(A) Mathematical thinking

There are four common mathematical ideas: function and equation, reduction, classified discussion and combination of numbers and shapes. 1. The function and equation function idea refers to analyzing, transforming and solving problems with the concept and properties of functions; The idea of equation is to start with the quantitative relationship of the problem, transform the conditions in the problem into a mathematical model with mathematical language, and then solve the problem by solving the equation (group). Functions are closely related to equations, such as unary linear function baxy+=, which can be regarded as binary equation 0 =-+ybax about x and y; The binary equation 0=-+ybax can be regarded as a linear function of x, and it can be said that the learning of the function cannot be separated from the equation. The characteristics of sequence equation, solving equation and studying equation are the embodiment of the idea of applying equation. 2. Transformation is to transform unfamiliar, irregular and complex problems into familiar, standardized and simple problems. It can be between numbers, between shapes, and between numbers and shapes. The elimination method, method of substitution, the combination of numbers and shapes, and the scope of evaluation all reflect the idea of reduction. For example, many quadrilateral problems can be transformed into triangular problems to study; Studying the positional relationship between two straight lines can be transformed into studying the quantitative relationship of angles; For example, after learning the arithmetic rules of rational numbers in senior one, we should understand them in combination with several arithmetic rules: subtraction and multiplication are converted into addition, and division and multiplication are converted into multiplication. For example, if we want to find the area of irregular figures, we can divide or supplement them and convert them into regular figures, and so on. 3. Classification Discussion When solving some mathematical problems, we sometimes encounter many situations and need to classify them one by one.

The mathematical concepts involved in the (1) problem are classified and defined. For example, the definition of |a| is divided into a >；; 0, a=0 and a<0. (2) The mathematical theorems, formulas, operational properties and rules involved in the problem are limited in scope or conditions, or given in categories. For example, the positional relationship between a point and a circle can be divided into three situations. (3) When solving problems with parameters, we must discuss them according to the different ranges of parameters. For example, when studying the opening direction of the image with quadratic function cbxaxy++=2, it is divided into a >: 0 and a; 0，△& gt； 0，△& lt； 0, △=0.(4) When solving some conditional open problems, we need to classify them according to several possible situations, such as? A little on one side of the triangle, make a straight line and split the original triangle in two, so that the cut triangle is similar to the original triangle. How many ways are there? There are four ways to classify the position of a straight line. For example, when proving the fillet theorem, the center of the circle is inside, outside and on the edge. When discussing classification, the principles to be followed are: determination of classification objects, unification of standards, no omission and no repetition. 4. The basic knowledge of junior high school mathematics combined with numbers and shapes can be divided into three categories: one is the knowledge about pure shapes, such as simple geometric figures, triangles, quadrilaterals, similar shapes, right triangles, circles and so on. One kind is about the combination of numbers and shapes, such as the corresponding relationship between points and numbers on the number axis, the definition of acute trigonometric function is defined by right triangle and so on. Use form to help numbers? And then what? Auxiliary shapes with numbers? In two aspects, its application can be roughly divided into two situations: or to clarify the relationship between numbers with the help of the vividness and intuition of shapes, that is, to use shapes as a means and numbers as the purpose, for example, to intuitively explain the nature of functions with the image of functions, or? Given the line segment AB=2cm, there is a point C and BC=6cm on the straight line AB. What is the length of the line segment AC? To solve this problem, you can draw a picture and find out two different positions of point C; Or we can clarify some properties of shapes with the help of the accuracy and rigor of numbers, that is, numbers are the means and shapes are the purpose. For example, we can accurately clarify the geometric properties of function images by using resolution functions, and judge the positional relationship between straight lines and circles according to the distance from the center of a circle to a straight line, or judge the positional relationship between two circles according to the quantitative relationship between the radius and the center of a circle.