教育王國
標題: 請教數學問題 [打印本頁]
作者: lillymarie 時間: 12-12-7 20:07 標題: 請教數學問題
和大囝計數，a - (b - c) = a - b + c.他不明白為何會是 “+ c"
我自己數學鈍只靠背書”負負得正“， 但囝囝好苦惱，因為他不明白為何負負得正？
請問有沒有朋友可以解答他的問題？又或者提供網站／書藉參考？
作者: cellon 時間: 12-12-7 20:30
lillymarie 發表於 12-12-7 20:07 
和大囝計數，a - (b - c) = a - b + c.他不明白為何會是 “+ c"
我自己數學鈍只靠背書”負負得正“， 但囝囝好苦惱，因為他不明白為何負負得正？ ...
作者: lillymarie 時間: 12-12-7 20:33 標題: 回覆：cellon 的帖子
升grade 8
作者: cellon 時間: 12-12-7 20:38
lillymarie 發表於 12-12-7 20:33
升grade 8
作者: lillymarie 時間: 12-12-7 20:48
cellon 發表於 12-12-7 20:38
Have you tried to teach him this concept using the "number line".
作者: laorenjia 時間: 12-12-7 21:34
回復 lillymarie 的帖子
The rule to remove the bracket is based on the distributiive law your son should have learned in P4 or P5. However, the proof of distributive law is not easy and normally we only explain to kids using a diagram (normally an area calculation example, e.g. 3 x (4+5) = 3 x 4 + 3 x 5. The following link may be helpful:
http://www.youtube.com/watch?v=vl-6G3VzlYE
作者: lillymarie 時間: 12-12-7 21:45 標題: 引用：回復+lillymarie+的帖子The+rule+to+remo
本帖最後由 lillymarie 於 12-12-7 22:07 編輯
原帖由 laorenjia 於 12-12-07 發表
回復 lillymarie 的帖子
The rule to remove the bracket is based on the distributiive law your son sh ...
Thanks! The rule to remove the bracket is also a puzzle to him. He also doesn't understand why a(b + c) = ab + bc.
Pity he hasn't learnt maths in HK since P4. Though his school taught algebra this year, they didn't touch these concepts. And pity I never asked why, now don't know how to explain. We borrowed some books to study about the concepts.
I'll let him watch the video tmr.
作者: cellon 時間: 12-12-7 22:30
laorenjia 發表於 12-12-7 21:34
回復 lillymarie 的帖子
The rule to remove the bracket is based on the distributiive law your son should have learned in P4 or P5. However, the proof of distributive law is not easy and normally we only explain to kids using a diagram (normally an area calculation example, e.g. 3 x (4+5) = 3 x 4 + 3 x 5. ...
a x (b+c) = a x b + a x c is based on Distributive Law.
But a - (b - c) = a - b + c is NOT based on Distributive Law.
作者: laorenjia 時間: 12-12-7 23:01
Most mainstrem textbooks prove a - (b-c) = a - b + c using distributive law:
a - ( b - c ) = a - 1 x ( b - c ) = a + (- 1) x b + (-1) x (-c) = a - b + c
What is your proof, btw?
作者: cellon 時間: 12-12-7 23:23
本帖最後由 cellon 於 12-12-8 01:50 編輯
laorenjia 發表於 12-12-7 23:01
Most mainstrem textbooks prove a - (b-c) = a - b + c using distributive law:
a - ( b - c ) = a - 1 x ( b - c ) = a + (- 1) x b + (-1) x (-c) = a - b + c
Of course, the above equation is correct, but it does NOT explain why (-1) x (-c) = + c
Read lillymarie's message: 「但囝囝好苦惱，因為他不明白為何負負得正？」
I suggest to use "Number Line" to illustrate (not "prove") this concept, because most math textbooks will use this method to teach a Grade 7 or 8 kid.
