請教數學問題

laorenjia

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.

laorenjia

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.

Elegant. But while conceptually I can understand, as you say, it is just moving both parts along the number line, is there a way to actually show it on the number line?

nadal

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 ------]

cstchan

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 ...

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待學得慢的同學，家長也不會投訴老師進度慢。看罷印象很深，因為覺得香港沒有這種文化。大家強調拔尖補，但只顧自己的需要。自己是尖，便著眼拔尖。自己是底，便緊張補底。
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。

Radiomama

There's a website:
'Khan academy' which helps a lot.
It's free to sign in.

cellon

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：

laorenjia

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.

laorenjia

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。

nadal

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.

If you agree to a demo using paper strips, then didn't you just also agree to a method utilizing the number line?

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.

laorenjia

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.

sillydad

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 ...

a-(b-c)
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

lillymarie

sillydad 發表於 12-12-20 14:56
a-(b-c)
let a=10, b=5,c=3
a-(b-c)

多謝你的建議。負負得正這問題，我和囝囝去看number line, 正就向右，負就轉方向往左，若再負（即再轉方向）就變成正了。這已經滿足了囝囝，再沒有問題了。既然他說沒問題，我亦沒有再和他去深究。

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。