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, so that the problem can be solved. 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 and shapes, we should pay attention to the following number-shape transformations: (1) the operation of set 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 (multi-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; With the help of analytic geometry, the commonly used methods to help modeling with numbers are: (1) the quantitative relationship followed by geometric trajectory; The idea of classified discussion is to analyze and solve it under different circumstances by combining the operation results with geometric theorems and according to the nature differences of the studied objects. Classification discussion questions cover more knowledge points, which is beneficial to examine students' knowledge, classification ideas and skills. 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, analyze and discuss classification at different levels, and not repeat or omit". Common classification situations are: classification by number; According to the range of letters; Classify events according to their possible occurrence; According to the position characteristics of graphics, etc. The method of classified discussion can penetrate into all chapters 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. The thought of function and equation is the most important mathematical thought, which occupies a large proportion in the college entrance examination. There are many comprehensive knowledge, many questions and many skills to be used. The function idea is simple. That is to say, the problems to be studied can be analyzed, transformed and solved by establishing functional relationships or constructing intermediate functions, combining the image and properties of elementary functions, such as evaluating, solving (proving) inequalities, solving equations, and discussing the range of parameters. The idea of equation is to transform the quantitative relationship in the problem into an equation model by using mathematical language. When applying the idea of function and equation, we should pay attention to the interrelation and transformation between function, equation and inequality, and should: (1) understand the properties of function f(x) deeply (monotonicity, parity, periodicity, maximum value, image transformation) and master the properties of basic elementary functions skillfully. This is the basis of solving problems by applying the thought of function. (2) Pay attention to three "secondary" problems. Quadratic function, quadratic equation and quadratic inequality are the important contents of middle school mathematics, which are rich in connotation and closely related. Master the basic properties of quadratic function, the distribution conditions of real roots of quadratic process, the transformation strategy of quadratic inequality, and the ideological transformation of conversion. It is to study and solve mathematical problems in a certain way, with the help of some functional properties, images, formulas or known conditions, and to transform the problems 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.