新知探究国际象棋起源于古代印度.相传国王要奖赏国际象棋的发明者,问他想要什么.发明者说:“请在棋盘的第1个格子里放上1颗麦粒,第2个格子里放上2颗麦粒,第3个格子里放上4颗麦粒,依次类推,每个格子里放的麦粒都是前一个格子里放的麦粒数的2倍,直到第64个格子.请给我足够的麦粒以实现上述要求.”国王觉得这个要求不高,就欣然同意了.假定千粒麦粒的质量为40克,据查,2016--2017年度世界年度小麦产量约为7.5亿吨,根据以上数据,判断国王是否能实现他的诺言.问题1:每个格子里放的麦粒数可以构成一个数列,请判断分析这个数列是否是等比数列?并写出这个等比数列的通项公式.是等比数列,首项是1,公比是2,共64项. 通项公式为〖a_n=2〗^(n-1)问题2:请将发明者的要求表述成数学问题.
二、典例解析例10. 如图,正方形ABCD 的边长为5cm ,取正方形ABCD 各边的中点E,F,G,H, 作第2个正方形 EFGH,然后再取正方形EFGH各边的中点I,J,K,L,作第3个正方形IJKL ,依此方法一直继续下去. (1) 求从正方形ABCD 开始,连续10个正方形的面积之和;(2) 如果这个作图过程可以一直继续下去,那么所有这些正方形的面积之和将趋近于多少?分析:可以利用数列表示各正方形的面积,根据条件可知,这是一个等比数列。解:设正方形的面积为a_1,后续各正方形的面积依次为a_2, a_(3, ) 〖…,a〗_n,…,则a_1=25,由于第k+1个正方形的顶点分别是第k个正方形各边的中点,所以a_(k+1)=〖1/2 a〗_k,因此{a_n},是以25为首项,1/2为公比的等比数列.设{a_n}的前项和为S_n(1)S_10=(25×[1-(1/2)^10 ] )/("1 " -1/2)=50×[1-(1/2)^10 ]=25575/512所以,前10个正方形的面积之和为25575/512cm^2.(2)当无限增大时,无限趋近于所有正方形的面积和
课前小测1.思考辨析(1)若Sn为等差数列{an}的前n项和,则数列Snn也是等差数列.( )(2)若a1>0,d<0,则等差数列中所有正项之和最大.( )(3)在等差数列中,Sn是其前n项和,则有S2n-1=(2n-1)an.( )[答案] (1)√ (2)√ (3)√2.在项数为2n+1的等差数列中,所有奇数项的和为165,所有偶数项的和为150,则n等于( )A.9 B.10 C.11 D.12B [∵S奇S偶=n+1n,∴165150=n+1n.∴n=10.故选B项.]3.等差数列{an}中,S2=4,S4=9,则S6=________.15 [由S2,S4-S2,S6-S4成等差数列得2(S4-S2)=S2+(S6-S4)解得S6=15.]4.已知数列{an}的通项公式是an=2n-48,则Sn取得最小值时,n为________.23或24 [由an≤0即2n-48≤0得n≤24.∴所有负项的和最小,即n=23或24.]二、典例解析例8.某校新建一个报告厅,要求容纳800个座位,报告厅共有20排座位,从第2排起后一排都比前一排多两个座位. 问第1排应安排多少个座位?分析:将第1排到第20排的座位数依次排成一列,构成数列{an} ,设数列{an} 的前n项和为S_n。
1.判断正误(正确的打“√”,错误的打“×”)(1)函数f (x)在区间(a,b)上都有f ′(x)<0,则函数f (x)在这个区间上单调递减. ( )(2)函数在某一点的导数越大,函数在该点处的切线越“陡峭”. ( )(3)函数在某个区间上变化越快,函数在这个区间上导数的绝对值越大.( )(4)判断函数单调性时,在区间内的个别点f ′(x)=0,不影响函数在此区间的单调性.( )[解析] (1)√ 函数f (x)在区间(a,b)上都有f ′(x)<0,所以函数f (x)在这个区间上单调递减,故正确.(2)× 切线的“陡峭”程度与|f ′(x)|的大小有关,故错误.(3)√ 函数在某个区间上变化的快慢,和函数导数的绝对值大小一致.(4)√ 若f ′(x)≥0(≤0),则函数f (x)在区间内单调递增(减),故f ′(x)=0不影响函数单调性.[答案] (1)√ (2)× (3)√ (4)√例1. 利用导数判断下列函数的单调性:(1)f(x)=x^3+3x; (2) f(x)=sinx-x,x∈(0,π); (3)f(x)=(x-1)/x解: (1) 因为f(x)=x^3+3x, 所以f^' (x)=〖3x〗^2+3=3(x^2+1)>0所以f(x)=x^3+3x ,函数在R上单调递增,如图(1)所示
