In the process of transformation, we should follow three principles:

1, the principle of familiarity, that is, transforming unfamiliar problems into familiar ones;

2. The principle of simplification, that is, turning complex problems into simple ones;

3. The principle of visualization, that is, abstraction is always concrete.

Strategy 1: Forward and reverse transformation

The title and conclusion of the proposition are dialectical unity of causality. When solving a problem, if thinking is blocked from below, we might as well start from the front and think backwards, and there is often another shortcut.

Example: 1: There are * *10 points at the vertex and midpoint of each side of a tetrahedron, among which four points that are not * * * faces are selected, and there are _ _ _ _ _ _ _ ways to take * * faces.

a、 150 B、 147 C、 144 D、 14 1

Analysis: The problem is more complicated from the front. If we consider it from the opposite side, it will be much easier to find the total number of four * * * faces before applying the idea of complementary sets.

There are many ways to select any four points in 10, among which four of the six points in the plane ABC are * * * planes, and there are also many other three planes, and there are also six kinds of * * * planes at the midpoint of each side and the opposite side, and there are three kinds of * * * planes at the midpoint of each side. There are three ways to select non-* * planes.

Strategy 2: the transformation from the local to the whole

It is a common way of thinking to analyze problems step by step, but for more complex mathematical problems, we should grasp things as a whole, don't entangle details, analyze problems from the system, and don't fight alone.

Example 2: If all sides of a tetrahedron are equal and all four vertices are on the same sphere, then the surface area of this sphere is ().

A, B, C, D,

Analysis: If we use the properties of the circumscribed sphere of a regular tetrahedron to construct a right triangle to solve it, the process is lengthy and easy to make mistakes. However, if the regular tetrahedron supplements to form a cube, then the center of the regular tetrahedron and the center of the cube and the spherical surface of the circumscribed sphere are * * *, because the side length of the regular tetrahedron is 1, so the radius of the circumscribed sphere is, and (a) should be selected.

Strategy 3: Turn the Unknown into the Known

Also known as analogical transformation, it is an important learning method to cultivate knowledge transfer ability. If we can grasp the known key information in the topic, lock the similarity and skillfully convert the analogy, the answer will come into being.

Example 3: In arithmetic progression, if, then there is an equation.

(hold, analogy of the above properties, in the geometric series, there is the equation _ _ _ _ _ _ _ _.

Analysis: arithmetic progression's,, must exist, so there is an analogy with geometric series, because, is established.

Second, the idea of logical division.

For example 1, the set A= is known, and the set B=, if B A, the set of values of the real number a.

Solution A=: It is discussed in two cases.

(1)B=￠, where a = 0；;

(2)B is a unary set, and B=. At this time, it is discussed in two situations:

(i) B={- 1}, then =- 1, a=- 1.

(ii)B={ 1}, then = 1, and a= 1. (secondary classification)

Based on the above requirements, it is set to.

Example 2, let the function f(x)=ax -2x+2, so as to satisfy 1? x？ All x values of 4 have f(x)? 0, the range of the real number a.

Example 3, known, try to compare the sizes.

analyse

Therefore, we can know that solving this problem must be classified and discussed, and its dividing point is.

Summary: The general steps of classified discussion:

(1) Clearly discuss the scope of object and object P (i.e. which parameter to discuss);

(2) Determine the classification standards, reasonably divide P, and unify the standards, which are neither heavy nor leakage, and shall not be discussed above the level.

(3) Discuss item by item and achieve phased results.

(4) Summarize and draw a comprehensive conclusion.

Summary and detailed explanation of eleven mathematical thinking methods

Mathematical thinking refers to the spatial form and quantitative relationship of the real world reflected in people's consciousness, which is the result of thinking activities. Mathematical thought is the essential understanding after summarizing mathematical facts and theories; The thought of basic mathematics is the basic, summative and most extensive mathematical thought embodied or should be embodied in basic mathematics. They contain the essence of traditional mathematical thought and the basic characteristics of modern mathematical thought, and are historically developed. Through the cultivation of mathematical thinking, the ability of mathematics will be greatly improved. Mastering mathematical thought means mastering the essence of mathematics.

1, the idea of functional equation

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 need to be transformed and interrelated to achieve the purpose of solving problems.

Descartes' equation thought is: practical problems? Math problems? Algebraic problems? 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; 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 thought of function, be good at excavating the implicit conditions in the problem, and construct the properties of distinguishing 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, set problems, sequence problems and some algebraic problems can also be transformed into related functional problems, 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 appropriate main variables and reveal their functional relationships; 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.

2. The combination of numbers and shapes.

"Numbers are invisible, not intuitive, and numerous shapes make it difficult to be nuanced", and the application of "combination of numbers and shapes" can make the problem to be studied difficult and simple. Combining algebra with geometry, such as solving geometric problems by algebraic method and solving algebraic problems by geometric method, is the most commonly used method in analytic geometry. For example, find the root number ((A- 1)2+(B- 1)2)+ root number (A 2+(B- 1)2)+ root number ((A- 1) 2+B).

Step 3 discuss ideas by category

When a problem may lead to different results because of different situations of a certain quantity or number, it is necessary to discuss the various situations of this quantity or number in categories. Such as solving inequality | a-1| >; 4. It is necessary to discuss the value of A in different categories.

