§2.4 基本不等式(二) 证明不等式
教学目标：
⒈ 通过师生互动，总结出利用基本不等式证明不等式的常用手段；
⒉ 通过一题多解培养学生解题的灵活性；
⒊ 通过辨析培养学生的批判性．
教学重点与难点：
不等式证明的方法．
教学过程：
一、复习基本不等式：
说明：① 基本不等式反映的是和积之间的不等式关系；
② 介绍算术平均数、几何平均数、平方平均数．
二、求证下列不等式：
(说明三者之间的关系)
四、课堂总结．
五、布置作业：《基础与发展》P65~66：基础性检测(B) ．
§3.2 函数关系的建立（一）
【教学目标】
⒈ 通过几个简单的实际问题建立两个变量间的函数关系，初步理解建立函数关系的步骤；
⒉ 通过师生合作，生生合作，突破将实际问题转化为数学问题的难点；
⒊ 体验数学知识与实际生活的关系，初步形成积极探究的态度、独立思考的习惯和团结协作的精神．
【教学重点】 建立实际问题中两个变量间的函数关系．
【教学难点】 将实际问题转化为数学形式的问题．
【教学过程】
高考数学专题―数学思想方法3
换元法及待定系数法
解数学问题时，通过一个或几个新变量代替原来的变量，使得代换后的问题中仅含这些新变量的方法称之为换元法。用这种方法解题的目的是变量研究，其实质是移问题至新对象的知识背景中去研究，达到化难为易，化繁为简的目的。
待定系数法的实质是方程的思想，把待定的未知数与已知数等同看待列式即得方程。
第一讲 换元法
例1、已知，求的最值。
分析：请看下面解法：
∵ ，
∴
得 的最大值为21，无最小值。
思考：上面解法是否正确？
正确解法：
解：由题意得：
故可设 ，
∵
∴当时，有最大值 ；
当时，有最小值 ；
例2、已知，求的最值；
解：可化为：
即
设
∴
∵
∴当 时，有最大值25；
当 时，在最小值 ；
例3、已知，，，求的值。
[分析] 此题条件中，的含义是，
，显然，按此递推公式求出，计算量较大，仔细观察条件中，的形式与正切的倍角公式相近。由此可得解法。
解：设 ，
∵
∴
┄┄┄┄┄
例4、在曲线：上求一点，使它到直线的距离取最小值。
解： ∵
设 ，
则
又设
则点在曲线上，到直线的距离为
∵ ，∴
∴ ，
∴ 当时，有最小值2 ；
由及，得
∴ 当点坐标为 时， 到直线的距离最小，最小值为2 ；
例5、已知集合，，
求集合；
解：令，
则可设，，
∴
，
关于的二次方程有实根的充要条件是
又∵
∴
∵
∴
解得；， ， ，
∴ 原方程为
∴
∴ 所求集合
练 习
1、已知，那么的值域是 ；
2、设实数满足，则的取值范围是 ；
3、设，求函数的最小值；
4、设，求证：，；
5、已知，且，求的最大值与最小值；
第二讲 待定系数法
例1、已知方程有一个根是解这个方程；
[分析] 根据实系数方程虚根成对原理，必有另一个根是，故方程等价于
，其中待定，求出后就可求同另二个根。
解： 设
令得, 令得；
∴,解得：，
∴原方程的根为。
例2、已知一个共100项的等比数列的前项的和，
若，求所有适合等式的值的和；
[分析] 中含有两个字母，直觉告诉我们，去确定之值，是解题中重要的环节。
解： ∵
又 是等比数列，
∴ ，又由知，
∴ ， ，
又 ，
由得：
∴ ，
∴
∴ ，
∴
例3、曲线：的图象与曲线：的图象关于点对称，求的值；
解：设是上任意一点，是关于对称的上的点，
则有
，
∴ ，
即 ①
①与应为同一方程，
即
比较系数得。
例4、设为常数，，，且方程有等根，
求之值；
若，求使成立的值；
解：由得 , 即 ，
又 ，故 ，
因此 或
方程有等根 ，故 ；
∵ ，
又 ，
∴且 ，
因此，将与代入得。
练 习
1、已知无穷等比数列前项和为，则所有项和等于
A、 B、 1 C、 D、 任意实数
2、满足< 500的的最大正整数是
A、 4 B、 5 C、 6 D、 7
3、在直角坐标系内有两点、，点在抛物线上，为抛物线的焦点，若，则的值为
A、 B、 C、 1 D、 不能确定
4、如果恒等式成立，则 ； ；
5、若方程的图象是两条直线，则 ；
6、函数的最大值为，最小值为，则的周期是 ；
7、已知函数的最大值为7，最小值为，求此函数的解析式；
8、已知抛物线，对任意实数均过定点， 求实数之值； 求抛物线焦点到准线距离的最大值；
面积问题和面积方法
基础知识
1．面积公式
由于平面上的凸多边形都可以分割成若干三角形，故在面积公式中最基本的是三角形的面积公式．它形式多样，应在不同场合下选择最佳形式使用．
设△，分别为角的对边，为的高，、分别为△外接圆、内切圆的半径，．则△的面积有如下公式：
（1）；
（2）
（3）
（4）
（5）
（6）
（7）
（8）
（9）
2．面积定理
（1）一个图形的面积等于它的各部分面积这和；
（2）两个全等形的面积相等；
（3）等底等高的三角形、平行四边形、梯形（梯形等底应理解为两底和相等）的面积相等；
（4）等底（或等高）的三角形、平行四边形、梯形的面积的比等于其所对应的高（或底）的比；
（5）两个相似三角形的面积的比等于相似比的平方；
（6）共边比例定理：若△和△的公共边所在直线与直线交于，则；
（7）共角比例定理：在△和△中，若或，则．
3．张角定理：如图，由点出发的三条射线，设，，，则三点共线的充要条件是：
．
例题分析
例1．梯形的对角线相交于，且，，求
