最新人教版高三英语选修9 Unit 4 Exploring plants 全单元教案

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高中英语 Unit4Exploring plants-Writing课件新人教选修9

h
6
Third listening
Listen to the first part again and complete the table of conditions that each dish of seeds was given.
h
7
Conditions
Dish A tap water, warm temperature,
TH: Right, let’s talk about what’s
happened in our experiment. What
are the results? Andrew?
A: The seeds in Dish A have started
growing but none of the others have.
3
1
5
4
2
h
3
Listen to the first part again and answer the questions.
1. What is the purpose of the experiment? 2. To find out three important
conditions that seeds need in order to start growing.
h
9
Did the Reasons why or seeds grow? why not
Yes Dish A
Dish B
No
Because they had water, warmth and oxygen.
Because it was too cold.
h
10
Did the Reasons why or why not

Unit 4 Exploring plants-Listening教学课件(人教选修9)

M: Sure. Today, I want to talk about an insect-eating plant called a pitcher plant.
Here’s a photo of it. Look at this part here. It’s shaped like a pitcher. It’s actually a leaf even though it looks like a flower. It’s really clever---the insects are attracted by the sweet smell coming from what they think is a flower. But it’s a trap.
4. In 40 years’ time we might not have
any rainforests left on earth. T
5. People can make more money from harvesting rainforest plants than from
cutting down the rainforests. T
E: Anne, we hear a lot these days about how quickly tropical rainforests are disappearing.
LISTENING TEXT
THE DESTRUCTION OF TROPICAL RAINFORESTS Ed Walker (E) is interviewing Dr Anne Butler (A) about the destruction of tropical rainforests.

人教新课标英语选修9Unit4Exploringplants--Section1教案

Unit 4 Exploring plantsPart 2 Teaching Resources第二部分教学资源Section 1 Backgrounds for Unit 4 Exploring plants1.Importance of PlantsPlants form the basic food staple for all life forms. They are the major source of food and oxygen on earth, since no animal can supply these necessary components without plants. The cattle we eat as beef, feed on grasses and the fish we eat, consume algae and are therefore dependent on plants for well being. Other important uses of plants include, providing shelter for animals, providing materials for clothing (cotton fibers), paper products, medicines and other chemicals, producing coal from once living plant material, reducing wind speed and noise levels, and reducing soil erosion and water runoff.2. The life cycle of a plantThe life cycle of a plant varies depending on the individual species. There are, however, certain requirements for life that most plants need. The growth of a plant is dependent upon light, water, oxygen, carbon dioxide, minerals in the soil, temperature and microbes in the soil. Light, water, and carbon dioxide are needed for photosynthesis which produces food for the plant. Oxygen is needed when it is dark, because the plant then needs oxygen to maintain itself.。

高中英语 Unit4Exploring plants-Listening课件新人教选修9

Look carefully at the photos of the plants and answer the questions.
living stone
Rafflesia
pitcher plants
Pitcher plant
Now listen to Mark, Gordon and Andrea telling their class about these plants. Which student talks about which plant?
5. People can make more money from harvesting rainforest plants than from cutting down theT rainforests.
LISTENING TEXT
THE DESTRUCTION OF TROPICAL RAINFORESTS Ed Walker (E) is interviewing Dr Anne Butler (A) about the destruction of tropical rainforests.
Expressions of disgust:
Oh, yuck! That’s disgusting!
Listening text
UNUSUAL PLANT
TH: Today Mark, Gordon and Andrea are going to tell us about some unusual plants. Mark, do you want to go first?
Tropical rainforests occur in three major regions: Asia, Africa and Central and South

人教版选修九unit4《exploring plants》教案(重点资料).doc

英语：Unit4《Exploring plants》教案（1）（新人教版选修9）教学内容： Plant exploration in the 18th and 19th centuries语言材料分析：本册书针对的是高三学生。

