§2.4 基本不等式(二) 證明不等式
教學(xué)目標(biāo)：
⒈ 通過師生互動，總結(jié)出利用基本不等式證明不等式的常用手段；
⒉ 通過一題多解培養(yǎng)學(xué)生解題的靈活性；
⒊ 通過辨析培養(yǎng)學(xué)生的批判性．
教學(xué)重點與難點：
不等式證明的方法．
教學(xué)過程：
一、復(fù)習(xí)基本不等式：
說明：① 基本不等式反映的是和積之間的不等式關(guān)系；
② 介紹算術(shù)平均數(shù)、幾何平均數(shù)、平方平均數(shù)．
二、求證下列不等式：
(說明三者之間的關(guān)系)
四、課堂總結(jié)．
五、布置作業(yè)：《基礎(chǔ)與發(fā)展》P65~66：基礎(chǔ)性檢測(B) ．
§3.2 函數(shù)關(guān)系的建立（一）
【教學(xué)目標(biāo)】
⒈ 通過幾個簡單的實際問題建立兩個變量間的函數(shù)關(guān)系，初步理解建立函數(shù)關(guān)系的步驟；
⒉ 通過師生合作，生生合作，突破將實際問題轉(zhuǎn)化為數(shù)學(xué)問題的難點；
⒊ 體驗數(shù)學(xué)知識與實際生活的關(guān)系，初步形成積極探究的態(tài)度、獨立思考的習(xí)慣和團(tuán)結(jié)協(xié)作的精神．
【教學(xué)重點】 建立實際問題中兩個變量間的函數(shù)關(guān)系．
【教學(xué)難點】 將實際問題轉(zhuǎn)化為數(shù)學(xué)形式的問題．
【教學(xué)過程】
高考數(shù)學(xué)專題―數(shù)學(xué)思想方法3
換元法及待定系數(shù)法
解數(shù)學(xué)問題時，通過一個或幾個新變量代替原來的變量，使得代換后的問題中僅含這些新變量的方法稱之為換元法。用這種方法解題的目的是變量研究，其實質(zhì)是移問題至新對象的知識背景中去研究，達(dá)到化難為易，化繁為簡的目的。
待定系數(shù)法的實質(zhì)是方程的思想，把待定的未知數(shù)與已知數(shù)等同看待列式即得方程。
第一講 換元法
例1、已知，求的最值。
分析：請看下面解法：
∵ ，
∴
得 的最大值為21，無最小值。
思考：上面解法是否正確？
正確解法：
解：由題意得：
故可設(shè) ，
∵
∴當(dāng)時，有最大值 ；
當(dāng)時，有最小值 ；
例2、已知，求的最值；
解：可化為：
即
設(shè)
∴
∵
∴當(dāng) 時，有最大值25；
當(dāng) 時，在最小值 ；
例3、已知，，，求的值。
[分析] 此題條件中，的含義是，
，顯然，按此遞推公式求出，計算量較大，仔細(xì)觀察條件中，的形式與正切的倍角公式相近。由此可得解法。
解：設(shè) ，
∵
∴
┄┄┄┄┄
例4、在曲線：上求一點，使它到直線的距離取最小值。
解： ∵
設(shè) ，
則
又設(shè)
則點在曲線上，到直線的距離為
∵ ，∴
∴ ，
∴ 當(dāng)時，有最小值2 ；
由及，得
∴ 當(dāng)點坐標(biāo)為 時， 到直線的距離最小，最小值為2 ；
例5、已知集合，，
求集合；
解：令，
則可設(shè)，，
∴
，
關(guān)于的二次方程有實根的充要條件是
又∵
∴
∵
∴
解得；， ， ，
∴ 原方程為
∴
∴ 所求集合
練 習(xí)
1、已知，那么的值域是 ；
2、設(shè)實數(shù)滿足，則的取值范圍是 ；
3、設(shè)，求函數(shù)的最小值；
4、設(shè)，求證：，；
5、已知，且，求的最大值與最小值；
第二講 待定系數(shù)法
例1、已知方程有一個根是解這個方程；
[分析] 根據(jù)實系數(shù)方程虛根成對原理，必有另一個根是，故方程等價于
，其中待定，求出后就可求同另二個根。
解： 設(shè)
令得, 令得；
∴,解得：，
∴原方程的根為。
例2、已知一個共100項的等比數(shù)列的前項的和，
若，求所有適合等式的值的和；
[分析] 中含有兩個字母，直覺告訴我們，去確定之值，是解題中重要的環(huán)節(jié)。
解： ∵
又 是等比數(shù)列，
∴ ，又由知，
∴ ， ，
又 ，
由得：
∴ ，
∴
∴ ，
∴
例3、曲線：的圖象與曲線：的圖象關(guān)于點對稱，求的值；
