In middle school mathematics, conversion is not only an important problem-solving thought, but also a basic thinking strategy. The so-called transformation thinking method is a method to solve mathematical problems by turning them into simple problems by some means. Turn difficult problems into easy-to-solve problems; In short, transformation is almost ubiquitous in solving mathematical problems, and its basic functions are: from unfamiliar to familiar, from complex to simple, from abstract to intuitive, from vague to clear. In the final analysis, the essence of transformation is to look at problems from the viewpoint of movement change and development, as well as the viewpoint that things are interrelated and mutually restricted, and to be good at transforming and transforming the problems to be solved, so as to solve problems.
Rosa Peter, a famous Hungarian mathematician, explained how mathematicians solved problems by conversion through a very vivid and interesting joke in his masterpiece Infinite Things. Someone asked, "Suppose you have a gas stove, faucet, kettle and matches in front of you. What do you want to do if you want to boil water?" Someone replied, "Fill the kettle with water, light the gas, and then put the kettle on the gas stove." The questioner affirmed the answer, but he asked, "What should you do if other conditions have not changed, but there is enough water in the kettle?" At this time, the questioner will definitely answer loudly and confidently, "Light the gas and put the kettle on." But a more perfect answer should be like this: "Only physicists will do what they just said, and mathematicians will answer," As long as the water in the kettle is emptied, the problem boils down to the above problems. "。
"Pour out the water", that is, the conversion method, which is commonly used by mathematicians. Turning to the history of mathematics development, there are countless examples, and the famous problem of the Seven Bridges in Konigsberg is a wonderful example. Great mathematician Euler's thinking program to solve this problem is:
This is a good application of transformation problem, from which we can easily sum up the thinking mode of transformation thinking method:
It can be seen that the strength of problem-solving ability lies in: 1, only keen insight can find the target model, and 2, only strong reduction ability can effectively transform the problem into the target model, and it is easier to solve it by using the inherent laws of the model.
In middle school mathematics, the common basic reduction forms are:
1, conversion between numbers. For example, calculate a formula to get a numerical value; Simplify an analytical formula to get the result; Solve the equation given by deformation; Transform the given inequality, solve the set and the mutual transformation among functions, equations and inequalities.
2. Conversion between forms. For example, using the knowledge of image transformation to make functional images; Use division, shape filling, folding and expansion as auxiliary lines and surfaces to deal with space graphics or plane graphics. , including turning a three-dimensional problem into a plane problem.
Example 2. As shown in the figure, in the regular triangular pyramid P-ABC, the length of each side is 2, e is the midpoint of the side PC, and d is any point of the side PB. Find the minimum perimeter of △ADE.
3. Digital-to-shape conversion. Digital-to-shape conversion is mainly based on the relationship between function and its image; Complex numbers and the geometric meaning of their operations; And the concepts of curves and equations in analytic geometry.
[Analysis]: This is an inequality proof problem with four unreasonable formulas, which is difficult to start with, and the reduction method can be applied. I noticed that the structure of the four irrational number formulas on the left is similar to Pythagorean theorem, so I thought of it and tried to reduce it to a geometric problem. This can be easily simplified to one: construct a square as shown in Figure 3, and it can be said that the inequality relationship is self-evident.
What do you mean by transformation and transformation? Give a simple example.
