2. Factorization Factorization is to turn a polynomial into the product of several algebraic expressions. Factorization is the basis of identity deformation. As a powerful tool and mathematical method, it plays an important role in solving algebra, geometry and trigonometric functions. 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. method of substitution method of substitution is a very important and widely used problem-solving method in mathematics. We usually refer to unknowns or variables as elements. 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. The discriminant of the roots of Vieta's theorem and the discriminant △=b2-4ac unary quadratic equation ax2 bx c=0(a, B, c∈R, a≠0) can be used not only to judge the properties of roots, but also as a method to solve algebraic deformation, equations (groups), inequalities, research functions and even analyze geometry. 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. When solving mathematical problems, the undetermined coefficient method is called the undetermined coefficient method. If it first judges that the obtained results have a certain form and contain some undetermined coefficients, then it lists the equations about the undetermined coefficients according to the problem setting conditions, and finally finds out the values of these undetermined coefficients or finds out some relationship between them, thus solving mathematical problems. It is one of the important methods commonly used in middle school mathematics.
6. Construction method When solving problems, we often use this method to construct auxiliary elements through the analysis of conditions and conclusions. It can be graphs, equations (groups), equations, functions, equivalent propositions, etc. 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. Reduction to absurdity is an indirect proof method. It is a way to put forward a hypothesis contrary to the conclusion of the proposition, and then proceed from this hypothesis and lead to contradictions through correct reasoning, 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 and trees 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
8. Equal (surface or volume) product method The area (volume) formula in plane (solid) geometry and the property theorems related to the calculation of area (volume) derived from the area (volume) formula can be used not only to calculate area (volume), but also to prove (calculate) geometric problems sometimes with half the effort. The method of using area (volume) relationship to prove or calculate geometric problems is called equal (surface or volume) product method, which is a common method in geometry. The difficulty in proving geometric problems by induction or analysis lies in adding auxiliary lines. The characteristic of equal (surface or volume) product method is to connect the known quantity with the unknown quantity by area (volume) formula, and to achieve the verification result through operation. Therefore, using equal (surface or volume) product method to solve geometric problems, the relationship between geometric elements becomes the relationship between quantities, which only needs calculation, sometimes without auxiliary lines, even if auxiliary lines are needed, it is easy to consider.
9. 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.
10. Method of solving objective questions Multiple-choice questions are a kind of questions that give conditions and conclusions and require correct answers according to certain relationships. Multiple-choice questions are ingenious in conception and flexible in form, which can comprehensively examine students' basic knowledge and skills, thus increasing the capacity and knowledge coverage of test papers. Fill-in-the-blank question is one of the important questions in standardized examination. Like multiple-choice questions, it has the advantages of clear test objectives, wide knowledge coverage, accurate and fast marking, and is conducive to examining students' analytical judgment and calculation ability. The difference is that the fill-in-the-blank question does not give an answer, which can prevent students from guessing the answer. Fast and accurate
Correctly solving multiple-choice questions and fill-in-the-blank questions requires not only accurate calculation and strict reasoning, but also methods and skills to solve multiple-choice questions and fill-in-the-blank questions.