The combination of numbers and shapes plays a very important role in the college entrance examination. The combination of numbers and shapes permeates each other, combining the accurate description of algebra with the intuitive description of geometric figures, transforming algebraic problems and geometric problems into each other, and organically combining abstract thinking with image thinking. The application of the combination of numbers and shapes is to fully investigate the internal relationship between the conditions and conclusions of mathematical problems, and not only analyze its algebraic significance but also reveal its geometric significance. By skillfully combining the quantitative relationship with the spatial form, we can find a solution to the problem and solve it. Using this mathematical idea, we should master the geometric meaning of some concepts and operations and the algebraic characteristics of common curves.
Applying the idea of combining numbers with shapes, we should pay attention to the following number-shape transformations: (1) set operation and Wayne diagram; (2) Functions and their images; (3) Function characteristics and function images of general terms and summation formulas of several series; (4) Equation (multivariate binary equation) and curve of equation.
The commonly used forms of help numbers are: using the number axis; With the help of functional images; With the help of the unit circle; With the help of the structural characteristics of numbers; By means of analytic geometry.
The commonly used numbered forms are: (1) quantitative relation followed by geometric trajectory; With the help of the combination of operation results and geometric theorems.
Classified discussion thinking
The idea of classified discussion is to analyze and solve it under different circumstances according to the nature differences of the studied objects. At the same time, it is diverse, logical and comprehensive. To establish the idea of classified discussion, we should pay attention to understanding and mastering the principles, methods and skills of classification, so as to "determine all objects, clarify the standards of classification, and analyze and discuss without repetition or omission"
The key to solving mathematical problems by applying the idea of classification discussion is how to classify correctly, that is, how to choose a classification standard correctly to ensure the scientific classification, no repetition and no omission. How to implement the correct classification requires us to first make clear the objects to be discussed and all the people to be classified, then determine the classification standards and methods, then discuss them item by item, and finally make a summary.
Common classification situations are: classification by number; According to the range of letters; Classify events according to their possible occurrence; Classification according to the position characteristics of graphics
The idea and method of classified discussion can penetrate into every chapter of senior high school mathematics. It classifies and solves problems according to certain standards, paying special attention to the principles of mutual exclusion, no leakage and simplicity.
Function and equation thought
The thought of function and equation is the most important mathematical thought, which accounts for a large proportion in the college entrance examination. It has many comprehensive knowledge, many questions and many application skills. The idea of function is very simple, which is to analyze, transform and solve the problem to be studied by establishing the function relationship or constructing the intermediate function, combined with the image and nature of the elementary function. Equation thought transforms the quantitative relationship in the problem into an equation model through mathematical language to solve it.
When using the idea of function and equation, we should pay attention to the mutual connection and transformation between function, equation and inequality, and should do the following:
(1) Deeply understand the properties of function f(x) (monotonicity, parity, periodicity, maximum and image transformation), and master the properties of basic elementary functions skillfully, which is the basis of solving problems by applying function thought.
(2) Pay close attention to the related issues of "three secondary". Three "quadratic" is an important content of middle school mathematics, which is rich in connotation and closely related. Master the basic properties of quadratic function, the distribution conditions of real roots of quadratic process and the transformation strategy of quadratic inequality.
Change and transform ideas
The idea of transformation is to study and solve mathematical problems in a certain way. With the help of some functional properties, images, formulas or known conditions, the problem is transformed through transformation, so as to achieve the idea of solving problems. Transformation is the process of transforming mathematical propositions from one form to another. Transformation means that through a certain transformation process, the problems to be solved are reduced to a class of problems that have been solved or are relatively easy to solve. Transformation and transformation is the most basic thinking method of middle school mathematics, which can be called the essence of mathematical thought. They have penetrated into every field of mathematics teaching content and every link in the process of solving problems. There are equivalent transformations and unequal transformations. The essence of the new problem after equivalent transformation is the same as the original problem. The unequal transformation partially changed the essence of the original object, and the conclusion needed to be revised.
The principle of applying the idea of transformation to solve problems should be to turn the difficult into the easy, turn the complex into the familiar, turn the complex into the simple and strive for equivalence. Common transformations are: positive and negative transformation, number transformation, equality and inequality transformation, whole and part transformation, space and plane transformation, complex and real number transformation, constant and variable transformation and mathematical language transformation.
There are more detailed points, such as:
Mathematical thought
The so-called mathematical thinking refers to the spatial form and quantitative relationship of the real world reflected in human consciousness, as well as 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 and talent of mathematics will be greatly improved. Mastering mathematical thought means mastering the essence of mathematics.
Theory and thought of function
A mathematical problem is expressed by function, and the general law of this problem is explored by function. This is the most basic and commonly used mathematical method.
A 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).
Classified discussion thinking
When a problem may lead to different results because of different situations of a certain quantity, it is necessary to classify and discuss the various situations of this quantity. Such as solving inequality | a-1| >; 4. It is necessary to discuss the value of A.
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.
Holistic 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.
Change idea
It is through deduction and induction that unknown, unfamiliar and complex problems are transformed into known, familiar and simple problems. Mathematical theories such as trigonometric function, geometric transformation, factorization, analytic geometry, calculus, and even rulers and rulers of ancient mathematics 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.
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.
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.
Modeling thinking
In order to describe an actual phenomenon more scientifically, logically, objectively and repeatedly, people use a language that is generally regarded as strict to describe various phenomena, which 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 carry out corresponding experiments. The experiment itself is also a theoretical substitute for the actual operation.
Change ideas
The idea of transformation is to turn the unknown into the known, the complex into the simple and the difficult into the easy. For example, fractional equations are transformed into integral equations, algebraic problems are transformed into geometric problems, and quadrilateral problems are transformed into triangular problems. The methods to realize this transformation are: undetermined coefficient method, collocation method, whole generation method and the transformation idea of turning dynamic into static and abstract into concrete.
Inductive reasoning thinking
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.
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