Welcome to K C Yan's Singapore Math blog!

Wanting to be updated on Singapore Math news or new Singapore Math? You have come at the right place! Please leave your comments before leaving. A googol thanks.

Singapore Math

Tuesday, March 11, 2014

Primes and Priests

If mathematics were a secular religion, then mathematicians could be regarded as its high priests. If Christianity were a scientific enterprise, then priests could be regarded as its prime managers.

Toying with "prime cubes"

Let’s look at some parallelisms between primes (or prime numbers, as British and Singaporeans like to call it) and priests (or pastors, depending on your denomination).

One unsolved problem in mathematics deals with the distribution of primes.
One unsolved spiritual problem in the Church involves the appointment of women-priests and women-bishops.


The prime number theorem states that the number of primes less 
than any real number x is approximately equal to x/(ln x). 
Graph from Paul Glendinning's Maths in minutes


Primes are the atoms of mathematics—all positive integers are composed of primes.
Priests are the ambassadors of Christ—God's chosen servants for His people, who are expected to exemplify a holy lifestyle.


All positive integers can be expressed as a product of primes. The fundamental theorem of arithmetic: “Every natural number greater than 1 can be written as a unique product of prime numbers.”
All born-again believers are members of a royal priesthood—they are all priests, in a spiritual sense.


Different proofs exist for the infinitude of primes—some 50 odd million known primes have been printed.        
Different proofs exist, which point to the Omnipresent, Omnipotent, and Omniscient God.


Pseudo-formulas for generating a few hundred or thousand primes exist.
Pseudo-Christs appear every now and then to deceive the believers, performing signs and wonders.


The number 1 is a pseudo-prime—its prime-like property fools the novice.
Cult leaders are pseudo-priests, out to control the lives of their followers.


A prime has four factors. For instance, the prime number 5 is divisible by 1, 5, –1, and –5.
A priest serves as a “surrogate’ to “forgive” sins, by acting as an earthly proxy for God.


Recruiting new priests is a challenging task for the Church, because of low pay and long hours, with many feeling unappreciated and undervalued.    
Finding new primes is a favorite brain-busting activity for math geeks, who spend unpaid hours on supercomputers to look for the next Mersenne prime—whose discovery is linked to the largest prime.


Different types of primes: twin primesemirpssexy primes, ....
Different types of priests: celibate, married, gay, women, ....


An order-3 magic square made up only of prime numbers,
with the smallest possible magic constant, 177


The prime number 2 is the only even prime, and this property is often used as a catalyst to pose many contests problems and mathematical quickies.
The High Priest is the only one allowed to offer sacrifices to God on behalf of the people every year in the Holy of Holies, as related in the Old Testament.


A magic prime: 73,939,133—it makes a new prime with each digit taken from the end.
A priest can act as a magician by using the surprising property of the Möbius strip to "explain" the concept of the Trinity.


Möbius strips for your Xmas decorations
Primes are used in cryptography—for security purposes, be it in banking or on-line transactions.
Priests are often consulted by world leaders for key decisions on complex or thorny issues—they anoint them to lead their nations with divine wisdom.


Divisibility tests and computer testing are signs that some numbers could potentially be primes—there is no foolproof method for finding primes.
Spiritual gifts and leadership qualities (speaking and interpreting in tongues, prophecy, vision, charisma, ...) are signs or criteria often used to select candidates for the priesthood.  


The idea of primeness or primehood has gone into the vernacular: prime time, prime target, prime locationprime ribs.
The idea of being, or behaving like, a priest has come to be associated with honesty, power, and authority.


Primehood denotes elements such as rarity, security, oddity, or money—coming up with a general formula for generating primes may make one rich!
Priesthood suggests qualities such as holiness, morality, chastity, spirituality, or respectability. You sound like a priest!


Escher goes topological!

The lure and challenge to factorize large numbers into prime factors has led security experts to design algorithms and to write programs to encrypt and decrypt digits-long numbers—it’s a multi-million dollar business.

Health-and-wealth priests, pastors, or ministers from mega-churches often live like kings, generating much income from the sales of their books, conference preaching, and the like—their prosperity gospel appeals to many materialistic believers.




Primes are used for survival (Darwinian weapons against predators)—less competition for food.  
Priests are God's ambassadors to bring healing and deliverance to entire tribes or nations—in recent years, there have been spiritual breakthroughs in countries like South Korea, Haiti, and Uganda.


The power of the Holy Spirit sweeps across nations, delivering peoples under the bondages of occultism and curses.
Primes are used to test the power of supercomputers, and the hunt for a formula for generating primes has indirectly yielded new knowledge in many unrelated branches of mathematics.


Harry Nelson was the first person to produce a 3 x 3 matrix containing only consecutive primes.


