A mathematical problem is expressed by function, and the general law of this problem is explored by function.

2 Number-shape combination thought

Combine algebra with geometry, such as solving geometric problems with algebra and solving algebraic problems with geometry.

3 holistic thinking?

Integral substitution, superposition and multiplication processing, integral operation, integral element setting, integral processing, complement in geometry, etc. It is the concrete application of integral thinking method in solving mathematical problems.

4 change ideas

It is through deduction and induction that unknown, unfamiliar and complex problems are transformed into known, familiar and simple problems.

5 analogical thinking?

Comparing two (or two) different mathematical objects, if they are found to have similarities or similarities in some aspects, it is inferred that they may also have similarities or similarities in other aspects. ?

Extended data:

Function thought refers to analyzing, reforming and solving problems with the concept and nature of function. The idea of equation is to start with the quantitative relationship of the problem, transform the conditions in the problem into mathematical models (equations, inequalities or mixed groups of equations and inequalities) with mathematical language, and then solve the problem by solving equations (groups) or inequalities (groups). Sometimes, functions and equations are mutually transformed and interrelated, thus solving problems.

Descartes' equation thought is: practical problem → mathematical problem → algebraic problem → equation problem. The universe is full of equality and inequality. We know that where there are equations, there are equations; Where there is a formula, there is an equation; The evaluation problem is realized by solving equations ... and so on; The inequality problem is also closely related to the fact that the equation is a close relative. Column equation, solving equation and studying the characteristics of equation are all important considerations when applying the idea of equation.

Function describes the relationship between quantities in nature, and the function idea establishes the mathematical model of function relationship by putting forward the mathematical characteristics of the problem, so as to carry out research.

It embodies the dialectical materialism view of "connection and change". Generally speaking, the idea of function is to use the properties of function to construct functions to solve problems, such as monotonicity, parity, periodicity, maximum and minimum, image transformation and so on. We are required to master the specific characteristics of linear function, quadratic function, power function, exponential function, logarithmic function and trigonometric function.

In solving problems, it is the key to use the function thought to be good at excavating the implicit conditions in the problem and constructing the properties of resolution function and ingenious function. Only by in-depth, full and comprehensive observation, analysis and judgment of a given problem can we have a trade-off relationship and build a functional prototype. In addition, equation problems, inequality problems and some algebraic problems can also be transformed into functional problems related to them, that is, solving non-functional problems with functional ideas.

Function knowledge involves many knowledge points and a wide range, and has certain requirements in concept, application and understanding, so it is the focus of college entrance examination.

The common types of questions we use function thought are: when encountering variables, construct function relations to solve problems; Analyze inequality, equation, minimum value, maximum value and other issues from the perspective of function; In multivariable mathematical problems, select the appropriate main variables, thus revealing the functional relationship between them.

Practical application of problems, translation into mathematical language, establishment of mathematical models and functional relationships, and application of knowledge such as functional properties or inequalities to solve them; Arithmetic, geometric series, general term formula and sum formula of the first n terms can all be regarded as functions of n, and the problem of sequence can also be solved by function method.

The main reasons for classified discussion are as follows:

① Classify and define the mathematical concepts involved in the problem. For example, the definition of |a| can be divided into three situations: a>0, a=0 and a<0. This kind of classified discussion questions can be called conceptual.

② Mathematical theorems, formulas, operational properties, laws, limited scope or conditions involved in the problem, or given by classification. For example, the formula of the sum of the first n terms of geometric series can be divided into two cases: q= 1 and q≠ 1. This kind of classified discussion questions can be called natural type.

③ When solving problems with parameters, we must discuss them according to the range of parameters. For example, solving the inequality ax> at 2 am>0, a=0 and a.

In addition, some uncertain quantities, the shape or position of uncertain figures, uncertain conclusions, etc. It is mainly discussed through classification to ensure their integrity and make them deterministic.

When discussing classification, we should follow the following principles: determination of classification objects, unification of standards, no omission and repetition, scientific classification, clear priority and no skipping discussion. The most important one is "no leakage and no weight".

When answering classified discussion questions, our basic methods and steps are as follows: first, we must determine the scope and the whole discussion object; Secondly, determine the classification standard, correct and reasonable classification, that is, the standard is unified, no duplication is missed, and the classification is mutually exclusive (no repetition); Then discuss the classification step by step and get the results in stages; Finally, a summary is made and a comprehensive conclusion is drawn.

References:

Baidu Encyclopedia-Mathematical Thinking Method