勾股定理教案
济水一中 李娜
一、指导思想与教学理念：
以学生为主体的讨论探索法
二、教学对象分析：
八年级学生好奇心强，学生对几何图形的观察，几何图形的分析能力已初步形成。能够正确归纳所学知识，通过学习小组讨论交流，
三、教材分析 ：
勾股定理是直角三角形的一条非常重要的性质，它将数与形密切地联系起来，揭示了一个直角三角形三边之间的数量关系，是后续学习解直角三角形的基础，是三角形知识的深化。
四、教学方法：
讲授法、讨论法
五、教学目标：
（1）知识与技能：了解勾股定理的产生背景，体验勾股定理的探索过程，掌握验证勾股定理的方法；了解勾股定理的内容；能利用已知两边求直角三角形另一边的长；
（2）过程与方法：在勾股定理的探索过程中，培养合情推理能力，体会数形结合和从特殊到一般的思想；
（3）情感与态度：在探索勾股定理的过程中，体验获得结论的快乐，培养合作意识和探索精神。
六、教学环境：
普通教室
七、教学用具：
黑板、粉笔、自制的方格纸、画笔
八、教学重、难点：
重点：探索和证明勾股定理
难点：用拼图方法证明勾股定理
九、教学过程：
一、创设情境，导入新课
1、出示问题，引发思考（用多媒体播放视频）“某楼房二楼失火，消防队员赶来救火，了解到每层楼高3米，消防队员取来6.5米长的云梯,如果梯子的底部离墙基的距离是2.5米,请问消防队员能否进入三楼灭火?”
2、引入新课：教师引导学生将实际问题转化成数学问题,也就是“已知一直角三角形的两边，如何求第三边？” 的问题。
二、探究勾股定理
1、探究等腰直角三角形的三边之间的特殊关系
（1）展示图片：（如图是一个行距、列距都是1的方格网。在方格网中投影显示出以格点为顶点等腰直角△ABC，并显示分别以三角形的各
边为边，向形外作正方形Ⅰ、Ⅱ、Ⅲ。）
提出问题：三个正方形面积SⅠ、SⅡ和SⅢ分别是多少？它们之间有怎样的关系？如用它们的边长表示，能得到怎样的式子？
（2）学生观察图片，分组交流.
（3）引导思考：等腰直角三角形的三边之间有怎样的特殊关系?
（4）归纳总结：等腰直角三角形的两条直角边平方的和等于斜边的平方.
2、探究一般直角三角形的三边之间的特殊关系
（1）展示图片（在行距、列距都是1的方格网中，再作一个格点不等腰直角△ABC，分别以三角形的各边为边，向形外作正方形Ⅰ、Ⅱ、Ⅲ。）让学生在课前备好的网格纸上画图，然后投影出图。
引导思考：1、三个正方形面积SⅠ、SⅡ和SⅢ分别是多少？（学生分组交流，展示求面积的不同方法，如：在正方形C周围补出四个全等的直角三角形而得到一个大正方形，通过图形面积的和差，得到正方形C的面积.或者，将正方形C分割成四个全等的直角三角形和一个小正方形，求得正方形C面积）。2、SⅠ、SⅡ和SⅢ是什么关系？
3、如用它们的边长a,b,c表示，能得到怎样的式子？[设计意图及设想]
（2）学生根据问题，分组交流
（3）引导学生思考：你们发现直角三角形三边的长有怎样的关系？能用简练的语言概括出来吗？
（4)归纳总结：直角三角形两条直角边的平方和，等于斜边的平方。
（5）介绍勾股定理的命名：.约 2000年前,代算书《周髀算经》中就记载了公元前1120年我国古人发现的“勾三股四弦五”.当时把较短的直角边叫做勾, 较长的直角边叫做股,斜边叫做弦.“勾三股四弦五”的意思是,在直角三角形中，如果勾为3,股为 4,那么弦为5.这里 .人们还发现,勾为6,股为8,那么弦一定为10.勾为5,股为12,那么弦一定为13等.所以我国称它为勾股定理.
西方国家称勾股定理为毕达哥拉斯定理。
三、证明勾股定理
1、介绍古今中外数学家和数学爱好者对勾股定理研究和证明的历史.
2、引导学生证明勾股定理：如图在直角△ABC中，∠C＝90°AB=C，BC=a, AC=b,
求证：a2+b2=c2
3、向学生介绍下列两种证明勾股定理的方法，激发学生的兴趣
方法一:：将四个全等的直角三角形拼成如图所示的正方形,
方法二：如图所示将两个直角三角形拼成直角梯形（美国总统的证明方法）
由
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得
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十、课堂小结：
1、通过这节课的学习，你有哪些收获？
2、你会用学过的内容解决课前的问题吗？
十一、布置作业：
课后作业1、2
十二、教材反思：
在课堂教学中，始终注重学生的自主探究能力，创设情境，由实例引入，激发学生的学习兴趣，然后通过动手操作、大胆猜想、勇于验证等一系列自主探究、合作交流活动得出定理，并运用定理进一步巩固提高。但本节课拼图验证的方法以前学生没接触过，稍嫌吃力。另在举勾股定理在生活中的例子时，学生思路不够开阔。
la�;\NU���� font-size:12.0pt;line-height:150%;font-family:"Times New Roman","serif"; mso-bidi-font-weight:bold'> 1. buy some drinks and snacks
2. borrow some money
3. clean your room
4. invite my friends to a party
5. go to the store
6. use your CD player
7. take out the rubbish
8. make your bed
Step 4.
