1, the combination of numbers and shapes: it is one of the most important and basic thinking methods in mathematics and an effective way to solve many mathematical problems. "Less is not intuitive, but more is difficult to be nuanced" is a famous saying of Professor Hua, a famous mathematician in China, which highly summarizes the role of the combination of numbers and shapes.

2. Transforming ideas: In the whole junior high school mathematics, transforming (transforming) ideas have been running through it. Transforming thinking is to transform an unknown (to be solved) problem into a solved or easy-to-solve problem, such as simplifying the complex, changing the difficult to the easy, changing the unknown to the known, and changing the high order to the low order. It is one of the most basic problem-solving ideas and one of the basic thinking methods of mathematics.

3. Classification idea: The classification of rational numbers, algebraic expressions, real numbers, angles, triangles and quadrangles, the positional relationship between points and circles, straight lines and circles, and the positional relationship between circles are all discussed through classification.

4. Overall thinking

Starting from the overall nature of the problem, we emphasize the analysis and transformation of the overall structure of the problem, find out the overall structural characteristics of the problem, and be good at treating some formulas or figures as a whole with the "overall" vision, grasping the relationship between them, and carrying out purposeful and conscious overall treatment.

5. Analogical thinking

Comparing two (or two) different mathematical objects, if they are found to have similarities or similarities in some aspects, it is inferred that they may also have similarities or similarities in other aspects.

6. Matching method

Try to form a flat formula, and then make the required transformation. This method is often used to solve the maximum problem of quadratic function, solve practical problems to save money and maximize profits.

7, undetermined coefficient method

When the mathematical formula we are studying has a certain form, to determine it, we only need to find the value of the undetermined letter in the formula. So we need to substitute the known conditions into the undetermined formula, and often we will get an equation or equation group with undetermined letters, and then we can solve the problem by solving this equation or equation group.