例2．在凸五边形中，设，求此五边形的面积．
例3．是△内一点，连结并延长与分别交于，△、△、△的面积分别为40，30，35，求△的面积．
例4．分别是△的边和上的点，且，求△的面积的最大值．
例5．过△内一点引三边的平行线∥，∥，∥，点都在△的边上，表示六边形的面积，表示
△的面积．求证：．
例6．在直角△中，是斜边上的高，过△的内心与△的内心的直线分别交边和于和，△和△的面积分别记为和．求证：．
例7．锐角三角形中，角等分线与三角形的外接圆交于一点，点、与此类似，直线与、两角的外角平分线将于一点，点、与此类似．求证：
（1）三角形的面积是六边形的面积的二倍；
（2）三角形的面积至少是三角形的四倍．
例8．在△中，将其周长三等分，且在边上，求证：．
例9．在锐角△的边边上有两点、，满足，作，（是垂足），延长交△的外接圆于点，证明四边形与△的面积相等．
三．面积的等积变换
等积变换是处理有关面积问题的重要方法之一，它的特点是利用间面积相等而进行相互转换证（解）题．
例10．凸六边形内接于⊙，且，，求此六边形的面积．
例11．已知的三边，现在上取，在延长线上截取，在上截取，求证：．
例12．在内，且∽，求征：
例13．在的三边上分别取点，使，，连相交得三角形，已知三角形的面积为13，求三角形的面积．
例14．为圆内接四边形的边的中点，于，于，于，求证：平分．
例15．已知边长为的，过其内心任作一直线分别交于点，求证：．
例16．正△正△，，，，，
，．求证：．
例17．在正内任取一点，设点关于三边的对称点分别为，则相交于一点．
例18．已知是正六边形的两条对角线，点分别内分，且使，如果三点共线，试求的值．
例19．设在凸四边形中，直线以为直径的圆相切，求证：当且仅当∥时，直线与以为直径的圆相切．
训练题
1．设的面积为10，分别是边上的点，且若，求的面积．
2．过内一点作三条平行于三边的直线，这三条直线将分成六部份，其中，三部份为三角形，其面积为，求三角形的面积．
3．在的三边上分别取不与端点重合的三点，求证：，中至少有一个的面积不大于的面积的．
4．锐角的顶角的平分线交边于，又交三角形的外接圆于，过作和边的垂线和,垂足是，求证：四边形的面积等于的 面积．
5．在等腰直角三角形的斜边上取一点，使，作交于，求证：．
6．三条直线互相平行，在的两侧，且间的距离为，间的距离为1，若正的三个顶点分别在上，求正的边长．
7．已知及其内任一点，直线分别交对边于（），证明：在这三个值中，至少有一个不大于2，并且至少有一个不小于2．
8．点和分别在的边和上，点和将线段分为三等分，直线和分别与边相交于点和，证明：．
9．已知P是内一点，延长分别交对边于，其中，，且，求之值．
10．过点P作四条射线与直线分别交于和，求证：
．
11．四边形的两对对边的延长线分别交，过作直线与对角线的延长线分别，求证：．
12．为的重心，过作直线交于，求证：．
课题：正弦公式
课型：新知课
目标：
1．知识目标：（1）会证明两角和与差的正弦公式，并能记住正弦公式。
（2）能够运用两角和与差的正弦公式。
2．隐性目标：（1）通过正弦公式的推导，进一步训练学生变形技巧；
（2）培养学生认识事物之间的普遍联系的哲学观点；
重点：两角和与差的正弦公式及应用
难点：两角和与差的正弦公式推导用应用
教学过程：
一、先行组织者：
1．回忆两角和与差的余弦公式，并求下列各式的值。
（1）cos (+) (2)cos (-)
2．已知cos72°=0.3090，则sin18°= ___________________ 。
cos24°=0.9135，则 sin66°= _____________。
sin3=0.1411，则cos (-3)=_____________。
二、新知：
1．尝试练习：试求sin (+)的值
2．两角和的正弦公式的推导：
两角差的正弦公式推导：
数学归纳法
基础知识
数学归纳法是用于证明与正整数有关的数学命题的正确性的一种严格的推理方法．在数学竞赛中占有很重要的地位．
1．数学归纳法的基本形式
（1）第一数学归纳法
设是一个与正整数有关的命题，如果
①当（）时，成立；
②假设成立，由此推得时，也成立，那么，根据①②对一切正整数时，成立．
（2）第二数学归纳法
设是一个与正整数有关的命题，如果
①当（）时，成立；
②假设成立，由此推得时，也成立，那么，根据①②对一切正整数时，成立．
2．数学归纳法的其他形式
（1）跳跃数学归纳法
①当时，成立，
②假设时成立，由此推得时，也成立，那么，根据①②对一切正整数时，成立．
（2）反向数学归纳法
设是一个与正整数有关的命题，如果
①对无限多个正整数成立；
②假设时，命题成立，则当时命题也成立，那么根据①②对一切正整数时，成立．
3．应用数学归纳法的技巧
（1）起点前移：有些命题对一切大于等于1的正整数正整数都成立，但命题本身对也成立，而且验证起来比验证时容易，因此用验证成立代替验证，同理，其他起点也可以前移，只要前移的起点成立且容易验证就可以．因而为了便于起步，有意前移起点．
（2）起点增多：有些命题在由向跨进时，需要经其他特殊情形作为基础，此时往往需要补充验证某些特殊情形，因此需要适当增多起点．
（3）加大跨度：有些命题为了减少归纳中的困难，适当可以改变跨度，但注意起点也应相应增多．
