Special topic of mathematical reverse thinking 1
Test of Fractional Equation in Reverse Analysis
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.
Special topic of reverse thinking in mathematics II
Attach importance to the reverse application of formulas and rules
The formula goes from left to right and from right to left, which is the embodiment of the ability to change from positive thinking to reverse thinking. Therefore, after teaching a formula and its application, we can give students a complete impression and broaden their thinking space by giving some examples of reverse application of the formula. In algebra, the reverse application of formulas can be seen everywhere. For example, the reverse application of the polynomial multiplication formula of factorization and the algorithm of the same base power can easily help us answer some questions. 5200 1; (2)2m？ 4m？ 0. 125m, etc. This set of problems is not only complicated, but even unsolvable. It will be a surprise to use the learned power operation flexibly. Therefore, reverse thinking can give full play to students' thinking ability, improve the efficiency of solving problems, and greatly stimulate students' subjective initiative in learning mathematics and their interest in exploring mathematical mysteries.
According to the inverse theorem of Pythagorean theorem, it can be known that △ABC is a right triangle.
Topic 3 of reverse thinking in mathematics
Strengthening the Teaching of Inverse Theorem
Every theorem has its inverse proposition, but it may not be true. The proof is the inverse theorem. Inverse proposition is an important way to discover new theorems. In plane geometry, many properties and judgments have inverse theorems, such as parallel lines, perpendicular lines of line segments, parallelograms, pythagorean theorems and inverse theorems. Pay attention to the relationship between conditions and conclusions, and deepen the understanding and application of the theorem. Paying attention to the teaching application of inverse theorem is of great benefit to broaden students' thinking horizons and activate their thinking. For example, in △ABC, a = 2n+ 1, b = 2n2+2n, c = 2n2+2n+ 1 (n > 0), which proves that △ABC is a right triangle.
It is proved that △ABC is a right triangle by analyzing three known sides, and the inverse theorem of Pythagorean theorem can be considered.
Prove ∵ n > 0
? 2 N2+2n+ 1 & gt； 2 N2+2n & gt； 2n+ 1 is c>b>a.
∫A2+B2 =(2n+ 1)2+(2 N2+2n)2 = 4n 4+8n 3+8 N2+4n+ 1。
C2 =(2 N2+2n+ 1)2 = 4n 4+8n 3+8 N2+4n+ 1
? a2+b2=c2
Special topic of reverse thinking in mathematics 4
Multipurpose? Reverse variant? Strengthen the training of students' reverse thinking
? Reverse variant? That is, under certain conditions, what is known and proved will be transformed into new problems similar to the original ones. For example, if you don't understand the equation, please judge the root of equation 2x2-6x+3=0. The variant is that the equation 2x2-6x+k=0 about x is known, and when k is taken, the equation has two unequal real roots. Often carry out these targeted? Reverse variant? Training and creating problem situations play a great role in the formation of reverse thinking.
Special topic of reverse thinking in mathematics 5
Inverse problem of mathematical concept
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|
According to the meaning of the question, it should be: 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 and x-4? 0
? The value range of x is: 1? x？ four