解：設(shè)是上任意一點，是關(guān)于對稱的上的點，
則有
，
∴ ，
即 ①
①與應(yīng)為同一方程，
即
比較系數(shù)得。
例4、設(shè)為常數(shù)，，，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，將與代入得。
練 習(xí)
1、已知無窮等比數(shù)列前項和為，則所有項和等于
A、 B、 1 C、 D、 任意實數(shù)
2、滿足< 500的的最大正整數(shù)是
A、 4 B、 5 C、 6 D、 7
3、在直角坐標(biāo)系內(nèi)有兩點、，點在拋物線上，為拋物線的焦點，若，則的值為
A、 B、 C、 1 D、 不能確定
4、如果恒等式成立，則 ； ；
5、若方程的圖象是兩條直線，則 ；
6、函數(shù)的最大值為，最小值為，則的周期是 ；
7、已知函數(shù)的最大值為7，最小值為，求此函數(shù)的解析式；
8、已知拋物線，對任意實數(shù)均過定點， 求實數(shù)之值； 求拋物線焦點到準(zhǔn)線距離的最大值；
面積問題和面積方法
基礎(chǔ)知識
1．面積公式
由于平面上的凸多邊形都可以分割成若干三角形，故在面積公式中最基本的是三角形的面積公式．它形式多樣，應(yīng)在不同場合下選擇最佳形式使用．
設(shè)△，分別為角的對邊，為的高，、分別為△外接圓、內(nèi)切圓的半徑，．則△的面積有如下公式：
（1）；
（2）
（3）
（4）
（5）
（6）
（7）
（8）
（9）
2．面積定理
（1）一個圖形的面積等于它的各部分面積這和；
（2）兩個全等形的面積相等；
（3）等底等高的三角形、平行四邊形、梯形（梯形等底應(yīng)理解為兩底和相等）的面積相等；
（4）等底（或等高）的三角形、平行四邊形、梯形的面積的比等于其所對應(yīng)的高（或底）的比；
（5）兩個相似三角形的面積的比等于相似比的平方；
（6）共邊比例定理：若△和△的公共邊所在直線與直線交于，則；
（7）共角比例定理：在△和△中，若或，則．
3．張角定理：如圖，由點出發(fā)的三條射線，設(shè)，，，則三點共線的充要條件是：
．
例題分析
例1．梯形的對角線相交于，且，，求
例2．在凸五邊形中，設(shè)，求此五邊形的面積．
例3．是△內(nèi)一點，連結(jié)并延長與分別交于，△、△、△的面積分別為40，30，35，求△的面積．
例4．分別是△的邊和上的點，且，求△的面積的最大值．
例5．過△內(nèi)一點引三邊的平行線∥，∥，∥，點都在△的邊上，表示六邊形的面積，表示
△的面積．求證：．
例6．在直角△中，是斜邊上的高，過△的內(nèi)心與△的內(nèi)心的直線分別交邊和于和，△和△的面積分別記為和．求證：．
例7．銳角三角形中，角等分線與三角形的外接圓交于一點，點、與此類似，直線與、兩角的外角平分線將于一點，點、與此類似．求證：
（1）三角形的面積是六邊形的面積的二倍；
（2）三角形的面積至少是三角形的四倍．
例8．在△中，將其周長三等分，且在邊上，求證：．
例9．在銳角△的邊邊上有兩點、，滿足，作，（是垂足），延長交△的外接圓于點，證明四邊形與△的面積相等．
三．面積的等積變換
等積變換是處理有關(guān)面積問題的重要方法之一，它的特點是利用間面積相等而進(jìn)行相互轉(zhuǎn)換證（解）題．
例10．凸六邊形內(nèi)接于⊙，且，，求此六邊形的面積．
例11．已知的三邊，現(xiàn)在上取，在延長線上截取，在上截取，求證：．
例12．在內(nèi)，且∽，求征：
例13．在的三邊上分別取點，使，，連相交得三角形，已知三角形的面積為13，求三角形的面積．
例14．為圓內(nèi)接四邊形的邊的中點，于，于，于，求證：平分．
例15．已知邊長為的，過其內(nèi)心任作一直線分別交于點，求證：．
例16．正△正△，，，，，
，．求證：．
例17．在正內(nèi)任取一點，設(shè)點關(guān)于三邊的對稱點分別為，則相交于一點．
例18．已知是正六邊形的兩條對角線，點分別內(nèi)分，且使，如果三點共線，試求的值．
例19．設(shè)在凸四邊形中，直線以為直徑的圓相切，求證：當(dāng)且僅當(dāng)∥時，直線與以為直徑的圓相切．
訓(xùn)練題
1．設(shè)的面積為10，分別是邊上的點，且若，求的面積．