（4）选择合适的假设方式：归纳假设为一定要拘泥于“假设时命题成立”不可，需要根据题意采取第一、第二、跳跃、反向数学归纳法中的某一形式，灵活选择使用．
（5）变换命题：有些命题在用数学归纳证明时，需要引进一个辅助命题帮助证明，或者需要改变命题即将命题一般化或加强命题才能满足归纳的需要，才能顺利进行证明．
5．归纳、猜想和证明
在数学中经常通过特例或根据一部分对象得出的结论可能是正确的，也可能是错误的，这种不严格的推理方法称为不完全归纳法．不完全归纳法得出的结论，只能是一种猜想，其正确与否，必须进一步检验或证明，经常采用数学归纳法证明．不完全归纳法是发现规律、解决问题极好的方法．
例题分析
例1．用数学归纳法证明：
（）
例2．已知对任意，，且，求证：．
例3．如果正整数不是6的倍数，则不是7的倍数．
例4．设都是正数，证明．
例5．已知函数的定义域为，对于区间内的任意两数均有．求证：对于任意，均有
．
例6试证：对一切大于等于1的自然数都有
．
例7试证：对一切自然数（）都有．
例8．证明：任一正方形可以剖分成任意个数多于5个的正方形．
例9．设，，，求证：对一切均有
例10．已知，，求证：对一切，都是整数．
例11．设，是否存在关于正整数的函数使等式对于的一切自然数都成立？并证明你的结论．
例12．设整数数列满足，，，且．证明：任意正整数，是一个整数的平方．
例13．设为正数（），证明：．
例14．已知，（），求证：．
例15．整数列（）满足，且有．求证：时，是奇数．
训练题
1．证明时，能被31整除．
2．设不小于6的自然数，证明：可以将一个正三角形分成个较小的正三角形．
3．用数学归纳法证明：
4．设为自然数，求证：．
5．对于自然数（），求证：．
6．已知，，求证：对于一切，是整数．
7．设有个球分成了许多堆，我们可以任意选甲、乙两堆来按照以下规则挪动：若甲戴盆望天的球数不小于乙堆的球数，则从甲堆拿个球放堆乙堆，这样算是挪动一次．证明：可以经过有限次挪动把所有的球合并成一堆．
8．已知数列满足：，，（），试证：．
数学高考基础知识、常见结论详解
一、集合与简易逻辑：
一、理解集合中的有关概念
（1）集合中元素的特征： 确定性 ， 互异性 ， 无序性 。
集合元素的互异性：如：，，A=B求；
（2）集合与元素的关系用符号，表示。
（3）常用数集的符号表示：自然数集 ；正整数集 、 ；整数集 ；有理数集 、实数集 。
（4）集合的表示法： 列举法 ， 描述法 ， 韦恩图 。
注意：区分集合中元素的形式：如：；；；；；
；
（5）空集是指不含任何元素的集合。（、和的区别；0与三者间的关系）
空集是任何集合的子集，是任何非空集合的真子集。
注意：条件为，在讨论的时候不要遗忘了的情况。
如：，如果，求的取值。
二、集合间的关系及其运算
（1）符号“”是表示元素与集合之间关系的，立体几何中的体现 点与直线（面）的关系 ；
符号“”是表示集合与集合之间关系的，立体几何中的体现 面与直线(面)的关系 。
（2）；；
（3）对于任意集合，则：
①；；；
② ； ；
； ；
③ ； ；
（4）①若为偶数，则 ；若为奇数，则 ；
②若被3除余0，则 ；若被3除余1，则 ；若被3除余2，则 ；
三、集合中元素的个数的计算：
（1）若集合中有个元素，则集合的所有不同的子集个数为_________，所有真子集的个数是__________，所有非空真子集的个数是 。
（2）中元素的个数的计算公式为： ；
（3）韦恩图的运用：
四、满足条件，满足条件，
若 ；则是的充分非必要条件；
若 ；则是的必要非充分条件；
若 ；则是的充要条件；
若 ；则是的既非充分又非必要条件；
五、原命题与逆否命题，否命题与逆命题具有相同的 ；
注意：“若，则”在解题中的运用，
如：“”是“”的_______ 条件。
六、反证法：当证明“若，则”感到困难时，改证它的等价命题“若则”成立，
步骤：1、假设结论反面成立；2、从这个假设出发，推理论证，得出矛盾；3、由矛盾判断假设不成立，从而肯定结论正确。
矛盾的来源：1、与原命题的条件矛盾；2、导出与假设相矛盾的命题；3、导出一个恒假命题。
适用与待证命题的结论涉及“不可能”、“不是”、“至少”、“至多”、“唯一”等字眼时。
正面词语
等于
大于
小于
是
都是
至多有一个
否定
正面词语
至少有一个
任意的
所有的
至多有n个
任意两个
否定
Teaching objectives
1. to talk about the schoolwork and after-class activities
2. to use adverbs of frequency
3. to ask for information
4. to express opinions
Schoolwork
Internet
Words and expressions:
chat with sb. .get along (with) have access to
download sometimes never
ago once / twice a week / a month
more and more search…for top
Sentence patterns:
How are you getting along…? ( Pretty well, thanks.)
