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河北省邯鄲一中2009屆高三上學(xué)期期末考試

英語(yǔ) 試題

第I卷（選擇題，共95分）

第一部分英語(yǔ)知識(shí)運(yùn)用（共三節(jié)，滿分50分）

第一節(jié) 語(yǔ)音知識(shí) （共5小題；每小題1分，滿分5分）

從A、B、C、D四個(gè)選項(xiàng)中，找出其劃線部分與所給單詞的劃線部分讀音相同的選項(xiàng)，并在答題卡上將該項(xiàng)涂黑。

1．stop A．lose B．woman C．shock D．rose

2．southern A．courage B．south C．youth D．trousers

3．wear A．near B．require C．cheer D．share

4．Society A．a(chǎn)chieve B．friend C．science D．believe

5．throw A．nowadays B．tomorrow C．crowd D．town

第二節(jié) 語(yǔ)法和詞匯知識(shí)（共15小題；每小題1分，滿分15分）

從A、B、C、D四個(gè)選項(xiàng)中，選出可以填入空白處的最佳選項(xiàng)，并在答題卡上將該項(xiàng)涂黑。

6．I don' t think we have met before. You' re taking me for ______.

A．some other B．someone else

C．other person D．one other

7．You must keep on working in the evening ______ you are sure you can finish the task in time.

A．a(chǎn)s B．if C．when D．unless

8．-- Shall we go to the art exhibition right away?

-- ______.

A．It' s your opinion B．I don' t mind

C．It' s all up to you D．That' s your decision

9．Kathy _____ a lot of Spanish by playing with the native boys and girls.

A．picked up B．took up C．made up D．turned up

10．After dinner he gave Mr. Richardson ______ ride to ______Capital Airport.

A．the; a B．a(chǎn); the C．不填; a D．不填; the

11．Helen _______ go on the trip with us, but she isn't quite sure yet.

A．shall B．must C．may D．can

12．The manager had fallen asleep where he______, without undressing.

A．was laying B．was lying

C．had laid D．had lied

13．My _______ of this weekend's activity is going out with some good friends.

A．idea B．opinion C．mind D．thought

14．That was really a splendid evening. It's years _______ I enjoyed myself so much.

A．when B．that C．before D．since

15．-- I'm terribly sorry I broke your glass.

-- ______.

A．That's right B．Bad luck C．Sorry D．You can forget it

16．John is the tallest boy in the class, ___________ according to himself.

A．five foot eight as tall as B．a(chǎn)s tall as five foot eight

C．a(chǎn)s five foot eight tall as D．a(chǎn)s tall five foot eight as

17．Great changes have taken place in that school. It is no longer ______ it was 20 years ago, _______ it was so poorly equipped.

A．what; when B．that; which C．what; which D．which; that

18．He is such a man who is always ________ fault with other people.

A．putting B．seeking C．finding D．looking for

19．I really can' t understand ______her like that.

A．you treat B．you to treat C．why treat D．you treating

20．-- Maggie had a wonderful time at the party.

-- ______ and so did I.

A．So she had B．So had she C．So she did D．So did she

第二節(jié)：完形填空(共20小題；每小題1.5分，滿分30分)

閱讀下面短文，從短文后所給各題的四個(gè)選項(xiàng)(A、B、c和D)中，選出可以填入空白處的最佳選項(xiàng)，并在答題卡上將該項(xiàng)涂黑。

I did very badly at school. My headmaster thought I was 21 and when I was 14 he said, "You’re never going to be 22 but a failure. "

After five years of 23 jobs, I fell in love with a very nice middle-class girl. It was the best _24 that could have happened to me. 25 I wanted to do something positive with my life because I wanted to prove to 26 that what people said about me was 27 . Especially her mother, who had said to me, "Let's 28 it, you've failed at everything you've ever done. " So I tried hard with my 29 and went to college. My first novel (小說) 30 while I was at college.

