Equations and equations are important basic knowledge for solving practical problems, practical problems and many mathematical problems, and they are widely used. Many mathematical problems, especially geometric problems with unknowns, need to be solved by equations or their knowledge. When solving a problem, an unknown quantity is set as an unknown quantity. According to related properties, theorems or formulas, the equivalent relationship between unknown quantity and known quantity is established, and equations or equations are listed for solving. This is the idea of equation. The idea of equation can solve the unknown elements or quantities in the problem well, and it has a wide range of applications in algebra and geometry.

(A) the use of equation thinking to solve the problem of related functions

The basic types are: to find the undetermined coefficient through a series of equations or equations, and then to find the resolution function; Study the intersection of function images and solve the intersection of function images and coordinate axes.

Example 1, (Heze City, Shandong Province, 2005) The motorcycle fuel tank can hold up to 5 liters of oil, and the remaining fuel quantity Q (liter) in the fuel tank is linearly related to the driving distance s(km), and its image is shown in the figure.

(1) Find the functional relationship between q and s;

(2) How many kilometers can a motorcycle travel after it is fully fueled?

Pointing: we can mainly find out the two points that the image passes through, and then list the equations to find the undetermined coefficients to get the resolution function; The motorcycle runs out of fuel just at the intersection with the X axis, so y=0.

(2) Solving geometric problems with the idea of equations.

The so-called solving geometric problems with equation thinking is to fully explore the quantitative relationship implied in the problem setting and conclusion, and seek the equivalent relationship between known quantities and unknown quantities with the help of the intuition of graphics, so as to establish equations or equations, and then solve geometric problems by using equation theory and solution method.

Example 2. (Heze City, Shandong Province, 2005) A right-angled triangular wooden board with a right-angled side length of 1.5m and an area of1.5m2. The master wants to process it into a square desktop with the largest area, and two students from Party A and Party B are requested to design and process it. Party A's design scheme is shown in figure 1, and Party B's design scheme is shown in figure 2.

Figure 1

E

G

B

A

C

The fourth note of the major scale

D

A

C

B

D

E

F

Figure 2

Map No.2 1

Which classmate do you think designed the scheme better? Try to explain why. (The machining loss is neglected, and the score can be retained in the calculation results. )

Second, the idea of conversion.

The idea of transformation requires us to grasp the essence of the problem condescendingly, analyze the problem dialectically when encountering more complicated problems, simplify the complicated problems through certain strategies and means, familiarize the unfamiliar problems and concretize the abstract problems. For example, fractional equations are transformed into integral equations, geometric problems are transformed into algebraic problems, and quadrilateral problems are transformed into triangular problems. The methods to realize this transformation are: undetermined coefficient method, collocation method, whole replacement method, changing dynamic into static, changing abstract into concrete and so on.

Third, the idea of classified discussion.

Classification discussion is to divide mathematical objects into different kinds of mathematical thinking methods according to the similarities and differences of their essential attributes. Classified discussion is a very important mathematical thinking method in solving problems. Mastering the idea of classified discussion is helpful for students to improve their ability to understand, organize and acquire knowledge independently. We should pay attention to two points in solving mathematical problems in this way of thinking: one is that it cannot be omitted, and the other is that it cannot be repeated. Common knowledge points that need to be classified and discussed are: the absolute value of algebra, the definition of equations and roots, the definition of functions, and the quadrant where points (coordinates are not given) are located. Geometry has the positional relationship of various figures, and the congruence or similarity of the corresponding relationship may be unclear.

Fourth, the combination of numbers and shapes.

The image of geometric figures is intuitive and easy to understand, the algebraic method is universal, the problem-solving process is mechanized, the operability is strong and it is easy to master. Therefore, the combination of numbers and shapes is an important thinking method in mathematics, which can cultivate students' flexibility, visualization and profundity. The so-called combination of numbers and shapes is to consider the combination of numbers and shapes when studying problems, or to transform the quantitative relationship of problems into the nature of graphics, or to transform the nature of graphics into the quantitative relationship, thus simplifying complex problems and concretizing abstract problems.

Mathematical thinking method is the most widely used knowledge in people's lives. There are many mathematical thinking methods in junior high school mathematics textbooks. The formation of students' mathematical thinking is a subtle process, which is formed on the basis of repeated understanding and application. This requires our teachers to study the teaching materials carefully, infiltrate the teaching of mathematical thinking methods, create scenarios and strengthen the problem-solving training of applying mathematical thinking methods.