How often do you …? ( About once a month./ Not very often.)
I’m sorry to hear that.
Chat with…on the Net
Did you log on?
Do you have time for fun?
I take part in hobby group activities.
Lucky you. I have very little time for fun.
Structures:
Adverbs of frequency:
1. often, always, usually, sometimes, never
2. once a week, twice a month, three times a yeay
3. every two weeks / months / years
1. to do tuning in and to learn new words and text
2. to learn notes to the text and discovering language
3. to do developing skills―listening and speaking
4. to do developing skills―reading and writing
5. to go over the text, do some dictations and to check exercises
I. Look at the picture and guess what the boys are doing now.
II.Questions:
1. Have you got any computers at home / in your school?
2. What do you do on the computer?
3. How long do you do the things above?
III.Look at the pictures about the text and then answer the following questions.
1. What is the girl doing now?
2. Where is she?
3. Where is the boy doing now?
4. Are they chatting each other on the Net?
1. Review the phonetic symbols. Show them the cards with the phonetic symbols of the new words they are going to learn.
2. Read after the tape.
3. Read by the students themselves. (T: Correct the pronunciation.)
4. Ask one student to read the words and the others should follow him or her.
5. Give the students a few minutes to remember the Chinese meaning of the words.
(T: Say English S: Say Chinese)
6. Read by the students themselves again.
I.. Listen to the tape of the text.(two times) and complete the tasks.
Task 1: Answer the questions:
A.What are the boy and girl doing?
B.What is the nationality of the girl?
C.Is she Lijie’s friend or classmate?
D.What are they talking about?
E.How often does the boy do projects?
F.What does he do for fun?
G.Does the girl have little time or much time for fun?
Task 2: Work Book P26 Comprehension check on the text
II. Listen to the tape and read after the tape. (Books open)
III.Read the text by the students themselves.
IV.Role-play the text.
Written work:
1. Copy the new words of Part 1 and text.
2. Do comprehension check on SB.
Oral work:
1. Read and recite new words.
2. Read the text fluently.
Warming-up
Role-play the text.
