答案是 9 還是 729

健偉

算術題：

162÷2(6+3)=？

不懂是哪一年的PMR算術題，有個人一直說答案是9，另一批人又說是729。哪位仁兄大姐有答案，可否計算思路寫出來。

多普勒效应

我还记得这题是我初三时候(2003)的PMR题目,
考试过后有一个自以为是的uncle跑出来登报纸讲他发现双重答案..
个人觉得,这是notation的问题,那个uncle也无聊到极点,还登报纸说"证明"了题目有两个答案..

还记得那时是这样的:

那个uncle说:
162÷2(6+3) = 162÷2*9 因为要重左做到右所以是729.
然后他又讲:
162÷2(6+3) = 162÷(12+6) = 9 .

所以"证明"了这个式子有两个答案..

个人认为,这是出题者的notation模糊不清,不是像uncle说的,有两个答案 =.=

G.O.D.

原帖由 健偉 于 7-11-2007 11:33 AM 发表
算術題：

162÷2(6+3)=？

不懂是哪一年的PMR算術題，有個人一直說答案是9，另一批人又說是729。哪位仁兄大姐有答案，可否計算思路寫出來。

(i)     162
     ---------- = 9
       2（6+3）

(ii)    162
     ----------（6+3）=729
         2



如果出题当时是横式的：162÷2(6+3)=？
那么肯定是(i)
意思是：162除于2个(6+3)

mathlim

原帖由 健偉 于 7-11-2007 11:33 AM 发表
算術題：
162÷2(6+3)=？
不懂是哪一年的PMR算術題，有個人一直說答案是9，另一批人又說是729。哪位仁兄大姐有答案，可否計算思路寫出來。

这是符号运用的问题！

若如上所示，则
162÷2(6+3)=162÷2(9)=162÷18=9。

若是162÷2×(6+3)=？，则
162÷2×(6+3)=81×9=729。

健偉

有一個uncle（不知道是不是同一個uncle）告訴我，這個算術題沒有問題。

他的解釋是：先乘除，後加減的運算規則沒錯，但是2(6+3)是一個單位，所以必須先計算2(6+3)。

另一位說答案是729的uncle（也不知道是不是你說的同一個uncle）就不肯把他的計算思路說出來。

那，到底有答案的沒有？

chan1314

9。
因为 parenthesis 得 priority 比较高。
虽然意思是乘， 但是比普通的乘还高一级

~HeBe~_@

我觉得是9,因为 parenthesis 得 priority 比较高。
应该先解决括号里的运算，才乘2，然后把162除以它，所以是9。

162÷2(6+3)=162÷2(9)=162÷18=9

健偉

現在就有3批人，3種不同的說法：

1. 先乘除，後加減。所以應該是從左到右做起。即：162÷2x(6+3)=729

2. 2(6+3)屬一個單位，此單位的()雖然可以用x來計算，但不代表()等於x。舉例說：2^2≠2x2，但是卻可以用2x2的方法來計算。

3. 這算術題本來就有問題。

計算機的答案是：

162÷2(6+3)=9

162÷2x(6+3)=729

~HeBe~_@

其实这问题几年前有刊登在报章上。
这是2003年PMR的考题。
那位Uncle就是中学老师。
引起了大家的主意。
结果，有位工程师也登了报纸来驳回Uncle.

i) 162÷2(6+3)=9

ii)162÷2x(6+3)=729

以上两个的做法都是对的,
那年PMR的问题是 (i) 162÷2(6+3)，
经过工程师的解释后， 162÷2(6+3)的答案是729，不可能是9。


我先来解 (i):
162÷2(6+3)=9

为何？

因为我们要先做括号里的运算，                                    162÷2(6+3) = 162÷[2(9)])
然后,解决括号,(在独中的初中课本上有教过)，
若括号旁边有数字，就先乘上它                                                        = 169 ÷ 18
才来用從左到右做起的方式。                                                            = 9


我先来解 (ii):
ii)162÷2x(6+3)=729

为何？
因为我们要先做括号里的运算，                                   162÷2x(6+3) = 162 ÷ 2 x (9)
然后,解决括号（在独中的初中课本上有教过),
但这个括号旁边没有数字，只好去掉括号，                                     = 162 ÷ 2  x  9
才来用從左到右做起的方式。                                                             = 729


所以PMR的问题只有一个答案，就是162÷2(6+3)=729
工程师也是如此的解释。

大家明白了吗？

[ 本帖最后由 ~HeBe~_@ 于 8-11-2007 04:38 PM 编辑 ]

