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.

1. Function idea: Function idea is to express some mutually restrictive variables in a certain change process, study the mutually restrictive relationship between these quantities, and finally solve the problem;

2. Using the function idea, establishing the functional relationship between variables is the key step to solve the problem, which can be roughly divided into the following two steps: (1) establishing the functional relationship between variables according to the meaning of the problem, and transforming 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;

3. 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.

Second, the combination of numbers and shapes.

The combination of numbers and shapes is one of the four important ways of thinking in middle school mathematics. For the algebraic problems studied, we can sometimes study the properties of the corresponding geometry to solve the problems (with the help of shape). Or for the learned geometric problems, we can solve the problems with the help of the quantitative relationship of the corresponding figures (with the help of numbers). This method of solving problems is called the combination of numbers and shapes.

1. The purpose of number-shape combination and number-shape conversion is to give full play to the visualization and intuition of shape and the standardization and rigor of number thinking. The two complement each other and foster strengths and avoid weaknesses.

2. Engels defined mathematics as follows: "Mathematics is a science that studies the relationship between quantity and spatial form in the real world". In other words, the combination of numbers and shapes is the essential feature of mathematics, and everything in the universe is harmonious and unified with numbers and shapes. Therefore, highlighting the combination of numbers and shapes in mathematics learning is precisely to fully grasp the essence and soul of mathematics.

3. The essence of the combination of numbers and shapes is that the nature of geometric figures reflects the quantitative relationship, which determines the nature of geometric figures.

4. Mr. Hua once pointed out: "A small number is less intuitive, and a small number is difficult to be nuanced; The combination of numbers and shapes is good, and nothing is right. " As a mathematical thinking method, the application of the combination of numbers and shapes can be roughly divided into two situations: either by means of the accuracy of numbers to clarify some properties of shapes, or by means of geometric intuition of shapes to clarify some relationship between numbers.

5. The combination of numbers and shapes as a means is mainly embodied in analytic geometry, which has been examined in the answers of college entrance examinations over the years (that is, studying geometric problems by algebraic methods). The combination of number and shape appears in the objective questions of college entrance examination with the help of shape.

6. The combination of numbers and shapes should grasp the following points:

(1) The problem of distance, angle or area can be directly solved geometrically;

(2) Studying the problem of function, equation or inequality (maximum) can be solved through the image of function (the zero and vertex of function are the key points), thus transferring and comprehensively applying knowledge;

(3) Attention should be paid to the following problems: The purpose of solving problems can be achieved by constructing distance function, slope function, intercept function, point x2+y2= 1 on the unit circle and cosine theorem.

Third, the mathematical thought of classified discussion

Classified discussion is an important mathematical thinking method. When the objects of the question cannot be studied uniformly, it is necessary to classify the research objects, then study each category separately, give the results of each category, and finally get the answer of the whole question by synthesizing all kinds of results.

1. Mathematical problems related to classified discussion need to be solved by using the idea of classified discussion, and the reasons that cause classified discussion can be roughly summarized as follows:

The mathematical concepts involved in (1) are classified and discussed.

(2) The applied mathematical theorems, formulas, or operation properties and laws are classified and given;

(3) There are many situations or possibilities for the conclusion of the solved mathematical problem;

(4) There are parameter variables in mathematical problems, and different values of these parameter variables lead to different results;

(5) More complicated or unconventional mathematical problems need to be discussed and solved by classification.

2. Classification discussion is a logical method, which is widely used in middle school mathematics. According to different standards, there are different classification methods, but the classification must start from the same standard, so as not to repeat or omit, to cover all kinds of situations, and to be conducive to the study of problems.

Fourth, the idea of transformation and transformation.

The so-called reduced thinking method is a method to solve mathematical problems by transforming them into transformations by some means. Generally, complex problems are always transformed into simple problems, difficult problems into easy-to-solve problems, and unsolved problems into solved problems.