Revision
Spell the new words and expressions on the blackboard. (2 students)
Language focus
Key expressions:
1. chat with sb…与某人聊天/闲谈
Who did you chat with on the Internet? 你和谁在网上聊天？
David and Jane chatted on the Internet about their school life. 戴维在网上与简聊学校生活的情况。
2. have access to 有接近（进入或使用）……的机会（权利）
Do you have access to the Internet? 你能使用国际互联网了吗？
Students have access to the library after school. 学生们放学后可在图书馆借阅图书。
Children have access to the books. 孩子们可以使用这些书籍。
You can have access to the information on the hot line. 你可以使用热线上的资料。
3. download vt. 下载
You can download some information about your hobbies. 你可以下载有关你业余爱好的资料。
I downloaded some interesting stories from the Internet. 我从国际互联网上下载了一些有趣的故事。
4. sometimes ad. 有时，间或
I sometimes send e-mails to my friends at weekends. 有时候我在周末给我的朋友们发电子邮件。
Sometimes I chat with my Net-pals about football. 我有时候与网友侃足球。
5. never ad. 从未；永不；决不
I never play computer games. 我从来不玩电脑游戏。
I was never late for school. 我上学从未迟到过。
I never tell lies. 我从不说谎。
6. ago ad. 以前，以往
Linda called you 10 minutes ago. 琳达十分钟前给你来过电话。
We
visited the
7. once / twice a week / month 每星期/月一次/两次
We have computer lessons once a week. 我们每周上一次电脑课。
We have hobby group activities twice a week. 我们每星期有两次兴趣小组活动。
We go to the cinema once a month. 我们每月看一次电影。
8. get along (with)…
How are you getting along at school?
9. How often do you…?
A:How often do you do projects? 你多久做一次课题？
B:About once a month. 大约每月一次。
A:How often do you surf the Internet? 你多久上网（浏览国际互联网）一次？
B:Every day. 我每天都上网。
A:How often do you send e-mails to your brother? 你多久给你哥哥/弟弟发一次电子邮件？
B:Every two weeks. 两周一次。
10. I’m sorry to hear that. 听了这话/消息我很难过。
A:Zhang Ying is very ill. 张英病得很厉害。
B:I’m sorry to hear that. 听了这消息我很难过。
A:Li Hong’s granny died yesterday. 李红的奶奶昨天去世了。
B:Really? I’m sorry to hear that. 真的吗？听了这消息我很难过。
11. pretty well 相当好
pretty ad. 相当，非常 又如：
He’s a pretty good student. 他是个相当不错的学生。
He studies pretty hard. 他学习非常努力。
I’m getting along pretty well. 我近况很好。
12. hobby groups 兴趣小组
hobby group activities 兴趣小组活动
13. I have very little time for fun. 我很少有时间玩。
Have time for fun 有时间玩
注意little和a little的区别。little a. [表示否定]（数量或程度上）不多的，微少的，少到几乎没有的；a little [表示肯定]（数量或程度上）一些，一点点。如：
He knows little English. 他不大懂英语。
He knows a little English. 他懂一点儿英语。
14. dull a. 愚笨的；枯燥的，如：
a dull boy 呆头呆脑的男孩。
These books are dull. 这些书籍很无趣。
This film is very dull. 这部电影太无聊。
15. log on
Did you just log on?
16. lucky
You are lucky. / Lucky you.
17. Notes to the text (SB P156)
Discovering language
I. Look at the charts and answer the questions below. (Text Book)
II.Practice ( Make sentences by students themselves )
Homework
Oral work:
1. Recite the text
2. Make up short dialogues according to WB/P27 III.
Written work:
1. Do language focus on WB.
Period 3
Warming-up
Act out the dialogue of yesterday’s oral work.
Revision
Topic: School life / English Corner / Favourite food and sports / family / English Evening / Birthday Party
Additional words
I. Vocabulary of Part 2 (Show them the cards with the phonetic symbols of the new words.)
II.homeage 主页 index 索引
network 网络 browse 浏览
password 密码，口令 browser 浏览器
code 代码 input 输入
chat room 聊天室 output 输出
Developing skills
Listening & Speaking
1) Listen to the tape twice and complete the questionnaire.
2) Act out the conversation. (2 students)
Suppose you work with the School Students’Union, you want to find out whether high school juniors have enough time for fun. Conduct a survey of your classmates and then write a report for your headmaster.
Task 1:Work out a questionaire in groups of four
Task 2:Use the completed questionnair to interview your schoolmates, and make a note of their answers.
Task 3:Write a report on the survey results for your headmaster and then e-mail him.
Homework
Oral work:
Written work:
Quick responses:
1.What do you always do?
2.When do you usually get up?
3.Do you often watch TV?
4.Do you sometimes visit your friends?
5.Do you often go to the cinema?
6.Do you ever tell lies?
7.Do you never come to school late?
Go over the new words of Part 2 and recite the text.
1. Read the passage and decide if the following statements are true or faise. Tick the right boxes.
2. Notes to the reading
1) Now more and more students have access to the Internet.
