⑴ Being able to judge the questions: Whether you can judge the questions is the basis of whether you can see the meaning of the questions clearly. In teaching, we should first cultivate students' habit of carefully examining questions; Secondly, we should teach students the general steps of examining questions:

(1) When you get a question, first find out what main concepts appear in the question and recall their definitions.

(3) Some topics need to be mapped according to the meaning of the topic, or what is known and verified should be written with mathematical symbols and numerical terms, that is, the common language should be "translated" into topics expressed in mathematical language to make the content of the topic clearer and the proof process clearer. For geometry problems, we should pay special attention to the implicit conditions of the problem, such as the edges and corners of the equal relationship, the shape of the figure and so on. ⑵ Will change: In the teaching arrangement, we should pay attention to two points:

① It should be taught to students in two stages. The first stage: it is only required that the original proposition will become an inverse proposition. The second stage: it will change with each other. (2) in the usual study, to give students a variety of inspiration and opportunities.

⑶ Address: Address refers to understanding the meanings of "sufficient conditions" and "necessary conditions" and applying them.

Second, master the format of reasoning proof

The basis of mathematical proof is concept, axiom and theorem, which are the basic knowledge in mathematics. We should not only understand it correctly, but also remember it firmly and use it flexibly.

In order to reason and prove correctly, it is not enough for us to "see the meaning of the problem clearly" and be familiar with the basis. That is to say, although we will use the known conditions and related mathematical concepts, axioms and theorems to gradually deduce the verification conclusion, it is still not enough. You also need to master some basic proof methods and reasoning formats, and be good at expressing your thinking process in mathematical language.

There are five common reasoning formats: comprehensive positive proof format, analytical reverse deduction format, three-step reduction to absurdity format, limited column discussion format and so on. In plane geometry, there are also overlapping methods. ⑴ The sequential proof format of comprehensive method is based on known conditions and follows deduction: from "known" to "inferred" and from "inferred" to "unknown", and the verification conclusion is gradually deduced, which is the format of sequential deduction.

The synthesis method is the most common reasoning proof method. Its written expression is often "⊙ ∴" or "= >" and so on. ⑵ The analytical method is just the opposite of the comprehensive method. It starts with proving the conclusion, and goes back to analysis, from unknown to known (known conditions, definitions, theorems, axioms, formulas, laws, etc. The key of this proof method is to ensure that every step of the analysis process is reversible. Its written expression is usually used as "proof" ... just ... "

It must be pointed out that "the above steps can be reversed."

When we usually do mathematical proof problems, we usually don't use analytical methods to reverse the format to write the process of expressing proof, but often use comprehensive methods to prove the format. Follow the proof in a comprehensive way (that is, from known to verified). Sometimes the idea is not necessarily good. Therefore, we often use analytical methods to "think" on the draft paper, find the idea of proof, and then use a comprehensive method according to the format of proof. This method is often called "reverse reasoning". (3) The format of reduction to absurdity is sometimes difficult to directly prove the proposition, and the negative proposition equivalent to the original proposition can be corrected. This is the basic idea of reducing to absurdity.

The general steps of using reduction to absurdity are as follows:

(1) Make an assumption contrary to the verification conclusion.

(2) Based on this assumption, we draw some conclusions with correct reasoning methods.

(3) Point out that the conclusion is inconsistent with the original intention (or related definitions, theorems, formulas, etc.). ), and this contradiction can assert that the hypothesis contrary to the verification conclusion is incorrect. So the conclusion verified in the original question is correct. Finally, the conclusion of contradiction is the result of reasoning with "law of excluded middle".