Writing Wall
Show our writings on the wall, watch a movie, read a story, please write down something, what you want to let others know……There are no endings, only beginnings!
No.1 from a Year 6 student
ROAD’S END
TASK: Today, you will independently write a recount of events as ‘a fly on the wall’.
You may use Passage A and the notes on cohesion as a guide. You should show an understanding of ordering events appropriately, as well as include some cohesive devices.
Please write your ‘fly on the wall’ recount below.
Why is a weirdly dressed man standing in the middle of a desert? Why was he touching his cars wheels? How can he survive? How is he alive from the deadly car crash? Questions were racing around my head as the driver approached the man like cars on a race track. Soon the weirdly dressed man stuck out his thumb, and the driver down, without a word, the man climbed in, and slammed the door close.
As the drove further, the driver kept asking him the same questions, but no words came out of his mouth. It seemed like the heat from the engine was boiling up his patience, he looked at his watch, TIME. The driver asked him the penultimate question on the penultimate turn on the penultimate minute of his one and only life. The man just gave him a sarcastic stare, the driver gave a stare in return. Then through the mirror like reflection of the man’s goggles, the driver was finally seeing what the man saw, his penultimate second. The car’s door swung open, the man dashed out, and the car flew down the cliff in a heartbeat, probably the driver’s last heartbeat.
How did he do that? Is he indestructible? Is he a ghost? Is he a murderer? Questions were racing around my head as the NEXT driver approached the man like cars on a race track.
No. 2 from ……maybe you?!
Critical Thinking and Writing: Is mathematics invented or discovered?
watch below video and write down your opinion!
from a year 6 student:
Maths is a subject, we either like it or dislike it. Maths is almost the basics of everything, from physics to chemistry, so it is also one of the most important subjects that we learn in school. Maths has a long history, stretching all the way back to thousands and thousands of years ago. It has such of a long history to make us doubt that if it ́s invented by humans, or if it ́s discovered by humans. There are a unimaginable amount of different theories, but there isn’t one that is correct yet. So this leads us to the brain draining question of is maths discovered or invented.
Just like everyone else, we can only convince other people to think what we think, but not force them to think what we think, so we can only use what we know and what has has already proven to us to prove and give evidence to what we think.
For me, I think that maths is already here there in the nature before and animas with high intelligence existed. For example, were there 2 rocks before animals evolved into human? Yes. Were there 2×2 rocks before we existed? Yes. Maybe other types of creatures or animals might also have a way of understanding maths that we didn’t study or know about.
Secondly, what makes me think that it existed before us is that maths is not a physical thing, it is just like a type of notation that exists and taught to us and stays in our minds. Sometimes, the way we write numbers gets too familiar with us, therefore it almost seems like a language. A universal language that we can all understand. Just like everything else, there is a different way to notate, I guess that maths was just being notated in a different form before we existed.
In addition form the last point, I think that maths is discovered is that because people were using clock like tools that we now use numbers to represent long before any maths theories that was spread worldwide was written. For example, a long time ago, people drew a clock on the floor where they used shadows from the standing object in the middle as the clock hands. The art were used around it to represent different times, maybe those were another generation of mathematical notation, maybe they used animals or different art to represent the number that we all know today. That is just another way of notating, so people then have a different way, so before that, there is a chance that there was another way.
Everybody can have different thoughts about the birth of maths, but I strongly believe that maths is discovered by humans, and not invented. Those are just my opinions, everybody can have different opinions.
Teacher’s comment:
12/05/2020:
Around the end of last year (2019), I discovered a unique way to find the n-th term of any diagonal/ oblique series of binomial coefficients in the Pascal/ Yang Hui Triangle and invented the General Formula to do so by inventing an ‘up factorial’ concept as opposed to the already known and used ‘down factorial’ concept – by using user-friendly numbers, bracket and punctuation marks on computer keyboard to represent the General Formula.
I told myself that I must write about it during the CNY (Chinese New Year) holidays.
This is what I wrote on 27/01/2020 in both my physics blog (http://physicsf45spm.blogspot.com) and Add-Maths blog (http://addmathsf45.blogspot.com).
‘27.01.2020 (2nd update, PJ Selangor, Malaysia).
This post is for math enthusiasts – not for students preparing to sit for IB math or IGCSE Math 0580, 0606, 0607 or CIE Math 9709 or Math 9231 unless the students would like to know beyond their exam syllabuses.
It is about the general formula /equation, N = (n…n+(p-1))!/p!, to work-out:
1) the individual formula for the oblique/diagonal number series in Pascal/Yang Hui triangle; and,
2) the value of any n-th term in the oblique/diagonal number series in the triangle.