After college I taught during the 31 in high schools and attended evening classes at London University, where I got a 32 in history. I became a lecturer at a college and was thinking of 33_ that job to write full time 34 I was offered a part-time job at Leeds University. I began to feel proud of myself -- 35 was a working-class boy who'd 36 school early, now teaching at the university.

My writing career (職業(yè)) took off when I discovered my own style. Now I' m rich and 37 , have been on TV, and met lots of film stars. 38 what does it mean? I 39 wish all the people that have put me down had _40 : "I believe in you. You'll succeed."

21．A．bright B．useless C．simple D．hopeful

22．A．a(chǎn)nything B．something C．everything D．nothing

23．A．low B．poor C．good D．useful

24．A．support B．happiness C．surprise D．thing

25．A．a(chǎn)dmitted B．decided C．planned D．told

26．A．me B．them C．her D．it

27．A．wrong B．right C．stupid D．faulty

28．A．see B．know C．understand D．face

29．A．experiment B．practice C．writing D．composition

30．A．came on B．came in C．came out D．came back

31．A．day B．night C．month D．year

32．A．graduation B．pass C．degree D．success

33．A．giving in B．giving back C．giving out D．giving up

34．A．while B．if C．when D．or

35．A．there B．here C．it D．that

36．A．left B．a(chǎn)ttended C．changed D．graduated

37．A．tired B．calm C．nervous D．famous

38．A．And B．But C．However D．Well

39．A．just B．exactly C．so D．very

40．A．praised B．said C．a(chǎn)nswered D．a(chǎn)dvised

第二部分閱讀理解（共25小題，第一節(jié)每小題2分，第二節(jié)每小題1分；滿分45分）

第一節(jié)，閱讀下面短文，從每題所給的四個(gè)選項(xiàng)（A、B、C和D）中，選出最佳選項(xiàng)，并在答題卡上將該項(xiàng)涂黑。

A

We were on tour a few summers ago, driving through Chicago, when right outside of the city, we got pulled over. A middle-aged policeman came up to the car and was really being troublesome at first. Lecturing us, he said, "You were speeding. Where are you going in such a hurry?" Our guitarist, Tim, told him that we were on our way to Wisconsin to play a show. His way towards us totally changed. He asked, "Oh, so you boys are in a band (樂隊(duì))?" We told him that we were. He then asked all the usual broad questions about the type of music we played, and how long we had been at it. Suddenly, he stopped and said, "Tim, you want to get out of this ticket, don't you?" Tim said, "Yes." So the officer asked him to step out of the car. The rest of us, inside the car, didn't know what to think as we watched the policeman talk to Tim. Next thing we knew, the policeman was putting Tim in the back of the police car he had parked in front of us. With that, he threw the car into reverse(倒車), stopping a few feet in back of our car. Now we suddenly felt frightened. We didn't know if we were all going to prison, or if the policeman was going to sell Tim on the black market or something. All of a sudden, the policeman's voice came over in a loudspeaker. He said, "Ladies and gentlemen, for the first time ever, we have Tim here singing on Route 90.” It turned out the policeman had told Tim that the only way he was getting out of the ticket was if he sang part of one of our songs over the loudspeaker in the police car. Seconds later, Tim started screaming into the receiver. The policeman enjoyed the performance, and sent us on our way without a ticket.

41．The policeman stopped the boys to ______.

A．put them into prison B．give them a ticket

C．enjoy their performance D．a(chǎn)sk some band questions

42．The policeman became friendly to the boys when he knew they ____

A．had long been at the band B．played the music he loved

C．were driving for a show D．promised into a performance

43．The boys probably felt ______ when they drove off.

A, joyful B．calm C．nervous D．frightened

B

Collections were the inspiration(靈感) for a project at Thomas Tallis School, which formed part of the Imagine Children's Literature Festival last autumn. Each child (aged 12-13) beatified （美化）a box and wrote a story on the subject of collections to throw inside it. The boxes were spread within the Royal Festival Hall's Ballroom. Some were left empty to encourage visitors to write their own stories.

