小学
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四年级奥数教程-速算与巧算(一)
速算与巧算(一)
计算是数学的基础,小学生要学好数学,必须具有过硬的计算本领。准确、快速的计算能力既是一种技巧,也是一种思维训练,既能提高计算效率、节省计算时间,更可以锻炼记忆力,提高分析、判断能力,促进思维和智力的发展。
我们在三年级已经讲过一些四则运算的速算与巧算的
方法
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,本讲和下一讲主要介绍加法的基准数法和乘法的补同与同补速算法。
例1 四年级一班第一小组有10名同学,某次数学测验的成绩(分数)如下:
86,78,77,83,91,74,92,69,84,75。
求这10名同学的总分。
分析与解:通常的做法是将这10个数直接相加,但这些数杂乱无章,直接相加既繁且易错。观察这些数不难发现,这些数虽然大小不等,但相差不大。我们可以选择一个适当的数作“基准”,比如以“80”作基准,这10个数与80的差如下:
6,-2,-3,3,11,-6,12,-11,4,-5,其中“-”号表示这个数比80小。于是得到
总和=80×10,(6-2-3,3,11-
,800,9,809。
实际计算时只需口算,将这些数与80的差逐一累加。为了清楚起见,将这一过程表示如下:
通过口算,得到差数累加为9,再加上80×10,就可口算出结果为809。
例1所用的方法叫做加法的基准数法。这种方法适用于加数较多,而且所有的加数相差不大的情况。作为“基准”的数(如例1的80)叫做基准数,各数与基准数的差的
和叫做累计差。由例1得到:
总和数=基准数×加数的个数+累计差,
平均数=基准数+累计差?加数的个数。
在使用基准数法时,应选取与各数的差较小的数作为基准数,这样才容易计算累计差。同时考虑到基准数与加数个数的乘法能够方便地计算出来,所以基准数应尽量选取整十、整百的数。
例2 某农场有10块麦田,每块的产量如下(单位:千克):
462,480,443,420,473,429,468,439,475,461。求平均每块麦田的产量。
解:选基准数为450,则
累计差=12,30,7,30,23,21,18,11,25,11
,50,
平均每块产量=450,50?10,455(千克)。
答:平均每块麦田的产量为455千克。
求一位数的平方,在乘法口诀的九九表中已经被同学们熟知,如7×7,49(七七
20的平方,而21,99的平方四十九)。对于两位数的平方,大多数同学只是背熟了10,
就不大熟悉了。有没有什么窍门,能够迅速算出两位数的平方呢,这里向同学们介绍一种方法——凑整补零法。所谓凑整补零法,就是用所求数与最接近的整十数的差,通过移多补少,将所求数转化成一个整十数乘以另一数,再加上零头的平方数。下面通过例
题
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来
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这一方法。
22例3 求29和82的值。
2解:29=29×29
,(29,1)×(29-1),12
,30×28,1
,840+1
,841。
2 82,82×82
2 ,(82,2)×(82,2),2
,80×84,4
,6720+4
,6724。
由上例看出,因为29比30少1,所以给29“补”1,这叫“补少”;因为82比80多2,所以从82中“移走”2,这叫“移多”。因为是两个相同数相乘,所以对其中一个数“移多补少”后,还需要在另一个数上“找齐”。本例中,给一个29补1,就要给另一个29减1;给一个82减了2,就要给另一个82加上2。最后,还要加上“移多补少”的数的平方。
2 由凑整补零法计算35,得
2 35×35,40×30,5=1225。这与三年级学的个位数是5的数的平方的速算方法
结果相同。
这种方法不仅适用于求两位数的平方值,也适用于求三位数或更多位数的平方值。
22例4 求993和2004的值。
2解:993=993×993
2 ,(993,7)×(993-7)+7
,1000×986,49
,986000,49
,986049。
2 2004=2004×2004
,(2004-4)×(2004+4),42
,2000×2008,16
,4016000,16
,4016016。
下面,我们介绍一类特殊情况的乘法的速算方法。
请看下面的算式:
66×46,73×88,19×44。
这几道算式具有一个共同特点,两个因数都是两位数,一个因数的十位数与个位
数相同,另一因数的十位数与个位数之和为10。这类算式有非常简便的速算方法。
例5 88×64,,
分析与解:由乘法分配律和结合律,得到
88×64
,(80,8)×(60,4)
,(80,8)×60,(80,8)×4
,80×60,8×60,80×4,8×4
,80×60,80×6,80×4,8×4
,80×(60,6,4),8×4
,80×(60,10),8×4
,8×(6,1)×100+8×4。
于是,我们得到下面的速算式:
由上式看出,积的末两位数是两个因数的个位数之积,本例为8×4;积中从百位起
前面的数是“个位与十位相同的因数”的十位数与“个位与十位之和为10的因数”的十位
1)。 数加1的乘积,本例为8×(6,
例6 77×91,,
解:由例3的解法得到
由上式看出,当两个因数的个位数之积是一位数时,应在十位上补一个0,本例为
7×1,07。
用这种速算法只需口算就可以方便地解答出这类两位数的乘法计算。
练习1
1.求下面10个数的总和:
165,152,168,171,148,156,169,161,157,149。
2.农业科研小组测定麦苗的生长情况,量出12株麦苗的高度分别为(单位:厘米):
26,25,25,23,27,28,26,24,29,27,27,25。求这批麦苗的平均高度。
3.某车间有9个工人加工零件,他们加工零件的个数分别为:
68,91,84,75,78,81,83,72,79。
他们共加工了多少个零件,
4.计算:
13,16,10+11,17,12,15,12,16,13,12。
5.计算下列各题:
222 (1)37; (2)53; (3)91;
222 (4)68: (5)108; (6)397。
6.计算下列各题:
(1)77×28;(2)66×55;
(3)33×19;(4)82×44;
(5)37×33;(6)46×99。
练习1 答案
1.1596。 2.26厘米。
