Middle school students skillfully solve mathematical problems by reverse thinking (1)
First, the inverse problem of mathematical concepts
Example 1 If the result of simplification | 1-x |- is 2x-5, find the value range of x.
Analysis: original formula =| 1-x|-|x-4|
X- 1-(4-x)=2x-5.
Considering from the opposite direction of the concept of absolute value, the conditions are as follows:
1-x≤0，x-4≤0
The value range of ∴x is: 1≤x≤4.
Second, the inverse process of algebraic operation
Example 2 has four rational numbers: 3,4-6, 10. These four numbers are added, subtracted, multiplied and divided (each number is used only once) to get the result of 24. Please write the formula that meets the requirements.
Analysis: Imagine 3×8=24 first, and then consider how to calculate 8 from 4,6, 10, so as to find the desired formula:
3(4-6+ 10)=24
Similarly: 4-(-6×10) ÷ 3;
10-(-6×3+4); 3( 10-4)-(-6) and so on.
Third, the reverse application of inequality properties.
Example 3 If the inequality is about x (a-1) x >; The solution set of a2-2 is X.
Analysis: According to inequality property 3, from the reverse direction analysis, we get:
a- 1 & lt； 0，a2-2=2(a- 1)
The value of ∴a is a=0.
Fourth, the test of reverse analysis of fractional equation
Example 4 It is known that the equation -= 1 has an increasing root, so find its increasing root.
Analysis: The root of this fractional equation may be x= 1 or x=- 1.
The denominator of the original equation is removed and sorted to get x2+mx+m- 1=0.
If you substitute x= 1, you can get m = 3；;
If x=- 1 is substituted, m cannot be found;
The value of ∴m is 3 and the root of the original equation is x= 1.
Fifth, the inverse problem of graphic transformation.
△ABC, AB in Example 5
Analysis: We once cut the trapezoid into triangles, that is, let a part of the trapezoid rotate around the midpoint of a waist 180. This problem is just the opposite. Inspired by this, and applying the properties of isosceles trapezoid, the following practices are obtained:
Let AD⊥BC, the vertical foot is point D, cut DE=BD on BC and connect AE, then ∠ AEB = ∠ B.
M passing through AC midpoint is MP∑AE, BC is in P, and MD is the required shear line. Cut off △MPC and you can make an isosceles trapezoid ABPQ.
Training of Reverse Thinking (2)
(A) bipolar inversion method
In general, when we meet or know one of the two poles, we might as well know the opposite pole again, and a new world may look forward to us.
Lu people do shoes and hats business.
There is a story in Han Feizi: There is a man in the state of Lu who knows how to knit, and his wife is also an expert at weaving silks and satins. They are going to do business in Yue together. Someone advised him, "Don't go, or you will fail." Lu asked, "Why?" The man replied, "You are good at knitting shoes, but Vietnamese people are used to walking barefoot;" Your wife is good at weaving silks and satins to make hats, while Vietnamese people are used to long hair and never wear hats. It's no use going to Vietnam with your good skills. Can you not fail? " As a result, the Lu people did not change their original intention. Three or five years later, instead of failing, he became a famous monopoly.
The success of many things and the solution of problems often benefit from reverse thinking, and so does the success of this Lu nationality.
Shandong people should of course go to areas that need shoes and hats, not to Vietnam, where they are not used to wearing shoes and hats. However, Lu people broke this habitual way of thinking and thought that it was because the Vietnamese people did not wear shoes and hats that they had broad market prospects and huge sales potential. As long as the rude habits of Vietnamese people are changed, Vietnam will become the largest market for shoes and hats. This is the secret of Lu's success, and reverse thinking has helped him a lot.
(2) Intermediate fusion method
Facing the two poles, people can neither hold this pole nor that pole, but make the two poles merge in the middle, and there is an either-or intermediate state. Many innovative ideas are produced in the process of integration. For example, women wear flat-heeled shoes and walk comfortably and easily; They wear high heels and walk tall and imposing. However, why can only be flat shoes or high heels, so that the opposing poles can merge in the middle? As a result, wedge shoes were developed. It is neither flat shoes nor high heels, but it has the advantages of both flat shoes and high heels.