作者: laorenjia 時間: 12-12-7 23:40
本帖最後由 laorenjia 於 12-12-8 01:37 編輯
- x - = + is a separate issue from removing the bracket. It per se cannot solve the “removing the bracket” problem. Suggest googling “remove the bracket in algebra” first. Again, what's your method？
作者: laorenjia 時間: 12-12-7 23:43
本帖最後由 laorenjia 於 12-12-8 01:37 編輯
How can the number line be utilised here？I am intrigued.
作者: cellon 時間: 12-12-8 01:50
本帖最後由 cellon 於 12-12-8 10:43 編輯
laorenjia 發表於 12-12-7 23:43
How can the number line be utilised here？I am intrigued.
Try to read a Math textbook for Grade 7 or 8.
作者: JustAParent 時間: 12-12-8 12:05 標題: 引用：和大囝計數，a+-+(b+-+c)+=+a+-+b+++c.他不
原帖由 lillymarie 於 12-12-07 發表
和大囝計數，a - (b - c) = a - b + c.他不明白為何會是 “+ c"
我自己數學鈍只靠背書”負負得正“， 但囝 ...
作者: laorenjia 時間: 12-12-8 12:27
回復 cellon 的帖子
Please show me yourself or show me a website which shows how to use the number line to do the "removing the brackets" as all I can see are websites using the distributive rule. Otherwise, I'm afraid I have to put an end to our dialogue as clearly it is heading nowhere. Sorry, my fault.
作者: petline 時間: 12-12-8 21:30
孩子程度唔高，建議唔好用大人思維去諗，用數線只會使慨念上亂上加亂，因不明白的學生多數抽象思維唔多掂。
死記是第一步，慢慢等仔仔大個自然會明，無謂太急強求。
其次，你可以用實例令到死記過程易入口D
例如 10 - (1+2) = 7 (先計括號，唔好講拆括號)
但 10 - 1 + 2 = 11 10 -1 -2 = 7
所以拆完後面要寫負，好似新年派例是，屋(括號) 內每個細路 (數字) 都有一封。
其後做10條(由數字到代數)，灌個CONCEPT入去，之後再係括號內寫三個數字睇下佢識唔識拆順手讚佢叻仔。 應該十五分鐘可以教完。
作者: cellon 時間: 12-12-9 00:57
petline 發表於 12-12-8 21:30
孩子程度唔高，建議唔好用大人思維去諗，用數線只會使慨念上亂上加亂，因不明白的學生多數抽象思維唔多掂。
死記是第一步，慢慢等仔仔大個自然會明，無謂太急強求。
其次，你可以用實例令到死記過程易入口D
例如 10 - (1+2) = 7 (先計括號，唔好講拆括號)
但 10 - 1 + 2 = 11 10 -1 -2 = 7
所以拆完後面要寫負，好似新年派例是，屋(括號) 內每個細路 (數字) 都有一封。 ...
But please understand that we are talking about teaching a Grade 8 student.
作者: lillymarie 時間: 12-12-9 07:08
本帖最後由 lillymarie 於 12-12-9 07:34 編輯
多謝 大家的建議。
a - (b - c) = a - b + c
用number line去示範，告訴囝囝凡遇到subtraction就要轉方向。他說他明白接受了。
a(b - c) = ab - ac
我看書這似乎是一個規則（distributive rule），講了幾次，囝囝仍說不明白，但又講不出不明白甚麼。睇書話algebra要多做題目才能明白當中概念，所以接下來會讓囝囝多做algebra的題目。
作者: lillymarie 時間: 12-12-9 08:43
本帖最後由 lillymarie 於 12-12-9 08:47 編輯
有本書作者給予一些guidelines for effective algebra study, 以下是我覺得可以參考的幾點：
1。starting on the very first day of classes, systematically work problems of all types untill you are confident that you understand all concepts.
