Tisch
Teaching objectives:
1. Understand the significance of deductive reasoning.
2. Can correctly use deductive reasoning for simple reasoning.
3. Understand the connection and difference between perceptual reasoning and deductive reasoning.
Teaching emphasis: Correct use of deductive reasoning and simple reasoning.
Difficulties in teaching: Understanding the connection and difference between rational reasoning and deductive reasoning.
Teaching process:
First, review: reasonable reasoning
Inductive reasoning from special to general
Analogical reasoning from special to special
Starting from specific problems-observation, analysis and comparison, association-induction. Analogy-make a guess
Second, the problem situation.
Observe and think
All metals conduct electricity.
Copper is a metal,
Therefore, copper can conduct electricity.
2. All odd numbers are not divisible by 2.
(2 100+ 1) is an odd number,
Therefore, (2 100+ 1) is not divisible by 2.
3. Trigonometric functions are all periodic functions,
Tan is a trigonometric function,
So tan is a periodic function.
Ask a question: Is this reasoning reasonable?
Second, student activities:
1. All metals can conduct electricity.
Copper is a metal, minor premise.
Therefore, copper can conduct electricity-conclusion
2. All odd numbers are not divisible by 2.
(2 100+ 1) is odd, ←-minor premise.
Therefore, (2 100+ 1) is not divisible by 2. -Conclusion
3. Trigonometric functions are periodic functions.
Tan is a trigonometric function.
So tan is a periodic function. -Conclusion
Third, structural mathematics.
Definition of deductive reasoning: from the general principle, a special conclusion is derived, which is called deductive reasoning.
1. Deductive reasoning is from general to special reasoning;
2. "Syllogism" is a general model of deductive reasoning; include
(1) major premise-known general principles;
(2) Minor premise-the special situation studied;
(3) Conclusion-Judgment of special circumstances according to general principles.
The basic format of syllogism
M-p (m is p) (major premise)
minor premise
Standard & Poor's (conclusion)
3. The basis of syllogism reasoning is understood from the point of view of set:
If all elements in the set M have the attribute P and S is a subset of M, then all elements in S also have the attribute P. ..
Fourth, the use of the number *
Example 1. "The image of function y=x2+x+ 1 is a parabola" is reduced to a complete syllogism.
Solution: The image of a quadratic function is a parabola (major premise).
The function y=x2+x+ 1 is a quadratic function (minor premise).
So the image of function y=x2+x+ 1 is a parabola (conclusion).
Example 2, known as lg2=m, calculate lg0.8.
Solution: (1) lgan = nlga (a > 0)- major premise
Lg8 = lg23-minor premise
Lg8 = 3lg2-conclusion
LG(a/b)= LGA-lgb(a & gt； 0, b & gt0)- major premise
LG 0.8 = LG(8/ 10)- minor premise
LG 0.8 = LG(8/ 10)- conclusion
Example 3, as shown in the figure; In acute triangle ABC, AD⊥BC, BE⊥AC,
D and e are vertical feet, which proves that the distance from M in AB to D and e is equal.
Solution: (1) Because a triangle with right angles is a right triangle,-major premise.
In △ABC, AD⊥BC, that is ∠ADB = 90- minor premise.
So △ABD is a right triangle-conclusion
(2) Because the median line on the hypotenuse of a right triangle is equal to half of the hypotenuse,-major premise
Because DM is the center line on the hypotenuse of a right triangle.
So DM = ab- conclusion
Similarly, EM=AB
So DM=EM.
Exercise: Exercise 1, 2, 3, 4 on page 35.
Verb (abbreviation of verb) review summary:
Deductive reasoning has the following characteristics: page 33 of the textbook.
The main reasons for deductive reasoning errors are as follows
1. The major premise is not established; 2. The minor premise does not meet the conditions of the major premise.
Homework: Exercise 5 on page 35. Exercise 2. 1 question 4.
extreme
Teacher: Please answer the following questions (for example):
(1) Observe the sequence 1, 1+2, 1+2+3, 1+2+3+4, …, and guess the formula an=.
(2) The midline of the triangle is parallel to the third side and equal to half of the third side. If you extend into space, what conclusion will you get?
(3) As shown in figure ∠ 1=∠2, what is the positional relationship between straight lines A and B? Why?
Health 1, (1) An = 1+2+3+…+n=.
(2) The middle section of the cone is parallel to the bottom surface, and its area is equal to that of the bottom surface.
Health 2, (3) A ∨ B.
Reason: As shown in Figure ∠2=∠3,
∵∠ 1=∠2,
∴∠ 1=∠3.
∴a∥b.
Teacher: (1)(2) What reasoning is used in the process of drawing a conclusion from a small problem?
