數學題 5x3 原來並不等於5+5+5

cow

5x3 can be 5 times 3, and can also be 5 multiplied by 3.

shadeslayer

原帖由 cow 於 15-10-31 發表
5x3 can be 5 times 3, and can also be 5 multiplied by 3.

No, 5x3 in words are five time three or 3 multiplied by 5. Not 5 multiplied by 3. This is English. Google it if you don't believe.

cow

Fundamental Concepts of Mathematics
The symbol of multiplication "x" or "." , is read "times" or "multiplied by"

Paperback Oxford English Dictionary
The sign x, used to indicate that one quantity is to be multiplied by another.

Discover Multiplication
5x3 = 15
Five multiplied by three is equal to fifteen

Ks1 Success Revision Guide MathsHow many fish are there altogehter in 3 bowls?
3 lots of 4 is the same as 4+4+4 which is 12
This means 4 multiplied by 3, or 4 x 3 = 12
Remember that the sign (x) means 'multiplied by'


A treatise on algebra by Elias Loomis p6
The sign X (an inclined cross) is employed to denote the multiplication of two or more numbers. Thus, 3X 5 signifies that 3 is to be multiplied by 5, making 15. In like manner, aXb signifies a multiplied by b

shadeslayer

原帖由 cow 於 15-10-31 發表
Fundamental Concepts of Mathematics
The symbol of multiplication "x" or "." , is read "times" or "mu ...

There has never been any questions on the meaning of "x" - multiplied by.

eviepa

其實我不知為什麼要辯論？從所有大學、中學、小學老師的評分標準，以至日常的用法，5x3都可以用作表達5+5+5或者3+3+3+3+3。

看judy的例子：

應用題A：橙每個5元，今買3個，要多少元？
絕大部分學生答此題時，都會列出：
5 x 3 = 15
原因很簡單，5在前，3在後。
我未見過，亦不能想像有老師質疑過這種答法。


應用題B：一個人買了5個價值3元的橙，他共付出了多少錢？
絕大部分學生答此題時，都會列出：
5 x 3 = 15
原因很簡單，5在前，3在後。
我未見過，亦不能想像有老師質疑過這種答法。

例A 是 5+5+5
例 B 是 3+3+3+3+3
但都是對的列式

中學、大學不會出這種簡單的題目，但大部分應用題，所列的式都包含乘數，而列出5x3 或者3x5，從來都是學生的權利，從未聽說過因為次序的問題而被扣分。

在日常生活中，「橙每個5元，今買3個，要多少元？」這樣的數，我從來都自由地用5x3或者3x5去和朋友溝通，從來未被別人指正過。

MASAYA

eviepa 發表於 15-10-31 20:54
其實我不知為什麼要辯論？從所有大學、中學、小學老師的評分標準，以至日常的用法，5x3都可以用作表達5+5+5 ...

如果問題冇規定, 我覺得5x3同3x5都可以, 香港學校都好似冇規定, 但寫既人應該要清楚自己係寫(5x)3定5(x3), 答案0岩冇問題既, 如果只係求分數係絕對冇問題

eviepa

符號的意義是約定俗成的，例如，639/3必定等於213而不可是1/213，前者是被除數，後者是除數，是清清楚的。學生搞錯了，老師就必定扣分。

如果約定俗成的結果是5x3必定要等於3+3+3+3+3+3，不能等於5+5+5的話，小學中學老師、大學教授對乘法的評分標準就不會是任由學生將任意安排次序。既然所有老師都對5x3的先後次序不理，照樣當作正確答案，就是默認了5x3可以表示3+3+3+3+3+3或5+5+5。


**PS: 大約十年前，阿女問我：
「639/3=?」
「213」
「不是，是9。」
「何解？」
「綠衫狗除衫，咪等於狗囉。」

jane-wang

这个橙子的例子，虽然交换结果一样，从乘法的数学概念而言，是不一样的，特别是刚开始接触乘法概念，老师初规定或强调这个顺序是有必要的，熟练到一定程度，无论是老师讲解或自己发现，相乘的两个数原来是可以交换的。不过这个和初规定N*M是N个M相加（或另一种情况）。这种约定俗成是对数学思维是有帮助的，即3+3+3+3+3和5+5+5数学现象是不一样的，虽然结果一样

MrBeast

哥有3隻橙，弟有2隻，他們共有多少隻？
可以咁答嗎？ 1+4=5

MASAYA

eviepa 發表於 15-10-31 21:30
符號的意義是約定俗成的，例如，639/3必定等於213而不可是1/213，前者是被除數，後者是除數，是清清楚的。 ...

