1. The combination of numbers and shapes is one of the most important and basic thinking methods in mathematics, and it is an effective idea to solve many mathematical problems. It is a famous saying of Professor Hua Luogeng, a famous mathematician in China, that "the number is less intuitive when it is missing, and it is difficult to be nuanced when it is countless", which highly summarizes the function of the combination of number and shape.

2. Transformation thought: In the whole junior middle school mathematics, transformation (conversion) thought runs through it all the time. Transforming thought is to solve an unknown (to be solved) problem into a solved or easy-to-solve problem, such as simplifying the complex, making the difficult easy, turning the unknown into the known, turning the high order into the low order, etc. It is one of the most basic ideas for solving problems, and it is one of the basic thinking methods of mathematics.

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

4. Holistic thinking

Starting from the holistic nature of the problem, it highlights the analysis and transformation of the overall structure of the problem, discovers the overall structural characteristics of the problem, and is good at treating some formulas or figures as a whole with an "integrated" vision, grasping the relationship between them, and carrying out purposeful and conscious holistic treatment.

5. Analogy

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