請教數學問題

lillymarie

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？

cellon

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？ ...

請問令郎讀幾年級？

lillymarie

升grade 8

cellon

lillymarie 發表於 12-12-7 20:33
升grade 8

Have you tried to teach him this concept using the "number line".

lillymarie

cellon 發表於 12-12-7 20:38
Have you tried to teach him this concept using the "number line".

Thanks for your suggestion! Just found the concept in an algebra book and explained to him. Actually he has heard about it but forgot, and the problem was I myself didn't know why either. Now both of us know, hahaha!

laorenjia

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

lillymarie

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.

cellon

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.

laorenjia

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?

cellon

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.

laorenjia

- 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？

laorenjia

How can the number line be utilised here？I am intrigued.

cellon

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.

JustAParent

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 ...

你所問的是去括號，不是數學上的「負負得正」。亦沒有理由把易的概念化成難的概念去理解。

laorenjia

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.

petline

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。

cellon

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。 ...

Your "新年派例是" method may be suitable for small kids.
But please understand that we are talking about teaching a Grade 8 student.

lillymarie

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。

lillymarie

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！

petline

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。