The basic knowledge of middle school mathematics is divided into three categories: one is the knowledge of pure numbers, such as real numbers, algebraic expressions, equations (groups), inequalities (groups), functions and so on. One kind is about pure form knowledge, such as plane geometry, solid geometry and so on. One is the knowledge about the combination of numbers and shapes, which is mainly embodied in analytic geometry.

The combination of numbers and shapes is a mathematical thinking method, which includes two aspects: "helping numbers with shapes" and "helping shapes with numbers". Its application can be roughly divided into two situations: one is to clarify the relationship between numbers with the help of the vividness and intuition of shapes, that is, to use shapes as a means and numbers as the purpose, such as using the image of functions to intuitively explain the nature of functions; Or clarify some properties of a shape with the help of the accuracy and rigor of numbers, that is, with numbers as the means and shape as the purpose, such as using the equation of a curve to accurately clarify the geometric properties of a curve.

Engels once said: "Mathematics is a science that studies the relationship between quantity and spatial form in the real world." The combination of numbers and shapes is to analyze its algebraic meaning and reveal its geometric intuition according to the internal relationship between the conditions and conclusions of mathematical problems, so as to skillfully and harmoniously combine the accurate description of numbers with the intuitive image of spatial forms, make full use of this combination, find out the solution to the problem, and simplify the complex. In this way, the problem is solved. "Number" and "shape" are a pair of contradictions, and everything in the universe is the unity of their contradictions.

The essence of the combination of numbers and shapes is to combine abstract mathematical language with intuitive images. The key is the mutual transformation between algebraic problems and figures, which can make algebraic problems geometric and algebraic. When analyzing and solving problems with the idea of combining numbers and shapes, we should pay attention to three points: first, we should thoroughly understand the geometric meaning of some concepts and operations and the algebraic characteristics of curves, and analyze the geometric and algebraic significance of conditions and conclusions in mathematical topics; The second is to set parameters reasonably, use parameters reasonably, establish relationships, and transform numbers from numbers to shapes. The third is to correctly determine the range of parameters.

Some knowledge of mathematics itself can be regarded as the combination of numbers and shapes. For example, the definition of acute trigonometric function is defined by right triangle; Define trigonometric function of any angle with rectangular coordinate system or unit circle.