2. Be sure to read the discussion given in the text of the sections covered on a given day. Work your way through all examples in the text. Have a pencil and paper close by and fill in any missing details in the examples. If there are parts of the examples that you do not understand, ask your instructor to help you fill in the details.
3. Do not get behind in the class. Once you get behind in the class, the snowball effect follows. The new concepts that you encounter usually depend on your understanding of the ones you are behind.
關於 algebra,
1. Algebra is not learned by osmosis. you will not automatically absorb algebra by simply attending class. You must work a lot of problems to fully understand the concepts.
2. Algebra is not a spectator sport. You must be an active participant in the learning process.
有一點我睇咗本書才知道：
The topics and concepts in algebra are sequentially dependent. This means that the topic you are studying today is dependent on topics you learned yesterday, and the topics you study tomorrow will depend on the ones you learn today.
阿囝走唔甩，做數，做數，做數！
作者: petline 時間: 12-12-9 09:00
cellon 發表於 12-12-9 00:57
Your "新年派例是" method may be suitable for small kids.
But please understand that we are talking ...
work定係唔work，係要睇學生feedback，唔係口講，不如等結果說話吧。
我教了十幾年數，只知道 algebra 要靠模仿同慣性去做，正如閣下做拆括號嘅時候唔會用 number line 想一遍先至做。
說服嘅過程唔好用大人思維去諗，或者用大人以前嘅經驗 去諗(相信好多家長都top band 1 學生)，好多時候佢地覺得唔舒服，用新工具說服佢只會令人更一頭霧水。 舊例子做幾次反而效果好d。
作者: petline 時間: 12-12-9 09:05
lillymarie 發表於 12-12-9 08:43
有本書作者給予一些guidelines for effective algebra study, 以下是我覺得可以參考的幾點：
1。starting ...
但係用ahead schedule 呢一招真係可以令學生谷底反彈，先教一次令佢上堂易 d 聽得明，信心返左黎，學生士氣同正面動機會好好多，但家長/老師就要花多d功夫啦。
作者: lillymarie 時間: 12-12-9 09:30
petline 發表於 12-12-9 09:05
都係基本學習習慣姐。日做日清，緊密跟進。
但係用ahead schedule 呢一招真係可以令學生谷底反彈，先教一 ...
小朋友怕悶怕重覆，若 algebra 必需靠多做多練才能掌握得好， 這就是考家長的地方。
作者: ANChan59 時間: 12-12-9 10:07
lillymarie 發表於 12-12-9 09:30
我現在認為囝囝的問題，是他做數做得不夠，同時他學校教的又太淺。其實老師給他的題目，他說“10秒”就做完 ...
作者: laorenjia 時間: 12-12-11 16:14
本帖最後由 laorenjia 於 12-12-11 16:17 編輯
所以拆完後面要寫負，好似新年派例是，屋(括號) 內每個細路 (數字) 都有一封。
I like your 派利是 method. However, you are more biased to "how" instead of "why". In a class of students, there are always one or two who are not only curious about "how" but also about "why". I was slow in mathematics (I gather I still am). I was always the last one in class to understand what the teacher taught, but I hated most was when the teacher told me that I did not need to know why yet. The beauty of maths is that with just a couple axioms, you can almost construct the whole mathematical palace youself step by step. Unlike other subjects such as chemistry or physics, there are nothing in primary and secondary school maths which cannot be traced back to those few axioms.
We learn distributive law in primary school; we learn operation of negative numbers in F1. With both, we can understand why the brackets are removed this way. Of course, we need your 派利是 method to remember the rule.
作者: lillymarie 時間: 12-12-11 20:59
我大囝正正就是想知道why, 他說若不明白怎能記住？
不過，我們暫時把這問題放下，上網找到一本教algebra的免費書，由pre-algebra學起。
作者: cstchan 時間: 12-12-11 23:07
本帖最後由 cstchan 於 12-12-11 23:15 編輯
lillymarie 發表於 12-12-7 20:07
和大囝計數，a - (b - c) = a - b + c.他不明白為何會是 “+ c"
我自己數學鈍只靠背書”負負得正“， 但囝 ...
