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10.2空间两条直线之间的位置关系_图文


10.2空间两条直线的位置关系

本节地位

大纲要求

我的理解
理解空间两条直线的 位置关系是关键,正 确判断空间两条直线 的位置关系是重点, 尤其是异面直线的判 定。教学中要做好从 平面到空间的过度, 借助平面来判断直线 的位置关系,完成从 平面到空间的过度。

直线与直线的位置 关系是“线线关系、 线面关系、面面关 系”中最重要、最 基本的一种,也是 后面两种位置关系 的基础,是我们研究 的重点。

通过观察实物和模型, 归纳出直线位置关系 的判定与性质;培养 学生的空间想象能力, 数学思维能力;重点 掌握好直线位置关系 的判定。

教材分析

知识储备

思想方法

学习态度

已经学习了平面的

基本概念,直线与
平面的位置关系。 掌握了常见平面和 直线的画法。

已经初步建立了空 间图形的概念和形 象,具有一定的空 间图形的想象能力 和判断能力。

立体几何直观形象 学生掌握起来比较 容易,有别于代数 的抽象复杂,有较

高的学习兴趣。

学情分析

知识与技能

过程与方法

情感态度与价值观

(1) 理解空间两条直 线的位置关系。 (2)会用平面衬托来画 异面直线。 (3)掌握并会应用平行 公理。 (4)会用异面直线所成 的角的定义找出或作出 异面直线所成的角,会 在三角形中求简单异面 直线所成的角。

在直线的位置关系 的判断过程中,掌 握借助平面判断空 间两条直线位置关 系的方法

(1) 感受到掌握空间两直 线关系的必要性,提高 学生的学习兴趣。 (2) 增强动态意识,培养 观察、对比、分析的思 维,通过平移转化渗透 数学中的化归及辩证唯 物主义思想。 (3) 通过探究增强学生的 合作意识、动脑意识和 动手能力。

教学目标

教学重点

教学难点

异面直线的判断

异面直线所成角的推证与求解

教学手段 学法指导 教法分析
电子白板 想 引 象 学 观 察 析 想 练 象 自主探究 小组合作 实物模拟

实物模型

— — —

— —

学生自主探究

学生分组协作

教师过程引导

学生按照老师创设的情境、提出的问题自主探究、合作学习。 对于问题,分组交流,相互补充。

教师参与小组讨论,解疑。

奇妙的线条——Philipp Klinger摄影作品
波兰籍德国摄影师Philipp Klinger从
2007年5月开始接触摄影。极有天分的

Philipp Klinger在摄影过程中,融入自己
对城市、建筑、景象的独特见解,以独特取 景视角,在作品中为我们呈现常常被我们忽 略的线条美感。

课前导学
? ? ? ?

在同一个平面内,两条直线的位置关系有几种? 根据什么分类的?

分别是什么?
在空间内任意两条直线这个结论是否还成立?

课前导学
D
C A B

课前导学

新课导学
? 1.异面直线
? 什么是异面直线? ? 定义中有哪些关键字?

新课导学
? 2.空间两直线的位置关系
? 空间两条直线的关系怎么分类? ? 分类的标准是什么?

合作探究
1.在正方体ABCD-EFGH中,和AE相交、平行 、异面的直线分别有哪些?

A B E F G C

D

H

合作探究
2.如图是一个正方体的展开图,如果将它还原为 正方体, 那么 AB , CD , EF , GH 这四条线段所在 直线是异面直线的有 对?
A C B G

D

F

H

E

新课导学
? 3.异面直线的判断
? 异面直线的判断需要平面的辅助,怎么寻找辅助

的平面呢?

A
B E F G C

D

H

新课导学
? 4.异面直线的画法

b

b

b

A
a

B A
a

a

合作探究
分别在两个平面内的两条直线是否一定异面?

b
a

a

M

b

a

b

?

?

?

?

?

?

a与b是异面直线

a与b是相交直线

a与b是平行直线

新课导学
5.异面直线所成的角
(1)复习回顾
同一平面内是如何刻画两条相交直线的位置关系的?

(2)问题提出
怎么刻画两条异面直线的不同位置关系呢?

(3)问题猜想
能否将异面直线平移到一起来解决?

(4)理论支持
公理4:在空间平行于同一条直线的两条直线互相平行.

(5)解决问题
已知两条异面直线 a , b , 经过空间任一点O作直线 a′∥a , b ′∥b 则把 a ′与 b ′所成的锐角(或直角)叫做异面直线所成的角。

5.异面直线所成的角
b

b′ a′

a

O

5.异面直线所成的角
【问题引导】
1.异面直线所成的角最大可以是多少?最小呢?它 的取值范围是多少? 2.两条异面所成的角可以是直角吗? 3.在求作异面直线所成的角时,怎样作角比较简单?

例题选讲
例1. 如图,已知空间四边形ABCD中,点E、F、G、H分别是边 AB、BC、CD、DA的中点,试判断四边形EFGH是什么四边形 ,并证明你的结论。

合作探究
在例1中,若加上条件AC=BD,那么这个 四边形是什么四边形?

例题选讲
例2.如图,正方体ABCD-EFGH中, (1) 求AB与EH所成的角? (2) 求BE与AC所成的角? E

H F
D

G

C

A

B

例题选讲
问题点拨
如何求解异面直线所成的角?可以分成几个步骤?

随堂训练
B 1.如图正方体中 (1)说出以下各对线段的位置关系?
①EC和BH是 ②BD和FH是 ③BH和DC是 直线 直线 直线

A
C E F
条?

D

H G

(2)与棱 A B 所在直线异面的棱共有 (3)如图,已知正方体ABCD-EFGH中,
①求BC 和EG 所成的角是多少度? ②求AH和EG 所成的角是多少度?

课堂小结
1. 2. 3. 4. 5. 6.

异面直线的定义: 不同在 任何 一个平面内的两条直线叫做异面直线。 空间两直线的位置关系:相交直线、平行直线、异面直线 异面直线的画法:用平面来衬托

异面直线所成的角:平移,转化为相交直线所成的角
公理4(平行公理):在空间平行于同一条直线的两条直线互相平行. 异面直线所成角的求法: 一作(找)二证三求


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