Candidates form certain obstacles when answering college entrance examination questions. Mainly manifested in two aspects, one is that we can't find the breakthrough point to solve the problem, and the other is that although we have found the breakthrough point to solve the problem, we can't do it. How to solve these two obstacles?
First, start with solving (proving)-find the basic method of solving problems. When we encounter a difficult problem, we will find that the questioner has set up various obstacles. Starting from the known, there are many forks in the road, and it is more and more complicated to push forward, so it is difficult to get the answer. If we start with the problem, we will find out what we have to do to get what we want. After finding the "need to know", we will take the "need to know" as a new problem until we can communicate with the "knowable" we can get from the "known" and solve the problem. In fact, the "analysis method" used in inequality proof is the full embodiment of this kind of thinking, which we call "reverse thinking"-necessary thinking.
Second, the deformation of mathematical formula-the key to complete the problem-solving process. The second obstacle encountered in college entrance examination math problems is the deformation of mathematical formulas. In order to complete the process from known to conclusion, a mathematical comprehensive problem must go through a lot of mathematical formula deformation, and it can't be completely mastered only by a lot of problem-solving processes. Many candidates have the experience that they can't solve a complicated problem, but when they look back at the answer, they suddenly realize that the original solution is so simple that they regret it. Why did they complain that they were so confused that they didn't change the formula again?
In fact, every step of reasoning and operation to solve mathematical problems is essentially transformation (deformation). However, the purpose of transformation (deformation) is to solve the problem better and faster, so the direction of transformation must be to simplify the complex, abstract the concrete, and turn the unknown into the known, that is, to create conditions for transformation in a direction conducive to solving the problem. It must also be noted that all transformations must be equivalent, otherwise the solution is wrong.
Solving mathematical problems is actually to build a bridge between known conditions and conclusions to be solved, that is, to reduce and eliminate these differences on the basis of analyzing the differences between known conditions and conclusions to be solved. Seeking difference is the principle of deformation dependence, and some regular things in deformation need to be summarized. In the next chapter, we list some thinking modes, which are summarized under the guidance of mathematical thought. In solving college entrance examination questions, mathematical deformation is always from complex to simple, which is also the way of thinking of transformation and mathematical formula deformation: always pay attention to the difference between what is sought and what is known.
Third, return to textbooks-lay a solid foundation.
1) Reveal the law-master the method of solving problems. No matter how difficult the college entrance examination questions are, they can't escape the thinking methods and laws revealed by textbooks. When we talk about returning to textbooks, we don't simply sort out knowledge points. The process of deducing theorems and formulas in textbooks contains important methods. However, many candidates do not completely expose their thinking process and find out the rules of their inner thinking, but hope to "realize" some truths through the tactics of asking questions. As a result, the problem is not short of bubbles, but it always fails to achieve results. In the end, I can only stay in a place with superficial understanding, only mechanical imitation and low thinking level. Therefore, we should focus on the analysis of basic concepts and basic theories, so as to make constant changes.
2) Building the Network —— The function of textbooks has many important conclusions. Many students rely on rote memorization because of their poor understanding ability, which eventually leads to poor memory and loss of points in exams.
For example:
If f(x+a)=f(b-x), then f(x) is symmetric. How to understand? Let X 1 = A+X, X2 = B-X, then F (X 1) = F (X2), X 1+X2 = A+B, = constant, that is, the sum of two independent variables is constant, and their corresponding function values are equal, thus understanding the essence of symmetry. It is very simple to remember this conclusion by using the abscissa of the coordinates of the points in analytic geometry as a fixed value, or by using the images of special functions and quadratic functions. As long as x 1+x2=a+b and = constant f(x 1)=f(x2), it can be written as f (x) = f (a+b-x) and other forms.
If f(x) = f (2a-x) and f (x) = (2b-x), then the period of f(x) is t = 2 | a-b |||| How to understand and remember this conclusion? Let's compare the trigonometric function f(x)=sinx. From the sine function diagram, we can know that x =/2 and x = 3/. This is the embodiment of the combination of numbers and shapes with abstract and concrete ideas. Thought refinement and summary play a key role in the review process. A similar conclusion is that if f(x) is symmetric about points A(a, 0) and B(b, 0), the period of f(x) is T=2|b-a|, and if f(x) is symmetric about point a (a, 0) and X = b, the period of f(x) is t =
In this way, you can do it from coarse to fine in the chapter of function, and you don't need to recite anything. At the same time, learn to use these conclusions in reverse.
Example: when two symmetrical axes x=a and b = 2a, x = b(b >；; A) is an even function. Similarly, the symmetry point B(B, 0), the symmetry axis x = a and b = 2a are all odd function.
3) Strengthen understanding-improve review ability and really return to the track of attaching importance to basics. No foundation, no ability The basis here does not refer to repeated mechanical training, but refers to understanding the basic principles and methods, experiencing the formation process of knowledge, and understanding and feeling the essential meaning of knowledge. Only by deeply understanding the concept can we grasp the essence of the problem and build a knowledge network.
4) Thinking mode-the steps of solving problems are fixed. There are certain rules to follow in solving math problems. The problem-solving operation should have clear ideas and goals, and the way of thinking should be achieved.
The so-called patterning is the immobilization of problem-solving steps, and the general thinking process is divided into the following steps:
A, the key to the examination is to find out what is required (certified) first. What are the known conditions? What is the conclusion? Whether the expression of conditions can be transformed (number-shape transformation, symbol-graph transformation, text expression into mathematical expression, etc.). ), and what are the characteristics of the given graphs and formulas? Can you use graphics (geometry, function or schematic diagram) or mathematical formulas (text questions) to express the problem? What are the implied conditions? What knowable terms and conditions can be deduced from known conditions? What must I do to ask an unknown conclusion? What conditions do you need to know?
B, clear the goal of solving the problem. Pay attention to the gap between what is known and what is sought, and carry out mathematical formula deformation (transformation) to bridge what needs to be known and what is known (missing)
1) Can you simplify the complicated formula in the problem?
2) Can we divide the conditions and turn a big problem into several small problems?
3) Can variable substitution (substitution) and identity transformation be carried out to make the form of the problem more obvious?
4) Is it possible to perform algebraic sub-geometric transformation (combination of numbers and shapes)? Solving algebraic problems by geometric methods? Or use algebraic (analytical) methods to solve geometric problems? Can mathematical languages be converted? (Vector expression is converted into solution expression, etc. )
5) Ultimate goal: transform the unknown into the known.
C, the problem-solving requirements are clear, concise and correct, the reasoning is rigorous, the operation is accurate, and the steps are not skipped; To express normative and step-by-step analytical thinking and problem-solving thinking can be summarized as: objective analysis, condition analysis, difference analysis, structural analysis, reverse thinking, reduction, intuition, special transformation, principal component transformation and substitution transformation.