导语在必修第一册中,我们研究了函数的单调性,并利用函数单调性等知识,定性的研究了一次函数、指数函数、对数函数增长速度的差异,知道“对数增长” 是越来越慢的,“指数爆炸” 比“直线上升” 快得多,进一步的能否精确定量的刻画变化速度的快慢呢,下面我们就来研究这个问题。新知探究问题1 高台跳水运动员的速度高台跳水运动中,运动员在运动过程中的重心相对于水面的高度h(单位:m)与起跳后的时间t(单位:s)存在函数关系h(t)=-4.9t2+4.8t+11.如何描述用运动员从起跳到入水的过程中运动的快慢程度呢?直觉告诉我们,运动员从起跳到入水的过程中,在上升阶段运动的越来越慢,在下降阶段运动的越来越快,我们可以把整个运动时间段分成许多小段,用运动员在每段时间内的平均速度v ?近似的描述它的运动状态。
课前活动:分成四组,对南京三个住房地段进行调查,新街口夫子庙、板仓、仙林,对住房居民进行问卷调查,自行设计调查问卷,分析该地段的房价要求,居民的要求,居民为何要在该地段购买住房,基础设施设置等等其它与居民购房有关的因素。◆设计意图:利用课前小组探究形式对所学问题进行相关调查,不仅让学生掌握知识了解知识来源于社会还能培养学生深入生活交际表达能力以及合作探究能力;3.问题设计同学们,不知道你们在调查过程中是否发现一个问题,郊区的房子,特别是别墅,都是些高档居所,许多有钱人的居住场所,那为什么会这样呢?伴随着城市化进程中居民都往城里挤,为什么还会有许多有钱人往郊区搬呢?这就是我们今天要研究的第二个问题:逆城市化。◆设计意图:承转过渡知识,从购房选择因素的学习过渡到郊区化的学习,同时培养学生发现问题的能力,给学生以启迪。
在学习语文经验交流会上,季老师举着我的《采花酿蜜集》,对大家说:“人日积月累辛勤采撷,终于酿出了知识的琼浆。大家都应这样,争做知识的富户啊!”老师有点激动,低低地爬在鼻梁上的眼镜突然滑了下来,正好落在那集子上。大家笑了,季老师也笑了。就这样,我的写作有了进步,好几篇作文登上了班级《学作园地》。从此,我爱上了语文,更深深地爱上了季老师。高中升学考前那个星期天的夜晚,季老师旧病复发,累倒了。半夜,老师们把他送进了公社卫生院。第二天,同学们都悄悄去卫生院看望。我去的时候,季老师正在挂滴流。可是,下午季老师又出现在讲台上,他脸色憔悴,声音沙哑……我手捧《采花酿蜜集》走近季老师,思绪的溪水从远方流了回来。“季老师”,我把本子捧给老师,深情地叫了声。季老师接过本子,仔细翻阅着,脸上露出了笑容,像是闻到了郁郁芳香的蜜汁似的。“进步不小呀!”季老师说着,又在本子扉页上题了
◆设计意图:培养学生独立思考、合作学习的能力。分别说明市场、交通、劳动力、机械和政策对农业生产的影响,让学生切实地考虑,拓展学生思路。教师激发和维持学生学习动机、引导学生、帮助学生自主发现、探索知识,达到巩固所学知识,检验学生的实践应用能力。D.分析教材,识别图片,理解农业地域阅读、分析教材,看图识别,研究案例《澳大利亚地混合农业生产》、思考问题、解决问题◆设计意图:图片展示能清楚直观地说明问题,通过案例分析,了解澳大利亚的混合农业生产,培养学生独立思考问题、解决问题的能力,开拓学生思路。3.课堂小结:◆设计意图:帮助学生回顾课堂、巩固所学知识。4.反馈练习:◆设计意图:知识与能力强化训练,巩固本课所学知识,提高应用能力。
4,教师重点点拨以下几个问题⑴廊坊战役:义和团浴血奋战,对敌人毫不退缩, 其胜利,挫败了西摩尔联军进犯北京的计划,沉重打击了侵略者的气焰,粉碎了所谓“西方军队无敌”的神话。这是对学生进行爱国主义的重要素材之一,教师重点点拨突出其情感教育的重要作用。⑵让学生(阅读教材小字和材料,结合插图,认清八国联军对中国人民犯下的滔天罪行。)并让学生分组探讨自己的看法,一方面培养学生的语言表诉能力,另一方面通过学生的诉说激发学生的爱国情感,树立落后就要挨打的观念.⑶指导学生通过对条约主要内容进行了分析,并自然得出结论:清政府完全成为帝国主义统治中国的工具,中国完全陷入半殖民地半封建社会。让学生逐步学会分析和归纳能力(4)进行拓展训练出示中国近代史上列强的大规模的侵华战争和中国人民的反抗斗争
