1 Four ways of thinking in high school mathematics
Learning a knowledge and studying its core mainly means learning its thoughts and methods, which is the essence of learning. The same is true of learning mathematics, which is divided into mathematical thoughts and mathematical methods.
2 Number-shape combination thought
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 (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; With the help of the combination of operation results and geometric theorems.
3 transformation and transformation thought
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 be as equivalent as possible. Common transformations include: inverse transformation, number transformation, equality and inequality transformation, whole and part transformation, space and plane transformation, complex and real number transformation, constant and variable transformation, mathematical language transformation and so on.
4 Classification and integration ideas
The idea of classified discussion is a way of thinking to classify mathematical objects and seek solutions. Classification principle: classification is neither heavy nor leakage. The steps of classification: ① determine the object of discussion and its scope; (2) Determine the classification criteria for classified discussion; (3) classified discussion; (4) Summary and comprehensive conclusion. The key to classified discussion is to break the whole into parts and reduce the difficulty through local discussion. Common types: discussion caused by mathematical concepts, classified discussion of concepts such as real number, rational number, absolute value, position relationship between point (straight line, circle) and circle;
Discussions caused by mathematical operations, such as whether both sides of inequality are multiplied by positive or negative numbers; The discussion caused by the restrictive conditions of properties, theorems and formulas, such as the discussion caused by applying the formula of finding the root of a quadratic equation with one variable; Discussion caused by the uncertainty of graphic position, such as discussion caused by related problems in right-angle, acute-angle and obtuse-angle triangles. Classification discussion caused by the influence of some letter coefficients on the equation, such as the influence of the number of letters in the quadratic function on the image, the influence of the quadratic term coefficient on the image opening direction, the influence of the linear term coefficient on the vertex coordinates, and the influence of the constant term on the intercept.
5 function equation thought
The thought of function equation is a problem-solving way of thinking by using the viewpoint and method of function and equation to deal with the relationship between variables or unknowns, and it is a very important mathematical thought. Function thought: using function relation to express some variables that are mutually restricted in a certain change process, studying the mutual restriction relation between these quantities, and finally solving the problem, this is function thought; It is a key step to solve the problem and establish the functional relationship between variables by using the function idea.
Generally, it can be divided into the following two steps: (1) establish the functional relationship between variables according to the meaning of the problem, and transform the problem into a corresponding functional problem; (2) Construct functions according to needs and solve problems by using relevant knowledge of functions; (3) Equation idea: In a certain change process, it is often necessary to determine the values of certain variables according to certain requirements. At this time, the equations or (equations) of these variables are often listed and solved by solving the equations (or equations). This is the idea of equation; Function and equation are two closely related mathematical concepts, which permeate each other. Many equation problems need to be solved with the knowledge and methods of functions, and many function problems also need the support of equation methods. The dialectical relationship between function and equation forms the idea of function equation.
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