本节为本单元第二课时，主要介绍了欧洲18和19世纪的植物探险。

本文简要地介绍一些著名的探险家和他们所取得的成果。

文章属于总─分的结构。

通过本文的学习，学生可以扩大视野，了解一些植物探险方面的知识。

但是这篇文章篇幅上很长，信息量很大对学生理解该文造成了一定的难度。

因此，在本课前要考虑这一问题，帮助高三学生提高阅读速度和对细节信息的分析。

Teaching goals:nguage target:A. Key words and phrase:exotic, conflict, container, restriction, date back to, appeal to B. Key sentencesa. Collecting “exotic” plants, as they are called, dates bake to the earliest times.b. However, it was not until the eighteenth and nineteenth centuries that the exploration of the botanical world began on a large scale.c. One of the collectors was Father Farges, who collected 37 seeds from a tree that has appealed to him.d. The European classes took great interest in collecting plants.e. Large numbers of seeds failed to grow after long sea voyages.2.Ability goalsA.Students get some information on plant exploration.B.Students get the main idea, find and infer information accordingto the reading material.C.Students express their ideas after reading.D.Students help, cooperate and communicate with each other.Teaching important pointsTo enable students to understand the reading material by using predicting, skimming, scanning, analyzing, summarizing and inferring after reading.Teaching difficult points1. To enable students make use of given information to express their opinions on the passage.2. Help students observe language points and prepare for the grammar revision.Teaching methodsTask-based learning approach, competition, group work, discussionTeaching aidsMulti-media computer platform, group cardsTeaching proceduresStep 1. Lead-in (3’)A. Divide the whole class into small groups.B. A video clipFocus: To arouse students’ interest and give the task to students. Step 2. Pre-reading (3’)Predict according to the title.The author may tell us _____ in this passage.A. some plants from other countriesB. the events happening in plant explorationC. how to explore plantsD. famous plant collectorsFocus: Students can make prediction after observing the title carefully.Step 3. While readingPart 1. Fast readingRead for the main idea and solve the task given in pre- reading. (Students work in groups and work out the questions).Part 2. Detailed reading (20’)(Students write down the words, phrases and sentences they can’t understand and work in groups to solve them and work out the multiple choices questions).Task 1. Scanning1. Plant collectors met many difficulties except_______.A. diseases and near-starvationB. the conflicts with local peopleC. restrictions on the movement of EuropeansD. lack of money2. Who is not a collector mentioned in the passage?A. Father d’Incarville.B. The Queen of England.C. Robert Fortune.D. E H Wilson.3. The following plants were taken to Europe except_______.A. the tree of HeavenB. the Dove TreeC. the tea plants4.The first recorded plant exploration happened _______.A. in the 18th centuryB. about 1500 years agoC. about 3500 years agoD. about 2500 years agoFocus: To develop students’ability to search for detailed information.Task 2. Analyzing1. The passage is organized in the order of ______.A. timeB. placeC. characterFocus: To develop students’ ability to analyze the whole passage.Task 3. Inference1. What do you think the author will talk about if he adds a paragraph to this passage?A.The effects of the plant exploration in the 18th and 19thcenturies.B. The plant exploration in the 20th century.rmation about the Dove Tree.Focus: To help students read beyond the reading material and make their own judgment according the reading material.Task 4. Language observation1. What is the meaning of the underlined part in the sentence “Oneof the collectors was Father Farge, who collected 37 seeds froma tree that had appealed to him”?A. that surprised himB. that he had never seen beforeC. that attracted himD. that he thought was very valuable2. The earliest record of plant exploration in history dates from 1500 BC.The earliest record of plant exploration in history ____ ____ ____1500 BC.3. People didn’t begin to explore plants on a large scale until the 18th and 19th centuries.It was ____ ____ the 18th and 19th centuries that people began to exploreplants on a large scale.4. The parts in red are used as______ in the sentences.A. the subjectB. the objectC. the predicative1.He collected seeds of trees.2. French Catholic missionaries were beginning to set themselves up in China.3. The European classes took great interest in collecting plants.4. Large numbers of seeds failed to grow after long sea voyages.5. … often we do not realise that many of them come from countries far away.Focus: To help students infer the meanings according to the context. To help students observe the feature of the object and prepare for the grammar learningStep 4. Post-readingHelp Liang Qiming find out who made the greatest contribution in plant exploration in the 18th and 19th centuries.1.Discussing in groups2.Giving opinions.Step 5. Assignments1.Try to add a short paragraph as the end of the passage.2. Get some information about plant exploration in the 20th and 21st centuries on the Internet.。

高中英语(人教版)选修九 Unit4 Exploring plants同步课件教案课件教案和测试(参考)教案U4S2

Unit 4 Exploring plantsPart 2 Teaching Resources第二部分教学资源Section 2 Notes to difficult sentences from Unit 4Exploring plants1．Collecting “exotic” plants，as they are called，dates back to the earliest times．收集“异国”植物，人们这样称呼他们，可以追溯到远古时期。