另一方面,其余的人反对这个计划,因为它可能会导致一些不好的影响。7.I hold the belief that space exploration not only enable us to understand how the universe began but also help us survived well into the future.我坚信探索太空不仅能够使我们了解宇宙的起源而且能够帮助我们更好地走进未来。8.I think we should spend more time and money exploring space so as to provide new and better solutions to people's shortterm and longterm problems.为了给人类的短期和长期问题提供更新和更好的解决方法,我认为我们应该花更多的时间和金钱来探索太空。9.From my point of view,it is wrong of young people to depend on their telephones too much,which may do harm to both their physical and mental health.在我看来,年轻人过度依赖手机是不对的,因为它们可能会对他们的身心健康都有害。最近你班同学就“人类是否应该进行宇宙探索”这个问题进行了激烈的讨论。有人认为,探索宇宙不仅让人类更好地了解宇宙的发展,还可以用来指导农业生产,以及把一些探索太空的高新技术用于现实生活;也有一些人认为探索太空花掉了大量的人力物力;影响了人们的生活水平。请你根据以下情况写一篇报告并发表自己的观点。注意:1.写作内容应包括以上全部要点,可适当发挥,使上下文连贯;
2. 您能看到, 我头发太长了。You can see that my hair is much too long.3. 无论什么时候, 只要您想回来就回来。Please come back whenever you want.4. 您仅有很少的头发要理! You only have too little hair to cut !5. 为您服务是我的荣幸!It is my honour to serve you!Step 9 Writing(Henry is walking down the street when he sees a sign for a place that cuts hair. He decides to have it cut. )H=Henry B=BarberH: Good afternoon, I’d like to have my hair cut, if I may. (The barber looks at Henry’s hair and continues cutting another man’s hair. ) Er, I’d really like a haircut. As you can see it’s much too long. B: (in a rude manner) Yes, I can see that. Indeed, I can. H: Fine, well, I’ll have a seat then. (He sits in one of the barber’s chairs. The barber turns to look at Henry. )B: It’s quite expensive here, you know! Are you sure you can afford it?H: Yes. I think so. (After his hair is cut, the barber tells Henry how much he must pay. Henry shows the barber the bank note. )B: Why Mr. . . (looks shocked)H: Adams. Henry Adams. I’m sorry. I don’t have any change. B: Please don’t worry! (wearing a big smile) Nothing to worry about! Nothing at all! Please come back whenever you want, even if you only have too little hair to cut! It will be my honour to serve you!Step 10 Pair workExchange drafts with a partner. Use this checklist to help your partner revise his/her draft.1. Are all the elements of a play included and in good order ?2. Do the character use suitable language ?3. Are the stage directions clear and useful ?4. Is the plot clear and exciting enough ?
