Knowledge points of mathematical reasoning in senior high school
? 1, inductive reasoning: As the name implies, an inductive process. For example, there are apples, pears, grapes, strawberries and so on in a basket, and then you find that apples are fruits, pears are fruits, grapes are fruits, strawberries are fruits, and then you guess: the basket is filled with fruits. This reasoning is a process from special to general, which may or may not be right. If the basket is really full of fruit, then you are right. If there are carrots in the basket, you are wrong. That's why there's evidence.
2. Analogical reasoning: As the name implies, it is also a process of analogy. For example, you know that apples are moist and sweet, pears are moist and grapes are moist and sweet, so it is obviously wrong for you to infer that bananas are moist and sweet. Bananas have no water. But if you deduce that litchi is rich in water and sweet, this is correct. (This example refers to normal fruit) Obviously, this way of reasoning is a process from special to special, which is not necessarily correct.
3, deductive reasoning: generally push special, must be right. For example, f(x)= 1, then f( 1)= 1.
Knowledge points of mathematics proof in senior high school
1, synthesis method: that is, our normal proof process is pushed down from conditions.
For example, 1 pineapple weight =4 apples weight, 1 apple weight =20 grapes weight, which proves that: the weight of 2 pineapples = 160 grapes weight.
Proof: Because 1 pineapple =4 apples, 1 apple =20 grapes.
_ _ _ _ _ _ _ _ _ _ So 1 weight of pineapple =4_20 grapes =80 grapes.
_ _ _ _ _ _ _ _ _ _ So 2 pineapple weight = 160 grape weight.
2. Analysis method: Deduce the equivalent conclusion from the conclusion and prove the validity of this equivalent conclusion.
The same proof of the above example: prove that the weight of two pineapples = 160 grapes, that is, prove that the weight of two pineapples = 2_ 1 grape, that is, prove that the weight of one pineapple =80 grapes.
Because 1 pineapple weight =4 apples, 1 apple weight =20 grapes.
So 1 pineapple weight =4_20 grape weight =80 grape weight, and the original formula is proved.
3, reduction to absurdity: first assume the opposite conclusion, then deduce it according to what is known, and finally find it inconsistent with what is known, accept it! This is a process of defeating yourself!
4, mathematical induction:
Problem solving process:
Proposition A holds when n= 1 (or n0), which is the basis of recursion;
B. suppose that the proposition holds when n=k;
C. it is proved that the proposition also holds when n=k+ 1.
Mathematical reasoning and proof in senior high school
I. Axioms, Theorems, Inferences and Inverse Theorems:
1. The accepted true proposition is called axiom.
2. Prove the correctness of other true propositions by reasoning, and the proved true propositions are called theorems. 3. A theorem directly derived from an axiom or theorem is called the inference of this axiom or theorem. 4. If the inverse proposition of a theorem is true, then this inverse proposition is called the inverse theorem of the original theorem.
Second, analogical reasoning:
A mathematical problem consists of three elements: known conditions, solutions and conclusions to be proved, which can be regarded as the attributes of mathematical examination questions. If two mathematical problems are similar in a series of attributes, or one comes from another problem, then we can infer that the attributes of one problem also have the same or similar attributes in the other problem by analogy.
Third, prove that:
1. The process of reasoning about a proposition is called proof, including cognition, verification and proof.
2. The general steps of proof:
(1) Review the meaning of the topic, and clarify the conditions and conclusions;
(2) Draw a picture according to the meaning of the question;
(3) according to the conditions and conclusions, combined with graphics, write the known verification;
(4) Analysis conditions and conclusions;
(5) According to the analysis, write the proof process.
3. Common methods of proof: synthesis, analysis and reduction to absurdity.
Fourth, the application of auxiliary lines in proof:
In the proof of geometric problems, sometimes, in order to prove, some lines are added to the original graphics. These line segments are called auxiliary lines and are usually represented by dotted lines. Write the addition process at the beginning of the proof, and the auxiliary lines added in the proof can be used as known conditions to participate in the proof.
Common inspection methods
(1) Use basic knowledge flexibly for reasoning, and use synthesis method and analysis method to prove it from two aspects: conditions and conclusions;
(2) In the senior high school entrance examination, we should examine analogical reasoning. First, we should design a question with certain conditions and conclusions, and then take it as the analogy object, and then carry out transformation. For example, the variation of graphics, adding some new attributes or changing some attributes, and inferring the conclusions and solutions of new problems by comparing with the original problems.
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