请教數學題

laorenjia

kenwong888 發表於 15-2-17 23:56
I really do not understand your logic!
Maybe modern people need more training!

抵唔住頸，試多次:

周界32，若是正方形，邊長就等於32/4=8。面積就係8x8=64。跟手就證明呢個係最大面積，若果圖形係長方形，長加濶等於32/2=16。簡單啲用整數諗，長同闊嘅組合有9x7, 10x6, 11x5, ..., 14x2, 15x1, 面積一路細落去，所以正方形就最大。

kenwong888

laorenjia 發表於 15-2-18 16:05
抵唔住頸，試多次:

周界32，若是正方形，邊長就等於32/4=8。面積就係8x8=64。跟手就證明呢個係最大面積， ...

Does the question require integer answer?

kenwong888

laorenjia 發表於 15-2-18 10:47
都話妻子氹我啦。唔用代數或微積分講得你明，唯有另請高明。

I suppose you can try mathematical induction but you need to establish the base case, not easy too right???

laorenjia

The integer answer is the result, not the requirement. I only used integers to show the change in the area. It is not mathematical induction. You've confused yourself with all the techniques in high school maths. I thought we were talking how to explain to a primary school kid.

kenwong888

laorenjia 發表於 15-2-18 16:37
The integer answer is the result, not the requirement. I only used integers to show the change in th ...

Then you need to show the logic is also correct for real number right???
Seems your logic is assuming something which is not stated explicitly, a very common mistake to take all for granted right???

compsognathus

As an S1 student in SIS, I would like to try to tackle this problem.

Firstly, there is the problem of whether the child's textbooks consider a square a rectangle. I will now explain why a square is a rectangle.

-By definition, a quadrilateral is a polygon with four sides. Therefore, one can infer:
-A parallelogram is a quadrilateral with opposite sides parallel. Therefore, one can infer:
-A rectangle is a parallelogram with four right angles. Therefore, one can infer:
-A square is a rectangle with four congruent sides. Therefore, a square is a rectangle.

Secondly, there is the problem of whether one can only use integers. If one can only use integers, you can simply utilize guess and check to find the answer, even if a square is not considered a rectangle by the child's textbooks. If all real numbers are permitted and a square is not a rectangle, then one starts dealing with calculus. The value of f would then be “infinitesimally close to 64 square feet”.

Completing the square has assumed the rectangle was a square. If one assumes the rectangle is a square, the method holds in theory, but there is a much easier method which has been covered in the second post by jasonsuen. Though, a primary school student may not be able to prove that the square has the greatest area for any quadrilateral with perimeter x.

Therefore, I recommend the guess and check method for a primary student to approach this question. Guess and check is also a learning experience in this case, as students will discover that the square is the quadrilateral that has the largest area for any given perimeter x.

c1234

回覆 laorenjia 的帖子

真是好方法

laorenjia

kenwong888 發表於 15-2-18 22:26
Then you need to show the logic is also correct for real number right???
Seems your logic is assumin ...

A third and the final attempt.


Actually, the proof is vigorous enough as  the function x(16-x) is a continuous function over the domain (0, 16). But this is for you only as you seem to be still obseesed with whether the solution is vigorous enough with the question on real numbers which a normal primary kid will not concern himself with.

kenwong888

laorenjia 發表於 15-2-20 03:14
A third and the final attempt.

Well, you now demand the primary student to have the concept of function, continuity, singularity, and etc. right???
Amazing!!!

laorenjia

kenwong888 發表於 15-2-20 08:06
Well, you now demand the primary student to have the concept of function, continuity, singularity, a ...

Stupid me. I left out an explanation. Now added.

kenwong888

laorenjia 發表於 15-2-20 09:58
Stupid me. I left out an explanation. Now added.

I think good primary students know real number right???
Fractional and decimal number is real number right???
Rational number is real number they need to grasp right???

laorenjia

kenwong888 發表於 15-2-20 10:02
I think good primary students know real number right???
Fractional and decimal number is real numbe ...