Finish 1b. Use the phrases in 1a to make conversations.
Step 5 Pair work
1. Parents: Could you please…?
Child: Yes, sure. / Sorry, I can’t. I have to ...
2. Child: Could I …?
Parents: Yes, you can. / No, you can’t. You ...
Step 6 Listening
Finish 1c. Listen to a conversation between Sandy and her mom. Check (√ ) the things in 1a that you hear.
1. buy some drinks and snacks
2. borrow some money
3. clean your room
4. invite my friends to a party
5. go to the store
6. use your CD player
7. take out the rubbish
8. make your bed
Finish 1d. Listen again. Fill in the chart. http://ww w.xkb1.co m
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What are they going to do? |
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Sandy’s mom |
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Sandy |
invite her friends |
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Sandy and Dave |
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Step 7 Finish 1e.
You are having a party. Invite your partner to come your party and ask for help with
these things. So, talk about the following things with your partner.
e.g. A: Would like to come to my birthday party?
B: Yes, I’d love to./Sorry, I can’t. I have to…
A: Could you please take out the rubbish?
B: Yes, sure./No, I can’t. I have to do…
Step 8 Reading
1. Warming up
2a. Discuss the questions with your partner
1) What do you often do to help your parents at home?
2) Do you think kids should help out with chores at home?
2. New words
stress n. 精神压力; 心理负担
waste n. 浪费; 垃圾 v. 浪费; 滥用
depend v. 依靠; 依赖
develop v. 发展; 壮大
fairness n.公正性; 合理性
since conj. 因为; 既然
prep., conj. & adv. 从……以后; 自……以来
neighbor(= neighbour) n. 邻居
drop v. 落下; 掉下X Kb1 .C om
3. Skimming
2b. The Sunday Mail magazine invited parents to write about whether they think young people should do chores at home. Skim the following letters. Which one agrees and which one disagrees?
了解阅读策略：
Skimming 意为“快速掠过，从中提取最容易取得的精华”。这种读法包含有原词的所有意思——快速读过去，取出读物中关键性的东西。因此，我们可以把这种读法理解为快速浏览课文，领会文章大意。一般而言，通过标题可知道文章的主题。对文章的首段和末段要多加注意，以便发现作者的观点。
Answer the questions.
1) What is Ms. Miller’s opinion?
2) What is Mr. Smith’s opinion?
4. Reading
2c. According to Mr. Smith and Ms. Miller, what are the pros and cons about kids
doing chores
Read the letters again and finish the following tasks.
I. 完成句子，每空词数不限。
1. Ms. Miller thinks children should spend their time on schoolwork in order to _ ____________________________________________.
2. Mr. Smith thinks these days children depend on _______________________.
II. 判断正(T)误(F)。
( )3. Ms. Miller thinks doing chores is not difficult.
( )4. Mr. Smith’s neighbors’ son looked after himself well during his first
year in the college.
2d. Write one sentence with each phrase from the letters.
Discuss the questions with partner.
Step 9 Language points
1. It is the parents’ job to provide a clean and comfortable environment at home for their children.
此处代词it仅为形式上的主语, 真正的主语是to provide … for their children。我们也可将It is one’s job (duty, …) to do something.视为一个固定的句型, 表示“做某事是某人的工作(职责等)”。如:
It’s every teacher’s job to explain things clearly to the students.
把东西给学生讲明白是每一位老师应做的事情。
provide 作动词，意为“ 提供；供应”。
provide sb. with sth.或provide sth. for
sb.意为“为某人提供某物”。如：
The movie theater provides us with good service.
His school provided a house for him.
汉译英。他们提供给他钱和衣服。
They provided him with money and clothes.
They provided money and clothes for him.
2. And anyway, I think doing chores is not so difficult.
反正我觉得干点家务也不难。
anyway是一个副词, 若位于句首, 其后往往会有逗号将其与句子的其他内容隔开。anyway在此句中的作用是追加评论, 相当于汉语的“反正; 仍然; 依然”。如:
Sam didn’t get the job, but he’s not unhappy because it didn’t pay well anyway.
萨姆没有得到那份工作, 但他并没有闷闷不乐, 反正薪酬也不算高。
anyway还可用来表示“不管怎样; 无论如何”之意。例如:
It’s just a cold. But anyway, you should still see the doctor.