现在越来越多的学生能使用国际互联网。
more and more 越来越多；后接名词，如：
more and more cars 越来越多的汽车，more and more highrises 越来越多的高楼
more and more… 越来越……；后接形容词或副词，如：
more and more beautiful 越来越美丽，more and more expensive 越来越昂贵，如果是单音节的形容词或副词，则是：
faster and faster 越来越快，closer and closer 越来越靠近，better and better越来越好
2) Still others use the Net to study. 还有人在网上学习。top a. 最上面的；最好的；最重要的 如：
the top 顶层，floor the top price 最高价，a top student 优秀学生，尖子生
top n. 顶部，最好的部分 如：still ad. 还有，此外还有 又如：
I still have another question. 我还有一个问题。
3) You can search these websites for information.
你可以搜寻这些网站寻找资料。
search for 搜寻；搜索；搜查，search此处用作不及物动词，如：
The policemen are searching for the thief. 警察正在搜寻窃贼。
search…for… 在……中搜寻（搜索）……，search此处用作及物动词，如：
The policemen searched the building for the thief. 警察在大楼里搜寻窃贼。
4) You can also access many of the world’s top libraries.
你也可以有机会进入世界上一流的图书馆。
The house is at the top of the hill. 这所房子在山顶上。
She is always at the top of the class. 她在班上总是名列前茅。
Writing
Write about Li Ming’s weekly activities in eight to ten sentences. Use the word and table cues.
1. Give the students a few minutes to finish the passage.
2. Let some students read their compositions.
3. Write some common mistakes on the blackboard and ask some students to correct.
Do writing on WB.
1. Recite the text.
2. Diction
1) How are you getting along at school?
2) I logged on about half an hour ago.
3) I take part in hobby group activities twice a week.
4) We do a school project every four month.
5) Lucky you. You have a lot of time for fun.
6) All work and no play makes Jack a dull boy.
1. Do listening on WB.
2. Check other exercises.
Go over Lesson 2 and preview Lesson 3.
Unit Four Housing Estates and Surroundings
Lesson One Choosing a flat
1. to talk and write about rooms and flats
2. to make comparisons
3. to ask for and express opinions
Flats/ Rooms
Talking about flats
Asking for and expressing opinions
Making comparisons
Words and expressions:
From the text
From the reading
talk about…
in the center of
need
get… together
face south
messy
have a look (at)
write out
this weekend
Sentence patterns:
Shopping is the most convenient.
The bedroom faces south.
What do you think? (I think so, too.)
What kind of …?
I think …
Structures:
Revisions of comparatives and superlatives of adjectives and adverbs
Time: 5 periods
1. to do tuning in and to learn new words
2. to learn text and notes to the text
3. to learn discovering language and to do developing skills―listening and speaking
4. to do developing skills―reading and writing
5. to go over the text, do some dictations and to check exercises
1. Quick responses:
1) Let’s read the text, shall we/
2)
How do you say 第
3) Do you do any warm-up exercises in a PE class?
4) Do you ever do push-ups?
5) Do you ever do waist bends?
6) How much is eleven plus fourteen?
7) How much is seventeen minus five?
8) How much is six times nine?
9) How much is a hundred divided by twenty-five?
10) Which country is the third biggest in the world?
11)
What’s the capital of
2. Numbers (Cardinal numbers and Ordinal numbers)
3. Comparatives and superlatives
1. Answer the questions:
1) How do you like your house/flat?
2) Do you want to move into a bigger house/flat?
3) Do you think public transport is important in your everyday life?
4) Which flat is the biggest of the three?
5) Which flat has the largest sitting room? How large is it?
6) Do you like a flat with a bigger sitting room or a smaller one?
7) Which flat do you like best? Why?
2. A survey on housing conditions. Questions:
1) Where do you live, in a flat or a house?
2) How many rooms does it have?
3) How large is your flat/house?
4) Is public transport convenient?
Name
Flat or house
No. of rooms
Size
Public transport
Tom
flat
4
105
convenient
III. New words
1. Read after the tape.
2. Read by the students themselves. (Correct the pronunciation.)
3. Ask some students to read the words.
4. Let the students remember the Chinese meaning of the words.
(Teacher says English and the students say Chinese.)
5. Read by the students themselves again.
Written work:
Copy the new words and text.
Oral work:
1. Read and recite new words.
2. Preview the text.
Daily talk: Say something about your house/flat.
1. Numbers
2. Comparatives and superlatives
3. New words
1. Answer the questions according to the pictures of the text.
1) How many people are there in the picture?
2) What are they talking about?
3) Does Mrs. Wang think shopping is the most important?
2. Listen to the tape and read after the tape.
3. Read the text by the students themselves.
4. Role-play the text.
IV. Under the useful expressions and phrases. Explain the key points.
1. talk about 谈论
The boys in our class often talk about football, but the girls like to talk about singers.
我们班男生常常谈论足球，而女生喜欢谈论歌唱演员。
What are you talking about? We are having a lesson now. Please listen attentively.