师:在科学发展过程中,前一个理论体系的不完善之处,往往是新的研究和新的发现的突破口。开普勒之后,意大利天文学家伽利略创制了天文望远镜,用更加精确的观察继续发展和验证哥白尼创立的新天文学理论。除了用望远镜进行天文观察以外,伽利略还开始进行自然科学的实验研究,哪位同学能给大家讲一讲伽利略在比萨斜塔上所作的关于物体自由下落的实验?生:(讲述这一实验)师:所以,伽利略在科学方面更加重要的贡献是奠定了近代实验科学的基础。(2)实验科学和唯物主义师:伽利略从实践上开辟了实验科学的方法,而英国唯物主义哲学家培根则从理论上阐述了实验科学的方法——归纳法。培根和伽利略同被称为实验科学之父,培根还有一句影响深刻的名言:“知识就是力量”,表明了他注重知识,尊崇科学的精神。我们再来概括一下意大利哲学家布鲁诺的唯物主义思想,是否有同学可以简述布鲁诺的生平事迹?
4.That was an experience that frightened everyone. →That was _____________________. 答案:1. taking 2. being discussed 3. in the reading room 4. a frightening experienceStep 6 The meaning and function of V-ing as the predicative动词-ing形式作表语,它通常位于系动词后面,用以说明主语“是什么”或“怎么样”一种表示主语的特质、特征和状态, 其作用相当于形容词; 另一种具体说明主语的内容, 即主语等同于表语, 两者可互换。The music they are playing sounds so exciting. 他们演奏的音乐听起来令人激动。The result is disappointing. 结果令人失望。Our job is playing all kinds of music. 我们的工作就是演奏各种音乐。Seeing is believing. 眼见为实。Step 7 Practice1. It is ________(amaze) that the boy is able to solve the problem so quickly.2. Buying a car is simply _______(waste) money. 3. Please stop making the noise—it’s getting ________(annoy). 4. complete the passage with the appropriate -ing form.La Tomatina is a festival that takes place in the Spanish town Bunol every August. I think many food festivals are __________ because people are just eating. however, this festival is _________ because people don't actually eat the tomatoes. Instead, they throw them at each other! the number of people ________ part in this tomato fight, can reach up to 20,000, and it is a very __________ fight that lasts for a whole hour. The _______ thing is how clean Bunol is after the tomatoes are washed away after the fight. this is because the juice form tomatoes is really good for making surfaces clean!答案:1. amazing 2. wasting 3. annoying4. boring interesting taking exciting amazing
The topic of this part is “Discover the reasons for festivals and celebrations.The Listening & Speaking & Talking part aims at talking about the experiences and feelings or emotions about the festivals and celebrations. This section aims at detecting the reason why the people celebrate the festivals, the time, the places, the types and the way of celebrations. It also explains why some traditions in the old celebrations are disappearing, like the firecrackers in the big cities and some new things are appearing like the prosperity of business or commerce. 1. Students can talk about what festivals they know and the reasons and the way of celebrating them.2. Students should learn the reading skills such as the headline and get the topic sentences, the structures of articles.3. Students can understand the past, the present situation of some festival around the world and why there are some changes about them. 4. Students can have the international awareness about the festivals.1. Students should learn the reading skills such as the headline and get the topic sentences, the structures of articles.2. Students can understand the past, the present situation of some festival around the world and why there are some changes about them.Step 1 Lead in---Small talkWhat festival do you like best ? Why ?I like the Spring Festivals because I can set off the fireworks, receive the lucky money and enjoy the Gala with my families.Step 2 Before reading---Pair workWhy do people celebrate different festivals ?The Spring Festivals is to celebrate the end of winter and the coming of spring and new life.The Mid-autumn Day is to celebrate the harvest and admire the moon.
