Nine problem-solving skills of mathematics in senior one.
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
How to study with poor math foundation in senior one?
First, quickly master the basic knowledge
For students with weak foundation, textbooks are the magic weapon they need to master in the first step. If you want to improve your math scores, you need to memorize every knowledge point in the math textbook, understand every example and master it chapter by chapter.
You can recite the formula first, then study the examples, and finally do exercises after class, using examples and exercises to think about how to solve them. Don't rush to calculate, just think about it first, and then read the textbook to see how the formula theorem is derived, especially the process and application cases. For the typical problems in textbooks, we should have a deep understanding and learn to reflect after solving the problems. Only in this way can we deeply understand this problem and jump out of the strange circle of the sea of questions.
Take notes on the wrong questions, record the mistakes that are easy to make, analyze the causes of the mistakes, and find the correct methods. Don't do the questions blindly, it is useful to do them on the basis of clear concepts.
Second, learn to use basic knowledge.
While mastering the basic knowledge of mathematics, we should learn to use it, so that we can get a point in the exam. The characteristics of high school mathematics learning are: high speed, large capacity and many methods. For students with poor foundation, sometimes they can't remember after listening, or they can't solve the problem after remembering. At this time, you need to take good notes. We don't need to write down what the teacher said word for word. We just need to write down the key ideas and conclusions. After class, we will sort out our review notes, which is also a process of re-learning.
If you want to learn math problems well, you must do more problems. Only by doing certain problems can we learn mathematics well, and doing problems is the main theme of high school mathematics learning. But the problem here is not to do it blindly, but to think. Learning to think, reflect and summarize is the king of learning mathematics.
In fact, it is not difficult to solve math problems. We should analyze the problem, dig out the known conditions, find out the relationship between these conditions and draw useful conclusions. This conclusion is the key to solving problems and the form of solving problems in mathematics. Therefore, if you want to learn mathematics well, you mainly rely on the way of answering questions, not the way of doing a certain problem.
Grading skills of mathematics in senior one.
First, preview is a smart choice.
The teacher had better specify the preview content, which should not exceed ten minutes every day. The purpose of preview is to force memory of basic concepts.
Second, the basic concept is fundamental.
Basic concepts should be understood and memorized word by word, and the connotation and extension of basic concepts should be accurately grasped. Only when thinking goes in can we understand the connotation, and when thinking diverges can we understand the extension. Only when the concept passes, can the problem be written quickly and accurately.
Third, homework can consolidate what you have learned.
It is a good thing to do your homework carefully, save time and steps, and not check your homework to fully expose the existing problems.
Fourth, the problem should be completed independently.
If you want to get high marks, you must pass this difficult problem. The key to the problem is to learn how to skillfully convert three languages. (written language, symbolic language, graphic language)
Five, double decreasing training method
Through training, you can gradually adjust to the best state of the exam from psychology, energy and accuracy. Training must be carried out under the guidance of professionals, otherwise it will not achieve results.
6. Don't do new questions before the exam.
Find the test paper you have done recently before the exam and repeat the wrong questions. This is a targeted review method.
Seven, a good attitude
Candidates should be confident and have objective test objectives. Pursue normal performance, don't expect your performance for a long time, so your mentality will be very peaceful. Calm and calm, but also moderately nervous, so that the brain is in the best state of activity.
Eight, the exam begins with the exam.
We should avoid the two bad habits of "guessing" and "missing" when reviewing the questions, so we should read the sentences word for word and then the sentences.
Learn to use calculus paper
Calculus paper should be a part of the examination paper, neat and orderly, and the title number should be clearly written for easy inspection.
Ten, correctly treat the problem.
The problem is used to open the score. No matter your level, you should learn to avoid difficult problems and do them at last. Don't be confused by difficult problems. Only in this way can you be guaranteed to be among the best in any exam.
Nine problem-solving skills articles in senior one mathematics;
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