这只不过是感冒, 但不管怎样, 你还是应该看看医生。
3. ill & sick
◆相同点
ill 与sick 都可以表示“生病的”，都可作表语。如：
Alice was ill / sick yesterday.
◆不同点
表示“生病”时，sick 可作定语，但ill 通常不作定语。如：
Could you help the sick girl?
【运用】根据句意，用ill或sick填空。
(1) The driver sent the ________ baby to the hospital.
(2) My brother is ________. I have to look after him at home.
4. The earlier kids learn to be independent, the better it is for their future.
孩子们越早学会独立, 对他们的未来就越好。
“the+比较级+……, the+比较级+……”是英语中一个常用的句型结构, 表示“越……, 就越……”。如:
The more he explained, the better we understood.
他解释得越多, 我们就理解得越透彻。
Many people believe that the more a person reads, the wiser he will become.
很多人相信一个人书读得越多就会变得越聪明。
Step 10 Exercise
I. 根据句意及所给首字母提示，补全句中所缺单词。
1. He was i_____ but nobody took him to the hospital.
2. If you don’t study hard, your grades will d______.
3. I like Old Henry because he often p_______ us with hot water.
4. We can d______ on him for help. 新|课 | 标|第 | 一| 网
5. Mother thinks playing computer games is a w______ of time.
II. 根据句意，用括号内所给单词的适当形式填空。
1. He _________(develop) a farm from nothing.
2. In fact, young people have too much ______ (stress) today.
3. He didn’t mind ________(help) me with my English.
4. Mr. Li and Mr. Wang are both my _________ (neighbor).
5. Why did the man make the little boy ________(carry) water?
Step 11 Homework
After class, please make some new dialogues with the phrases in 1a.
Section B 2 (3a—Self check)
Step 1 Revision
Discuss the questions with a partner. And take notes.
Do you think children should do some chores at home?
Why or why not?
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Children should do chores because… |
Children should not do chores because… |
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Step 2 How to write a letter
1. 信头：发信人地址和日期。
这部分中有时可省去发信人地址，但一般要写日期。放在信纸的右上方。
2. 称呼：对收信人的尊称。
最普通的是Dear，但与Mr，Mrs，Miss等不同的是：Dear后一般用名字，而Mr，Mrs，Miss后不能单独用名字，而只能是姓或姓与名，比如：对John Smith的称呼，应该是Mr Smith，Mr John Smith或 Dear John.称呼后面可用逗号或者冒号。
3. 信的正文。http://ww w.xkb1.co m
4. 结尾：发信人对收信人的谦称或敬语。放在信的右下方。
这部分要特别注意英语习惯，不可根据汉语意思写上诸如“此致敬礼”、“祝您健康”之类的中国式的结尾语。常用的英语书信结尾语如下：
写给同龄人或年长的朋友： Yours sincerely，Sincerely yours，Yours等等。
写给好朋友： Love，With love， Best wishes等等。
写给亲戚：Love，With love，Affectionately yours，Your （ever） loving son（sister等等）。
5. 签名。
签名一般在结尾语的下面一行。
6. 常见的私人信件的开头方式：
高兴：I was so glad/pleased/happy to receive your letter.
感谢：Thank you for your wonderful gift / your interesting letter.
关心与询问：How are you these days？/ How are you getting on these days？/ How are you getting on with your work/studies？
抱歉：I am sorry that I did not write to you sooner but I have been very busy these days.
一般在表示道歉没有及时给对方写信时，还要简述原因或写上安慰的话。
遗憾：I was sorry to learn that you did not do well in the examination. / I was so upset to hear that you are ill these days. I do hope you are getting better.
Step 3 Writing
1. Write a letter to the Sunday Mail and express your opinion.
Dear Sir or Madam,
I think /believe that __________. I agree/disagree that ________________.
I think it is fair/unfair for children to __________________________________.
I think children should/should not ____ because ________________________.
For example, they should/should not_________ because ___________________.
Yours truly,
__________
2. Useful phrases.
help with housework and chores at home
have enough stress from school
spend the time on school work in order to…
There is no need for them to…
It is important for sb. to do sth.
It’s not enough to…
develop children’s independence
The more…, the more…
Step 4 Self check w W w . X k b 1.c O m
1. Make a list of chores using these verbs.
1. do _______________________
2. clean _____________________
3. make _____________________
4. fold _______________________
5. sweep ____________________
6. take out ___________________
Answers: do the dishes clean the living room make your bed
fold your clothes sweep the floor take out the rubbish
2. Are these polite requests or permissions? Write the numbers in the correct
places in the chart.
1. Could you please do your homework?
2. Could I watch TV?
3. Could you take out the rubbish first?
4. Could I use your computer?
5. Could I leave now?
6. Could you come back before nine?
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Requests |
Permissions |
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2 4 5 |
1 3 6 |
Step 5 Homework
Use the questions in activity 2 to write a conversation.
A: _______________________
B: _______________________
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