2．過內(nèi)一點作三條平行于三邊的直線，這三條直線將分成六部份，其中，三部份為三角形，其面積為，求三角形的面積．
3．在的三邊上分別取不與端點重合的三點，求證：，中至少有一個的面積不大于的面積的．
4．銳角的頂角的平分線交邊于，又交三角形的外接圓于，過作和邊的垂線和,垂足是，求證：四邊形的面積等于的 面積．
5．在等腰直角三角形的斜邊上取一點，使，作交于，求證：．
6．三條直線互相平行，在的兩側(cè)，且間的距離為，間的距離為1，若正的三個頂點分別在上，求正的邊長．
7．已知及其內(nèi)任一點，直線分別交對邊于（），證明：在這三個值中，至少有一個不大于2，并且至少有一個不小于2．
8．點和分別在的邊和上，點和將線段分為三等分，直線和分別與邊相交于點和，證明：．
9．已知P是內(nèi)一點，延長分別交對邊于，其中，，且，求之值．
10．過點P作四條射線與直線分別交于和，求證：
．
11．四邊形的兩對對邊的延長線分別交，過作直線與對角線的延長線分別，求證：．
12．為的重心，過作直線交于，求證：．
課題：正弦公式
課型：新知課
目標(biāo)：
1．知識目標(biāo)：（1）會證明兩角和與差的正弦公式，并能記住正弦公式。
（2）能夠運用兩角和與差的正弦公式。
2．隱性目標(biāo)：（1）通過正弦公式的推導(dǎo)，進(jìn)一步訓(xùn)練學(xué)生變形技巧；
（2）培養(yǎng)學(xué)生認(rèn)識事物之間的普遍聯(lián)系的哲學(xué)觀點；
重點：兩角和與差的正弦公式及應(yīng)用
難點：兩角和與差的正弦公式推導(dǎo)用應(yīng)用
教學(xué)過程：
一、先行組織者：
1．回憶兩角和與差的余弦公式，并求下列各式的值。
（1）cos (+) (2)cos (-)
2．已知cos72°=0.3090，則sin18°= ___________________ 。
cos24°=0.9135，則 sin66°= _____________。
sin3=0.1411，則cos (-3)=_____________。
二、新知：
1．嘗試練習(xí)：試求sin (+)的值
2．兩角和的正弦公式的推導(dǎo)：
兩角差的正弦公式推導(dǎo)：
數(shù)學(xué)歸納法
基礎(chǔ)知識
數(shù)學(xué)歸納法是用于證明與正整數(shù)有關(guān)的數(shù)學(xué)命題的正確性的一種嚴(yán)格的推理方法．在數(shù)學(xué)競賽中占有很重要的地位．
1．?dāng)?shù)學(xué)歸納法的基本形式
（1）第一數(shù)學(xué)歸納法
設(shè)是一個與正整數(shù)有關(guān)的命題，如果
①當(dāng)（）時，成立；
②假設(shè)成立，由此推得時，也成立，那么，根據(jù)①②對一切正整數(shù)時，成立．
（2）第二數(shù)學(xué)歸納法
設(shè)是一個與正整數(shù)有關(guān)的命題，如果
①當(dāng)（）時，成立；
②假設(shè)成立，由此推得時，也成立，那么，根據(jù)①②對一切正整數(shù)時，成立．
2．?dāng)?shù)學(xué)歸納法的其他形式
（1）跳躍數(shù)學(xué)歸納法
①當(dāng)時，成立，
②假設(shè)時成立，由此推得時，也成立，那么，根據(jù)①②對一切正整數(shù)時，成立．
（2）反向數(shù)學(xué)歸納法
設(shè)是一個與正整數(shù)有關(guān)的命題，如果
①對無限多個正整數(shù)成立；
②假設(shè)時，命題成立，則當(dāng)時命題也成立，那么根據(jù)①②對一切正整數(shù)時，成立．
3．應(yīng)用數(shù)學(xué)歸納法的技巧
（1）起點前移：有些命題對一切大于等于1的正整數(shù)正整數(shù)都成立，但命題本身對也成立，而且驗證起來比驗證時容易，因此用驗證成立代替驗證，同理，其他起點也可以前移，只要前移的起點成立且容易驗證就可以．因而為了便于起步，有意前移起點．
（2）起點增多：有些命題在由向跨進(jìn)時，需要經(jīng)其他特殊情形作為基礎(chǔ)，此時往往需要補(bǔ)充驗證某些特殊情形，因此需要適當(dāng)增多起點．
（3）加大跨度：有些命題為了減少歸納中的困難，適當(dāng)可以改變跨度，但注意起點也應(yīng)相應(yīng)增多．