The number 73 as a magical prime filled with numerical curiosities.
Melchizedek as the High Priest, as reported in the Old Testament—the one who blessed the patriarch Abraham.  


Art and math for young children


Clay's millennium prize (unsolved) problems, one of which is the yet-to-prove Riemann hypothesis, which is related to the distribution of primes.
Unsolved spiritual problems, such "How can Jesus be both man and God?"; "What comes before God?".


Many contests and security problems tap on the properties of primes and prime factorization
Many real-world problems have their solutions in the Bible, as priests interpret God's Word in a modern-day context—applications of His Word to solve practical problems.



The Riemann hypothesis deals with the distribution of primes. 
Riemann initially established that there are trivial zeros for the 
negative even integers, which don't contribute much to the 
overall series of the Riemann zeta function. His hypothesis was: 
The remaining zeros all include a real part equal to 1/2—they 
should lie on a line expressed as 1/2 + ix, where x is a real 
number and i is √–1. Source: Paul Glendinning's Maths in minutes


Numerologists deify prime numbers. In some superstitious milieux, as a divination tool, prime magic squares may be used as an omen to ward off evil and reduce birth pain.
Priests in some quarters often condone the worship of saints among believers, although this idolatrous practice isn't advocated in the Holy Scriptures.


Prime life cycles of insects are used as a camouflage to fool the predators. For example, there are different species of periodical cicadas, some with a 13-year life cycle and others with a 17-year life cycle. The cicadas benefit from the lengthy rotation rate, since the different-cycle cicadas compete for food less frequently.
Priests are pretty busy during these two seasons every year: Lent season leading to Easter, to commemorate the resurrection of Christ; and weeks-long caroling leading to Christmas, to celebrate the birthday of Christ.


It's now your turn to share some commonalities between primes and priests with the mathematical brethren.


References

Pickover, C. A. (2002). The zen of magic squares, circles, and stars. Princeton & Oxford: Princeton University Press.

Schwartz, R. E. (2010). You can count on monsters. A K Peters/CRC Press.


© Yan Kow Cheong, March 11, 2014.

Sunday, December 15, 2013

The Lighter Side of Singapore Math (Part 4)


A Citizenship Proficiency Test

Below is a sample of math-related questions that could be posed in a "Singapore Citizenship Proficiency Test" paper meant for future naturalized citizens.

1. Singapore is
A. the fourth largest city in China, after Guangzhou, Shanghai, and Beijing.
B. the capital of Malaysia.
C. an island situated near the equator.
D. an island off the East coast of Taiwan.
E. none of the above.

2. The Singapore model method, or bar method, is a naturalized math product. Where did it originate from?
I. China  
II. Israel
III. Japan
IV. Russia
V. United States

A. I only
B. I and III only
C. I, III, and IV
D. II and IV
E. I, III, IV, and V

3. Singapore math textbooks are currently used in
A. the United States and in fewer than 10 Asian countries.
B. more than 50 countries around the world.
C. fewer than 20 Commonwealth countries.
D. over 120 American schools.
E. none of the above.

4. The "Math in Focus" series of books, published in Singapore and printed in China, is an adaptation of a local popular math series. What is the name of this math series?
A. Shaping Maths
B. Discover Maths
C. My Pals Are Here
D. Everyday Maths
E. Maths in Action
  
5. Which country came top in the Trends in International Mathematics and Science Study (TIMSS) for three consecutive times?
A. China
B. Japan
C. Finland 
D. Singapore 
E. Germany


The Art of Model Drawing: The Right Model

Based on a grade two word problem, the diagrams below show the solutions of four students. Which one best represents a suitable model drawing?


The right [or write?] model is ____.


An [Unofficial] PISA's Ranking for World's Cities

Following the recent PISA ranking, with Shanghai coming up first in math, science, and reading, one mainland Chinese commented online:

"If China were to submit test results for all 30 of its provinces then they will rank from #1 to #30 with Korea and Japan squeezed in between." Z

The comment is anything but far-fetched, as it's an open secret that it's much harder for a high-school graduate to be admitted to the top universities in China than to secure a place in the top two universities in Singapore.

Here's a possible PISA or TIMSS ranking, should they decide to compare city-state Singapore with the world's best cities in mathematics or science.

Toyko
Tel Avi
...
Beijing
Shanghai

Seoul
Taipei
...
Hong Kong
Guangdong
Moscow
Mumbai
Tehran
Singapore
Bangalore
Pyongyang
Chennai
New York
London
Washington


Will Singapore be around in the next century?
 
In the sixties, to control the population growth, the slogan then to [primarily non-college-] graduates couples was: "Stop at two kids!"

Today, to address the declining birthrate, the dangerously unwritten slogan in some postmodern or liberal circles is: "Seek two… wives!" Or, maybe it's: "Stick to one wife, but sleep with a few concubines!"