句中as they are called是非限制性定语从句，as是关系代词，起连接作用，同时作从句中的主语补语。

请看关系代词as的其它例句：I received the same grade as you did. Those as want to can come with me. I received the same grade as you did. Those as want to can come with me. 2．However，it was not until the eighteenth and nineteenth centuries that the exploration ofthe botanical world began on a large scale．然而，直到十八、十九世纪植物学世界的探险活动才开始大规模进行。

句中it was not until…that…意思是：在...以前一直(不...),...以后才。

例如：It was not until he was thirty that he started to paint. 他直到三十岁才开始绘画。

3．Large numbers of seeds failed to grow after long sea voyages or trips across land between Asia and Europe. 大量的种子在跨越亚洲和欧洲之后没能成功地生长。

Unit 4 Exploring plants-Reading task教学课件(人教选修9)

The female wasps visit the orchid flowers to collect the chemical that the male wasps were attracted to earlier. The female wasps do not have a role in pollination since the flowers are dying when they visit them.
The Traveller’s palm and the Madagascar black lemur:
If the Traveller’s palm died out, the Madagascar black lemur would not die out as it can find other sources of food.
2.Can you guess why the Traveller’s Palm has that name?
The Traveller’s Palm has this name because travellers used to drink its nectar when they were thirsty.
Discuss the question: What’s the relationship between plants and animals?
Pre-reading
Before you read the text, glance quickly through it and answer these questions.
s
flower can be disadvantage for
palm
opened by the the palm only if

人教版高三英语选修9 Unit 4 Exploring plants 全单元教案

Unit 4 Exploring plantsPeriod oneAim:To train the students’ ability of listening and speaking.Contents:Step I Warming upQ1 What do you know about plants?Q2 Do you know what kind of plant don’t have flowers?Q3 Do you know what kind of plant have roots that live in the air?Q4 Do you know what kind of plant need animals to pollinate them?Q5 Do you know what kind of plant grow on other plants?Q6 Do you know that people buy plants that come from other countries for their gardens?Q7 Do you know that a new plant does not always grow from a seed?Step II Listen and discuss:First give some photos of the plants and then ask students answer the following question.(Page 38)Q1 How would you describe each plant?Q2 What do you think is unusual about each plant?Q3 One of the pictures shows pitcher plants and another shows “Living stones”which picture shows which plant?Now listen to Mark，Gordon and Andrea telling their class about these plants.Which students talk about which plants?Listen again and complete the table with the correct information.Period twoAim:To make students understand the text.Plant exploring in the 18th And 19th centuries.Contents:Q1 What is each paragraph about?Q2 Write down the topic of each paragraph in your own words.Q3 What happened in these years? Complete the sentences on page 33.Q4 Answer there question in groups.1) Why was there a lot of plant collecting in the 18th and 19th centuries?2) How did French missionaries come to be plant collection?3) What were some of the negative aspects of plant collecting expeditions?4) Transporting plants over long distances was a big problem, why was this?5) Why did many plants fail to reach their destinations alive? Can you thinkof any other reasons?6) What important the survival rates of plants transported long distances? 0Period threeAim:To make students master the language points in this text.Contents:Phrases:1 date back to: go back to/date fromeg: This town dates back to the Rome time2 on a large scale3 take / find/ have/ show/ find interest in.4 take the opportunity of doing sth5 go on botanical expeditions6 come into conflicts withbe in conflicts with7 accompany sb: keep sb company8 on purpose / by design / by intentionwith the purpose of9 come acrossrun into / acrossmeet with10 keep sth alive11 make a trip to .12 dress as13 in…style14 result inlead tobring about15 be excited with16 have a vast variety of17 n ot only…but alsonot alone…but alsonot merely/ simply… but alsoA as well as BA andB as well18 distant adj. far away in space or time 远的，遥远的19 distance n. in the distancefrom the distanceat a distancekeep sb. a distance20 scale n. 1) balance for weighing磅秤2) relative size, extent, scope规模，程度on a large / small scale大/小规模on the scale of … to…按…的比例to a scale按一定比例to a scale of … to…按…的比例21 appeal to 1）恳求，恳请I am appealing on behalf of the refugees.我代表那些难民提出恳求He appeal to us for help.他恳求我们的帮助2）诉诸，诉之于Appeal to the law / a higher court3）事物引起（人的）兴趣，有吸引力，投入所为Does the film appeal to you?n. make an appeal to 吸引, 诉之于Sentence structures1.It was not until the eighteenth and nineteenth centuries that the explorationof the botanical world began on a large scale.2.Not only did Fortune introduce over 120 species of plants to western gardensbut he also shipped 20000 tea plants from…Period FourAim:To make students understand the textFlowers and their animal pollinatorsContents:I. Questions:1)What is the text about?2)What do the pictures show you?3)What is the chart about?4)What is the advantage for animal of visiting flowers?5)Who do some plants need animals?6)How does an animal pollinate a flower?7)What is nectar?8)What kinds of animals are most flowers pollinated by?II. Language points1.over time 随着时间的推移over the weekend 整个周末里over these years 在这些年里2.reward sb with sth用…回报某人3.attach… to依附于…上4.pass on to把…往下传5.adapt to 使…适应The spokesman adapts his speech to suit the interests of his audience.He has adapted to the climate here.6.describe … as 把…描述成7.guide sb to .把…领到…nd on 登陆9.provide sb. with sth向…提供provide sth for sb.provide for 养活，规定III. Difficult sentences1)Pollen becomes attached to the animals during its visit to a flower and is thenpassed on to another plant’s blossom on its next visit.动物接触一朵花后,花粉就附着在它身上,动物接触下一种植物的花时就把这些花粉传到花朵上.2)Through evolution, most flowers have become adapted to attract specific typesof pollinators.通过进化,大部分花朵变得适于吸引特殊种类的花粉传播者.。

