Nine problem-solving skills in college mathematics
1, matching method
Using the method of constant deformation, the method of matching some items of an analytical formula into the sum of positive integer powers of one or more polynomials to solve mathematical problems is called matching method. The most commonly used matching method is completely flat method, which is an important constant deformation method in mathematics. It is widely used in factorization, simplifying roots, solving equations, proving equality and inequality, finding extreme values of functions and analytical expressions.
2, factorization method
Factorization is to transform a polynomial into the product of several algebraic expressions, which is the basis of identity deformation. As a powerful mathematical tool and method, it plays an important role in solving algebra, geometry and trigonometry problems. There are many methods of factorization, such as extracting common factors, formulas, grouping decomposition, cross multiplication and so on. Middle school textbooks also introduce the use of decomposition and addition, root decomposition, exchange elements, undetermined coefficients and so on.
3. Alternative methods
Method of substitution is a very important and widely used method to solve problems in mathematics. Usually, unknowns or variables are called variables. The so-called method of substitution is to replace a part of the original formula with new variables in a complicated mathematical formula, thus simplifying it and making the problem easy to solve.
4. Discriminant method and Vieta theorem.
The method for judging the roots of unary quadratic equation ax2bxc=0(a, B, C belongs to R, a≠0) and△ = B2-4ac is not only used to judge the properties of roots, but also widely used as a method for algebraic deformation, solving equations (groups), solving inequalities, studying functions and even solving problems in geometric and trigonometric operations.
Vieta's theorem not only knows one root of a quadratic equation, but also finds another root. Knowing the sum and product of two numbers, we can find the symmetric function of the root, calculate the sign of the root of quadratic equation, solve the symmetric equation and solve some problems about quadratic curve. , has a very wide range of applications.
5, undetermined coefficient method
When solving mathematical problems, it is first judged that the obtained results have a certain form, which contains some undetermined coefficients, then the equations about undetermined coefficients are listed according to the problem setting conditions, and finally the values of these undetermined coefficients or some relationship between these undetermined coefficients are found. This method is called undetermined coefficient method to solve mathematical problems. It is one of the commonly used methods in middle school mathematics.
6. Construction method
When solving problems, we often use this method to construct auxiliary elements by analyzing conditions and conclusions, which can be a figure, an equation (group), an equation, a function, an equivalent proposition and so on. And establish a bridge connecting conditions and conclusions, so that the problem can be solved. This mathematical method of solving problems is called construction method. Using construction method to solve problems can make algebra, trigonometry, geometry and other mathematical knowledge permeate each other, which is beneficial to solving problems.
7. Find the area method
The area formula in plane geometry and the property theorems related to area calculation derived from the area formula can be used not only to calculate the area, but also to prove that plane geometry problems sometimes get twice the result with half the effort. The method of proving or calculating plane geometric problems by using area relation is called area method, which is commonly used in geometry.
The difficulty in proving plane geometry problems by induction or analysis lies in adding auxiliary lines. The characteristic of area method is to connect the known quantity with the unknown quantity by area formula, and achieve the verification result through operation. Therefore, using the area method to solve geometric problems, the relationship between geometric elements becomes the relationship between quantities, and only calculation is needed. Sometimes there may be no auxiliary lines, even if auxiliary lines are needed, it is easy to consider.
8, geometric transformation method
In the study of mathematical problems, the transformation method is often used to transform complex problems into simple problems and solve them. The so-called transformation is a one-to-one mapping between any element of a set and the elements of the same set. The transformation involved in middle school mathematics is mainly elementary transformation. There are some exercises that seem difficult or even impossible to start with. We can use geometric transformation to simplify the complex and turn the difficult into the easy. On the other hand, the transformed point of view can also penetrate into middle school mathematics teaching. It is helpful to understand the essence of graphics by combining the research of graphics under isostatic conditions with the research of motion.
Geometric transformation includes: (1) translation; (2) rotation; (3) symmetry.
9. reduce to absurdity
Reduction to absurdity is an indirect proof method. First, a hypothesis contrary to the conclusion of the proposition is put forward, and then from this hypothesis, through correct reasoning, contradictions are led out, thus denying the opposite hypothesis and affirming the correctness of the original proposition. The reduction to absurdity can be divided into reduction to absurdity (with only one opposite conclusion) and exhaustive reduction to absurdity (with more than one opposite conclusion). The steps to prove a proposition by reduction to absurdity can be roughly divided into: (1) reverse design; (2) return to absurdity; (3) conclusion.
