Introduction to mathematical thinking methods

High school mathematics is a line, algebra and geometry are two beads;

Keep in mind the three basics, four can not be idle.

Five routines are practiced every day, and six strategies change from time to time.

Intensive reading of seven thoughts on mathematics is fun to learn.

A line: the main line of function (running through the textbook)

Two beads: algebra and geometry (focusing on the intersection of knowledge)

Three foundations: method (familiarity), knowledge (firmness) and skill (dexterity)

Four abilities: conceptual operation (accurate), logical reasoning (rigorous),

Spatial imagination (rich), problem decomposition (flexible)

Five methods: method of substitution, collocation method, undetermined coefficient method, analysis method and induction method.

Six strategies: simply control complexity, deal with difficulties, retreat for progress, change differences into similarities, replace trees with flowers, calm down and move.

Seven thoughts: Functional equations are the most important, and classified integrals are often used.

The combination of numbers and shapes is as good as ever, and the transformation is inseparable;

The limited self will be described infinitely, or will be expressed inevitably.

Special and general dialectical, knowledge gradually cross.

2. On mathematical knowledge and methods;

Set and logic

Set logic is reciprocal, and subsets and complete sets intersect and complement each other.

Distinguishing right from wrong is difficult, and distinguishing right from wrong is a proposition;

Whether the crisscross is primitive or not, there are four relations that are necessary and sufficient.

When it is true or false, the false is not true, or the operation of true and false is strange.

Function and sequence

Mother and son of sequence function, arithmetic difference ratio is self-contained

What is the sum of series? The idea of general recursion is open;

There is no difference between the separation of variables and the synthesis of functions.

The same increase and different decrease monotony, and the interval digs the maximum.

trigonometric function

The triangle defines the birth ratio, and the radians are fused with each other.

From the same point of view, the flexibility of three kinds of good induction is twice as different.

If there are three equilibria before the solution, there will be a pulse after the solution;

The calculation of angle value is large or small, and the chord tangent is also different.

Equality and inequality

Unequal roots of functional equations often lead to parameter ranges;

One positive, two definite and three equal phases, the mean value theorem is the best.

The parameter ratio is uncertain, and the two formulas are different.

Equality and inequality are not absolute, but variable separation is constant.

Analytic geometry

The intersection setting of simultaneous equations does not require clever discrimination;

Vieta's theorem indicates the chord length, and the slope is converted into the midpoint.

Select parameters to model the trajectory and calculate the distance symmetrically;

The moving point is related to the definition, and the static auxiliary analysis is obtained during the moving process.

solid geometry

Multi-point * * * lines cross on both sides, and the multi-line * * * surface is ingenious;

The vertical chord in space is large and the spherical arc is small.

Line-to-line relationship line opposite search, face-to-face angle line table;

Equal product transformation is continuous projection and can cut bridges.

Arrangement and combination

Step by step, classification, multiplication and addition, if you want to be adjacent, you need to tie it up and insert it once in a while;

Orderliness leads to disorderly groups, and difficulty leads to exclusion.

Repeated multiplication of elements, you first take special elements;

Average grouping factorial division, I am a master of diversity.

binomial theorem

How much binomial power knows, the source of Wan Li is the general term;

Expand the trinomial index system, and the combination coefficient is Yang Hui angle.

Divisibility proves that the bottom is wonderful, binomial and unique;

Who is the most symmetrical at both ends? The main peak is all the other peaks that look short in the sky. ..

Probability and Statistics

Probability and statistics have the same root, random occurrence and so on;

Mutually exclusive events is a show, fighting for independence at the same time.

The whole sample sampling test, binary point independent repetition;

Distribution table, a random variable, false expectation variance theory.