(The Pascal /Yang Hui triangle:
Image result for pascal triangle – by Google)
You see, currently, students of binomial theorem know that the horizontal series of numbers /binomial coefficients can be found by using binomial theorem or, to a limited extent, the Pascal /Yang Hui triangle.
What about the terms in the oblique/diagonal series of binomial coefficients?
As you can see, the binomial, say (x + y) or (a + b):
– to the power 1 yields an oblique number /binomial coefficient (BC) series called ‘natural numbers’: 1 + 2 + 3 + 4 + … + n, where the n-th term = n;
– to the power 2 yields an oblique number /BC series called triangular number series of: 1 + 3 + 6 + 10 + 15 + … + n, where the n-th term = n(n+1)/2;
– to the power 3 yields a diagonal number /BC series called tetrahedral number series of: 1 + 4 + 10 + 20 + … + n, where the n-th term = n(n+1)(n+2)/6;
– to the power 4 yields a diagonal number /BC series of: 1 + 5 + 15 + 35 + … + n, where the n-th term = n(n+1)(n+2)(n+3)/24.
The n-th term formula for each oblique/diagonal BC series is currently analysed]and work-out individually – but I saw a pattern that led me to the discovery and invention of the aforesaid general formula that can be used:
1) to work-out the individual formula for the n-th term of any oblique/diagonal BC number series; and hence
2) to find the value of any n-th term in any oblique/diagonal BC series in the Pascal / Yang Hui triangle.
I have googled /searched far and wide and it seems that no one had found it this general formula before! If you are aware of anyone having discovered and invented this general formula earlier than me, kindly let the world know through the comment section of this post – kindly give citation to your source material(s).
The General Formula
N = (n…n+(p-1))!/p!
where,
1) n refers to the n-th term of the diagonal BC series generated by the binomial to the power p;
2) (n…n+(p-1))! refers to an ‘up factorial” idea introduced n developed by me, where,
– an up factorial that starts with n and ends with n is:
(n…n)! = n
– an up factorial that starts with n and ends with (n+1) is:
(n…n+1)! = n(n+1)
– an up factorial that starts with n and ends with (n+2) is:
(n…n+2)! = n(n+1)(n+2); and, so on.
3) p! refers to the conventional ‘down factorial’ which starts with p and ends with 1. Thus,
p! = p(p-1)(p-2)x…x3x2x1
5! = 5x4x…x2x1
1! = 1
0! = 1
In the Pascal/Yang Hui triangle (please see Google results):
When the power p=1, a natural number series (1+2+3+4+5+…+n) is generated. And, using the general formula:
N = (n…n+(p-1))!/p! = (n…n+(1-1))!/1! = (n…n)!/1! = n i.e. N = n;
When p=2, a triangular number series (1+3+6+10+15+21+…+n) is generated. And, using the general formula:
N = (n…n+(p-1))!/p! = (n…n+1)!/2! = n(n+1)/(2×1) = n(n+1)/2 i.e. N = n(n+1)/2
When p=3, a tetrahedral number series (1+4+10+20+35+…+n) is generated. And, using the general formula:
N = (n…n+(p-1))!/p! = (n…n+2)!/3! =n(n+1)(n+2)/6 i.e. N = n(n+1)(n+2)/6
When p=4, a number series (1+5+15+35+70+…+n) is generated. And, using the general formula:
N = (n…n+(p-1))!/p! = (n…n+3)!/4! = n(n+1)(n+2)(n+3)/24 i.e. N = n(n+1)(n+2)(n+3)/24
And, so on, for p = 5, p = 6,…
Hence, the General Formula, N = (n…n+(p-1))!/p!, is capable of:
1) finding out the individual n-th term formula for any oblique/diagonal binomial coefficient series in the Pascal /Yang Hui triangle – by just substituting the p value into my general formula; and,
2) working out the value of any n-th term in any oblique/diagonal series of binomial coefficients in the Pascal /Yang Hui triangle – by substituting the n-th value into the individual formula obtained by substituting p!
You’ll note that to generate the General Formula, I have to introduce or invent the idea and notation of an ‘up factorial’.
[This idea of ‘up factorial’ can be used to introduce the idea of partial ‘down factorial’: for example, 9x8x7x6 can be expressed as (9…6)! So, the binomial coefficient (BC) which is usually expressed as: BC = p!/(n!(p-n))! can now be easily expressed as a ‘partial down factorial / another partial down factorial’.]
Hahaha, is it not obvious: that mathematics is both a discovery and an invention?
Critical comments, anyone?
Signing off,
Douglas Tan