你们在谈论什么？ 现在在上课，请专心听讲。
2. have a look (at) 看一看
Come and have a look at my new computer.
来看看我的电脑。
What’s happening there? Shall we go and have a look?
那儿发生了什么事了？ 我们去看一看，好吗？
3. a small flat一个小套间
flat 是英国英语，美国英语是apartment.
4． We need to buy a bigger one. 我们需要买一个大一点的套间。
need 既可用作情态动词，又可用作表意动词，如：
You needn’t go now. 你现在不用去。（情态动词）
I need to find a job. 我需要找一份工作。（表意动词）
5. What do you think? I think so, too. 你认为怎样？（用于征求对方的意见时） 我也这样认为。
A: Let’s go there by light rail. It’s both cheap and fast. What do you think?
我们乘轻轨去那儿，这样既便宜又快。你觉得如何？
B: I think so, too.
我也这样认为。
6. What kind of…?
What kind of T-shirts do you like, the tight ones or the loose ones?
你喜欢什么样的T恤衫？ 紧身的还是宽松的？
A: What kind of books are you interested in?
你对那种书感兴趣？
B: I’m interested in detective stories.
我对侦探小说感兴趣。
7. I think… 我认为…
I think the room with a balcony is better than the one without.
我认为有阳台的那间房比没有阳台的那见房好。
I don’t think living in the suburbs is inconvenient.
我认为住在郊区没有什么不方便的地方。
I don’t think watching TV is a waste of time.
我认为看电视不是浪费时间。
1. Learn the text by heart.
2. Do comprehension check on WB and SB.
I.Revision
1.Recite the text. (whole class)
2.Role-play the text.
3.Check comprehension check on WB and SB.
II.Additional words
decorate the room 布置房间 store room 储藏室
furnish the kitchen/bathroom 给厨房/卫生间安装设备配备家具
neighborhood 居民点，街坊，四邻 community 社区
III.Discovering language
1.Review the use of “can, may, must, need”.
2.Read the sentences on p115.
3.Answer the questions.
4.Fill in the blanks with should, need, can or may.
IV.Developing skill―Listening
1.Listen to the conversation between Wang Qiang and his parents once.
2.Answer the questions:
1)Is Wang Qiang’s room big or small?
2)Is Wang Qiang a big boy or a small boy?
3)Do they need a bigger flat?
4)Does Wang Qiang father agree with Wang Qiang and his mother?
3.Listen to the conversation again.
4.Answer the four questions on p115.
V.Developing skill―Speaking
Wang Qiang and his father at the housing agency. They are talking about the flats. Suppose one of you is Wang Qiang or the agency and the other is Wang Qiang’s father. Make up short dialogues about the flats according to the following models. Use the word and phrase cues in the box.
VI.Homework
Do language focus on WB.
Fourth Period
I.Revision
Recite the text
II.A survey
Step 1 Conduct a survey on your group members’ housing conditions in the past and at present with the help of the following questions, and then fill in the survey below.
Step 2 Fill in the blanks with proper words, and then tell your classmate about your survey results.
III.Reading
1.Read the passage and decide if the following statements are true or false. Tick the right boxes.
2.Answer the questions:
1)When did the Whites buy a new house?
2)Is the public transport more convenient than around their old house?
3)When are they going to move to it?
4)Why are the sitting room and the bedroom messy?
5)Is father looking for his jacket?
6)Why are they busy?
7)Does Jack know how to pack books?
3.Explain the key points and underline the useful phrases.
1)get … together 把…收拢
Please get all the newspapers together and throw them into the recycling box.
把所有的报纸收拢来，然后扔进回收箱里。
Can you help me get all the books together?
你能帮我把所有的书都收拢起来吗/
2)messy 乱七八糟
Look at your study. It’s messy. 看你的书房，乱七八糟。
After the party, the garden was messy. 聚会后，花园里乱七八糟。
3)write out 写出， 列出
The Chinese teacher wrote out a list of books for us to read on the first day of school.
开学第一天，语
Let’s write out all the things we are going to buy. I can’t remember so many things.
我们把要买的所有东西都写出来，我记不住那么多。
4)and the public transport is more convenient than that around their old house.
that 指public transport。同一句中为避免重复往往用代词that, those, one 或ones 指代前面的主语
The weather in
上海的天气比北京的好。
The apple on the plate is larger than the one in the basket.
盘子里的苹果比篮子的大。
5)Maybe it’s in a bag, I’m not sure. 大概在包里，我不肯定。
maybe 也许，大概；用于猜测或对某事不肯定。
Maybe Xiao Li lives on the fifth floor, I’m not sure.
也许小李住在5楼，我不能肯定。
Let’s try another library. Maybe we can borrow the book there.
我们到另一家图书馆去试试，也许在那儿能借到这本书。
IV.Homework
1.Writing on SB.
