The existence of mathematical thinking method and mathematical knowledge, the guiding role of mathematical thinking in mathematical activities, and the teaching law that recognized thinking methods can only be truly mastered through repeated use all determine that successful thinking methods can only be consciously infiltrated into ordinary teaching, especially in the widely used mathematical thinking teaching, such as the idea of combining numbers with shapes. Number and shape are the most studied objects in elementary mathematics, and the combination of number and shape is a mathematical feature. Understand the shape with numbers, and deepen the understanding of the position relationship of graphics by using the abstraction of numbers. In short, it is the coordinate of the position problem of the figure and the graphic relationship of the number. The combination of numbers and shapes is an important problem-solving strategy. Many problems in mathematics, such as the discussion of solutions of equations and inequalities, or the proof of inequalities, are limited by numbers. Although they can solve the problem, the process is tedious and even difficult. If we can consider the internal relationship between the conditions and conclusions of the problem and reveal the geometric meaning of numbers, that is, the combination of numbers and shapes, the problem will be solved. As a middle school math teacher, we should be good at digging examples of the combination of numbers and shapes, refining the idea of the combination of numbers and shapes, revealing and transforming the relationship between numbers and shapes, and often guiding students to study the problems of numbers intuitively with numbers, and making a richer, more accurate and profound discussion on the nature of numbers with numbers. At the same time, in teaching, we often teach students to think of "shape" when they see numbers (shapes), and "shape" when they see numbers (shapes). The sentence "number is not intuitive when it is invisible, but it is difficult to be nuanced when there are countless shapes" illustrates the dialectical relationship between number and shape. The combination of numbers and shapes plays an important role in enlightening ideas, understanding problems, analyzing and thinking, and judging feedback.

Refining and infiltrating the thinking method of combining numbers and shapes in mathematics, and giving full play to the role of thinking of combining numbers and shapes in teaching, is of great benefit to improving students' mathematical quality, improving their ability to analyze and solve problems, and cultivating students to analyze things with dialectical materialist viewpoints that are interrelated and transformed.

1 is a combination of numbers and shapes.

That is to say, with shape as the means and number as the purpose, with the help of geometric intuition of shape, some relationship between numbers is clarified. The application of this combination of numbers and shapes is common, and the key is to dig out the geometric meaning of number expressions.

Explanation: When the geometric intuition of "shape" is used to clarify a certain relationship between "numbers", different representations of "shape" can simplify the problem.

Deformation is digital, digital communication.

"The distortion of numbers and the communication between numbers and shapes" refers to clarifying some properties of shapes with the help of the accuracy of numbers. The solution to this kind of problem is usually to coordinate graphics. For example, many problems such as plane analytic geometry are studied by numbers. Some geometric problems, if we use different methods to deal with numbers, are directly related to the difficulty of solving them.

Based on the above discussion, we have a little understanding of some basic relations between number and shape in junior high school mathematics and how to use these basic relations to deal with mathematical problems. But from a higher point of view, as long as there is a coordinate system as the medium, any algebraic mathematical problem should have a geometric mathematical problem corresponding to it. Studying the relationship between these two corresponding forms and their mutual transformation and combination is a problem that we often have to study, so the idea of combining numbers with shapes is a very important one.

Please read the original text in PDF format for the charts, notes and formulas involved in this article.

This article is the full text of the original. Users who don't have a PDF browser should download and install the original text first.