win the college entrance examination, you must go through three levels: strength, writing and volunteering. And writing is the most important level.

The above "eight transformations" are magnificent and seemingly meticulous, but in fact there are leaks. For example, the scoring strategy in the exam, psychological problems in the exam, and "question types" are difficult to discuss in detail because of space constraints, so it is very simple to write, but readers should know its extreme importance. Moreover, everyone's situation is very different, and the countermeasures vary from person to person, each with its own emphasis. In short,

Note: Don't spread this information, but you can keep it for future reference.

2065438+ revised in March 2002 1

Author Li Guo, inventor and developer of self-help universal learning tool for solid geometry.