The subject chosen by Luren was an imaginative one. "It's a sort of Cinderella (灰姑娘) story," she told me, inspired by a collection of letters from her cousin, in the story these become love letters, burned by a cruel stepmother. Lauren's best friend Charlotte is the stepmother. "I'm in Charlotte's story too," says Lauren, "and I get run over." Charlotte's tale was inspired by the girls' coin collection. "We've collected foreign coins for years ? since our families went on holiday to Tenerife." she explains. "That was before the Euro, so we put pesetas in." Lauren continues: "I find a coin in the road, go to get it and get run over. I'm in hospital and then I die." Charlotte adds: "Or she might not die. I haven't decided yet."

Millie Murray, who is a teen-novel author, thinks that setting the subject of collections was a useful inspiration to their creativity rather than a restriction(限制). "In the beginning I thought, 'Will the children be able to do it?'" she says. "But it's been fruitful. Some have their own collection, some have parents who do, and some have written complete stories. It's made them think about something they wouldn't have otherwise, which can only be a good thing."

44．What were the children asked to do in the project?

A．To meet friends at Thomas Tallis School

B．To write stories on the subject of collections.

C．To encourage visitors to write their own stories.

D．To have their friends for characters in the stories.

45．The underlined word "pesetas" in Paragraph 2 is a kind of _______.

A．Charlotte hurt herself when getting a coin

B．both of them developed their imagination

C．both of them will die in each other's stories

D．Latwen's cousin posted her some love letters

46．From the stories by Lauren and Charlotte, we know that _____ .

A．Charlotte hurt herself when getting a coin

B．both of them developed their imagination

C．both of tram will die in each other's stories

D．Latwen's cousin posted her some love letters

47．Millie Murray thinks ________.

A．collections could inspire writing creativity

B．it was good for parents to have collections

C．inspirations were very useful in writing stories

D．setting collection subjects restricted inspirations

C

Paula Radcliffe, chasing (角逐) a third London marathon title(冠軍), says she has become a stronger person after her terrible experience at the 2004 Athens Games.

Radcliffe, who failed to complete the Olympic marathon and the 10,000m last August, said: "Athens made me a stronger person and it made me care less about criticism (批評(píng))"

"In the past I wanted to please everyone, but now I am going to listen even more to the people around me."

She didn't care about criticism made at the weekend by Liz McCulgan, who felt Radcliffe should have rested and let her body recover after her failure in Athens.

"Liz is someone I look up to but she hasn't spoken to me since last year and if she really cared for me, I'm sure she would have contacted (聯(lián)系) me."

Instead Radcliffe won the New York City marathon just 11 weeks after Athens.

"In New York I wasn't in my best state but I did know I was good enough to win the race. Radcliffe insisted her only goal in Sunday's race would be winning a third title and not chasing world records.

However, RadcIiffe has not ruled out(排除) in the future chasing her "final" world record time and questioned sayings that marathon runners have the ability in their career to produce only four or five world-class times.

"I don't think that -- although I can't put a number on it," said RadcLiffe. "That changes from person to person."

Radcliffe is sure she can better her winning London 2003 performance at some point in the future. Following a successful three-month training period in the United States, the 31-year-old will chase a third title on Sunday after her first victory in 2002 and again 12 months later.

Radcliffe clocked a time of 2:18:56 in her first 42.2-kilometre race three years ago.

Afterwards she set a "mixed course" mark of 2:17:18 five months later in Chicago before lowering that to a time of 2:15:25 in the 2003 London event.

48．Radcliffe's failure in Athens made her ________

A．develop respect for Liz B．love people around her more

C．rest for five months D．face criticism calmly

49．Which of the following is true according to the passage?

A．Radcliffe broke the world record in the New York City marathon.

B．Radcliffe didn't fully recover before the New York City marathon.

C．Radcliffe won her first marathon title in the New York City marathon.