(3) Rebound Pipa Method
Reduction to absurdity is the way to play the pipa. Its characteristic is not to prove the proposition directly, but to demonstrate it from the opposite side. That is, assuming that the conclusion of the original proposition cannot be established, put forward the opposite conclusion, and then prove that this opposite conclusion cannot be established, thus proving that the original conclusion is correct. The steps of applying reduction to absurdity are as follows: ① Make a negative conclusion on the proposition to be proved. (2) Starting from this negative conclusion, we use logical methods to reason, thus reaching contradictory conclusions; Or contradicts the conditions of the proposition, or contradicts the tentative hypothesis, or contradicts the known axioms, definitions and theorems. ③ Eliminate the negative conclusion and affirm the original conclusion of the proposition.
Sun Bin is cleverer than Wei Huiwang.
Sun Bin was a famous strategist in the Warring States Period. He went to Russia to apply for a job. Wei Huiwang is narrow-minded and jealous of his talent. He deliberately made things difficult for Sun Bin, saying, "I heard that you are quite talented. If you can let me leave my seat, I will appoint you as a general. " Wei Huiwang thought, I just can't get up, so what can you do? Sun Bin wants Wei Huiwang to stay in his seat. I can't drag him down by force. Pulling the emperor off the stage is a capital crime. What shall we do? Only by thinking backwards can he come down automatically. So Sun Bin said to Wei Huiwang, "I really can't let the king step down, but I can let you sit on the throne." . Wei Huiwang thought, this is not-what's the matter, I just don't sit down, you can bear with me again! He stepped down from his seat happily. Sun Bin immediately said, "Although I can't let you sit back now, I have let you get down from your seat." Wei Huiwang knew that he had been cheated, so he had to be appointed general.
"Playing the Pipa" is actually using the complementary relationship of opposites to implement circuitous tactics. A king always boasts that he is not only the supreme authority, but also the "merciful" savior. Before executing prisoners, he should give them a chance to draw a life-and-death ticket. If the word "live" is drawn, they will be saved from death. Once, a prisoner was about to be executed, and his enemy bribed the jailer and wrote "death" on two pieces of paper. Unexpectedly, someone leaked the news to the prisoner. The prisoner smiled and said, "Ah! I can survive. " After the king announced the draw, the prisoner smoked a lot and swallowed it without saying anything. Now everyone present is in a panic, because no one knows whether the prisoner swallowed "death" or "life" I heard the king shout loudly, "Idiot, all you have to do is look at the rest of the paper." Obviously, what was left was the signboard of "death", which proved that the prisoner had swallowed the signboard of "living". The clever prisoner skillfully applied the reduction to absurdity and survived.
(4) transposition method
Transposition method is a creative method, which can reverse the investigation proposition and invent new things.
In zoos, it is common for tourists to keep animals in cages and watch animals in parks. However, in the wildlife park, animals are eating grass. In order to prevent liger from attacking people, tourists are allowed to sit in a closed car for sightseeing, but it is not interesting.
Manufacturers of toys generally pursue bright colors and beautiful shapes to win customers' love. However, the American genius company has designed a toy dog with wrinkled and ugly skin. This unconventional idea is an ugly dog with a different style. There is a trace of innocence in the ugliness, which arouses people's curiosity. It is worthwhile to spend a few money to take a strange dog home. As expected, the wrinkled dog became the best-selling product in the market.
Langmuir, who invented the inflatable light bulb, also adopted this method. At that time, the light bulb had a fatal wound, the tungsten wire was brittle after being electrified, and the bulb wall would turn black soon after being used. It is generally believed that to overcome this problem, the vacuum degree of light bulb must be greatly improved. Langmuir thinks differently. He was not busy improving the vacuum degree of the bulb, but filled the bulb with hydrogen, nitrogen, carbon dioxide, oxygen and water vapor to study their interaction with tungsten filament at high temperature and low pressure. When he found that nitrogen can reduce the evaporation of tungsten wire, he made a judgment that it is possible for tungsten wire to work in nitrogen for a long time at atmospheric pressure. 1928, won the Parkin medal for his outstanding contribution to the invention of inflatable light bulbs and the study of chemical reactions at high temperature and low pressure.
Thinking method to a 180 degree turn, sometimes can achieve unexpected results. Many scientists in history have made great discoveries and inventions by using the method of reverse thinking.
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