（4）選擇合適的假設(shè)方式：歸納假設(shè)為一定要拘泥于“假設(shè)時命題成立”不可，需要根據(jù)題意采取第一、第二、跳躍、反向數(shù)學(xué)歸納法中的某一形式，靈活選擇使用．
（5）變換命題：有些命題在用數(shù)學(xué)歸納證明時，需要引進(jìn)一個輔助命題幫助證明，或者需要改變命題即將命題一般化或加強(qiáng)命題才能滿足歸納的需要，才能順利進(jìn)行證明．
5．歸納、猜想和證明
在數(shù)學(xué)中經(jīng)常通過特例或根據(jù)一部分對象得出的結(jié)論可能是正確的，也可能是錯誤的，這種不嚴(yán)格的推理方法稱為不完全歸納法．不完全歸納法得出的結(jié)論，只能是一種猜想，其正確與否，必須進(jìn)一步檢驗或證明，經(jīng)常采用數(shù)學(xué)歸納法證明．不完全歸納法是發(fā)現(xiàn)規(guī)律、解決問題極好的方法．
例題分析
例1．用數(shù)學(xué)歸納法證明：
（）
例2．已知對任意，，且，求證：．
例3．如果正整數(shù)不是6的倍數(shù)，則不是7的倍數(shù)．
例4．設(shè)都是正數(shù)，證明．
例5．已知函數(shù)的定義域為，對于區(qū)間內(nèi)的任意兩數(shù)均有．求證：對于任意，均有
．
例6試證：對一切大于等于1的自然數(shù)都有
．
例7試證：對一切自然數(shù)（）都有．
例8．證明：任一正方形可以剖分成任意個數(shù)多于5個的正方形．
例9．設(shè)，，，求證：對一切均有
例10．已知，，求證：對一切，都是整數(shù)．
例11．設(shè)，是否存在關(guān)于正整數(shù)的函數(shù)使等式對于的一切自然數(shù)都成立？并證明你的結(jié)論．
例12．設(shè)整數(shù)數(shù)列滿足，，，且．證明：任意正整數(shù)，是一個整數(shù)的平方．
例13．設(shè)為正數(shù)（），證明：．
例14．已知，（），求證：．
例15．整數(shù)列（）滿足，且有．求證：時，是奇數(shù)．
訓(xùn)練題
1．證明時，能被31整除．
2．設(shè)不小于6的自然數(shù)，證明：可以將一個正三角形分成個較小的正三角形．
3．用數(shù)學(xué)歸納法證明：
4．設(shè)為自然數(shù)，求證：．
5．對于自然數(shù)（），求證：．
6．已知，，求證：對于一切，是整數(shù)．
7．設(shè)有個球分成了許多堆，我們可以任意選甲、乙兩堆來按照以下規(guī)則挪動：若甲戴盆望天的球數(shù)不小于乙堆的球數(shù)，則從甲堆拿個球放堆乙堆，這樣算是挪動一次．證明：可以經(jīng)過有限次挪動把所有的球合并成一堆．
8．已知數(shù)列滿足：，，（），試證：．
數(shù)學(xué)高考基礎(chǔ)知識、常見結(jié)論詳解
一、集合與簡易邏輯：
一、理解集合中的有關(guān)概念
（1）集合中元素的特征： 確定性 ， 互異性 ， 無序性 。
集合元素的互異性：如：，，A=B求；
（2）集合與元素的關(guān)系用符號，表示。
（3）常用數(shù)集的符號表示：自然數(shù)集 ；正整數(shù)集 、 ；整數(shù)集 ；有理數(shù)集 、實數(shù)集 。
（4）集合的表示法： 列舉法 ， 描述法 ， 韋恩圖 。
注意：區(qū)分集合中元素的形式：如：；；；；；
；
（5）空集是指不含任何元素的集合。（、和的區(qū)別；0與三者間的關(guān)系）
空集是任何集合的子集，是任何非空集合的真子集。
注意：條件為，在討論的時候不要遺忘了的情況。
如：，如果，求的取值。
二、集合間的關(guān)系及其運算
（1）符號“”是表示元素與集合之間關(guān)系的，立體幾何中的體現(xiàn) 點與直線（面）的關(guān)系 ；
符號“”是表示集合與集合之間關(guān)系的，立體幾何中的體現(xiàn) 面與直線(面)的關(guān)系 。
（2）；；
（3）對于任意集合，則：
①；；；
② ； ；
； ；
③ ； ；
（4）①若為偶數(shù)，則 ；若為奇數(shù)，則 ；
②若被3除余0，則 ；若被3除余1，則 ；若被3除余2，則 ；
三、集合中元素的個數(shù)的計算：