試試跟孩子用以下情景來計算 10 - (2 + 1) 和 10 - 2 - 1︰
籃子內有10個橙。媽媽打算給哥哥2個橙、弟弟1個橙。
情況一︰媽媽先把2個給哥哥的橙和1個給弟弟的橙放在膠袋內才取走分給兩兄弟，所以籃內剩下 10 - (2 + 1) 個橙。
情況二︰媽媽先取2個橙直接給哥哥，然後再取1個直接交給弟弟，所以籃內剩下 10 - 2 - 1 個橙。
拆括號等於不用膠袋，所以括號內所有數值前的正負號都要倒轉。
其實在 a - (b - c) = a - b + c 中，不是單單由 -c 變 +c，還有 +b 變 -b！
作者: cstchan 時間: 12-12-11 23:14
本帖最後由 cstchan 於 12-12-11 23:15 編輯
lillymarie 發表於 12-12-9 07:08
a(b - c) = ab - ac
我看書這似乎是一個規則（distributive rule），講了幾次，囝囝仍說不明白，但又講不出不明白甚麼。睇書話algebra要多做題目才能明白當中概念，所以接下來會讓囝囝多做algebra的題目。
至於乘法分配性質，可以考慮計長方形面積，先以 a(b + c) = ab + ac 較容易理解︰
ab + ac 是用來計算兩分別「長b闊a」和「長c闊a」的長方形面積總和。
因為兩個長方形有相同的闊，所以可以合成一個「長(b+c)闊a」的長方形，其面積可直接以 a(b + c)來計算。
作者: laorenjia 時間: 12-12-12 17:34
cstchan 發表於 12-12-11 23:07
試試跟孩子用以下情景來計算 10 - (2 + 1) 和 10 - 2 - 1︰
籃子內有10個橙。媽媽打算給哥哥2個橙、弟弟1 ...
Your analogy works fine with 10 - (2 + 1). Can you think of one for a - (b - c) as well?
作者: cstchan 時間: 12-12-12 20:19
本帖最後由 cstchan 於 12-12-12 20:19 編輯
laorenjia 發表於 12-12-12 17:34
Your analogy works fine with 10 - (2 + 1). Can you think of one for a - (b - c) as well?
Again, we can consider a basket with 10 oranges.
At first, mother would like to give 2 oranges to her son. So she picks up 2 oranges and puts them into a plastic bag. However, she changes the idea and would like to give 1 orange to her son. So she takes out 1 orange and put it into the basket again. The number of oranges in the basket will be 10 - (2 - 1) where (2 - 1) is the number of oranges in the plastic bag.
If there's no plastic bag, then mother will pick up 2 oranges to her son. When she changes her mind, she will get back 1 from her son and put it back into the basket. The number of oranges will be 10 - 2 + 1.
作者: nadal 時間: 12-12-12 23:21
回復 laorenjia 的帖子
The proof is easy.
a - (b -c)
= a + c - (b - c + c) (effectively this is just moving both numbers along the number line)
= a + c - (b - 0)
= a + c - b
Done.
作者: cellon 時間: 12-12-12 23:41
本帖最後由 cellon 於 12-12-12 23:43 編輯
interesting...
作者: Annie123 時間: 12-12-13 01:16
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作者: lillymarie 時間: 12-12-13 06:34 標題: 引用：Quote:lillymarie+發表於+12-12-7+20:07+和
原帖由 Annie123 於 12-12-13 發表
負負得正 is a rule that students learn when they learn multiplication of negative numbers.
And this ...
The number line concept is good enough for him.
Now we focus on pre-algebra first, just to make sure he won't miss any important concepts.
