數(shù)形結(jié)合思想及其應(yīng)用
Concerning the number form combining ideas and applications

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摘要： 數(shù)學(xué)是一門以研究客觀世界的數(shù)量關(guān)系與空間形式為基礎(chǔ)的科學(xué)，數(shù)是形的抽象概念，形是數(shù)的直觀表現(xiàn)。數(shù)形結(jié)合就是把抽象的數(shù)學(xué)語(yǔ)言、數(shù)量關(guān)系與直觀的幾何圖形、位置關(guān)系結(jié)合起來(lái),通過“以形助數(shù)”或“以數(shù)助形”即通過抽象思維與形象思維的結(jié)合,可以使復(fù)雜問題簡(jiǎn)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(xué)生學(xué)會(huì)解決問題是數(shù)學(xué)課堂教學(xué)的一項(xiàng)重要任務(wù)，也是數(shù)學(xué)教學(xué)和數(shù)學(xué)學(xué)習(xí)的最終目的，它也是用來(lái)檢驗(yàn)教師教和學(xué)生學(xué)這兩方面的標(biāo)準(zhǔn)，因此它便是數(shù)學(xué)課堂教學(xué)的重要部分，也是數(shù)學(xué)教學(xué)的主旋律。
如果教師在教學(xué)的過程中能夠培養(yǎng)學(xué)生的數(shù)形結(jié)合思想，這樣便可以有利于提高學(xué)生的數(shù)學(xué)解題能力，以此來(lái)達(dá)到我們想要的高效課堂教學(xué)效果。教師在數(shù)學(xué)課堂教學(xué)的實(shí)踐中如果能夠運(yùn)用數(shù)形結(jié)合思想來(lái)進(jìn)行教學(xué)，不但可以提高整節(jié)課的教學(xué)質(zhì)量，而且還可以幫助學(xué)生提高他們的解題能力。采用數(shù)形結(jié)合思想教學(xué)對(duì)實(shí)現(xiàn)學(xué)生的素質(zhì)教育也有非常大的積極作用。
數(shù)形結(jié)合方法不是一種工具，它是一種思想，在教學(xué)過程中，應(yīng)該給學(xué)生灌輸?shù)氖撬枷攵皇呛?jiǎn)單的方法，只有當(dāng)學(xué)生真正的掌握了這種思想之后，他們?cè)诮忸}的時(shí)候才能有效的應(yīng)用這種方法，真正的提高他們的數(shù)學(xué)解題能力。

關(guān)鍵詞：數(shù)學(xué)思想方法 數(shù)形結(jié)合 解題能力 應(yīng)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(shù)形結(jié)合的介紹 4
2.1 數(shù)形結(jié)合的概念 4
2.2 數(shù)形結(jié)合思想方法的應(yīng)用類型 4
2.3 形結(jié)合思想方法的運(yùn)用準(zhǔn)則 6
第三章 數(shù)形結(jié)合思想在教學(xué)中的應(yīng)用 9
3.1利用數(shù)形結(jié)合方法解決集合問題 9
3.2 利用數(shù)形結(jié)合方法解決方程與不等式問題 10
3.2.1 利用數(shù)形結(jié)合思想解決方程問題 10
3.2.2 利用數(shù)形結(jié)合思想解決不等式問題 11
3.3 利用數(shù)形結(jié)合方法解決函數(shù)問題 11
3.3.1 利用數(shù)形結(jié)合思想方法解決函數(shù)最值問題 12
3.3.2 利用數(shù)形結(jié)合思想方法解決函數(shù)性質(zhì)問題 13
3.4 利用數(shù)形結(jié)合方法解決三角函數(shù)的問題 14
3.5 利用數(shù)形結(jié)合方法解決線性規(guī)劃問題 15
3.6 利用數(shù)形結(jié)合方法解決數(shù)列問題 16
3.7 利用數(shù)形結(jié)合思想解決向量問題 17
3.8 利用數(shù)形結(jié)合方法解決幾何問題的方法 17
第四章 反思數(shù)形結(jié)合方法 19
4.1 利用數(shù)形結(jié)合方法解題的誤區(qū) 19
4.2 如何解決數(shù)形結(jié)合方法存在的問題 20
結(jié) 論 21
致 謝 22
參考文獻(xiàn) 23