4. Equal thinking

When a problem may be related to an equation, we can solve it by constructing the equation and studying its properties. For example, when proving Cauchy inequality, Cauchy inequality can be transformed into a discriminant of quadratic equation.

5. Overall thinking

Starting from the overall nature of the problem, we emphasize the analysis and transformation of the overall structure of the problem, find out the overall structural characteristics of the problem, and be good at treating some formulas or figures as a whole with the "overall" vision, grasping the relationship between them, and carrying out purposeful and conscious overall treatment. The holistic thinking method is widely used in simplification and evaluation of algebraic expressions, solving equations (groups), geometric proof and so on. Integral substitution, superposition multiplication, integral operation, integral demonstration, integral processing and geometric complement are all concrete applications of integral thinking method in solving mathematical problems.

6, return to thought

It is through deduction and induction that unknown, unfamiliar and complex problems are transformed into known, familiar and simple problems. The mathematical theories of ancient mathematics, such as trigonometric function, geometric transformation, factorization, analytic geometry, calculus and even ruler drawing, are permeated with the idea of transformation. Common transformation methods include: general special transformation, equivalent transformation, complex and simple transformation, number-shape transformation, structural transformation, association transformation, analogy transformation and so on.

Transformation thinking can also be called transformation thinking in a narrow sense. The idea of transformation is to transform the problem A to be solved or difficult to solve into the problem B with a fixed solution or easy to solve by some transformation means, and solve the problem A by solving the problem B. ..

7. Implicit conditional thinking

Conditions that are not explicitly stated but can be inferred from existing explicit expressions, or conditions that are not explicitly stated but are routines or truths. For example, in an isosceles triangle, a line segment is perpendicular to the base, so the line in which this line segment is located also bisects the base and vertex.

8. 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.

9. Modeling ideas

In order to describe an actual phenomenon more scientifically, logically, objectively and repeatedly, people use a language that is generally considered rigorous to describe various phenomena. This language is mathematics. What is described in mathematical language is called a mathematical model. Sometimes we need to do some experiments, but these experiments often use abstract mathematical models as substitutes for actual objects, and the experiments themselves are also theoretical substitutes for actual operations.

10, inductive reasoning thought

Some objects of a certain kind of things have certain characteristics, and all objects of this kind of things have the inference of these characteristics, or the inference that generalizes general conclusions from individual facts is called inductive reasoning (induction for short). In short, inductive reasoning is from part to whole, from individual to general reasoning.

In addition, there are mathematical ideas such as probability statistics, for example, probability statistics refers to solving some practical problems through probability statistics, such as the winning rate of lottery tickets, the comprehensive analysis of an exam and so on. In addition, some area problems can be solved by probability method.

Let me give you an example. There is an angular bisector in the picture, which can be perpendicular to both sides.

You can also look at the picture in half, and there will be a relationship after symmetry.

Angle bisector parallel lines, isosceles triangles add up.

Angle bisector plus vertical line, try three lines.

Perpendicular bisector is a line segment that usually connects the two ends of a straight line.

It needs to be proved that the line segment is double-half, and extension and shortening can be tested.

The two midpoints of a triangle are connected to form a midline.

A triangle has a midline and the midline extends.

A parallelogram appears and the center of symmetry bisects the point.

Make a high line in the trapezoid and try to translate a waist.

It is common to move diagonal lines in parallel and form triangles.

The card is almost the same, parallel to the line segment, adding lines, which is a habit.

In the proportional conversion of equal product formula, it is very important to find the line segment.

Direct proof is more difficult, and equivalent substitution is less troublesome.

Make a high line above the hypotenuse, which is larger than the middle term.

Calculation of radius and chord length, the distance from the chord center to the intermediate station.

If there are all lines on the circle, the radius of the center of the tangent point is connected.

Pythagorean theorem is the most convenient for the calculation of tangent length.

To prove that it is tangent, carefully distinguish the radius perpendicular.

Is the diameter, in a semicircle, to connect the chords at right angles.

An arc has a midpoint and a center, and the vertical diameter theorem should be remembered completely.

There are two chords on the corner of the circle, and the diameters of the two ends of the chords are connected.

Find tangent chord, same arc diagonal, etc.

If you want to draw a circumscribed circle, draw a vertical line in the middle on both sides.

Also make a dream circle with inscribed circle and bisector of inner angle.

If you meet an intersecting circle, don't forget to make it into a string.

Two circles tangent inside and outside pass through the common tangent of the tangent point.

If you add a connector, the tangent point must be on the connector.

Adding a circle to the equilateral angle makes it not so difficult to prove the problem.

The auxiliary line is a dotted line, so be careful not to change it when drawing.

If the graph is dispersed, rotate symmetrically to carry out the experiment.

Basic drawing is very important and should be mastered skillfully.

You should pay more attention to solving problems and often sum up the methods clearly.

Don't blindly add lines, the method should be flexible.

No matter how difficult it is to choose the analysis and synthesis methods, it will be reduced.

Study hard and practice hard with an open mind, and your grades will soar.

1 1, extreme thoughts

The idea of limit is the basic idea of calculus, and a series of important concepts in mathematical analysis such as continuity, derivative and definite integral of function are defined by means of limit. If you want to ask, "What is the theme of mathematical analysis?" Then it can be summed up as follows: "Mathematical analysis is a subject that studies functions with extreme ideas".