What is the concept of equation? There is no consensus at present. But what is certain is that the concept of equation is not only the solution of equation. It should include: as a model, for some practical (mathematical) problems, equation models can be constructed; List the equations and solve them. The key point of learning equation is to analyze the problem with the "viewpoint" of the equation and construct the model with mathematical thought. Solving the equation is another aspect. Therefore, the concept of equation can be said to be a way of thinking to solve problems by communicating known and unknown through equations and equations, and it is also an applied consciousness of mathematics.
The study of the equation, we advocate:
1. Embody the concept of modeling
Let students experience the whole process of "problem situation-establishing equation model-solving equation-explaining". The purpose of "problem situation-establishing equation model" is to make students realize that equation is an effective mathematical model to describe the real world, which is the primary aspect of equation concept.
2. Application of expression equation
Basically, every course has some application problems. It is necessary to integrate the equation, solve the equation and explain the solution of the equation, rather than deal with them separately, so that students can experience a mathematical problem abstracted from a specific situation, then study and solve it, and then explain the whole process of the actual problem with mathematical knowledge, so as to realize the application value of the equation and understand the relationship between mathematics and the real world.
3. The key point is to find the solution (exact solution or approximate solution) of the equation by various methods.
"Standard" emphasizes that we can seek reasonable and simple operation ways and methods according to the subject conditions, strengthen estimation, encourage the use of calculators and encourage the diversification of algorithms. There are no uniform steps to solve the equation, so we can choose the appropriate solution according to the problem, but we emphasize the general methods, such as solving the quadratic equation with one variable, taking collocation method as the most basic method and solving the quadratic equation with two variables as the basic thinking method in the textbook. Finding the approximate solution of the equation is an important step in solving the problem, because in the process of solving practical problems, we often do not find the exact solution of the equation, but the approximate solution of the equation that satisfies a certain precision. For example, a quadratic equation with one variable. There are also many methods to estimate the approximate solution, such as finding the approximate solution of binary linear equations according to the images of linear functions and finding the approximate solution of univariate quadratic equations using the images of quadratic functions. Here we pay more attention to the development beyond knowledge and skills.
The textbook does not discuss "the discriminant of roots" and "the relationship between roots and coefficients". First of all, they only discuss the univariate quadratic equation of digital coefficient based on the standard, and at the same time, they are based on the basic and phased consideration of compulsory education. Of course, there is no absolute right or wrong in the choice of course content, which mainly depends on the identification of the value of each part of the content, that is, the role for students' development and the reasonable arrangement of students' limited study time. Teachers can also make appropriate expansion and extension here according to the students' situation.