Inverse problem of mathematical concepts 1
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.
Characteristics of reverse thinking II
1. General
Reverse thinking is applicable to various fields and activities. Because the law of unity of opposites is universal, the forms of unity of opposites are various, and there are forms of unity of opposites, and correspondingly there are forms of unity of opposites.
Reverse thinking
Thinking angle, so there are infinite forms of reverse thinking. For example, the transformation of opposite poles in nature: soft and hard, high and low; Interchange and inversion of structure and position: up and down, left and right, etc. Process reversal: gas becomes liquid or liquid becomes gas, electricity becomes magnetism or magnetism becomes electricity, etc. Either way, as long as you think of the opposite from one aspect, it is reverse thinking.
criticalness
Reverse is compared with positive, and positive refers to conventional, common sense, recognized or habitual ideas and practices. The reverse thinking is anti-traditional, anti-conventional and anti-common sense.
Reverse thinking
Rebellion is a challenge to tradition It can overcome the mindset and get rid of the rigid cognitive model caused by experience and habit.
3. novelty
Although it is simple to follow the law of thinking and solve problems in the traditional way, it is easy to make thinking rigid and rigid, and you can't get rid of the shackles of habits, and you often get some common answers. In fact, everything has many attributes. Influenced by past experience, people often see the familiar side and turn a blind eye to the other side. Reverse thinking can overcome this obstacle, which is often unexpected and gives people a refreshing feeling.
What are the articles about the topic of reverse thinking in college mathematics?
★ Reverse thinking training questions and classic cases
★ Examples of reverse thinking in mathematics
★ reverse thinking training topic
★ Mathematical problems of reverse thinking
★ Reverse thinking practice
★ What are the skills for the grand finale of liberal arts mathematics in the college entrance examination?
★ Divergent thinking questions
★ Topic training and answers about divergent thinking
★ What are the math problem-solving skills of the college entrance examination?
★ How can we improve the scores of big math questions?