數學題 5x3 原來並不等於5+5+5

1234ats

我小學時學乘數，老師教
5個X3=15個
5個是被乘數，有名數（個），
3是乘數（沒有名數，代表倍），
15是積，名數與被乘數必須一樣（個）。

5個X3=3個X5都是等於15個
但意思不一樣。

一樓所示的  5X3 是代表 5 groups by 3，所以答案是3 + 3 + 3 + 3 + 3 也是正確。

他們的定義與我們所學的定義次序相反，只要學生清楚，都可接受。

MrBeast

judy 發表於 15-10-30 12:05
5x3=5+5+5
3x5=3+3+3+3+3.

My understanding also.
學完enopi 去傳統小學實比人叩分

1234ats

我不同意第二題扣分。

按他們的定義 4 X 6 等于 4 groups by 6。問題是可以將 one column 看成 one group，也可將 one row 看成 one group，所以兩個 array 都應該正確。

shadeslayer

At the end, multiplication is a arithmetic operation, a way to arrive at a number. And I keep saying it is commutative.  Who cares if the "meaning" is not correct.  Why is the Meaning of multiplication (if it exists) important in primary maths.

JustAParent

shadeslayer 發表於 15-10-30 20:02
At the end, multiplication is a arithmetic operation, a way to arrive at a number. And I keep saying ...

Why do you think the lexical meaning of 5x3 is 5 groups of 3 ?  Just by convention, right ?  That is exactly one of the goal Common Core is trying to achieve -- to standardize the conventions, to standardize the reading of problems.

The Common Core experts explained that the standardization of conventions helps students when they get into advanced mathematics like multi-variable calculus.  This point is a moot.  However, at least, I don't like students writing such array conventions column x row / row x column interchangeably and arbitrarily.

You can criticize Common Core.  You can also say that the strict interpretation of multiplication is pedant.  However, there is absolutely nothing related to the communicative property of multiplication.  The teacher is not a moron.  He must be aware of 5 x 3 = 3 x 5 in the real number domain.

Why is the meaning (semantics) of multiplication important to primary students ?  I would say, if you are serious about mathematics, getting the solution right is not all that matters, even for a grade 3 student.

eviepa

judy 發表於 15-10-30 12:05
5x3=5+5+5
3x5=3+3+3+3+3.

//5x3=5+5+5
3x5=3+3+3+3+3.

我理解係咁。

應用題：橙每個5元，今買3個，要多少元。係唔係應列式： 5*3=15 (元) 呢。//

以我從小學到大學考試的經驗，應付類似的數，列式做5*3 或者3*5都是正確，未試過因此而被扣分，先後次序是學生的自由。

eviepa

有老師覺得，5*3 不等如 3*5。原來在某種情況下，4+3 可以是大過 3+4的！


宋国有一个养猴的老人，喜欢猴子，把它们成群养着，他可以理解猴子的意
思，猴子也可以理解老人的心意。养猴的老人宁可减少他与家人的食物也要满足猴子的需求。不久，他家里的粮食缺乏了，他将限定猴子的食物的数量。但又怕猴子不顺从自己，就先欺骗猴子说：“给你们橡实，早上三颗然后晚上四颗，够吗？”猴子们都站了起来并且十分恼怒。他又说：“给你们橡实，早上四个，晚上三个够了吧？”猴子都非常高兴然后一个个都趴在地上。

shadeslayer

原帖由 JustAParent 於 15-10-30 發表
Why do you think the lexical meaning of 5x3 is 5 groups of 3 ?  Just by convention, right ?  That i ...

Thanks for explaining. I am not saying the teacher is a moron, I am saying when you explain to the students the meaning of 5x3 is not the same as 3x5, how do you explain commutative property of the operation at the same time, i.e. They mean different things but they are eventually the same.

shadeslayer

原帖由 shadeslayer 於 15-10-30 發表
Thanks for explaining. I am not saying the teacher is a moron, I am saying when you explain to the s ...

Or have I underestimated a year 3 student.

akys

eviepa 發表於 15-10-30 22:07
//5x3=5+5+5
3x5=3+3+3+3+3.

試下問阿仔，佢答法跟你一樣，話學校係咁教， 要分被乘數和乘數。係咪英數和中數嘅分別？話時話，我中學讀英數又唔會考呢啲嘅。

shadeslayer

英文："Five times three" 意思上確是 3+3+3+3+3.

