摘    要：众所周知，在中学阶段，我们主要研究数量关系与空间形式这两类基本内容，它们既相互独立，又相互渗透。数形结合思想作为数学思想的一个重要组成部分，它可以分为“用图形解决几何问题”和“用代数解决几何问题”两个方面。只要能熟练地掌握数形结合的中心思想，就可以在图像与代数之间自由切换，从而降低解题的难度。

本文在查阅了大量文献的基础上，结合初高中数学教材，围绕数形结合的中心思想，结合本人在初三的实习经验，以数形结合思想的基本内容及其在中学数学教材中的体现为核心内容进行阐述。本文旨在通过解决集合问题、函数与方程问题、不等式问题、数列问题、向量问题、几何问题这六大方面对数形结合的应用进行介绍，并结合典型例题进行析。在文章最后对如何培养学生数形结合思想总结了一些方法。

关键词：数形结合; 以形助数; 以数解形; 中学教学; 教学应用 

Abstract：As we all know, in the secondary school stage, we mainly study the two basic contents of quantitative relationship and spatial form. They are independent and infiltrate each other. As an important part of mathematics thoughts,the combination of number and shape can be pided into two aspects: "resolving geometric problems with graphs" and "geometric problems with algebra." As long as we can master the central idea of combination of numbers and shapes, we can freely switch between images and algebra, thus reducing the difficulty of solving problems.

This article mainly uses the literature research method to consult a large amount of literature, based on the middle school mathematics textbooks,relying on the combination of number and shape, combined with own teaching experience in junior high school, systematically introduced the combination of number and shape and its application in mathematics teaching. This paper mainly introduces its application in  solving set problems, solving function and equation problems, solving inequalities problems, solving the problems of sequence of number, solving vector problems, and solving geometric problems. Through the analysis of typical examples, the importance of the application of the combination of number and shape in solving problems is obtained. In the end, it introduces the ways to train students' ideas of number and shape.

Keywords: the combination of number and shape; Using Geometry to Solve Algebra Problems; Using algebraic methods to solve geometric problems; Middle school mathematics; Teaching application 

目    录

摘要........I

Abstract II

1.引言 1

1.1 问题研究的背景 1

1.2 问题研究目的及意义 1

1.3 国内外研究现状 2

2.对数形结合思想方法的认识 2

2.1数形结合思想概念 2

2.2数形结合思想的表现形式 3

2.3数形结合思想的基本原则 3

3.数形结合思想在中学数学教学中的应用 3

3.1数形结合思想解在集合问题中的应用 3

3.2数形结合思想在函数与方程问题中的应用 4

3.2.1数形结合思想在函数的定义域、值域或最值问题中的应用 4

3.2.2数形结合思想在函数的零点个数问题中的应用