咪話題目冇指定, 我覺得兩個寫法都可以囉, 不過改卷老師突然話唔得, 睇下你告唔告上校長度或教局度喇
我只係想講, 3+3+3+3+3定5+5+5學生自己要知囉, 佢可以唔駛畀人知, 但自己要知, 好似寫程式咁, 你唔駛話畀人知用咩方式寫, 但自己要清楚, 唔係既話玩得深入少少自己玩完

shadeslayer

我明老師點解話錯。題目

5x3 = ，用英文讀出來的時候 （five times three equals)，如果你寫 5+5+5，即係孩子未掌握 five times three equals 的意思。

但，有無需要咁嚴謹呢？ ok, 好明顯老師覺得要。

Artie

shadeslayer 發表於 15-11-1 11:19
但，有無需要咁嚴謹呢？ ok, 好明顯老師覺得要。

唉，所以話，知道老師要求，下次跟就得啦。
當然，我都會話比小朋友知， 3x5 和 5x3 ，其實答案是一樣。
例如 3+5 和 5+3 一樣。
但 math expression ，大部分先後次序好重要，例如減數除數，先後不可對掉，例如 5 -3  和 3-5 就不同。
因此老師咁執著？

Artie

其實，我地自己來看 3x5 和 5x3 就覺得一樣。
但是小朋友初學 5x3 ，老師一定有教是點樣做。
可能老師已教，5x3 是 5 times 3，即是將 3 "寫 5 次"  (5 times 嘛)，因此 5x3 = 3+3+3+3+3 ，就係咁囉。
鬼佬教小學生數，教得好慢，我仔當年學乘數，係橫跨兩年學，日日上堂做 exercise ( 冇 homework ，但好多 classwork )，其實都應該知老師要D乜。

3KIDSMAMMY

我見亞女P.2嘅數學書係咁教，所以衣家好亂...究竟點至岩呢？

15-11-1 17:20 上傳

下載附件 (10.83 KB)

cow

3x4 = 3 times 4 = 4+4+4=12
3x4 = 3 multiplied by 4 = 3+3+3+3=12

shadeslayer

原帖由 cow 於 15-11-01 發表
3x4 = 3 times 4 = 4+4+4=12
3x4 = 3 multiplied by 4 = 3+3+3+3=12

3x4, 4 is the multiplicand, and 3 is the multiplier.
So when we say it, it should be:
1) three times four. Four is the multiplicand and the 3 is the multiplier.
2) since 4 is the multiplicand, it can also be said as 4 multiplied by 3.

The math expression is 3x4 and the 4 is the multiplicand. The way we say it (whether we use "times" or "multiplied by") does not change the meaning of the expression.

https://en.m.wikipedia.org/wiki/Multiplication

https://en.m.wiktionary.org/wiki/multiplicand

https://www.mathsisfun.com/definitions/multiplicand.html

I agree it is confusing and is probably more trouble than it is worth.

JustAParent

原帖由 3KIDSMAMMY 於 15-11-01 發表
我見亞女P.2嘅數學書係咁教，所以衣家好亂...究竟點至岩呢？

其實件事嘅背景應該搞清楚，呢次爭論嘅重點其實唔係5個3或者3個5, 而係common core中有D根本冇mathematical logic既規則要同學遵守，有人覺得無謂甚至"教壞"學生，但有專家同學者認為把一些東西系統化規範化對學生長遠學習數學有幫助，誰是誰非，真係再拗幾年都拗唔完。而事實上，類似今次有關 common core既爭抝常有發生，只不過以前既討論涉及較深既數學，一般人根本睇唔明，今次你講5ｘ3，就咁囉。

Obiepapa

學野

cow

回覆 shadeslayer 的帖子

3(multiplier) x 4(multiplicand) = 3 times 4 = 4+4+4=12
3(multiplicand) x 4(multiplier) = 3 multiplied by 4 = 3+3+3+3=12

There is no common agreed definition saying that the first factor must be multiplier or multiplicand. If you search "multiplier multiplicand" in google image, you will see either multiplier or multiplicand is used as the first factor.

shadeslayer

原帖由 cow 於 15-11-01 發表
回覆 shadeslayer 的帖子

3(multiplier) x 4(multiplicand) = 3 times 4 = 4+4+4=12

We don't need google to unite. I don't have common core information so i googled a bit and gave you the links. There are definitely more definitions that favour the first number being the multiplier. However, it does not matter. The only thing that matters is the definition that common core use as it all started with the common core curriculum. And it is pretty clear what definition they use.

If the teacher teach students the first number being the multiplier as part of common core in class, the students should not say "Teacher, but some sites in Google says otherwise, I can use either, right?"  What is more, the "times" way of saying this aloud "gives" you which one is the multiplier (again the first number). The "times" way of saying is used much more often in practice.