Step 3 Analyzing article structureActivity 31. Teachers raise questions to guide students to analyze the chapter structure of this diary and think about how to describe the festival experience. (1)What should be included in the opening/body/closing paragraph(s)?(2)How did the writer arrange his/her ideas?(3)What kind of interesting details did the writer describe?(4)How did the writer describe his/her feelings/emotions during the event?2. Students read and compare the three sentence patterns in activity 2. Try to rewrite the first paragraph of the diary with these three sentence patterns. After that, students exchange corrections with their partners. Such as:●This was my first time spending three days experiencing the Naadam Festival in China’s Inner Mongolia Autonomous Region and it was an enjoyable and exciting experience. ●I'll never forget my experience at the Naadam Festival because it was my first time to watch the exciting Mongolian games of horse racing, wrestling, and archery so closely. ●I'll always remember my first experience at the Naadam Festival in China’s Inner Mongolia Autonomous Region because it was so amazing to spend three days witnessing a grand Mongolian ceremony. Step 4 Accumulation of statementsActivity 41. Ask the students to read the diary again. Look for sentences that express feelings and emotions, especially those with the -ing form and the past participle. Such as:● …horse racing, wrestling, and archery, which are all so exciting to watch. ● some amazing performances● I was surprised to see…● I was a little worried about. . . ● feeling really tiredOther emotional statements:●I absolutely enjoyed the archery, too, but the horse races were my favourite part. ●I'm finally back home now, feeling really tired, but celebrating Naadam with my friend was totally worth it. ●He invited me back for the winter to stay in a traditional Mongolian tent and cat hot pot. I can’t wait!2. In addition to the use of the -ing form and the past participle, the teacher should guide the students in the appreciation of these statements, ask them to memorize them, and encourage them to use them reasonably in writing practice.
The topic of this part is “Write about your festival experience”.During the Listening and Speaking and Talking, students are just asked to say out their festival experiences such as the Spring Festival, Mid-autumn Day, but this part students will be asked to write down their own festival experiences. During the reading part, it introduces the Naadam Festival in Inner Mongolia Autonomous Region, which can give students a good example to imitate. Students not only learn the festival, but touch and feel the Inner Mongolian’s character, the spirit and cultural atmosphere, which can help students form the cultural awareness and learn to enjoy and value the diversity of Chinese culture.Concretely, the dairy tells the experience that the author spent the Naadam Festival in Inner Mongolia Autonomous Region with his/her friend. The structure is clear. In the opening paragraph, it introduces the topic of the Naadam Festival and the whole feeling. Then it introduces the items of the festival like the ceremony, wrestling and horse racing. Finally, it summarizes this experience. Because this part is a travel journal, we must guide students pay more attention to these details: 1. use the first person. 2. use the past tense to tell the past thing and use the present or future tense to describe the scenery. 3. use the timeline to tell the development. 4. be careful for the author’s psychology, emotion and feeling, etc.1. Read quickly to get main idea; read carefully to get the detailed information about Naadam Festival.2. Learn the structure of the reading article and language.3. Write an article about a festival experience4. Learn to use the psychology, emotions and feeling in the writing.1. Write an article about a festival experience.2. Use the structure of the reading article and language.
1. 表示时间。Hearing these stories, I’m skeptical about the place. = When I heard these stories. . . 2. 表示原因。Not knowing his address, I can’t send this book to him. = Because/Since/As I don’t know his address. . . 3. 表示结果。His father died, leaving him a lot of money. =. . . and left him a lot of money4. 表示条件。Going straight down the road, you will find the department store. = If you go straight down the road. . . 5. 表示让步。Being tired, they went on working. =Although they were tired. . . 6. 表示行为方式、伴随情况或补充说明。He lay on the grass, staring at the sky for a long time. =. . . and stared at the sky for a long time注意:非谓语动词作状语时, 如所提供的动词不能和句子中的主语保持一致, 动词-ing形式必须有自己的逻辑主语, 通常由名词或代词来担任, 这就是独立主格结构。The last bus having gone, we had to walk home. (having gone的逻辑主语是the last bus, 而不是we)Weather permitting, the football match will be played on Friday. (permitting的逻辑主语是time, 而不是the football match)Step 7 Practice1. ________(study) hard, you are sure to get first prize. 2. People use plastic in their daily life, _______(leave) large amounts of waste. 3. ________(work) hard at your lessons, you are to succeed. 4. The old man, ____________(work) abroad for twenty years, is on the way back to his motherland. 5. ______________(finish) his homework, he was playing on the playground. Answers: 1. Studying 2. leaving 3. Working 4.having worked 5. Having finishedStep 8 HomeworkFinish the homework on Page 22.
