If you find that most of your energy is spent on problems that can be solved by a calculator during your study, it is obviously a mistake.
Transforming thinking is a common way of thinking in the process of mathematics learning, and it is one of the basic ideas and methods to solve mathematical problems. Stories that have been told for a hundred years, such as Sima Guang's "Breaking the Jar" and Cao Chong's "Talking about Elephants", have all successfully used the strategy of breaking the whole into parts.
Change the way of thinking
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Transformation and transformation is one of the most basic mathematical ideas in middle school mathematics and the core of all mathematical thinking methods.
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Transformation is an objective existence, and transformation thinking is a reflection of subjectivity to objectivity. The idea of transformation can be seen everywhere in mathematics, and the process of solving mathematical problems is actually the process of solving problems through transformation.
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The combination of number and shape embodies the transformation of number and shape; The idea of function and equation embodies the mutual transformation between function, equation and inequality; The idea of classified discussion embodies the mutual transformation between the part and the whole, so the above three ideas are also the concrete manifestations of transformation and transformation.
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When using the idea of transformation, we should pay attention to deformation, quantitative change and qualitative change to ensure that transformation is only constant deformation or equivalent deformation. Once the transformation causes the change of constraint conditions, it should be checked in time.
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What problems have been solved?
Except some basic problems, we should directly solve them with relevant definitions, theorems and rules. Usually, we should transform conditions and conclusions, from recessive to dominant, from decentralized to centralized, from multivariate to unitary, from high order to low order, from unknown to known or through general and special transformation.
Number and shape are transformed into each other, motion and static are transformed into each other, and part and whole are transformed into each other. From unfamiliarity to familiarity, the problem to be solved is transformed into a solved problem, and the problem is solved.
When studying mathematical problems, the principle of transformation is:
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The connotation of transformation is very rich. Equivalent transformation and non-equivalent transformation, known and unknown, quantity and graph, graph and graph can all be transformed to solve problems.
The idea of transformation inspires us to look at problems from multiple angles and directions when solving mathematical problems.
Specific application method
Common conversion methods:
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① Direct transformation method: directly transform the original problem into a basic theorem, a basic formula or a basic graphic problem;
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(2) method of substitution: using "method of substitution" to transform formulas into rational formulas or algebraic expressions into idempotents, and to transform complex functions, equations and inequalities into basic problems that are easy to solve;
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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;
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(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;
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⑤ specialization method: transform the form of the original problem into a specialized form, prove the specialized problem, and make the conclusion suitable for the original problem;
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⑥ construction method: "construct" a suitable mathematical model to turn the problem into an easy-to-solve one;
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⑦ Coordinate method: Using coordinate system as a tool to solve geometric problems by calculation method is also an important way of transformation method.
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Analysis of middle school test sites
Paying attention to the flexible use of mathematical thinking methods is an important tool to solve the finale of the senior high school entrance examination, and it is also an important prerequisite to ensure that the finale can be solved correctly, completely and aesthetically.
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The essence of mathematics is that it can constantly change problems, turn complex problems into simpler ones, and turn unfamiliar problems into familiar ones.
Dangdang Dangdang ~ Qingguo is going to get welfare again. Below, Mr. Chang Hengjun, Dean of Qingguo Education Research Institute, combined with specific test sites, specially designed the following classic questions for the specific application of "transforming ideas" to share with you.
I hope the students will understand carefully, do a problem, learn a problem, and take the road of Wan Li.
When solving mathematical problems, we should keep (knowledge) unchanged and keep exploring. Sometimes we can use special values to verify the conclusion, so that we will have a general direction, and then solve mathematical problems by constantly transforming problems.
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In a word, transforming thoughts is the core of all mathematical thinking methods. There is no such thing, because transforming ideas is a general method to solve mathematical problems to some extent, and any problem can be solved in this way.