Anti-design is the basis of reduction to absurdity. In order to make correct anti-design, we need to master some commonly used negative expressions, such as: yes/no; Existence/non-existence; Parallel/non-parallel; Vertical/not vertical; Equal to/unequal to; Large (small) inch/small (small) inch; Both/not all; At least one/none; At least n/ at most (n-1); At most one/at least two; Only/at least two.
Reduction to absurdity is the key to reduction to absurdity. There is no fixed model in the process of derivation of contradiction, but it must be based on reverse design, otherwise the derivation will become passive water without roots. Reasoning must be rigorous. There are the following types of contradictions: contradictions with known conditions; Contradicting with known axioms, definitions, theorems and formulas; There are dual contradictions; Contradictions
Answering Strategies in College Mathematics
First, learn to examine questions before solving them.
Many candidates do not pay enough attention to the examination of the topic, and often rush to write without seeing the topic clearly. Examining questions well is the key to doing them. When reviewing the topic, you must read it word for word, and make sure whether the topic is easy to miss and make mistakes. Only by carefully examining the questions can we get more information from them. Only by excavating the hidden conditions in the problem, enlightening the thinking of solving the problem and reminding us of the common misunderstandings and mistakes that are easy to occur can we improve our problem-solving ability. Only by careful examination and cautious attitude can we accurately figure out the questioner's intention and find more information, so as to quickly find the direction to solve the problem.
Before the exam, we should keep a clear head, abandon distractions, constantly give positive psychological hints, create a relaxed atmosphere, create mathematical situations, and then brew mathematical thinking, calm down, comfort ourselves with self-confidence, and prepare for the exam with a steady, confident and positive attitude. This requires us to be good at observation.
Second, do simple questions first, and then do difficult problems.
From our psychological point of view, we are generally nervous after getting the test paper. At this time, don't rush to solve the problem. We can browse the number, distribution and difficulty of the questions from beginning to end. When you do the problem, you should make it easy before it is difficult, so that you know what you are doing. Generally, short answer questions account for 60% of the whole paper. This is a very important part of the score. When you see simple questions, you should solve them seriously, try to use mathematical language, be more rigorous, cheer up and review well.
If it takes time to do the questions in order and you can't get points, the questions you can do will be delayed again. So do the simple questions first. Years of experience tell us that when you don't solve the problem smoothly, you should be calm, calm and just hold your horses. According to your actual situation, skip the questions that you can't do decisively and finish the simple ones. If we can get this part of the score, we will have won the battle, and then concentrate on the more difficult problems. With the confidence to win, be patient and don't worry when facing more difficult questions. Get the score you deserve. It is best to have the ability to make difficult problems simple.
Third, do more exercises to improve your ability.
Generally speaking, if you want to do well in the college entrance examination mathematics, you must do a lot of exercises and have a solid theoretical foundation, supplemented by problem-solving skills on this basis, so that there will be no shortage of examination time, and there will be no time to do the problems that can be done in the end, which is not worth the candle. On the basis of a lot of practice, we are required to sum up the ideas of equation, combination of numbers and shapes, function and so on. , master the laws of various topics.
We also require candidates not only to do questions, but also to answer them accurately and quickly. It is necessary to master problem-solving skills through practice, and use problem-solving skills to solve problems quickly. Practice makes perfect. This is the purpose of our practice. Concentrate on the problem, which is the guarantee of success in the exam. Sometimes when you are nervous, the problems you can do will become impossible. Usually, it is necessary to train some difficult problems in a targeted manner, which is conducive to positive thinking and confidence building.
Therefore, it is very necessary for most college entrance examination students to strengthen training, develop accurate problem-solving habits and master problem-solving skills.
Fourth, the problems that can be done are guaranteed to be done correctly.
This is very important. In practice, it is found that the loss rate of the questions we will do in the exam is 10%, that is to say, because of carelessness, everyone will lose so many points in every exam. How to turn your problem-solving strategy into a scoring point, although the problem-solving idea is correct, even ingenious, may be wrong in the end, which is often ignored by some candidates. However, because we are not good at turning graphic language into a language we understand, we will make mistakes on paper in many cases. Our own estimate. For example, in solid geometry argument, more than one-third of the total score will be lost, and in algebraic argument, the score will be even less pitiful. Therefore, it is necessary to check whether the thinking of solving the problem is correct while doing it, and then check it carefully after finishing it. Not only to complete the topic, but also to ensure accuracy. If you can do it, be sure to do it right and get points.