（1）若集合中有個元素，則集合的所有不同的子集個數(shù)為_________，所有真子集的個數(shù)是__________，所有非空真子集的個數(shù)是 。
（2）中元素的個數(shù)的計算公式為： ；
（3）韋恩圖的運用：
四、滿足條件，滿足條件，
若 ；則是的充分非必要條件；
若 ；則是的必要非充分條件；
若 ；則是的充要條件；
若 ；則是的既非充分又非必要條件；
五、原命題與逆否命題，否命題與逆命題具有相同的 ；
注意：“若，則”在解題中的運用，
如：“”是“”的_______ 條件。
六、反證法：當(dāng)證明“若，則”感到困難時，改證它的等價命題“若則”成立，
步驟：1、假設(shè)結(jié)論反面成立；2、從這個假設(shè)出發(fā)，推理論證，得出矛盾；3、由矛盾判斷假設(shè)不成立，從而肯定結(jié)論正確。
矛盾的來源：1、與原命題的條件矛盾；2、導(dǎo)出與假設(shè)相矛盾的命題；3、導(dǎo)出一個恒假命題。
適用與待證命題的結(jié)論涉及“不可能”、“不是”、“至少”、“至多”、“唯一”等字眼時。
正面詞語
等于
大于
小于
是
都是
至多有一個
否定
正面詞語
至少有一個
任意的
所有的
至多有n個
任意兩個
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…與某人聊天/閑談
Who did you chat with on the Internet? 你和誰在網(wǎng)上聊天？
David and Jane chatted on the Internet about their school life. 戴維在網(wǎng)上與簡聊學(xué)校生活的情況。
2. have access to 有接近（進(jìn)入或使用）……的機(jī)會（權(quán)利）
Do you have access to the Internet? 你能使用國際互聯(lián)網(wǎng)了嗎？
Students have access to the library after school. 學(xué)生們放學(xué)后可在圖書館借閱圖書。
Children have access to the books. 孩子們可以使用這些書籍。
You can have access to the information on the hot line. 你可以使用熱線上的資料。
3. download vt. 下載
You can download some information about your hobbies. 你可以下載有關(guān)你業(yè)余愛好的資料。
I downloaded some interesting stories from the Internet. 我從國際互聯(lián)網(wǎng)上下載了一些有趣的故事。
4. sometimes ad. 有時，間或
I sometimes send e-mails to my friends at weekends. 有時候我在周末給我的朋友們發(fā)電子郵件。
Sometimes I chat with my Net-pals about football. 我有時候與網(wǎng)友侃足球。
5. never ad. 從未；永不；決不
I never play computer games. 我從來不玩電腦游戲。
I was never late for school. 我上學(xué)從未遲到過。
I never tell lies. 我從不說謊。
6. ago ad. 以前，以往
Linda called you 10 minutes ago. 琳達(dá)十分鐘前給你來過電話。
We
visited the
7. once / twice a week / month 每星期/月一次/兩次
We have computer lessons once a week. 我們每周上一次電腦課。
We have hobby group activities twice a week. 我們每星期有兩次興趣小組活動。
We go to the cinema once a month. 我們每月看一次電影。
8. get along (with)…
How are you getting along at school?