D．Radcliffe had a 3-month training before the New York City marathon.

50．By saying "I can't put a number on it," Radcliffe means she's not sure____.

A．if she has the ability to set a new world record

B．if she can win another race though she has won many tunes

C．how many times a marathon runner can set the world record

D．if she has the ability to produce four or five world-class times

51．According to the text, Redcliffc has won ____ London marathon title(s).

A．four B．three C．two D．one

52．What can we learn from Radcliffe's story?

A．Practice makes perfect.

B．Well begun is half done.

C．A friend in need is a fried indeed.

D．Where there is a will there is a way.

D

From Mr. Ward Hoffman.

Sir, I was halfway through Professor Raj Persaud's article “What's the tipping point"

(Financial Times Weekend, April 9-l0) when it occurred to me that what I was reading was not ironic(諷刺的). If Prof Persaud wants to know why Americans tip in restaurants, he needs only to ask the first American he meets in London.

Americans tip(小費(fèi)) in restaurants for one reason, and one reason only: we tip to supplement (補(bǔ)貼) the salary of restaurant workers. Quality of service does not enter into it, beyond the fact that one may tip a bit less for poor service, or a little more for good service.

Not tipping at all in a non-fast-food restaurant is not a choice. In the US, one used to tip about 15 per cent for dining in a family-style restaurant or in an up-market (高檔的) restaurant. Here, in San Francisco Bay area restaurants, we are encouraged to tip 20 per cent or more, to help restaurant workers live in this very expensive area.

After eating at an Italian restaurant in my city, I left a tip of 20 per cent on the non-tax part of our dinner bill. It was expecteD．There is nothing more complicated (復(fù)雜的) than that about Americans tipping in restaurants.

Ward Hoffman,

Palo Alto, CA 94306, US

* * *

From Mr. Philip McBride Johnson.

Sir, I agree with most of Raj Persaud's opinion about the doubtful value of tipping, but with one exception(例外). Tips can be very useful when one is a repeat customer or diner.

It is only when the tipper is a stranger and likely to remain so that the system does not work to his or her advantage. But frequent a hotel or a restaurant, always tip a bit more, and the difference in service and treatment will be easily felt.

Phfiip McBnde Johnson,

Great Falls, VA 22066, US

53．What can we learn from Hoffrnan's letter?

A．Quality of service determines tipping in the US.

B．Americans don't tip in non fast-food restaurants.

C．Tipping in US market restaurants is unnecessary.

D．How to tip in the United States is not complicated.

54．Johnson's letter shows ________.

A．a(chǎn) stranger in a restaurant is likely to tip a bit more

B．diners receive better service if they frequent a restaurant

C．repeat diners may get good service if they tip a bit more

D．the tipping system works to the advantage of new customers

55．From the two letters, we can learn Professor Raj Persaud ______ .

A．feels doubtful about the value of tipping

B．believes tipping improves quality of service

C．wants to ask Hoffman about tipping m the US

D．thinks tipping a bit mom one can get good service

56．The two letters most probably appears in a ______.

A．notice B．handbook

C．book review D．newspaper

E

At Dallm/Fort Worth Airport, the lights are controlled by sensors(傳感器) that measure sunlight. They dim immediately when it's sunny and brighten when a passing cloud blocks the sun.

A wall of windows at a University of Pennsylvania engineering budding has built-in blinds (百頁(yè)窗) controlled by a computer program that follows the sun's path.

Buildings are getting smarter -- and the next generation of building materials expected to do even more.

Windows could catch the sun's energy to heat water. Sensors that measure the carbon dioxide breathed out by people in a room could determine whether the air conditioning needs to be turned up.

Many new materials and technology have been designed in the last 15 years. They are now being used in a wave of buildings designed to save as much energy as possible. They include old ideas, like "green roofs," where a belt of plants on a roof helps the building keep heat in winter and stay cool in summer, and new ideas, like special coating for windows that lets light in, but keeps heat out.