Two more expensive-cheat Singapore math books—titles which over-promise and under-deliver, or whose contents are worth a fraction of their published prices—according to dozens of local teachers, tutors, lecturers, and parents. Order them from Amazon.com if you've extra cash to spare!


 
References

Yan, K. C. (2011). The Lighter Side of Singapore Math (Part 3). May 1, 2011. http://www.singaporemathplus.com/2011/05/lighter-side-of-singapore-math-part-3.html

Yan, K. C. (2010). The Lighter Side of Singapore Math (Part 2). Sep. 25, 2010. http://www.singaporemathplus.com/2010/09/lighter-side-of-singapore-math-part-2.html

Yan, K. C. (2010). The Lighter Side of Singapore Math (Part 1). April 1, 2010. 

© Yan Kow Cheong, December 15, 2013.

Friday, December 13, 2013

13 Math Tidbits on Friday 13


Friday 13 is one of the beloved or lucky days in the Gregorian calendar year for most recreational mathematicians and math bloggers, who can't wait to share some musings about the number 13 and Friday the thirteenth to their non-mathematical friends, fiends, and foes.

1. Triskaidekaphobia Isn't Hereditary

If you can read about the number 13, you probably don't suffer from triskaidekaphobia—the fear of the number 13… but your children and grandchildren still might.


2. Pseudoscience Triumphs Over Profit 

On a plane or in a theater, shouldn't you feel "unsafe" rather than safe because row "13" has often been intentionally mislabeled "14"?

Or, In a lift with no button number 13, would you feel "safer" that they've tried to "cheat death" by making triskaidekaphobes or paraskavedekatriaphobes (who are fearful of Friday the 13th) feel better? All in the name of profit or superstition!


3. Synesthesia AND Autism AND Math

What color and shape is the number 13? What is the taste of 13? 

© [Unknown]


This book may shed some light on the prime number 13!


4. The Closest of Two Friday the 13ths 

Can Friday the 13th occur in two successive months? It's not difficult to see that the shortest span between two consecutive Friday the 13ths is 28 days. How often do you think this event happens? Have you experienced one in your lifetime, or know of someone who rejoices in this kind of numerical (or numerological) oddity?


5. An Unsolved Arithmetic Problem

The number 13 is both a prime and a Fibonacci number. Whether there are infinitely many primes that are also Fibonacci still remains an unsolved problem.

Cracking this higher arithmetic nut may bring you some mathematical prestige or fame, as your proof graces the pages of a Number Theory publication. 


6. A Friday 13 Baby

A baby is born on Friday 13, 2013. When is the next time a baby will be born on the same day of the week in the same month, which also falls on a Friday 13? How often in a 400-year calendrical cycle does this happen?


7. A Friday 13 Survivor 

What are the odds that you will not survive on a Friday the 13th? Is the answer equivalent to finding the probability that a baby will be born on a Friday 13?


8. An Inauspicious Day 

Using the code A = 1, B = 2, C = 3, ...., show that the sum value of the phrase FRIDAY THE THIRTEENTH IS AN UNLUCKY DAY is a multiple of 13.


9. Birth and Death on a Friday 13

What are the chances that one born on a Friday 13 will also die on a Friday 13? How many thousands of folks every year experience this double whammy? 


A solution may be found from Aha! Math.


10. Bio of 13

Prepare a résumé for Number 13, showcasing some of its elementary numerical skills. For instance,

13 = 1 + 4 + 8

13 = 3 + 3¹ + 3²

12 × 13 = 156 and 21 × 31 = 651

13² = 169 and 31² = 961


11. The Unluckiest and the Happiest

List 13 parallelisms between Friday 13 and December 25—what common features the most fearful (or frightful) date of the calendrical year shares with the king of the public holidays. Get creative, by connecting the unconnected!

An ideal gift for numbers lovers & creative problem solvers

 
12. Social Media 13

Defriend 13 of your Facebook friends, or unfollow 13 of your Twitter followers, who have been inactive in the last 13 months. Send them a Friday 13 alert of your mean intention to save bits and bytes on your RAM to reduce your carbon footprint.


13. Problem Posing on a Friday 13

Pose at least 13 mathematical quickies and trickies, and share them with your on-line friends and fiends.

What is the angle between the hands of a clock at 13:13?

The sum of the ages of Casey and Ian is 13 years. If Casey is 1.3 months older than Ian, how old are they?


A Happy & Blessed Friday 13!


References

Yan, K. C. (2011). CHRISTmaths: A creative problem solving math book. Singapore: MathPlus Publishing.

Yan, K. C. (2006). Aha! math. Singapore: Panpac Education.


© Yan Kow Cheong, December 13, 2013.