2.Skill focus on WB.
I.Revision
Dictation
1. The most convenient thing here is shopping.
2. I want to choose a flat in the suburbs.
3. The kitchen has six square metres.
4. Both our sitting room and my bedroom face south.
5. There are many highrises in the center of town.
6. The light rail station is near the crossroads.
II.Ask some students to read out their compositions.
III.Check the exercise in the workbook.
Do skill focus―listening on WB.
IV.Phonetics.
Listen to the tape. Make the stressed syllables of these words.
V.Homework
Go over this lesson.
Lesson One Computers
To enable students to do the following:
1. talk about computers and the Internet
2. express ability with be able to
3. express obligation with ought to
4. use modal verbs
Words and expressions:
From the text
From the reading
ought to
be connected to
get familiar with
access
control
order
type in
book
do word processing
enable
in one word
Structures:
be able to
ought to
Revision of modal verbs
Sentence patterns:
What can you do with…?
What’s the use of…?
We use it for…
1. to learn new words, tuning in and discovering language
2. to learn text and notes
3. to learn listening and speaking
4. to learn reading and writing
5. to check workbook
My favorite football hero
Label the different parts of the computer and complete the following sentences.
Teach these new words according the pictures: main unit, monitor, keyboard, mouse, electronics, type in, do word processing
Answer the questions
1. What are the main parts of the computer?
2. What can you do with the mouse?
3. What’s the use of the keyboard?
be able to 的用法
Look at the table and complete the dialogues below with be able to in its proper form.
1. What does be able to mean? It means can.
2. What is the past / future / question form of be able to?
The past form is was able to.
The future form is will be able to.
3. What’s the negative form of be able to in the future tense?
The negative form in the future is won’t able to.
Do language focus Ex. 1 on WB.
Copy the new words
Recite the new words
Preview the text
What does a monitor of a computer do?
What does a keyboard do?
What helps to control the computer?
What can we do on the computer?
Where is the electronics?
Spell the new words
Step 1
Answer the question according to the pictures:
1. Who’s the man?
2. What are they doing?
3. What lesson are they having?
Step 2
Finish the comprehension check on WB after the tape
Step 3
Read after the tape
Step 4
Role-play the text
Step 5
Read the text by the students themselves
Notes to the text (Page 127 ?128)
Additional notes
get familiar with 熟悉
We must get familiar with our new neighbors.
The teacher is doing his best to get familiar with the new textbook.
control 控制
Now we can use the computer to control the public transport.
Can we use the keyboard to control the computer?
type in 打字输入
You need to type in the information yourself.
Type in your password.
do word processing 进行文字处理
We can do word processing on the computer.
Can you show me how to do word processing on the computer?
What can you do with…? 用……你可以做些什么？
A: What can you do with the computer?
B: A lot. I can do word processing, surf the Internet and send e-mail.
Copy the text
Recite the text
Do comprehension check on SB
Daily talk: Computer
Read the text
Recite the text
Do comprehension check in pairs
Do language focus Ex. 2 on WB
1. Listen to the dialogue between Julia and David, and find out about their use of computers. Tick the right boxes.
2. Work in pairs and talk about your computer skills after the model. Use the word cues below.
3. Suppose one of your classmates’ parents has written a letter to your headmaster or headmistress. Read the letter and complete the two tasks below.
Task 1 Discuss the advantages and disadvantages of the Internet in groups of four after the model. Use the word cues
Task 2 Complete the following reply to Mr. Green’s letter according to your discussion.
Do language focus Ex. 3 4
Period Four
Present your reply in class.
1. Read the passage
2. Complete the sentences.
3. Notes
be connected to 与……连接
That computer is connected to the Internet.
The line is connected to that machine.
access 接近 (v.) 享有权；接近权 (n.)
Can you access the central data bank?
Now we can access modern technology.
The students have access to many foreign books.
order 订购；点（菜）
Let’s order some fish and chips.
Tom has ordered a table in the restaurant.
book 预定
I’d like to book
two seats
Paul booked a room for two nights.
enable 使能够
The Internet enables us to get more information.
The computer enables us to do a better job.
in one word 简而言之
In one word, the Internet has made our lives more convenient.
In one word, the computer is a useful machine.
4. Questions
a. Do you like computers?
b. What can you do on the computer?
c. How long do you use the computer every week?
d. Do your parents often use the Net?
e. How do they like the computer?
Write six to eight sentences about your computer skills.
Read out the composition.
Do skill focus on WB
A children’s rhyme
netizen 网民
Net surf 网络漫游
computer virus 计算机病毒
cyber 计算机的，网络的
browse 浏览
cyberspace 网络空间
scroll 滚动显屏
access charge 接入费
cursor 光标
BBS 公告板服务
backer 黑客
chat room 聊天室
click 点击
Do skill focus (Listening)
Go over the lesson