高中英语 Unit4Exploring plantsListening课件新人教选修9

land surface.
F 6%
2. More than half of the rainforests
have disappeared. T
3. 150 acres of rainforest are destroyed every day. F Over 150 acres of rainforest are destroyed every minute..
4. In 40 years’ time we might not have any rainforests left on earth. T
5. People can make more money from harvesting rainforest plants than from cutting down the rainforests. T
S1: That’s amazing! S2: What fantastic plants! TH: Gordon. G: These are the plants I’m going to talk
about. They’re often called “living stones”, but even though they do look a bit like stones, they are in fact plants.
Listening
Listening on P36
1. How would you describe each plant? What do you think is unusual about each plant? 2. One of the pictures shows pitcher plants and another shows “living stones”. Which picture shows which plant? The name of the plant in the other picture is Rafflesia.

人教版英语选修九 unit 4 exploring plants reading(课件) (共57张ppt)

h结构： 1. 他躺在床上，门关着。 the door shut/closed He lay on the bed with___________________. 2. 汤姆不在，我一直感觉孤独。 With ___________, Tom away I always feel lonely. 3. 这个人双手被绑着被带进来了。 his hands tied The man was brought in with __________. 4. 有着太多问题要去处理，新经理担忧的吃不下任何东西。 many problems tonew settle Withso _________________, the manager was too worried to eat anything. 5. 机器开着，他开始工作了。 the machine running He started to work with _________________.
1. The castle dates back to the 14th
century. （某个时期）
2. My interest in NBA dates from my
schooldays. （一段时期）
date构成的短语 up to date / updated 最新的过时定下日子
out of date / outdated
expedition recorded in history was around 1500 BC when the Queen of Egypt sent ships away to gather plants, animals and other goods. plant collecting expedition ：收集植物的探险历史上有记载的第一个收集植物的探险是在公元前 1500 年，那时候埃及女王派出船只去收集植物、动物以及其他货物。

人教版高中英语选修9课件 Unit 4 Exploring plants-Reading task 课件

5. Why do the female wasps visit the orchid flowers? Do they have a role in pollination? The female wasps visit the orchid flowers to collect the chemical that the male wasps were attracted to earlier. The female wasps do not have a role in pollination since the flowers are dying when they visit them.
The Traveller’s palm and the Madagascar black lemur:
If the Traveller’s palm died out, the Madagascar black lemur would not die out as it can find other sources of food.
Discussion
Discuss what you think would happen to the animal pollinators of each of the plants described if these plants died out.
Suggested answers:
The yucca and the yucca moth: If yucca plants died out, the Yucca moth would also die out as the caterpillars need the yucca seeds for food.