作者: lillymarie 時間: 12-12-13 06:39 標題: 引用：+本帖最後由+cstchan+於+12-12-11+23:15+編
原帖由 cstchan 於 12-12-11 發表
本帖最後由 cstchan 於 12-12-11 23:15 編輯
作者: lillymarie 時間: 12-12-13 06:40 標題: 引用：+本帖最後由+cstchan+於+12-12-11+23:15+編
原帖由 cstchan 於 12-12-11 發表
本帖最後由 cstchan 於 12-12-11 23:15 編輯
作者: sell2japan 時間: 12-12-13 09:17
petline 發表於 12-12-9 09:00
work定係唔work，係要睇學生feedback，唔係口講，不如等結果說話吧。
我教了十幾年數，只知道 algebra 要 ...
「靠模仿同慣性去做」的確如此
作者: 1234ats 時間: 12-12-13 12:30
楼主提出的问题已有很多网友提供不同的解释方法。
但我觉得这些很基本的数学观念问题应该在学校里解决，而不应留给家长或补习社解释。为何学生遇到不明白时不发问？为何老师没有更好渠道了解学生是否真的明白？似乎大家都普遍接受由补习老师教导学生，那学校老师的角色又是什么？
我问过一些需要补习的学生为什么老师教完也不懂，几乎清一色的答案都是不明白老师讲什么，问也没用。
为何芬兰学生放学后不需要补习而我们家长却要花金钱精力做就这么多补习天王？
我不是埋怨谁，只是觉得我们今天的教育模式对一部分学生是非常无效和低效率的，教育界好应多思考这问题。
作者: awah112 時間: 12-12-13 15:11
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作者: lillymarie 時間: 12-12-13 16:26
老師在課堂上只能照顧到大多數學生的進度。要拔尖補底就靠自己了。
作者: ha8mo 時間: 12-12-13 16:34
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作者: laorenjia 時間: 12-12-13 17:10
cstchan 發表於 12-12-12 20:19
Again, we can consider a basket with 10 oranges.
At first, mother would like to give 2 oranges to ...
very good indeed. 
作者: laorenjia 時間: 12-12-13 17:21
本帖最後由 laorenjia 於 12-12-14 02:02 編輯
The proof is easy.
a - (b -c)
= a + c - (b - c + c) (effectively this is just moving both numbers along the number line)
= a + c - (b - 0)
= a + c - b
Done.
作者: nadal 時間: 12-12-14 10:25
Don't get too caught up with the 'number line', its nothing more than a tool for visualizing numbers.
If you really want to put the question to the test, here is how I think you can do it.
The numbers 'A', 'B' and 'C' can all be represented on the line as segments of different length.
Visually, you can now also see that 'B-C' is the difference between B and C (in red).
Therefore, to find A - (B-C), you are for the difference between A and "the difference between B and C".
Move segments A as below and you can visually see why A - (B-C) = A + C - B.
From:
[------- B ---------]
[----- C ------]
[------------- A ------]
To:
[------- B ---------]
[------------- A ------][----- C ------]
作者: cstchan 時間: 12-12-14 13:18
1234ats 發表於 12-12-13 12:30
楼主提出的问题已有很多网友提供不同的解释方法。
但我觉得这些很基本的数学观念问题应该在学校里解决，而 ...
教育界要反思之餘，整個香港社會也要反思。沒有芬蘭那種對人的包融、信任和尊重，是不可能得到芬蘭的成果。
作者: Radiomama 時間: 12-12-15 10:41 標題: 回覆：請教數學問題
There's a website:
'Khan academy' which helps a lot.
It's free to sign in.
作者: cellon 時間: 12-12-15 15:26
本帖最後由 cellon 於 12-12-15 15:29 編輯
cstchan 發表於 12-12-14 13:18
曾看書知道芬蘭都是近入學，學校沒有好壞之分。而老師會因為照顧弱的學生時而拖慢進度，但所有學生都願意等 ...
芬蘭不單只學校比香港好......