As technologies such as sensors become cheaper, their uses spread.

The elevators (電梯) at Seven World Trade Center, which is under construction in New York, use a system that groups people traveling to nearby floor into the same elevator, thus saving elevator stops. People who work in the building will enter it by swiping (刷) ID cards that will tell the elevators their floor; readouts will then tell them which elevator to use. The building also has windows with a coating that blocks heat while letting in light.

More new building materials and technology are in development. A Philadelphia building firm is now working on "smart wrap" that uses tiny solar collectors to catch the sun's energy and transmitters (傳輸器) the width of a human hair to move it. They are expected to change the face of the construction industry in the next ten years or so.

57．_______ will be developed and used in the construction industry.

A．"Green mops" that cool or heat buildings

B．"Smart wrap" that catches the sun's energy

C．Sunlight-measuring sensors that control lights

D．Window coating that lets light in, but keeps heat out

58．The elevators at Seven World Trade Center are special because they can ______.

A．send people to floors with fewer stops

B．teach people how to use their ID cards

C．make people stay very cool in summer

D．help people go traveling in the building

59．The underlined word "it' in the last paragraph refers to _____.

A．a(chǎn) human being B．smart wrap

C．the sun's energy D．a(chǎn) transmitter

60．What might be the most suitable title for the text?

A．Buildings Are Becoming Smarter

B．Buildings Are Getting More Sunlight

C．Buildings Are Lacking in Much Energy

D．Buildings Are Using Cheaper Materials

(二)根據(jù)對(duì)話情景和內(nèi)容，從對(duì)話后所給的選項(xiàng)中選出能填入每一空白處的最佳選項(xiàng)，并在答題卡上將該選項(xiàng)涂黑。選項(xiàng)中有兩個(gè)為多余選項(xiàng)。

(注意：61----65填涂到答題卡71----75處)

Sandy is at home. She is now calling Lisa, her classmate.

Sandy: Hi, Lisa. Have you finished helping your parents?

Lisa: I finished cleaning the living room a little while ago, but I haven’t cleaned my bedroom yet. 61

Sandy: I know what you mean. 62 Do you think you’ll be finished soon?

Lisa: I should be done in about an hour.

Sandy: 63

Lisa: I did the English homework last night. 64 I don’t understand it.

Sandy: Me neither. 65_ We can help each other work it out.

Lisa: OK. Then we can go and have our hair cut.

Sandy: Great. See you in a little while.

A．I don’t like math at all.

B．But I haven’t started my math yet.

C．Housework is tiring, and I’m tired

D．What do you think we should do then?