人教版高中英语选修9课件 Unit 4《Exploring plants》课件

5. date back to vt. 回溯到(开始于；从……的时候存在)
例句: Our partnership dates back to 1960. 我们从1960年就合伙了。
This castle dates back to Roman times. 这个城堡可追溯到罗马时代。
表示“追溯到”的短语还有go back to, trace back to, date from, date to等, 但to date表示“至今为止”。
4. What was invented in order to transport the plants safely? 4.The Wardian case.
Summary:
The plant exploration began on a large scale between the eighteenth and nineteenth centuries. Some missionaries and plant collectors introduced many exotic plants into different countries with the help of the Wardian case.
运用根据汉语意思完成下列英文句子。
⑴ 如果这样的话，你的观点一定会和他的冲突。 If so, your points of view are bound to ____ _c_o_n_f_li_c_t_w__it_h__h_i_s___.
⑵ 你的陈述同其余证据有矛盾。 Your statement is _i_n__c_o_n_f_li_c_t_w__it_h___ the rest of the evidence.

高中英语 Unit 4 Exploring plants课件新人教选修9

respiration is reduced and the cut flower will age more slowly.
• Another vital factor in keeping cut flowers is the quality of the water in which they are placed.Flowers find it difficult to “drink”
generates the energy the flower needs to give the flower its shape
and color.The making of seeds also depends on this energy.While all living things respire，flowers have a high level of respiration.A result of all this respiration is heat，and for flowers
water that is dirty or other wise polluted.Even when water looks and smells clean，it almost certainly contains harmful substances that can endanger the flowers.To rid the water of these unwanted substances，household chlorine bleach (含氯漂白剂) can be used in small quantities.It is recommended that 15 drops of chlorine bleach (at 4% solution) be added to each litre

新人教选修9_Unit4_Exploring_plants[全套课件]

【考查方向】考查date back to短语作谓语的情况。因“追溯”是从现在算起指向过去，所以该动词的时态通常用一般现在时，没有被动语态。
【考题预测】His interest in stamp collecting his school days.
A. is dated from B. dates back to
【解析1】动词-ing短语often facing many dangers including disease, near-starvation, severe environments and conflicts with the local people作谓语took the opportunity的伴随状语。
1 However，it was not until the eighteenth and nineteenth centuries that the exploration of the botanical world began on a large scale 然而，直到18和19世纪人们对植物世界大规模的探索才开始。
A. producing B. to produce C. having produced D. produced
【点拨】the first后面省略了名词company， the first后接动词不定式作后置定语。句意：这家公司是世界上第一家生产小收音机和录音机的公司。选B。
5 Although the missionaries collected large numbers of species, there was not enough material for particular species to growing in Western gardens. (Reading Para.9)