但題目是阿拉白數目字：5x3=？，咁可不可以寫成：5x3=3x5=5+5+5=15

JustAParent

shadeslayer 發表於 15-10-30 22:21
Thanks for explaining. I am not saying the teacher is a moron, I am saying when you explain to the s ...

Should we rephrase the question using our familiar terms ?  In 5x3, which is multiplicand ?  Which is multiplier ?  The distinction between multiplicand and multiplier still lies in the "assumed" meaning of multiplication.  How did we teach our children the commutative property ?  By examples ?  Or just brainwashed them that both multiplicand and multiplier are factors and factors can be multiplied by any order ?

goodenough

Yanamami  我覚得老師果个係3x5?  發表於 昨天 10:52
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除非老師出題目前有教乘數互換，否則，老師可堅持佢嗰個係啱，即是應該係5 X 3。

於是，扣分並不是源於數學上的原則性問題，因為同學可從其他途徑學到乘數互換，只是代表了同學上堂沒有聽書！

jane-wang

原帖由 judy 於 15-10-30 發表
5x3=5+5+5
3x5=3+3+3+3+3.

应该是3*5，即5+5+5，三个5相加，每个橙的价钱加起来。这是乘法最原始的概念，也是最重要的概念。将来到很抽象的代数，有些非实数的是不可以交换的，只是实数乘法满足交换率

jane-wang

我现在闲时和女儿玩就会告诉她2个5加起来就是2*5，然后继续告诉她3个5相加就是3*5，这就是乘法概念，后面她自然可以推到任何N*M就是N个M相加M+M+......+M，所以这样玩加法，就顺便玩乘法了，我没有教她乘法口诀，现在她做乘法就是用加法的方法，乘法口诀我就等学校老师教。我始终认为强化数学概念在生活中的实际背景，理解概念是最重要的

shadeslayer

原帖由 JustAParent 於 15-10-31 發表
Should we rephrase the question using our familiar terms ?  In 5x3, which is multiplicand ?  Which  ...

I am not a teacher. I don't know how to teach students the two are simultaneously different and the same to year 3 students.  And how to explain the practical implication of the difference.  And what might happen if we put things in a "wrong order".

As I said I might have underestimated 7 years olds.

judy

有老師覺得，5*3 不等如 3*5。原來在某種情況下，4+3 可以是大過 3+4的！

孩子小時，我也很他們解释為何4+3=3+4。加法很易理解，我說：孩子，假使你去買一個七元之蘋果，左邊袋放4元，右邊袋放3元。先掏左邊再掏右邊，和先掏右邊再掏左邊是不是一樣？4x3=3x4就難解释了，我要擺好棋子，每列幾個，分多少粒數一次，再轉90度，又數一次，才算解釋清楚。

再回到我的應用題：每天吃5個橙，3天吃了多少？是5嗰橙加3次，清清楚楚。你寫5x3 或 3x5 都可以，但理解上一定足5+5+5，不是3+3+3+3+3。 我之所以懷疑教師出錯，是因自小到大，5x3 我都寫成 5+5+5。

3+4 < 4+3 ?

是的，現實中，朝4暮3 可能比朝3暮4 更健康。

cow

我覺得common core定義得唔好。
橙每個5元，3個橙要多少錢？
5元x3= 15元
3個橙15元，每個橙要多少錢？
15元÷3 = 5元

如果寫成
3x5元 = 15元
那麼做乘除數時單位的位置就不同了


5^3 = 5x5x5
5x3 = 5+5+5

shadeslayer

原帖由 cow 於 15-10-31 發表
我覺得common core定義得唔好。
橙每個5元，3個橙要多少錢？
5元x3= 15元

Common core 係鬼佬野，要由英文考慮。5x3 is "Five times three" or "three multiplied by 5"

橙每個5元，3個橙要多少錢？不是寫成5x3，「應該」寫成 3x5.

I totally understand where the teacher is coming from but I am baffled as to how to teach students both this lexical meaning of multiplication and commutative at the same time. And why is the meaning the words so important.

On the other hand, the array question is totally different. The teacher was right and the student was wrong, period.

1234ats

Shadeslayer

They mean different things but they are eventually the same.
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I would say
They mean different things but they are equal, just like a ten dollar note is different from ten one-dollar coins( in weight, colour, size, volume etc) but they have the same values.