E. Why don’t we do it together this afternoon?

F. Have you looked at Monday’s homework yet

G. I also hate doing the cleaning around the house.

第Ⅱ卷（非選擇題,共55分）

第三部分寫作（共三節(jié)，滿分55分）

第一節(jié) 單詞拼寫（共10小題, 每小題1分，滿分10分）

66．______(十二月) is the twelfth month of the year. 66._____________

67．Now we have a growing population of senior______(公民). 67._____________

68．They are ____ (決心的) to solve problems of common people. 68._____________

69．My brother often comes to see me,_____(尤其) when he is free. 68.___________

70．A lovely bird______(飛)past me just now. 70._____________

71．Operated on in time, the ______(病人) is likely to recover soon. 71.___________

72．What he said ____(聽起來) so strange that I couldn’t believe it! 72.___________

73．Well, you are the______ (二十), that is , the last to come. 73._____________

74．I must ______(道歉) for not having written to you sooner. 74._____________

75．In summer, a sudden change in ______(溫度) could bring rain. 75.___________

When I was four years old , I got ill. I took medicine twice

a day. The medicine had so a bitter taste that I took it mixed in 76．________

orange juice . The problem was that I can still taste the medicine. 77．________

In a particular afternoon my mother brought the drink in. As 78．________

she gave it to me, when the phone rang and she went to answer 79．________

it. I looked at the orange drink and decided I couldn’t face them, 80．________

so I hid the glass behind a pile magazines . Of course, when 81．________

my mother asked, “ Have you already taken your medicine ? ” 82．_______

I said, “Yes. ” I felt very pleased with me. The next day, my 83．________

mother found drink when she was doing the housework. She 84．________

looked at me serious and said, “ It’s bad to tell lies ! ” 85．________

第二節(jié)：書面表達(dá)（滿分30分）

假設(shè)2008年2月12日是你父親的生日，下面三幅圖描繪的是你給父親買完禮物后乘坐地鐵回家時(shí)經(jīng)歷的一件事。請(qǐng)根據(jù)圖示，用英語(yǔ)寫一篇日記記敘這件事，并談?wù)勛约旱母惺堋?/p>

注意：

1．詞數(shù)不少于100；

2．可適當(dāng)加入細(xì)節(jié)，以使行文連貫。

參考詞匯：

地鐵（列車）----- subway train

February 12^th, 2008

________________________________________________________________________________________________________________________________________________________

_______________________________________________________________________________

試題詳情

章節(jié)復(fù)習(xí) 坐標(biāo)系

[教學(xué)目標(biāo)]

試題詳情

《數(shù)學(xué)實(shí)驗(yàn)：擺線探究》教學(xué)設(shè)計(jì)

教學(xué)目標(biāo)

知識(shí)目標(biāo)：

在數(shù)學(xué)實(shí)驗(yàn)平臺(tái)中，觀察圓在直線上滾動(dòng)時(shí)圓上定點(diǎn)的軌跡(平擺線)，了解平擺線的生成過程并能推導(dǎo)出參數(shù)方程．

了解動(dòng)圓在定圓上滾動(dòng)時(shí)形成的擺線（外擺線、內(nèi)擺線、環(huán)擺線)的生成過程，了解三種擺線的內(nèi)在聯(lián)系及相應(yīng)分類，會(huì)在具體刻畫中對(duì)變幅擺線進(jìn)行簡(jiǎn)單的分類．

了解擺線在實(shí)際中應(yīng)用的實(shí)例．

能力目標(biāo)：

提高學(xué)生的信息技術(shù)素養(yǎng)，提高學(xué)生的觀察能力和研究能力

情感目標(biāo)：

培養(yǎng)學(xué)生主動(dòng)探求知識(shí)、合作交流的意識(shí)，在學(xué)生的親身操作中，感受曲線的美感受，數(shù)學(xué)的力量，改善數(shù)學(xué)學(xué)習(xí)信念．

教學(xué)過程：

一、平擺研究

1、情境創(chuàng)設(shè)

問題：如圖，一個(gè)人的自行車外帶上沾了一點(diǎn)白色油漆，當(dāng)他騎車向前直行時(shí)，這個(gè)白色油漆斑點(diǎn)在空中會(huì)描出一條什么樣的曲線？

引導(dǎo)學(xué)生拖動(dòng)點(diǎn)C，使車輪在地面上滾動(dòng)，觀察點(diǎn)P的軌跡；學(xué)生會(huì)發(fā)現(xiàn)，點(diǎn)P的軌跡是一條以前沒有研究過的曲線．從而引入主題，即：一個(gè)圓沿著一條直線作無滑動(dòng)的滾動(dòng)時(shí)，圓周上的一個(gè)定點(diǎn)的軌跡叫做平擺線，又叫旋輪線.．

操作提示：先按【滾動(dòng)】或直接拖動(dòng)點(diǎn)C，使得圓在直線上無滑滾動(dòng)，初步認(rèn)識(shí)、猜想點(diǎn)P軌跡的形狀；繼而【追蹤】，直觀刻畫點(diǎn)P的軌跡形狀，