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Unit 4 Exploring plantsPeriod oneAim:To train the students’ ability of listening and speaking.Contents:Step I Warming upQ1 What do you know about plants?Q2 Do you know what kind of plant don’t have flowers?Q3 Do you know what kind of plant have roots that live in the air?Q4 Do you know what kind of plant need animals to pollinate them?Q5 Do you know what kind of plant grow on other plants?Q6 Do you know that people buy plants that come from other countries for their gardens?Q7 Do you know that a new plant does not always grow from a seed?Step II Listen and discuss:First give some photos of the plants and then ask students answer the following question.(Page 38)Q1 How would you describe each plant?Q2 What do you think is unusual about each plant?Q3 One of the pictures shows pitcher plants and another shows “Living stones”which picture shows which plant?Now listen to Mark，Gordon and Andrea telling their class about these plants.Which students talk about which plants?Listen again and complete the table with the correct information.Period twoAim:To make students understand the text.Plant exploring in the 18th And 19th centuries.Contents:Q1 What is each paragraph about?Q2 Write down the topic of each paragraph in your own words.Q3 What happened in these years? Complete the sentences on page 33.Q4 Answer there question in groups.1) Why was there a lot of plant collecting in the 18th and 19th centuries?2) How did French missionaries come to be plant collection?3) What were some of the negative aspects of plant collecting expeditions?4) Transporting plants over long distances was a big problem, why was this?5) Why did many plants fail to reach their destinations alive? Can you thinkof any other reasons?6) What important the survival rates of plants transported long distances? 0Period threeAim:To make students master the language points in this text.Contents:Phrases:1 date back to: go back to/date fromeg: This town dates back to the Rome time2 on a large scale3 take / find/ have/ show/ find interest in.4 take the opportunity of doing sth5 go on botanical expeditions6 come into conflicts withbe in conflicts with7 accompany sb: keep sb company8 on purpose / by design / by intentionwith the purpose of9 come acrossrun into / acrossmeet with10 keep sth alive11 make a trip to s.p.12 dress as13 in…style14 result inlead tobring about15 be excited with16 have a vast variety of17 not only…but alsonot alone…but alsonot merely/ simply… but alsoA as well as BA andB as well18 distant adj. far away in space or time 远的，遥远的19 distance n. in the distancefrom the distanceat a distancekeep sb. a distance20 scale n. 1) balance for weighing磅秤2) relative size, extent, scope规模，程度on a large / small scale大/小规模on the scale of … to…按…的比例to a scale按一定比例to a scale of … to…按…的比例21 appeal to 1）恳求，恳请I am appealing on behalf of the refugees.我代表那些难民提出恳求He appeal to us for help.他恳求我们的帮助2）诉诸，诉之于Appeal to the law / a higher court3）事物引起（人的）兴趣，有吸引力，投入所为Does the film appeal to you?n. make an appeal to 吸引, 诉之于Sentence structures1.It was not until the eighteenth and nineteenth centuries that the explorationof the botanical world began on a large scale.2.Not only did Fortune introduce over 120 species of plants to western gardensbut he also shipped 20000 tea plants from…Period FourAim:To make students understand the textFlowers and their animal pollinatorsContents:I. Questions:1)What is the text about?2)What do the pictures show you?3)What is the chart about?4)What is the advantage for animal of visiting flowers?5)Who do some plants need animals?6)How does an animal pollinate a flower?7)What is nectar?8)What kinds of animals are most flowers pollinated by?II. Language points1.over time 随着时间的推移over the weekend 整个周末里over these years 在这些年里2.reward sb with sth用…回报某人3.attach… to依附于…上4.pass on to把…往下传5.adapt to 使…适应The spokesman adapts his speech to suit the interests of his audience.He has adapted to the climate here.6.describe … as 把…描述成7.guide sb to s.p.把…领到…nd on 登陆9.provide sb. with sth向…提供provide sth for sb.provide for 养活，规定III. Difficult sentences1)Pollen becomes attached to the animals during its visit to a flower and is thenpassed on to another plant’s blossom on its next visit.动物接触一朵花后,花粉就附着在它身上,动物接触下一种植物的花时就把这些花粉传到花朵上.2)Through evolution, most flowers have become adapted to attract specific typesof pollinators.通过进化,大部分花朵变得适于吸引特殊种类的花粉传播者.四、排序不等式（一）概念【9】：设有两组实数n a a a ，，，⋅⋅⋅21 （1） n b b b ，，，⋅⋅⋅21 （2）满足n a a a ≤⋅⋅⋅≤≤21 （3）n b b b ≤⋅⋅⋅≤≤21 （4）另设n c c c ,21⋅⋅⋅，，（5）是实数组（2）的一个排列，记逆序积和1121b a b a b a S n n n +++=-乱序积和n n c a c a c a S +++=2211'似序积和n n b a b a b a S +⋅⋅⋅++=2211''那么'''S S S ≤≤且等式成立当且仅当n a a a =⋅⋅⋅==21或者n b b b =⋅⋅⋅==21证明【9】：1，预备知识引理1（Abel 变换）设（1）（2）为任意两组有序的实数组，令,010∑===ki i k b B B ，那么k n k n k k k n n k k B a a B a b a ∑∑=-=---=1111)(事实上：=-=∑∑==-n k nk k k k k k B B a b a 111)(112111)()(B a B B a B B a n n n n n n +⋅⋅⋅+-+------⋅⋅⋅-----=-------)()(2221111n n n n n n n n n n B a B a B a B a B a 112)(B a a -∑-=---=111)(n k k k k n n B a a B a引理2 设实数组（2）满足（4）式，实数组（5）是实数组（2）的任意一个排列，那么显然有∑∑∑=+-==≤≤k i i n k i i k i i b c b 1111 引理3 设实数组（2）满足（4），那么。