Example:One day, a poor boy who was trying to pay his way through school by sending newspapers door to door found that he only had one dime(一角)left. He was so hungry that he decided to beg for a meal at the next house.However, he lost his nerve when a lovely young woman opened the door. Instead of a meal he asked for a drink of water. She thought he looked hungry so she brought him a large glass of milk. He drank it slowly, and then asked, “How much do I owe you?” “You don’t owe me anything,” she replied, “Mother has taught me never to accept pay for a kindness.” “Then I thank you from the bottom of my heart.” With these words, Howard Kelly left that house.Years later the woman became badly ill and was finally sent to the hospital in a big city. Dr. Howard Kelly, now famous, was called in. When he heard the name of the town she came from, a strange light filled his eyes. Dressed in his doctor’s clothes, Dr. Kelly went into her room and recognized her at once. From that day on, he gave special attention to her, and decided to do his best to save her life.At last the woman was saved. Dr. Kelly asked the business office to pass the final bill to him. He looked at it and then wrote something on the side. The bill was sent to the woman’s room. She was afraid to open it because she was sure that it would take the rest of her life to pay for it off. Finally she looked, and the note on the side of the bill caught her attention. She read these words: “Paid in full with a glass of milk, Dr. Howard Kelly.” Tear of joy flooded her eyes.
The topic of this part is “Learn to make choices in life”.The Listening & Speaking & Talking part aims at the moral dilemmas, and this part is about making choices in life. The heroin is Lin Qiaozhi, a famous medical scientist, made a great contribution to our country’s medical care. Most importantly, her life experience can inspire our students whether in studying or the development of career. she had moral dilemmas and life choices, which are similar to the students who will step into society. Besides, Lin has quite good virtues like kindness, self-improvement, insistence, job-loving , generosity and responsibility, which is worth being learned.Concretely, this article is a biography about Lin Qiaozhi. The article tells her whole life according the timeline, among which the life choices is emphasized. For example, whether married or chased her dream, returned home or stayed abroad, family or public, her choices all reflected her faith, spirit, responsibility and devotion.1. Fast reading to get the detailed information about Lin Qiaozhi; careful reading to do the deductive information.2. Learn the reading skills--deductive judgement according the context.3. Study the structure features and language features. 4. Communicate about Lin’s life choices and reflect their own life choices.1. Learn the reading skills--deductive judgement according the context.2. Study the structure features and language features.3. Communicate about Lin’s life choices and reflect their own life choices.Step 1 Lead in---Small talkWhat are some important life choices?Importance choices: university study, jobs and marriage partners. Because they can determine our future.
1. 这个寓言是一个关于一位国王古寓言。 The fable is an old fable about a king.2.作者用这个故事让读者对于社区的问题负有个人责任的必要印象深刻。The author used the story to impress upon readers with the need to take personal responsibility for problems in the community.3. 这个故事十分成功的实现了它的目的。The story was quite successful in achieving its purpose.Step 7 WritingPlease write a review of the story according the outline above.The fable is an old fable about a king who thought his people are lazy, so he put a large stone in the middle of the road and hides and waited to see if anyone will try to move it.The author used this story to impress upon readers with the need to take personal responsibility for problems in the community. The story was quite successful in achieving its purpose, and I liked it because it had a clear moral.However, while the moral of the story is clear, the actions of the king seemed pointless to me, because none of the characters in the story learnt anything. For this reason, I think there are better stories that can be used to impress upon people with the need for personal responsibility.Step 8 Pair workExchange drafts with a partner. Use this checklist to help your partner revise his/her draft.1. Does the writer give a short description of the story ?2. Does the description include the most important details of the story ?3. Does the writer give his or her opinion about the character or their actions ?4. Is the review well-organised ? 5. Does the writer use the -ing form as the adverbial correctly in the writing ?6. Are there any grammar, spelling, or punctuation errors ?Step 9 HomeworkPut up your revised draft in the classroom or read it to your class.
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