2、方程探究

通過對(duì)平擺線的直觀研究，可以發(fā)現(xiàn)平擺線的一些性質(zhì)如：圖象是由一些呈周期性排列的拱組成，每個(gè)拱的拱高為2r，拱底長(zhǎng)為2πr．引導(dǎo)學(xué)生認(rèn)識(shí)到進(jìn)一步研究平擺線需要研究其曲線方程．

以問題：“如何才能實(shí)現(xiàn)動(dòng)圓在直線上無滑滾動(dòng)呢？”為主線引導(dǎo)學(xué)生在實(shí)驗(yàn)平臺(tái)中探究，學(xué)生容易發(fā)現(xiàn)在滾動(dòng)過程中保持線段AC及弧長(zhǎng)的長(zhǎng)度相等，從而進(jìn)一步嘗試方程的推導(dǎo)如下：

設(shè)是軌跡上任一點(diǎn)，

則

那么，從而得出相應(yīng)的軌跡參數(shù)方程．

操作提示：按【比較】會(huì)顯示線段AC及弧長(zhǎng)的動(dòng)態(tài)度量值，從而能幫助學(xué)生認(rèn)識(shí)其中的規(guī)律．

3、平擺變幅

在問題：“P點(diǎn)的軌跡是平擺線，若直線OP上另有一點(diǎn)Q，那么圓在直線上作無滑滾動(dòng)時(shí)，Q點(diǎn)的軌跡如何呢？” 的指引下，引導(dǎo)學(xué)生探究相應(yīng)的軌跡，從而得出下列結(jié)論：一個(gè)圓沿著一條直線作無滑動(dòng)的滾動(dòng)時(shí)，圓周上的一個(gè)定點(diǎn)的軌跡叫做平擺線，圓內(nèi)一個(gè)定點(diǎn)的軌跡叫做短幅平擺線；圓外一個(gè)定點(diǎn)的軌跡叫做長(zhǎng)幅平擺線．

操作提示：按紐【軌跡】會(huì)給出點(diǎn)Q的軌跡，拖動(dòng)點(diǎn)Q能幫助學(xué)生初步對(duì)變幅擺線形成動(dòng)態(tài)印象；按紐【比較】會(huì)給出兩條典型的短幅擺線和長(zhǎng)幅擺線，再次拖動(dòng)點(diǎn)Q可以幫助學(xué)生形成整體動(dòng)態(tài)認(rèn)識(shí)，從而能認(rèn)識(shí)到可以根據(jù)點(diǎn)Q相對(duì)于圓的位置對(duì)變幅擺線加以分類．

二、拓展研究

1、外擺線探究

有了平擺線的研究經(jīng)歷和基礎(chǔ)，學(xué)生對(duì)類似情境“動(dòng)圓在定圓外滾動(dòng)會(huì)得到什么樣的曲線呢”的研究會(huì)產(chǎn)生極大的興趣．學(xué)生在如圖所示的實(shí)驗(yàn)平臺(tái)中容易發(fā)現(xiàn)，軌跡是一條與平擺線相似的曲線，從而引出探究主題，即動(dòng)圓在定圓外無滑滾動(dòng)時(shí)，動(dòng)圓圓周上的一個(gè)定點(diǎn)的軌跡是外擺線．

操作提示：與平擺線的研究相類似，學(xué)生在外擺線的探究上也須經(jīng)歷一個(gè)由感性認(rèn)識(shí)到理性升華的過程，先讓學(xué)生在【滾動(dòng)】中猜想外擺線的形狀，通過【追蹤】驗(yàn)證直覺判斷，繼而通過【比較】度量值的大小，從而對(duì)動(dòng)圓在定圓外無滑滾動(dòng)有所感悟，從而對(duì)外擺線有個(gè)全面而深刻的認(rèn)識(shí)．