The thinking method of transformation is a method of transforming a problem into a simple problem by some means when studying and solving related mathematical problems. Generally, complex problems are always transformed into simple problems, difficult problems are transformed into easy-to-solve problems, and unsolved problems are transformed into solved problems through transformation. The idea of conversion occupies a very important position in the college entrance examination, and the solution of mathematical problems, conversion and conversion are always inseparable, such as the transformation from unknown to known, from new knowledge to old knowledge, from complex problems to simple problems, from different mathematical problems to each other, from practical problems to mathematical problems and so on. Various transformations and specific problem-solving methods are the means of transformation, and the transformed thinking method permeates all mathematics teaching contents and problem-solving processes. Principles of transformation and transformation? (1) Familiarity principle: Turn unfamiliar problems into familiar ones, so that we can solve them with familiar knowledge and experience. (2) Simplification principle: simplify complex problems into simple ones and solve them, so as to achieve the purpose of solving complex problems, or get some inspiration and basis for solving problems. (3) Visualization principle: turn more abstract problems into more intuitive ones to solve. (4) When a problem encounters difficulties, you can consider the negative side of the problem and try to explore it from the negative side of the problem, so that the problem can be solved. 2. Common transformation and reduction methods? When studying and solving mathematical problems, we often use the transformation and transformation of ideas. When thinking is blocked or seeking simple methods or switching from one situation to another, that is, switching to another situation to solve problems, this switching is an effective strategy to solve problems and a successful way of thinking. Common transformation methods are: (1) direct transformation method: the original problem is directly transformed into a basic theorem, a basic formula or a basic graphic problem. (2) method of substitution: Using "method of substitution" to transform formulas into rational formulas or algebraic expressions into idempotents, and to transform more complicated functions, equations and inequalities into basic problems that are easy to solve. (3) Number-shape combination method: study the relationship between quantity (analytical formula) and spatial form (figure) in the original problem, and obtain the transformation path through mutual transformation. (4) Equivalent transformation method: the original problem is transformed into an equivalent proposition that is easy to solve, so as to achieve the purpose of reduction. (5) specialization method: transform the form of the original question into a specialized form, and prove that the specialized question and conclusion are suitable for the original question.
Introduction to transformation ideas
Transformation thought refers to the process of a problem from difficult to easy, from complex to simple, from complex to simple, which is the abbreviation of transformation and solution.
In a word, tell me the difference between transforming thought and returning thought in mathematics.
In short, conversion is a purposeful transformation.
Transformation thought refers to the process of a problem from difficult to easy, from complex to simple, from complex to simple, which is the abbreviation of transformation and solution.
In the process of solving problems, mathematicians often do not directly solve the original problems, but transform and transform the problems until they are classified as solved or easily solved. After some changes, the problem to be solved is simplified to another problem *, and then the original problem can be solved by solving the problem * and applying the solution result to the original problem. This solution to the problem is called reduction.
Induction is the basic thinking method to analyze and solve problems. In mathematics, the usual practice is to simplify a non-basic problem into a familiar basic problem by decomposition, deformation, replacement, or translation, rotation and expansion. To find a solution. For example, after learning the knowledge of linear equation of one variable and factorization, we can learn quadratic equation of one variable through factorization. It is solved by simplifying it into a linear equation of one variable. Later, when we were studying a special one-dimensional higher-order equation, we solved it by simplifying it into one-dimensional linear equation and one-dimensional quadratic equation. We have a similar method for unary inequality. For example, in plane geometry, after learning the related theorems such as the calculation of the sum and area of the internal angles of triangles, the calculation of the sum and area of the internal angles of N-polygons is also solved by splitting and splicing into several triangles. After we have learned the most basic and simple knowledge of conic, the research on general conic is also realized by translating or rotating the coordinate axis into a basic conic (in the new coordinate system). Other examples, such as geometric problems turned into algebraic problems, solid geometric problems turned into plane geometric problems, and trigonometric functions of arbitrary angles turned into acute trigonometric functions, are even more. Mastering the thinking method of transformation is of great significance to mathematics learning. In short, the transformation principle is to transform the unknown into the known, the complex into the simple, the abstract into the concrete, the general into the special and the non-basic into the basic knowledge, so as to get the correct answer.
What is the difference between mathematical problems, transforming ideas and transforming ideas? 50 points.
It must be different,
The essence of the transformation thought of 1 is to reveal the connection and realize the transformation. Except for extremely simple mathematical problems, the solution of every mathematical problem is realized by transforming it into a known problem. In this sense, solving mathematical problems is the process of transforming from unknown to known. Transformation is the basic idea of solving mathematical problems, and the process of solving problems is actually a process of gradual transformation. Transformations in mathematics can be seen everywhere. Such as the transformation from unknown to known, from complex problem to simple problem, from new knowledge to old knowledge, from proposition to proposition, from number to form, from space to plane, from high dimension to low dimension, from multivariate to unitary, from high order to low order, from transcendence to algebra, from function to equation, etc.
There are equivalent transformations and non-equivalent transformations. Before and after the equivalent transformation is a necessary and sufficient condition, so make the transformation as equivalent as possible; If necessary, unequal transformation should be carried out, and restrictions should be attached to maintain equivalence, or the conclusions drawn should be verified.