Health 3: Be reasonable;
Teacher: Can you be more specific?
Health 3:( 1) Use inductive reasoning, (2) Use analogical reasoning.
Teacher: What are the characteristics of inductive reasoning and analogical reasoning?
All beings: inductive reasoning is from special to general; Analogical reasoning is from special to special.
Teacher: (3) Is the process of drawing a conclusion from a small question reasonable?
Sentient beings: No 。
Teacher: (3) The process of drawing a conclusion is not rational reasoning, so what is this way of reasoning? This is the subject we are going to study in this class-deductive reasoning.
(Type in blackboard writing or courseware: deductive reasoning)
Teacher: Let's look at another proposition:
Proposition: The two base angles of an isosceles triangle are equal.
A
B
C
D
Teacher: in order to prove this proposition, according to past experience, we must first draw a picture, write what is known and verify it. Ask a classmate to finish it?
Health 4. It is known that in △ABC, AB=AC,
Proof: ∠ b = ∠ c.
Teacher: Let one student prove it on the blackboard and the other students do it in the exercise book.
Health 5: Proof: As shown in the figure, the foot of AD⊥BC is D,
In Rt△ABD and Rt△ABC,
AB = AC，……P 1
AD = AD, P2
∴△ADB≌△ADC.P3
∴∠B=∠C.…………………………q
Teacher: Students, have a look. Is student 5' s certificate correct?
All beings: correct.
Teacher: Is there any other way to prove it?
Health 6: It can be used as the bisector of ∠BAC, AD, BC and D, or it can take the midpoint D of BC, connect AD, and then prove △ ADB △ ADC.
Teacher: Good (by the way, the teacher marked the main steps of student 5' s proof as P 1P2P3, q). Please look at student 5' s proof again. How was P3 obtained?
Health 7: According to the truth value of P 65438+P 2 and the judgment theorem of triangle congruence, P3 is deduced to be true.
Teacher: How did Q come from?
Health 8: Because P3 is true, according to congruent triangles's definition, Q is true.
Teacher: A reasoning method like this is called deductive reasoning. Let the students experience deductive reasoning and try to say what deductive reasoning is.
Health 9: Starting from the definition of concepts or some true propositions, the process of drawing correct conclusions according to certain logical rules is usually called deductive reasoning (this step should be completed under the guidance of teachers and the continuous improvement of students).
Teacher: Please think about it. Is the conclusion drawn by using reasonable reasoning in the previous study correct?
All beings: not necessarily.
Teacher: Deductive reasoning is different from perceptual reasoning. Its basic feature is that when the premise is true, the conclusion must be true.
Teacher: Let's look at the steps proved before P3 and Q. What is the basis for getting Q from P3?
All beings: the definition of triangular congruence
Teacher: Good. The above process of getting q from P3 can be written in detail as follows:
The angles corresponding to congruent triangles are equal (1).
△ADB?△ADC②
∠B =∠C……③
This is a typical syllogism reasoning, which is often used in deductive reasoning. Where ① is the major premise, ② is the minor premise and ③ is the conclusion.
Teacher: Please consider why a general syllogism can be expressed.
Health 10: M is p.
S is m
So, s is p.
Teacher: Good. What is "m is p" here? What is "s is m"? What is "S is P"?
Health10: "m is p" is the major premise-providing general principles and "s is m" is the minor premise-pointing out a special object and the conclusion that "s is p".
Teacher: By combining the major premise with the minor premise, we can draw the internal relationship between the general principle and the special object, and then draw the conclusion that "S is P".
In the practical use of syllogism, for the sake of brevity, the major premise or minor premise is often omitted, sometimes even omitted. For example, in the proof of "proposition: the two base angles of an isosceles triangle are equal", the major premise "the angles corresponding to congruent triangles are equal" is omitted when Q is obtained from P3, and the major premise "the top angles are equal" is omitted when ∠2=∠3 is obtained in the proof of the cited example (3), and the minor premise "∠2, ∠3" is.
Example 1: It is known that in the space quadrilateral ABCD, points E and F are the midpoint of AB and AD, respectively (as shown in the figure), which proves the EF∑ plane BCD.
(Solution: Ask one student to act it out, others do it in the exercise book, and then teachers and students comment together, and emphasize that writing math problems generally omits the "major premise". Unless the "major premise" is unfamiliar, students will form a good habit of rigorous writing, and teachers and students will sum up the basic methods of parallel lines and planes together. )
Example 2: Verification: When A > 1
㏒a(a+ 1)>㏒(a+ 1)a，
Teacher: Comparing the sizes of two logarithms, can you think of any knowledge and methods that are often used?
1 1: monotonicity of logarithmic function.
Teacher: Can you prove that this problem can be solved directly by monotonicity of logarithmic function?