~HeBe~_@

这里是我教小学的时候研究的功课。
只不过我的功课只是做到，
1)
perform Combined Operationsinvolving multiplication and addition;
2)
perform Combined Operationsinvolving multiplication and subtraction;
而已。
我的功课里有解释什么是混合运算，
什么是In ascending order for combined operations;
什么是In ascending order for the same level operation
什么是In ascending order for different level operation

我们可要先打好基础哦。。。
上课咯。。。

~HeBe~_@

Combined Operations

Learning Objective
After learning,student should be able to :
1)perform Combined Operations involving multiplication and addition;
2)perform Combined Operations involving multiplication and subtraction;
3)perform Combined Operations involving divison and addition ;
4)perform Combined Operations involvingdivision and subtraction;
5)perform Combined Operations involving brackets.
6)Resolve subjective.

Teaching Instruments

Computer (Microsoft Powerpoint)

Moral Value
Be careful、in ordering、fair、stand by regulations、proper on mind thinking、sparingly、hard working、gregarious.

Thought Skill

Relationship thinking、Comparing difference、planning、solving question.

Students learnt before of the knowledge and experiments

Students have learnt before simple combined operations involving addition andsubtraction; multiplication and division; multiplication andaddition;multiplication and subtraction; division and addition;division and subtraction.

Introduction Contents
1)Combined Operations involving multiplication and addition;
2)Combined Operations involving multiplication and subtraction;
3)Combined Operations involving division and addition ;
4)Combined Operations involving division and subtraction;
5)Combined Operations involving one bracket;
6)Combined Operations involving two brackets;
7)Subjective.

Teacher’s Reference materials
Combined operations: in a same computation, its involve 2 types or above different calculates’ symbolic, i.e.: plus、minus、times、divide or bracket,

In ascending order for combined operations

• In a computation combined operations, should calculate second leveloperation first, then calculate first level operation. (addition andsubtraction known as first level operation, whereas multiplication anddivision known as second level operation.)
• If there arebrackets, should perform computation operation in a small bracketfirst, then perform computation operation in a standard bracket before perform computation operation in a big bracket.
• If it’s a same level operation, hence should perform calculation from left to right.

In ascending order for the same level operation

Involving in a same level operation ,should follow in ascending order computation operation that calculate from left to right.
When teaching:

• First of all, give explanation for combined operations involving additionandsubtraction, then give explanation for combined operations involving multiplication and division,  after that conclude it: operationsinvolving addition and subtraction or  multiplication and division mustfollow computation ordering from left to right.
• Students are required use simple oral to calculate 2 steps operation, and then students are required
  touse method writing to calculate it. When use standard form tocalculate,in the mean time let students separate into 2 standard forms.
• Teacher should give more explanation for those students who usually make mistake in calculation.

In ascending order for different level operation.

If an operation consist of first level operation and second level operation in the same time, therefore perform the second level operation before perform the first level operation, i.e. times、dividefirst, then plus、minus.


When teaching:

• Teacher is required to explain combined 2 types level operation in ascending order, in order to avoid students misunderstand
  ‘times、divide first, then plus、minus’ as ‘times first then divide, plus first then minus’.
• Guide students apply specific mathematically language to read out 2 steps and 3 steps of computation.
• Guide students to use the proper operation with in ascending order to perform accuracy calculations.
• Teacher should rectify students who usually make common mistaken, such as 45 +0x 2 write wrongly as 45 x 2, 100 ÷ 25 + 25 x 0 write wrongly as 100 ÷50x 0 and so on.

Brackets

The symbol bracketis used to change the in ascending order operation. Usually weuse:small bracket “( )” , also known as parenthesis; standard bracket“[ ]”,also known as square bracket; big bracket “{ }”, also known as curlybracket. Primary students usually apply small bracket and stand bracket.
When involving brackets in computation,must following from internal bracket to external bracket, that is perform computation operation in a small bracket first, then perform computation operation in a standard bracket before perform computation operation in a big bracket.
If operation inside the bracket, we still following the rule of “times、divide first,then plus、minus”, whereas same level operation must calculate from left to right.

For Combined Operation Involving Multiplication And Addition Only Learning Objective

After learning, student should be able to perform Combined Operations involving multiplication and addition.


Teaching Instruments

Computer (Microsoft Powerpoint)

Moral Value

Be careful、in ordering.