3.711个。 4.147。
5.(1)1369; (2)2809; (3)8281;
(4)4624; (5)11664; (6)157609。
6.(1)2156; (2)3630; (3)627;
(4)3608; (5)1221; (6)4554。
Class: Name: ( )1. Good morning! ! A.Morning ! B.Hello ! C.Hi !
( )2. Nice to see you again ! . A.How are you ? B.Nice to see you , too . C.How do you do ?
( )3.Good night,mom !
A.Night ! B.Good night ! C.Good evening . ( )4.How do you do ? A.How are you ? B.Fine,thanks . C.How do you do ? ( )5.How many story books do you have ? A.I have 10. B.I can see 10. C.Thirty yuan. ( )6.Do you have new teachers? A.Yes,we do . B.Yes,we don’t. C.Yes,we have .
( )7.Who’s your art teacher ? A.Mr Zhu. B.Miss Zhu. C.He’s tall.
( )8.What’s he like?
s tall and strong . B.Yes,he is. C.Mr Zhu. A.He’
( )9.Is your English teacher young? A.No,she isn’t. B.Yes,she is . C.No,she is.
( )10. ? Her name is Chen Jie. A.What’s your name ? B.What’s she name ? C.What’s her name ?
( )11. ? I like Chinese,math and English. A.What classes do you like? B.What do you like? C.What are you like ?
( )12. ? We have English and P.E.
A.What do you have on Mondays ? B.What do you have ?
C.What do you have on Monday?
( )13. ? It’s Monday.
A.What is it today ? B.What day is it today ? C.What day is today ?
( )14. ? I watch TV and do my homework. A.What do you do ? B.What do you do in Mondays? C.What do you do on Sundays ?
( )15.May I have a look ? A.Sure.Here you are . B.Look ! C.Here you are . ( )16.Our math teacher is Canada. A.from B.in C.at
( )17.I three new teachers.
A.has B.am C.have
( )18.What’s Chinese teacher like ?
A.you B.your C.you’re
( )19.My P.E.teacher is thin.
A.too B.so C.very
)20.There are days in a week. (
A.six B.seven C.eight
( )21.There are month(月)in a year(年).
A.ten B.eleven C.twelve
( )22.I often watch TV Saturday . A.on B.in C.at
( )23.I like P.E. I don’t lime music.
A.but B.and C.so
( )24.This is apple. It is red apple. A.a , a B.an,an C.an, a
( )25.What do you like ?
A.classes B.class C.class’s
六年级英语测试题
Class: Name: ( )1. How are you ?
A.Fine,thanks. B.Yes,it is. C.How are you ? ( )2. Nice to meet you ! A.Fine,thank you. B.OK. C.Nice to meet you ,too ! ( )3. How do you go to school ? A.I go to Canada by plane. B.I go to school by bike. C.What about you ?
( )4. How do you go to the USA ? A.I usually go to school by bus. B.I go to England by ship. C.I go by plane.
( )5. ? My home is near the post office. A. Where is your home ? B.OK. C.See you then!
( )6. See you at 2 o’clock.
B.The fifth floor. C.It’s easy. A.See you then !
( )7. ? You can go by the No.15 bus. A.It’s not far. B. How can I get to Zhongshan Park ? C.Sure. ( )8. Where is the hospital?
A.Next to the cinema. B.Thank you . C.You’re welcome.
( )9. Excuse me ,is there a cinema near here ? A.Yes,there is . B.No,it’s not far. C.It’s near the post office.
( )10.? ? It’s near the post office.