9. How often do you…?
A:How often do you do projects? 你多久做一次課題？
B:About once a month. 大約每月一次。
A:How often do you surf the Internet? 你多久上網(wǎng)（瀏覽國際互聯(lián)網(wǎng)）一次？
B:Every day. 我每天都上網(wǎng)。
A:How often do you send e-mails to your brother? 你多久給你哥哥/弟弟發(fā)一次電子郵件？
B:Every two weeks. 兩周一次。
10. I’m sorry to hear that. 聽了這話/消息我很難過。
A:Zhang Ying is very ill. 張英病得很厲害。
B:I’m sorry to hear that. 聽了這消息我很難過。
A:Li Hong’s granny died yesterday. 李紅的奶奶昨天去世了。
B:Really? I’m sorry to hear that. 真的嗎？聽了這消息我很難過。
11. pretty well 相當(dāng)好
pretty ad. 相當(dāng)，非常 又如：
He’s a pretty good student. 他是個相當(dāng)不錯的學(xué)生。
He studies pretty hard. 他學(xué)習(xí)非常努力。
I’m getting along pretty well. 我近況很好。
12. hobby groups 興趣小組
hobby group activities 興趣小組活動
13. I have very little time for fun. 我很少有時間玩。
Have time for fun 有時間玩
注意little和a little的區(qū)別。little a. [表示否定]（數(shù)量或程度上）不多的，微少的，少到幾乎沒有的；a little [表示肯定]（數(shù)量或程度上）一些，一點點。如：
He knows little English. 他不大懂英語。
He knows a little English. 他懂一點兒英語。
14. dull a. 愚笨的；枯燥的，如：
a dull boy 呆頭呆腦的男孩。
These books are dull. 這些書籍很無趣。
This film is very dull. 這部電影太無聊。
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 主頁 index 索引
network 網(wǎng)絡(luò) browse 瀏覽
password 密碼，口令 browser 瀏覽器
code 代碼 input 輸入
chat room 聊天室 output 輸出
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.
現(xiàn)在越來越多的學(xué)生能使用國際互聯(lián)網(wǎng)。
more and more 越來越多；后接名詞，如：
more and more cars 越來越多的汽車，more and more highrises 越來越多的高樓
more and more… 越來越……；后接形容詞或副詞，如：
more and more beautiful 越來越美麗，more and more expensive 越來越昂貴，如果是單音節(jié)的形容詞或副詞，則是：
faster and faster 越來越快，closer and closer 越來越靠近，better and better越來越好
2) Still others use the Net to study. 還有人在網(wǎng)上學(xué)習(xí)。top a. 最上面的；最好的；最重要的 如：
the top 頂層，floor the top price 最高價，a top student 優(yōu)秀學(xué)生，尖子生
top n. 頂部，最好的部分 如：still ad. 還有，此外還有 又如：
I still have another question. 我還有一個問題。
3) You can search these websites for information.
你可以搜尋這些網(wǎng)站尋找資料。
search for 搜尋；搜索；搜查，search此處用作不及物動詞，如：
The policemen are searching for the thief. 警察正在搜尋竊賊。
search…for… 在……中搜尋（搜索）……，search此處用作及物動詞，如：
The policemen searched the building for the thief. 警察在大樓里搜尋竊賊。
4) You can also access many of the world’s top libraries.
你也可以有機(jī)會進(jìn)入世界上一流的圖書館。
The house is at the top of the hill. 這所房子在山頂上。
She is always at the top of the class. 她在班上總是名列前茅。
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 談?wù)?/p>
The boys in our class often talk about football, but the girls like to talk about singers.
我們班男生常常談?wù)撟闱�，而女生喜歡談?wù)摳璩輪T。
What are you talking about? We are having a lesson now. Please listen attentively.
你們在談?wù)撌裁矗? 現(xiàn)在在上課，請專心聽講。
2. have a look (at) 看一看
Come and have a look at my new computer.
