What are the skills and methods to do math problems?
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(1) observation method: intuitively and purposefully discover the laws, properties and solutions of mathematical objects.
(2) Experimental method: The experimental method is to create some mathematical objects that are beneficial to observation with purpose and simulation, and to visualize and simplify complex problems through observation and research. It has the important advantages of strong intuition, clear characteristics, trial solution and experimental conclusion.
2. Comparison and classification
(1) comparison method
It is a way of thinking to determine the similarities and differences of things. In mathematics, two mathematical objects must have a certain relationship before they can be compared. We often compare the similarities and differences between two kinds of mathematical objects or make a comprehensive comparison.
(2) Classification
Classification is a way of thinking based on the comparison and similarities and differences of mathematical objects, which classifies objects with the same nature into one category and objects with different properties into different categories. As shown in the above figure, the classification of k of a linear function is greater than zero and less than zero when it is not equal to zero, which embodies the principle of no repetition and no leakage.
3. Special and General
(1) specialization method
The method of specialization is to narrow the scope from a given area, even to a special value, special point, special figure, etc., and then consider the solution and rationality of the problem.
(2) Generalized method
4. Association and conjecture
(1) analogy association
Analogy is a way of thinking that another thing may have certain attributes according to the same or different attributes between two objects or things.
New knowledge can be discovered through analogy and association; Through analogy and association, find out the methods and ways to solve mathematical problems;
(2) Inductive conjecture
Newton said: Without bold conjecture, there would be no great invention. Guess can find truth and judgment; Conjecture can foresee the methods and ideas of proof. Junior high school mathematics is mainly to observe the conditions of propositions and draw conclusions, or to put forward solutions and methods to solve problems through observing conditions and conclusions.
Induction is a thinking process of drawing general conclusions from similarities or similarities contained in similar things. There are complete induction and incomplete induction. The conjecture obtained by complete induction is correct, and the conjecture obtained by incomplete induction may be correct or wrong, so it needs to be proved as a conclusion. The key is to make a reasonable and well-founded guess.
5. Substitution and formula
(1) replacement method
When solving a mathematical problem, we regard a formula as a whole and replace it with a variable, thus simplifying the problem. This is called substitution. The essence of substitution is transformation, the key is to construct elements and set elements, and the theoretical basis is equivalent substitution. The purpose is to change the research object, move the problem to the knowledge background of the new object, standardize non-standard problems, simplify complex problems and become easy to deal with.
Substitution method is also called auxiliary element method and variable substitution method. By introducing new variables, scattered conditions can be linked, implicit conditions can be revealed, or conditions can be linked with conclusions. Or turn it into a familiar form to simplify complicated calculation and derivation.
When using substitution method, we should follow the principle of facilitating operation and standardization. Pay attention to the selection of the new variable range after substitution, and make sure that the new variable range corresponds to the value range of the original variable, which cannot be reduced or expanded. You can observe the formula first, and you can find that there is always the same formula in this formula for method of substitution, and then use a letter instead of them to calculate the answer. Then, if there is this letter in the answer, bring the formula in and the calculation will come out.
(2) Matching method
Matching method is a technique of directional deformation of mathematical formula (matching into a "complete square"), and the relationship between known and unknown is found through the formula, thus simplifying the complex. When making a formula, it is necessary to make appropriate predictions and use the skills of "dividing", "adding", "matching" and "gathering" reasonably, so as to complete the formula. Sometimes called "matching method". The most common formula is identical deformation, so that the mathematical formula appears completely square. It is mainly suitable for discussing and solving known or unknown quadratic equations, quadratic inequalities, quadratic functions and quadratic algebras. The most basic formula used in the matching method is the binomial complete square formula (a+b) 2 = a 2+2ab+b 2. By using this formula flexibly, various basic formula forms can be obtained.
6. Construction method and undetermined coefficient method
(1) construction method The so-called construction method refers to the concepts and methods in mathematics, which can be defined and realized in a fixed way through limited steps. Common ones are constructors, graphs and identities. Adding auxiliary lines in plane geometry is a common drawing method. There are three ways to solve problems by construction method: direct construction, changing conditional construction and changing conclusion construction.
(2) The undetermined coefficient method: the polynomial is expressed as another new undetermined coefficient form, thus an identity is obtained. Then, according to the properties of identity, the equation or equation that the coefficient should satisfy is found, and then the coefficient to be solved is found by solving the equation or equation, or the relationship that some coefficients satisfy is found. This method of solving problems is called undetermined coefficient method.
7. Formulas and reduction to absurdity
(1) formula method
The method of solving problems by using formulas. The most commonly used method in junior high school is to find the root of a quadratic equation with one variable by formula; The method of complete square formula, etc. For example, the following set of problems is the application of the complete square formula:
(2) The reduction to absurdity is an "indirect proof method", that is, the title is affirmed and the conclusion is denied, which leads to contradictions, the correctness of the conclusion of the proposition is affirmed and the proposition is proved.