Monday, November 18, 2013

General Paper and Math Essays

A six-year series of past exam GP papers

In many Commonwealth countries, high school students sitting for the Cambridge G.C.E. 'A' Level/Higher School Certificate Examination are required to sit for the "General Paper," a paper that "tests the candidate's understanding and use of English and the extent to which he has achieved a maturity of thought appropriate to sixth-form (or high-school) students in their second year."

The three hours "General Paper," which is primarily not a test of general knowledge, is made up of two parts:

Paper 1 contains topics for composition on a number of disciplines, ranging from geography and history to literature and language to arts and crafts to mathematics and science. From a dozen questions, students choose one to write an essay between 500 and 800 words in length.

Paper 2, which lasts one hour 30 minutes, tests comprehension of one passage of continuous prose, or of two different passages that allow for comparative analysis.


University of Cambridge Math Essays

Let's look at some math-related topics that have appeared in Paper 1 of the General Paper in the last half century.

7. Consider the view that mathematics possesses not only truth, but supreme beauty. (2012)

12. Can mathematics be seen as anything more than a useful tool in everyday life? (2010)

9. Discuss the view that too much faith is placed in statistics. (2008)

?. Consider the view that the study of mathematics is intellectually satisfying, but of little practical use. (2005)

5. How important is numeracy in today's society? (2004)
 
?. Statistics measure everything but prove nothing. Discuss. (2003)


A ten-year series of past exam GP papers

7. Can mathematics be made fun, interesting and worthwhile? (2003)

10. 'An education is incomplete without a sound understanding of mathematics.' Do you agree? (2002)

6. What is the relevance of Mathematics? (2000)
 
5. What is the value of mathematics? (1992)

7. 'Mathematics is the most perfect language of all.' Discuss. (1991) 

6. How could the teaching of science and mathematics be improved in schools on your country? (1991)

5. How necessary is it for the non-scientist to have some knowledge of mathematics? (1987)

7. 'Statistics can be both helpful and misleading.' Discuss, with examples. (1984)

6. What mathematical knowledge should all young people have acquired by the time they leave school? (1979)

6. 'Neither Physics nor Chemistry could have reached its present level without Mathematics.' Explain this statement, giving examples from either Physics or Chemistry. (1969)

8. Write simply, in non-technical language as far as possible on one of the following:
(a) logarithms;  (b) genetic code;  (c) the internal combustion engine;  (d) the metric system. (1967)

 
Mathematical Writing vs. Mathematics Writing

The General Paper (GP) provides high-school math students an opportunity to write about their love for the language of science and of technology—they write about instead of on mathematics. In other words, they're to showcase their mathematical writing skills, as compared to professional mathematicians who focus on mathematics writing, which grace the pages of journals and periodicals.

One wonders what percent of GP students would choose to write on these math-related themes, even if they're doing well in the subject? How many pre-university students would be confident or motivated to write an essay about the beauty, utility, or ubiquity of mathematics? It would be interesting to get some information on the popularity of math essays among GP students, from the Cambridge Examining Board.
 

Conclusion
 
The General Paper also provides a good opportunity for both arts and science students to be mathematically cultured, as they write about the beauty and power of mathematics. Encouraging more students to write math essays would indirectly lead them to learn more about the story or history of mathematicshow mathematics and mathematical ideas have enriched the lives of humankind over the centuries. In other words, how the evolution and revolution of mathematical results or breakthroughs have helped shape civilization. At the least, GP math essay questions could help bring humanities, arts, and math closer.


Some Questions on GP Math Essays

1. Assuming that the essays are free of grammatical and spelling mistakes, what would make one's "math essay" stand head-and-shoulders above the rest of the competition?

2. Compare and contrast the GP essay with a 500-hundred-word college admissions essay. Which one promotes a higher degree of critical thinking?

3. How does the General Paper encourage students to explore and appraise mathematical, scientific, and technological issues?

4. Most GP or English language teachers are known not to like math. Would they give math composition a miss? Or, would they make an effort to learn more about the subject, so that they in turn would be confident to assign and mark these math-related essays?

References

Fairfield Book Publishers Pte Ltd. (2013). General paper: Answers with explanations. Singapore: Fairfield Books Publishers Pte Ltd.

Rajamanikam, J. (ed.) (1985). General Paper. Singapore: Redspot.

SEAB/UCLES (2005). General Paper Yearly Questions G.C.E. A-Level Nov. Examination Paper 1 & 2 2001-2005. Singapore: Dyna Publisher Pte Ltd.
 
Singapore Asia Publishers Pte Ltd. (2013). H1 A Level General Paper. Singapore: SAP Education.

Web Publications Pte Ltd (2004). A-Level General Paper Past Examination Questions. Singapore: Web Publications.


Past-exam papers with modeled solutions

© Yan Kow Cheong, Nov. 17, 2013.