Sentient beings:No..
Teacher: How to solve this problem? Please carefully observe the differences and characteristics of these two logarithms.
Health 12: First, the bases of these two logarithms are different. Second, the true number of the logarithm on the left side of inequality is greater than the base, and the true number of the logarithm on the right side of inequality is less than the base.
Teacher: Students, what inspiration can you get from it?
Health 13:∫A > 1,
∴㏒a(a+ 1)>㏒aa= 1，
㏒(a+ 1)a㏒a(a+ 1)
Teacher: How did you come to the final conclusion?
Sheng 13: the essence of inequality (transitivity)
Teacher: Would you please observe the proof of this problem?
Teacher: The inference rule used here is "If aRb, bRc, then aRc", where R stands for transitive relation. This inference rule is called transitive relation inference. Of course, some "relationships" are not transitive. Can students give some examples?
Sheng 14:"≦ "relation is not transitive. ∵1≠ 2,2 ≠1,but 1≠ 1 is wrong.
Student 15: The relationship between classmates cannot be transmitted.
Teacher: Good. Let's look at example 3.
Example 3: Prove that the value of the function f (x) = X6-X3+X2-X+ 1 is always positive.
Teacher: What should I do to prove that the value of a formula is always greater than zero?
Health 16: the formula is constantly changing.
Teacher: Would you please look at the deformation of f(x)?
Health17: f (x) = X6-x2 (x-1)-(x-1)
=x6+(x2+ 1)( 1-x)
Teacher: Please observe the formula obtained by student 17 transformation. How does it help us prove this problem?
For students 18: X6 ≥ 0, X2+ 1 > 0, only one more condition is needed to prove that the value of f(x) is constant.
1-X ≥ 0, that is, x≤ 1 is enough.
Teacher: Can you be more specific?
Sheng 18: when x≤ 1, x6≥0, (x2+ 1) (1-x) ≥ 0, and these two formulas cannot reach zero at the same time.
∴ when x≤ 1, X6+(x2+ 1) (1-x) > 0.
That is to say, the value of f(x) is constant.
Teacher: Have you finished this problem?
Sheng 19: No, it only proves that the value of f(x) is unchanged when x≤ 1; X > 1 has not been proved.
Teacher: how to prove when X > 1 Can it still be proved by the deformation formula of 17?
The deformation formula of sheng 20: sheng 17 cannot prove that when X > 1, it should return to the original formula.
Teacher: Please consider how to prove it and how to prove it.
(Later, the teacher asked a classmate to answer)
Healthy 2 1:∫x > 1, ∴x6≥x3, X2 ≥ X-(a)
∴x6-x3≥0,x2-x≥0
∴x6-x3+x2-x≥0
∴f(x)=x6-x3+x2-x+ 1≥ 1>0
Teacher: How did you come to the above conclusion (a)?
The properties of 2 1: exponential function.
Teacher: Do you understand?
Sentient beings: I understand.
Teacher: This problem has been solved. Please write a complete proof of this problem.
(and ask a classmate to perform. After the students perform, teachers and students will comment. )
Teacher: Have we used this way of thinking to solve problems before?
All beings: used.
Teacher: Like what?
All beings: discuss and solve by classification.
Teacher: In this proof, all possible values of X prove that f(x) is positive, so it is concluded that f(x) is always positive. This rule of deductive reasoning that takes all situations into account is called complete inductive reasoning.
Teacher: please give an example of a problem solved by complete inductive reasoning before?
Health 22: The proof of "the angle formed by a straight line and two parallel planes is equal".
Teacher: Good. This proof can be divided into three cases: ① The straight line L is perpendicular to a plane; ②l∑ or l, 3l and skew. no need to say any more Please do exercises a and B.
(Teachers and students exchange comments later)
Teacher: Let's sum up what we have learned in this lesson:
1, what is deductive reasoning? Syllogism?
2. What are the functions of deductive reasoning and rational reasoning?
3. Experiencing transitive reasoning and completely inductive reasoning.
4. What did you get after learning deductive reasoning and syllogism? (the rigor of writing)
Here, teachers guide students to summarize themselves, teachers and students improve together, and form a complete knowledge structure.
Teacher: (Conclusion): Syllogism reasoning (deductive reasoning) is often used in real life. For example, the conclusion that "you should abide by the school rules and regulations" omits the syllogism reasoning of the major premise that "students should abide by the school rules and regulations" and the minor premise that "you are a student". In fact, as long as we are good at observing and thinking, we can realize that there is mathematics everywhere in our lives, and mathematics is used everywhere in our lives. Arrange the following layout.
Homework: P62, Exercise 2- 1A, T 1, BT3, class dismissed.