The idea of combining numbers and shapes, combining numbers and shapes, permeates each other, combines the accurate description of algebra with the intuitive description of geometric figures, transforms algebraic problems and geometric problems into each other, and organically combines abstract thinking with image thinking. Applying the idea of the combination of numbers and shapes is to fully investigate the internal relationship between the conditions and conclusions of mathematical problems, analyze their algebraic significance and reveal their geometric significance, and skillfully combine the quantitative relationship with the spatial form to find a solution.
Applying the idea of combining numbers with shapes, we should pay attention to the following number-shape transformations: (1) set operation and Wayne diagram; (2) Functions and their images; (3) Function characteristics and function images of sum formulas of general terms and series; (4) Equation (multivariate binary equation) and curve of equation.
The commonly used forms of help numbers are: using the number axis; With the help of functional images; With the help of the unit circle; With the help of the structural characteristics of numbers; With the help of analytic geometry, the commonly used methods to help modeling with numbers are: (1) the quantitative relationship followed by geometric trajectory; With the help of the combination of operation results and geometric theorems.
Step 2 discuss ideas by category
The idea of classified discussion is to analyze and solve it under different circumstances according to the nature differences of the studied objects. At the same time, it is diverse, logical and comprehensive. To establish the idea of classified discussion, we should pay attention to understanding and mastering the principles, methods and skills of classification, so as to "determine all objects, clarify the classification standards, analyze and discuss the classification at different levels, and not repeat or omit".
The key to solving mathematical problems by applying the idea of classification discussion is how to classify correctly, that is, how to choose a classification standard correctly to ensure the scientific classification, no repetition and no omission. How to implement the correct classification requires us to first make clear the objects to be discussed and all the people to be classified, then determine the classification standards and methods, then discuss them item by item, and finally make a summary.
Common classification situations are: classification by number; According to the range of letters; Classify events according to their possible occurrence; According to the position characteristics of graphics, etc. The thinking method of classified discussion is based on certain standards, and special attention should be paid to the principles of mutual exclusion, no leakage and simplicity when classifying and solving problems.
3. Function and equation thought
The thought of function and equation is the most important mathematical thought, which has many comprehensive knowledge, many problems and skills. The idea of function is very simple, which is to analyze, transform and solve the problem to be studied from the aspects of evaluation, inequality (proof), equation, parameter value range and so on by establishing function relationship or constructing intermediate function and combining the image and nature of elementary function. The idea of equation is to use mathematical language to transform the quantitative relationship in the problem into an equation model to solve it. When applying the idea of function and equation, we should pay attention to the interrelation and transformation among function, equation and inequality. We should (1) deeply understand the properties of function f(x) (monotonicity, parity, periodicity, maximum, image transformation) and master the properties of basic elementary functions skillfully, which is the basis of solving problems by using function thought.
4. Transformation and regression of ideas
The idea of transformation and transformation is to study and solve mathematical problems in a certain way. With the help of some functional properties, images, formulas or known conditions, the problem is transformed through transformation, so as to achieve the idea of solving problems.
Transformation is the process of transforming mathematical propositions from one form to another, and transformation is the process of reducing the problems to be solved to a class of problems that have been solved or are relatively easy to solve. The thought of transformation is the most basic thinking method of middle school mathematics, which can be called the essence of mathematical thought. It permeates every field of mathematics teaching content and every link of problem-solving process. There are equivalent transformations and unequal transformations. The new problem after equivalent transformation is the same as the original problem. The unequal transformation partially changed the essence of the original object, and the conclusion needs to be revised. The principle of applying transformation and regression thinking to solve problems should be to change the difficult to the easy, the familiar to the familiar, the complex to the simple, and the equivalent transformation as far as possible.
Common transformations include: inverse transformation, number transformation, equality and inequality transformation, whole and local transformation, space and plane transformation, complex and real number transformation, constant and variable transformation, and mathematical language transformation.