Thought Skill

In arranging order、planning.

Teaching Suggestion

• Teacher gives revision that students have learnt before about simple steps forcombined operations involving addition and subtraction; multiplicationand division.
• Before starting, teacher is required to give explanation of  in ascending order for the same level operation and in ascending order for different level operation (see in “Teacher’s Reference materials”)
• Teacher shows and reads the ‘Learning Objective’ in powerpoint, in order to letstudents know learning objective is Combined operations involving multiplication and addition.
• Teacher gives explanation Step by step in powerpoint,
  such as we times first and then plus it from left to right.
• Teacher gives question 1 from Work example 1, shows the solution meanwhile use the rule of ‘in ascending order for different level operation’ to explain it.
• After that, teacher gives question 2 from Work example 1. Askstudents how to solve that question, obviously majority of them willsay perform calculation from left to right. In the mean time, teachershould rectify student’s errors, emphasizes students that they mustperform multiplication first.
• Eventually, give student try question 1 and 2 from Exercise example 1,
  give explanation on questions respectively.

For Combined Operation Involving Multiplication And Subtraction Only Learning Objective

After learning, student should be able to perform Combined Operations involving multiplication and subtraction.


Teaching Instruments

Computer (Microsoft Powerpoint)

Moral Value

Be careful.

Thought Skill

In arranging order、planning.

Teaching Suggestion

• Teacher gives revision that students have learnt before about simple steps forcombined operations involving addition and subtraction; multiplicationand division.
• Before starting, teacher is required to give explanation of  in ascending order for the same level operation and in ascending order for different level operation (see in “Teacher’s Reference materials”)
• Teacher shows and reads the ‘Learning Objective’ in powerpoint, in order to letstudents know learning objective is Combined operations involving multiplication and subtraction.
• Teacher gives explanation Step by step in powerpoint,
  such as we times first and then plus it from left to right.
• Teacher gives question 1 from Work example 2, shows the solution meanwhile use the rule of ‘in ascending order for different level operation’ to explain it.
• After that, teacher gives question 2 from Work example 2. Askstudents how to solve that question, obviously majority of them willsay perform calculation from left to right. In the mean time, teachershould rectify student’s errors, emphasizes students that they mustperform multiplication first.
• Eventually, give student try question 1 and 2 from Exercise example 2,
  give explanation on questions respectively.

[ 本帖最后由 ~HeBe~_@ 于 27-11-2007 12:45 AM 编辑 ]

jinqwem

英文教数理??
不过同意应该是9, priority 的问题...

~HeBe~_@

哈哈！不可以吗？

kaixun

不是9咩=。=，729的是想太多o？

G.O.D.

原帖由 ~HeBe~_@ 于 9-11-2007 03:38 PM 发表
哈哈！不可以吗？

哇~一大堆“鸡肠”...
我都懒得看了！！
重点是看priority，对吗？？

~HeBe~_@

重点是看precending和priority。

你不认为吗？

G.O.D.

原帖由 ~HeBe~_@ 于 9-11-2007 06:15 PM 发表
重点是看precending和priority。

你不认为吗？

对！！绝对赞同！！！
不然小学时我时常一个人坐在家里的白板前面对着数学题发呆是白白浪费时间的??...
（楼上这题当时也是我研究的其中之一。。。）

现在想起，数学就像是一种语言，最重要是了解“她”的意思~   

[ 本帖最后由 G.O.D. 于 11-11-2007 06:39 PM 编辑 ]

angel4896

原帖由 ~HeBe~_@ 于 8-11-2007 04:06 PM 发表
其实这问题几年前有刊登在报章上。
这是2003年PMR的考题。
那位Uncle就是中学老师。
引起了大家的主意。
结果，有位工程师也登了报纸来驳回Uncle.

i) 162÷2(6+3)=9

ii)162÷2x(6+3)=729

以 ...

2003?是我那一年的...
我做法是(i),
因为它用的是乘(),不是*,
那么说明应该是162/2个(6+3).
我用SCIENTIFIC CALCULATOR照这个程式打,
也是等于9.

fgh

很明显是9吧。
先乘除后加减
乘在除前面！

超级丑八怪[-_-]

原帖由 健偉 于 8-11-2007 01:22 AM 发表
他的解釋是：先乘除，後加減的運算規則沒錯，但是2(6+3)是一個單位，所以必須先計算2(6+3)。

同意这个论点