A.Yes,there is . B.No,it’s not far. C. Where is the library? ( )11. ? It’s next to the hospital.
A. Where is the cinema ,please ? B.Is it far ? C.Go straight. ( )12. How can I get to the hospital ? A.It’s next to the hospital. B.You can go by the No.201 bus. C.Thank you. ( )13.Thank you .
A.Thank you . B.OK. C.You’re welcome.
( )14. How can I get to the museum? A.Go straight.Then turn left. B.Thank you . C.It’s east of the cinema .
( )15. Where is the post office ? A.Go straight.Then turn left. B.Thank you . C.It’s east of the cinema .
( )16. What are you going to do this evening ? A.I’m going to the cinema . B.I go to school on foot. C.Yes,it is.
( )17. Is it far ? A.No,it is . B.Yes , it is . C.Yes,it isn’t.
( )18. do you go to school ?
B.Where C.How A.What
( )19. I go to school bike. A.on B. by C.get
( )20. Can I go foot ?
A.by B.at C.on
( )21.How can I to the post office ? A.near B. get C.for
( )22. me.
A. Excuse B. How C.next ( )23. birthday to you ! A. After B.Happy C.First ( )24. The hospital is the left. A. at B.in C.on
( )25. is the bookstore ?
A.Where B.How C.When
四年级英语
四年级上册英语教案四年级英语上单词默写四年级英语下册教案译林英语四年级上期末绍兴小学英语期末试卷
测试题
Class: Name: ( )1.Good afternoon ! A.Hello! B.Hi ! C.Afternoon ! ( )2.Nice to see you again ! A.How are you ? B.Hello ! C.Nice to see you , too.
( )3.Good night,moom ! A.Good evening ! B.Good night ! C.Night ! ( )4.How are you ?
A.Fine,thank you . B.I am 10. C.Nine . ( )5.How do you do ?
A.Fine ,Thank you . B.How are you ? C.How do you do ? ( )6.How old are you ?
A.How are you ? B.I am 11. C.I have 11. ( )7. ? I have 23.
A.How many books do you have ? B.How many books can you see? C.How many book do you have ?
( )8. ? I can see 6.
A.How many lights do you have ? B.How many lights can you see ? C.How many light can you see ?
( )9.May I have a look ?
A.Sure.Here you are . B.Look! C.Sure.Here are you . ( )10. ? 50 yuan.
A.How much is this schoolbag ? B.How many is this schoolbag ? C.How much are this schoolbag ?
( )11.Where’s my seat ?
A.It’s near the door. B.It’s on the door. C.It’s under the door.
( )12.Let’s clean the desks and chairs. A.All right ! B.Hello ! C.It’s nice .
( )13.What’s this ?
A.There is a board . B.It’s a bee. C.It’s bee.
( )14. ? My name is Mike.
A.Here are you ? B.What’s your name ? C.How do you do ?
( )15.Who’s the inventor of paper ?
A.Chinese people. B.Oh! Great ! C.Hello! ( )16.I a student. You a teacher. A.am ,am B.are, are C.am, are ( )17.This a boy.His name Zhang Peng. A.is, is B.am , is C.is , are
( )18.Let clean the fish bowl.
A.I B. me C.my
( )19.This is apple.It is red apple . A.an ,an B.a, a C.an, a
( )20.We a new classroom.
A.are B.have C.has
( )21.There a bee in our classroom. A. is B.are C.am
( )22. have a new schoolbag. schoolbag is heavy. A.I , my B.I ,My C.My , I
( )23.Put your English book your head. A.at B.in C.on
( )24.There many books in the desk. A.am B.is C.are
( )25.How many do you have ? A.pencil B.pencils C.pencils.
三年级英语测试题
Class: Name:
( )1.当向别人打招呼时,应该说:
A.Hello. B.Good morning . ( )2.How are you ? 的正确答语是:
A.name B.I’m fine,thank you .
( )3.当想知道别人的名字时,应该说:
A.What’s your name ? B.See you. ( )4.字母K的小写是:
A.k B.
( )5.Nice to meet you 的意思是:
A.见到你很高兴。 B.你好吗,
( )6.向别人告别时,应该说:
A.Good bye! B.Hi.
( )7.Jenny is a
A.boy B.girl
( )8.Li Ming is a
A.boy B.girl
( )9.Jenny lives in
A.Canada. B.China. ( )10.字母P的大写是:
A.P B.q
( )11.当向别人说谢谢时,应该说:
A.Nice to meet you . B.See you later. ( )12.早上好的正确答语是:
A.Good morning . B.See you later.’
( )13.What’s this , It’s a
A.desk B.book
( )14.This is a
A.teacher B.Danny
( )15.This is a boy.What’s name ?
A.his B.her