In HK the concept of real numbers is formally introduced when students learn about complex numbers but some teachers will talk about them in form 1 when they teach directed numbers n number line. The concept of rational numbers is learned at f2 or f3. Square roots n radical signs have been taken out of primary school maths for ages.

kenwong888

laorenjia 發表於 15-2-20 11:10
In HK the concept of real numbers is formally introduced when students learn about complex numbers b ...

OMG!
OK no real number!!!
You should know definition of rational number right???
Kind of representation of " M / N " is rational number M & N integer right???
Don't scare me if primary student don't know fractional number!!!

laorenjia

kenwong888 發表於 15-2-20 14:01
OMG!
OK no real number!!!
You should know definition of rational number right???

Mankind has been using real numbers since day one, but the concept of real numbers was not known to us until after the great Pythagoras. And why all a sudden a kid will understand what real numbers mean if he knows what a fractional number is?

kenwong888

laorenjia 發表於 15-2-20 14:36
Mankind has been using real numbers since day one, but the concept of real numbers was not known to ...

I am now saying no need to understand real number!!!
But they still know fractional number right???
Seems you have mixed up a lot of concept but just focusing on the "name" or "form"!!!
What is substantial is that they should know there exists fractional numbers but integers!!!
Should they consider the possible answers not limited to integer???

kenwong888

lamyeelok 發表於 15-2-20 15:11
分清楚real numbers 同natural numbers

Exactly true!

laorenjia

lamyeelok 發表於 15-2-20 15:11
分清楚real numbers 同natural numbers先啦！

Sorry for the late reply as the busy holiday routine caught up with me and prevented me from sitting quietly in front of the PC.

I'm puzzled why you have the idea why I've mixed up real numbers with natural numbers even though I had made reference to the number line.

Maybe you're referring to the first part of my statement "Mankind has been using real numbers since day one, but the concept of real numbers was not known to us until after the great Pythagoras." Actually it was not made up by me. Some scientists suggest real numbers are psychologically more primitive than natural numbers. The special role of natural numbers in the history of arithmetic has more with the discrete character of human language. if you are interested in such academic trivia,  you can google for more information.  My point is that primary school kids use fractions but it does not mean they know what real numbers mean, a mistake in logic kenwong888 made.


My table can take care of integers, fractional, irrational numbers (again a kid may not need to know this) depending on what numbers you get for the perimeter.

Mind you it is kenwong888 who first thought fit to use differentiation to tackle a primary school problem and later thought complete the square method also made do and suggested my method was not rigorous enough fot failing to consider real number solutions.

You can also find my following reply to 王家爸爸 useful.

Dear 王家爸爸

You're right about Europeans but...
There are many myths in mathematics, just like in many other things. It is only a myth that zero is a relative new toy for the humans. It's just that the zero symbol as we use today is relatively new but zero as a symbol (placeholder) was already used by the Babylonians and later by the Indians around 200 BC and the Chinese (孫髀算經, rather for the lack of a symbol). Zero as a number was first used in the sixth century AD by the Indians (It is the clever Indians who actually invented the Arabic numbers we use today, another myth in mathematics).
As to negative numbers, it was a new toy for Europeans but it appeared long before in China (孫髀算經 again, red for positive n black for negative) and in India.

My reference to Pythagoras is that only after Pythagoras irrational numbers started to be studied seriously and not just used as the Chinese did with 勾三股四弦五.

The mankind as a whole knew about the negative numbers and irrational numbers long long ago.

laorenjia

lamyeelok 發表於 15-2-23 11:50

毫無疑問人類對數字有概念一定始於"Natural Number"而不是"Real Number”，所以才命名為“Natural" 。其實又何需google找支持論據，試想想大家教導幼兒理解數字概念時，你估解釋何謂natural number容易啲定real number容易啲？

Natural numbers are called as such as they come naturally when we humans count things. No one can argue real numbers are much more sophisticated and more difficult to understand. However it doesn't contradict the proposition real numbers are used by humans even before we had language. Note I used the word "used" here and in the previous posts. Scientists even go as far as saying real numbers are used by animals like pigeons and rats. Again I encourage you to google for more information on this area if you're interested. I value more an academic curiosity in adults than in kids.