以下是芬蘭監獄的一個標準囚室：
作者: laorenjia 時間: 12-12-18 01:23
nadal 發表於 12-12-14 10:25
From:
[------- B ---------]
[----- C ------]
[------------- A ------]
To:
[------- B ---------]
[------------- A ------][----- C ------]
What you are using here is not the number line. However, I certainly agree we can demonstrate the concept to the children using paper strips of different length.
作者: laorenjia 時間: 12-12-18 01:33
本帖最後由 laorenjia 於 12-12-19 22:32 編輯
petline 發表於 12-12-8 21:30
所以拆完後面要寫負，好似新年派例是，屋(括號) 內每個細路 (數字) 都有一封。
諗真啲，派利是未夠形象。不如......
強盜揸刀 ( - 號)入村搜掠, 連屋 ( ) 都拆埋，重將屋裏面啲嘢「摷」到反轉哂，＋變－，－變＋：
故 - (b - c) = - b + c。
作者: nadal 時間: 12-12-18 11:41
laorenjia 發表於 12-12-18 01:23
What you are using here is not the number line. However, I certainly agree we can demonstrate the concept to the children using paper strips of different length.
A 'number line' is nothing but a tool to represent numbers as lengths on a line.
Additions and substations are simply lengths extending or contracting in opposite direction.
Brackets are merely extensions and/or contractions that must be performed ('calculated') before others.
There are many many ways to represent a length on a line. You could do it with walking steps, a drawn line on a piece of paper, a paper strip you cut out, etc. It really doesn't matter which way you choose.
To be honest, I did not know you were in an argument with cellon on 'how to utilize a number line to remove a bracket' until now. When I first made my post, I use the term 'number line' because it is perhaps the most used tools to demonstrate elementary addition/subtraction problems. Period.
So If you insist on arguing for the sake of arguing (or to save face?), then OK, I give up. 
作者: laorenjia 時間: 12-12-19 22:46
本帖最後由 laorenjia 於 12-12-20 15:51 編輯
nadal 發表於 12-12-18 11:41
If you agree to a demo using paper strips, then didn't you just also agree to a method utilizing the ...
You have misunderstood me. I meant literally my question. I am genuinely interested in knowing how to use the number line to remove the brackets, just like when I said your proof was elegant. I agree that the number line "is perhaps the most used tool(s) to demonstrate elementary addition/subtraction problems" but I don't know how to show a - ( b - c ) with the number line. Let us start at b, we move backforward by c to reach the point b - c. Using the number line, we then have to jump to the point - ( b - c ) on the other side of zero. Then we move forward by a. The process is clumsy and unconvincing. My understanding is that we normally use the number line to show x – y by starting at x but not starting at y as in our case. In addition, I have not seen any textbook using this method. That is why I added the note that using paper strips of different length is easier for kids to understand.
To win an argument is never my purpose of writing in this forum.
作者: sillydad 時間: 12-12-20 14:56
lillymarie 發表於 12-12-7 20:07
和大囝計數，a - (b - c) = a - b + c.他不明白為何會是 “+ c"
我自己數學鈍只靠背書”負負得正“， 但囝 ...
let a=10, b=5,c=3
a-(b-c)
=10-(5-3)
=10-2
=8
a-b+c
=10-5+3
=5+3
=8
so a-(b-c)=a-b+c
這樣的證明好像有點問題,因為只假設了一種情況
但足以令小朋友相信和明白"負負得正"
如果他看得出證明有問題,他的數學就係幾好,你都唔駛太擔心啦
作者: lillymarie 時間: 12-12-20 15:50
sillydad 發表於 12-12-20 14:56
a-(b-c)
let a=10, b=5,c=3
a-(b-c)
至於拆括號，這幾天囝囝重頭由HCF／LCM，分數點數，rational numbers/absolute numbers, value/number lines, equalities/inequalities等等再學過。慢慢建立上去，或許就不再有太大問題。
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