數形結合思想及其應用
Concerning the number form combining ideas and applications

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摘要： 數學是一門以研究客觀世界的數量關系與空間形式為基礎的科學，數是形的抽象概念，形是數的直觀表現(xiàn)。數形結合就是把抽象的數學語言、數量關系與直觀的幾何圖形、位置關系結合起來,通過“以形助數”或“以數助形”即通過抽象思維與形象思維的結合,可以使復雜問題簡單化,抽象問題具體化,從而起到優(yōu)化解題途徑的目的。讓學生學會解決問題是數學課堂教學的一項重要任務，也是數學教學和數學學習的最終目的，它也是用來檢驗教師教和學生學這兩方面的標準，因此它便是數學課堂教學的重要部分，也是數學教學的主旋律。
如果教師在教學的過程中能夠培養(yǎng)學生的數形結合思想，這樣便可以有利于提高學生的數學解題能力，以此來達到我們想要的高效課堂教學效果。教師在數學課堂教學的實踐中如果能夠運用數形結合思想來進行教學，不但可以提高整節(jié)課的教學質量，而且還可以幫助學生提高他們的解題能力。采用數形結合思想教學對實現(xiàn)學生的素質教育也有非常大的積極作用。
數形結合方法不是一種工具，它是一種思想，在教學過程中，應該給學生灌輸的是思想而不是簡單的方法，只有當學生真正的掌握了這種思想之后，他們在解題的時候才能有效的應用這種方法，真正的提高他們的數學解題能力。

關鍵詞：數學思想方法 數形結合 解題能力 應用

Concerning the number form combining ideas and applications
Abstract： Mathematics is an objective world to study the relationship between the amount of space in the form of basic science , the number is shaped abstraction form is intuitive performance numbers. The combination of symbolic and graphic is to abstract mathematical language , the number of relationships with intuitive geometry , the positional relationship together, through the " help to shape a few " or " several help shape" that is, through a combination of abstract thinking and thinking in images , you can make complex simplification of the problem , abstract issues concrete, which played optimization problem solving approaches purposes. Let students learn to solve problems is an important task for mathematics teaching , but also the ultimate goal of mathematics teaching and learning of mathematics , it is also used to test the teachers teach and students learn standard in these two areas , so it is important to classroom teaching of mathematics part is the main theme of mathematics teaching.
If the teacher in the teaching process can train students the combination of symbolic and graphic thinking so that we can help improve students' mathematical problem-solving ability in order to achieve efficient teaching effect we want. In practice math teachers in the classroom if they can use for teaching thinking the combination of symbolic and graphic , not only can improve the quality of teaching the whole class , but also can help students improve their problem-solving abilities. Using The combination of symbolic and graphic thought teaching students for achieving quality education also has a very large positive effect.
The combination of symbolic and graphic method is not a tool , it is an idea, in the teaching process , students should be given the ideological indoctrination rather than a simple method, only when the students really grasp this idea after their problem-solving when the application of this method to be effective , truly improve their problem-solving ability .
Key words：Mathematical thinking The combination of symbolic and graphic Problem-solving ability Application

目 錄
引 言 1
第一章 問題的提出 2
1.1 問題研究的背景 2
1.2 問題研究的意義 3
第二章 數形結合的介紹 4
2.1 數形結合的概念 4
2.2 數形結合思想方法的應用類型 4
2.3 形結合思想方法的運用準則 6
第三章 數形結合思想在教學中的應用 9
3.1利用數形結合方法解決集合問題 9
3.2 利用數形結合方法解決方程與不等式問題 10
3.2.1 利用數形結合思想解決方程問題 10
3.2.2 利用數形結合思想解決不等式問題 11
3.3 利用數形結合方法解決函數問題 11
3.3.1 利用數形結合思想方法解決函數最值問題 12
3.3.2 利用數形結合思想方法解決函數性質問題 13
3.4 利用數形結合方法解決三角函數的問題 14
3.5 利用數形結合方法解決線性規(guī)劃問題 15
3.6 利用數形結合方法解決數列問題 16
3.7 利用數形結合思想解決向量問題 17
3.8 利用數形結合方法解決幾何問題的方法 17
第四章 反思數形結合方法 19
4.1 利用數形結合方法解題的誤區(qū) 19
4.2 如何解決數形結合方法存在的問題 20
結 論 21
致 謝 22
參考文獻 23