First, exchange ideas.
The idea of substitution, also called variable substitution method, refers to skillfully setting new variables to replace some complex formulas or variables in the original expression according to the structural characteristics of formulas, and then finding the results of new variables.
Substitution method is to connect scattered conditions by introducing new variables, so as to achieve the purpose of simplifying the complex and turning the unknown into the known.
Second, the combination of numbers and shapes.
The idea of the combination of numbers and shapes is actually the combination of abstract mathematical language and intuitive graphics. Through the understanding of graphics, the transformation of the combination of numbers and shapes can cultivate the flexibility of thinking and make difficult problems simple and abstract problems concrete. The problem of using numbers can be solved by using shapes.
Third, transformation and transformation.
The reduction method means that when studying and solving related mathematical problems, the problems can be transformed by some means to solve them.
Usually complex problems become simple problems, difficult problems become easy problems, and unsolved problems become solved problems through transformation.
Conversion and conversion are the most basic thinking methods in mathematics. Any problem in mathematics is inseparable from transformation and classification.
The combination of numbers and shapes embodies the mutual transformation between numbers and shapes; Functions and equations embody the mutual transformation among functions, equations and inequalities; Classification and topics reflect the transformation of parts and whole.
There are many methods of transformation, such as transformation, analysis, reduction to absurdity, undetermined coefficient method, construction method and so on. Therefore, transformation and domestication are the soul of mathematical thinking methods.
Fourth, functions and equations
Functional thinking is to use the concept and properties of functions to construct functions reasonably through analogy, correlation and transformation, and then analyze, study, transform and solve problems.
Equation thinking refers to using the properties and theorems of equations to realize the mutual transformation between problems and equations through a series of thinking processes such as observation, analysis and judgment, so as to achieve the purpose of solving problems.
Five, classified discussion ideas
Classification discussion means that when the objects given by the question cannot be studied in a unified way, we need to classify the research objects, then study each category separately, draw the conclusion of each category, and finally synthesize all kinds of results to get the answer to the whole question.
Classification discussion is basically a strategy of breaking the whole into parts, breaking one by one, and then breaking the whole into parts. When discussing classification, we should pay attention to understanding and mastering the principles, methods and skills of classification, so as to determine the overall object and clarify the classification standards.