來看看我的電腦。
What’s happening there? Shall we go and have a look?
那兒發(fā)生了什么事了？ 我們?nèi)タ匆豢�，好嗎�?/p>
3. a small flat一個小套間
flat 是英國英語，美國英語是apartment.
4． We need to buy a bigger one. 我們需要買一個大一點的套間。
need 既可用作情態(tài)動詞，又可用作表意動詞，如：
You needn’t go now. 你現(xiàn)在不用去。（情態(tài)動詞）
I need to find a job. 我需要找一份工作。（表意動詞）
5. What do you think? I think so, too. 你認(rèn)為怎樣？（用于征求對方的意見時） 我也這樣認(rèn)為。
A: Let’s go there by light rail. It’s both cheap and fast. What do you think?
我們乘輕軌去那兒，這樣既便宜又快。你覺得如何？
B: I think so, too.
我也這樣認(rèn)為。
6. What kind of…?
What kind of T-shirts do you like, the tight ones or the loose ones?
你喜歡什么樣的T恤衫？ 緊身的還是寬松的？
A: What kind of books are you interested in?
你對那種書感興趣？
B: I’m interested in detective stories.
我對偵探小說感興趣。
7. I think… 我認(rèn)為…
I think the room with a balcony is better than the one without.
我認(rèn)為有陽臺的那間房比沒有陽臺的那見房好。
I don’t think living in the suburbs is inconvenient.
我認(rèn)為住在郊區(qū)沒有什么不方便的地方。
I don’t think watching TV is a waste of time.
我認(rèn)為看電視不是浪費時間。
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 布置房間 store room 儲藏室
furnish the kitchen/bathroom 給廚房/衛(wèi)生間安裝設(shè)備配備家具
neighborhood 居民點，街坊，四鄰 community 社區(qū)
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…收攏
Please get all the newspapers together and throw them into the recycling box.
把所有的報紙收攏來，然后扔進(jìn)回收箱里。
Can you help me get all the books together?
你能幫我把所有的書都收攏起來嗎/
2)messy 亂七八糟
Look at your study. It’s messy. 看你的書房，亂七八糟。
After the party, the garden was messy. 聚會后，花園里亂七八糟。
3)write out 寫出， 列出
The Chinese teacher wrote out a list of books for us to read on the first day of school.
開學(xué)第一天，語
Let’s write out all the things we are going to buy. I can’t remember so many things.
我們把要買的所有東西都寫出來，我記不住那么多。
4)and the public transport is more convenient than that around their old house.
that 指public transport。同一句中為避免重復(fù)往往用代詞that, those, one 或ones 指代前面的主語
The weather in
上海的天氣比北京的好。
The apple on the plate is larger than the one in the basket.
盤子里的蘋果比籃子的大。
5)Maybe it’s in a bag, I’m not sure. 大概在包里，我不肯定。
maybe 也許，大概；用于猜測或?qū)δ呈虏豢隙ā?/p>
Maybe Xiao Li lives on the fifth floor, I’m not sure.
也許小李住在5樓，我不能肯定。
Let’s try another library. Maybe we can borrow the book there.
我們到另一家圖書館去試試，也許在那兒能借到這本書。
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 打字輸入
You need to type in the information yourself.
Type in your password.
do word processing 進(jìn)行文字處理
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 與……連接
That computer is connected to the Internet.
The line is connected to that machine.
access 接近 (v.) 享有權(quán)；接近權(quán) (n.)
Can you access the central data bank?
Now we can access modern technology.
The students have access to many foreign books.
order 訂購；點（菜）
Let’s order some fish and chips.
Tom has ordered a table in the restaurant.
book 預(yù)定
I’d like to book
two seats
Paul booked a room for two nights.
enable 使能夠
The Internet enables us to get more information.
The computer enables us to do a better job.
in one word 簡而言之
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 網(wǎng)民
Net surf 網(wǎng)絡(luò)漫游
computer virus 計算機(jī)病毒
cyber 計算機(jī)的，網(wǎng)絡(luò)的
browse 瀏覽
cyberspace 網(wǎng)絡(luò)空間
scroll 滾動顯屏
access charge 接入費
cursor 光標(biāo)
BBS 公告板服務(wù)
backer 黑客
chat room 聊天室
click 點擊
Do skill focus (Listening)
Go over the lesson
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