The so-called classified discussion means that when studying and solving a mathematical problem, when the objects given by the problem cannot be studied uniformly, we need to divide the objects into different types according to the similarities and differences of the essential attributes of the mathematical objects, then study and solve them one by one, and finally synthesize various results to get the solution of the whole problem. This way of thinking is called "the idea of classified discussion".
1. The thinking method of classified discussion is one of the basic methods of middle school mathematics, and it is also the focus of college entrance examination over the years.
⑴ The idea of classified discussion has obvious logical characteristics;
(2) Classified discussion questions generally cover more knowledge points, which is beneficial to the investigation of students' knowledge;
(3) To solve the problem of classification discussion, students need to have certain analytical ability and classification skills;
The idea of classified discussion is closely related to production practice and advanced mathematics.
2. Discuss the essence of ideas by classification.
The essence of the idea of classified discussion is "breaking the whole into parts, breaking the whole into parts", which increases the problem-solving strategy of setting conditions for problems.
3. The basic steps of solving problems with the idea of classified discussion.
(1) Determine the object of discussion and the research field;
(2) Make a reasonable classification of the problems discussed (classification needs to be non-repetitive, non-omission, unified standard and clear hierarchy);
(3) classified discussion: discuss all kinds of problems in detail and solve them step by step;
⑷ Summarize the integration and draw a conclusion.
4. It is helpful to master the thinking method of classified discussion to solve problems by clarifying the reasons for the thought of classified discussion. The main reasons are:
The classification discussion caused by (1) mathematical concept, such as the definition of absolute value, the first n terms of geometric series and formulas, etc.
⑵ Classification discussion caused by mathematical operation requirements: such as the requirement of even square root being non-negative, the requirement of logarithm base and real number, and the influence of multiplying two sides of inequality by real number on the direction of inequality.
(3) Classification discussion caused by the limitations of function properties, theorems and formulas;
(4) Classification discussion caused by the uncertainty of the relative positions of points, lines and surfaces in geometric figures;
5. Classification discussion caused by parameter changes: For some problems with parameters, different parameter values will lead to different results, or different solutions or proof methods will be adopted for different parameter values;
Others are classified and discussed according to the concrete analysis of practical problems, such as arrangement and combination, practical application problems and so on.
5. Discuss the types of ideas by classification
(1) Classify and discuss the variables in the problem or the parameters to be discussed;
(2) The conditions in the problem are given by classification;
(3) The problem-solving process can't be described uniformly, and it must be discussed in categories;
(4) When geometric problems are involved, the changes of the shape and position of geometric elements need to be discussed in categories.