2、擺線拓展

在前面探究的基礎(chǔ)上，改變相應(yīng)的R、r的值，可以得到不同類型的擺線模型，初步認(rèn)識(shí)到擺線的形狀取決于R、r的比例．

操作提示：選中參數(shù)R或r的度量值，按動(dòng)小鍵盤上的“+”或“-”改變相應(yīng)參數(shù)的值，可以發(fā)現(xiàn)擺線的形狀發(fā)生變化，當(dāng)r值為負(fù)時(shí)表示動(dòng)圓在定圓內(nèi)滾動(dòng)；拖動(dòng)半徑r的兩端點(diǎn)可以改變圖形的大小，從而能在屏幕上得到合適的顯示．

3、擺線全景

在實(shí)驗(yàn)平臺(tái)中同時(shí)顯示四個(gè)圖，目的是讓學(xué)生對(duì)擺線形成整體認(rèn)識(shí)，即擺線包括三種類型：外擺線、內(nèi)擺線、環(huán)擺線，并在比較中歸納出擺線的分類（如下表，根據(jù)R、r的相對(duì)比例）：

操作提示：按動(dòng)按紐【選中R】將選中全部四個(gè)圖中參數(shù)R，這樣再按小鍵盤上的“+”或“-”就可以同時(shí)改變四個(gè)圖中的參數(shù)了，從而可以地實(shí)現(xiàn)對(duì)擺線的整體研究，按動(dòng)按紐【選中r】也有相類似的效果．

在R或r的值過大的情況下，對(duì)應(yīng)的圖形也會(huì)超出相應(yīng)的范圍，這樣不利于觀察圖形的特征，可以拖動(dòng)綠色點(diǎn)（與紅色點(diǎn)接近圖形縮小，反之則圖形放大）后再按按紐【統(tǒng)一長(zhǎng)度】即可解決問題．

4、擺線變幅

在順利研究了擺線的分類后，與平擺線的變幅相類似，學(xué)生在問題“P點(diǎn)的軌跡是擺線，若直線OP上另有一點(diǎn)Q，那么動(dòng)圓P在定圓O外（或內(nèi)）作無滑滾動(dòng)時(shí)，Q點(diǎn)的軌跡如何呢”的引領(lǐng)下，很自然地會(huì)想到研究擺線的變幅．

在相應(yīng)數(shù)學(xué)實(shí)驗(yàn)平臺(tái)的支持下，學(xué)生容易認(rèn)識(shí)到：當(dāng)Q點(diǎn)在圓P上時(shí)，Q點(diǎn)的軌跡是擺線；當(dāng)Q點(diǎn)在圓P內(nèi)時(shí)，Q點(diǎn)的軌跡是短幅擺線；當(dāng)Q點(diǎn)在圓P外時(shí)，Q點(diǎn)的軌跡是長(zhǎng)幅擺線．

操作提示：在顯示【軌跡】的基礎(chǔ)上，拖動(dòng)點(diǎn)Q，隨著點(diǎn)Q相對(duì)于動(dòng)圓的位置不同，將產(chǎn)生不同的變幅擺線；同時(shí)也可改變參數(shù)R或r，觀察不同類型的擺線的變幅，感受其異同之處，對(duì)擺線的變幅形成整體認(rèn)識(shí)．

5、哥白尼定理

在擺線全景中，學(xué)生會(huì)發(fā)現(xiàn)特殊情形下的擺線會(huì)成為一條直線，從而產(chǎn)生探究興趣．動(dòng)圓P在定圓O內(nèi)無滑滾動(dòng)且R=2|r|時(shí)，動(dòng)圓上一個(gè)定點(diǎn)的軌跡是定圓的一條直徑（哥白尼定理）．根據(jù)哥白尼定理，可以把旋轉(zhuǎn)運(yùn)動(dòng)變成往返的直線運(yùn)動(dòng)，這一定點(diǎn)機(jī)械設(shè)計(jì)上是很有用的．上述條件下，動(dòng)圓所在平面內(nèi)與動(dòng)圓固定地連接在一起的圓內(nèi)（或圓外）一點(diǎn)的軌跡是橢圓（卡丹轉(zhuǎn)盤）．