H2 Maths Tuition By White Group Mathematics

noreply@blogger.com (Mr Koh) — Fri, 28 Jan 2022 15:06:00 +0000

On This Very Page.......

Many a time less is more, hence I shall keep the introductions real short and sweet.

WELCOME.

"Everything should be made as simple as possible, but not simpler." - Albert Einstein

noreply@blogger.com (Mr Koh) — Fri, 28 Jan 2022 15:04:00 +0000

Problem 33: Determining Probability of Success For A Binomial Distribution With Known Mode

noreply@blogger.com (Mr Koh) — Tue, 27 Oct 2020 11:58:00 +0000

Problem 32: Vectors Coupled With Summation of Series

noreply@blogger.com (Mr Koh) — Sat, 14 Mar 2020 14:56:00 +0000

Problem 31: Algebraic Manipulation

noreply@blogger.com (Mr Koh) — Tue, 31 Dec 2019 14:43:00 +0000

Final Look at 2019: School, Science and Education

The badass looking six-seat Tesla cybertruck (Image Credit: Tesla)

The 2010s draws to a close on the heels of a rather tumultuous 2019: obnoxious American president Donald Trump got his ass impeached by the House of Representatives no thanks to one alleged “quid pro quo” phone call with Ukraine, the tit-for-tat trade war between US and China granted a mere reprieve courtesy of a slipshod 'phase 1' ceasefire, Brexit going full steam ahead after Boris Johnson thoroughly crushed Labour in the December British general election, Japanese emperor Naruhito's ascension to the throne (marking the beginning of the Reiwa era), state-owned Saudi Aramco oil processing facilities at Abqaiq and Khurais being severely hit by drone strikes, an undeterred Aramco proceeding to list on Riyadh’s Tadawul exchange months later - even attaining a record $2 trillion market capitalization (just for a few hours though), former Wall Street darling WeWork's IPO ambitions dealt a devastating blow over massive valuation cuts, still ongoing pro-democracy protests in Hong Kong pitting citizens against both lawmakers and the police, Indian Prime Minister Narendra Modi’s unpopular citizenship law seeing many take to the streets, a most violent weekend in May where Israel and Gaza militants battled each other to the tune of at least 23 deaths......

May celebrities alongside ordinary folks who departed prematurely owing to suicide (K-pop stars Sulli, Goo Hara and Cha In Ha), tragic gun violence incidents (victims of Virginia Beach, El Paso, Daytona Beach and Texas Church mass shootings) as well as natural disasters (those who perished in Bahamas' Hurricane Dorian, China's Typhoon Lekima, Japan's Typhoon Hagibis, New Zealand's White Island volcanic eruption, Mozambique's Cyclones Idai and Kenneth) among numerous others retire peacefully in the afterlife.

On a cheerier note, heartiest congratulations to Kenyan Eliud Kipchoge on becoming the world's first person to run a marathon in under two hours (at an absolutely blistering pace of 21.18 km/h), former world no.1 golfer Tiger Woods for capturing the 2019 Masters at Augusta (thus ending his 11-year majors drought), Liverpool for winning the Club World Cup, the Champions League, the European Super Cup and also emerging unbeaten in 50 successive top-flight matches at Anfield (clinching the English Premier League title also appears imminent at present) and tech billionaire Elon Musk on the unveiling of his really tough cybertruck (never mind those inconceivably fragile windows shall we?); then of course I mustn't neglect sneaking in a mention about tenacious youth climate activist Greta Thunberg who was named Time Magazine's Person of the Year.

How DARE you nearly forgot to pay tribute to moi??? (Image Credit: Reuters)

Hang on, "hats off" too to ultra chubby North Korean leader Kim Jong Un for procuring a handsome white stallion capable of tolerating his hamburger-sized frame without crumbling sideways. Hopefully poor horsey didn't end up as his supper thereafter.

This weird winter dictator bullshit is obviously photoshopped.......who cares anyways because it's so darn laugh out loud hilarious. (Image Credit: Thomas Gaulkin And Heribero Arribas Abato)

Without further ado, let's review developments around the globe surrounding science and education this past year.

For the first time ever, immediate surroundings of galaxy M87's supermassive black hole were captured by a network of eight telescopic sites under the Event Horizon Telescope (EHT) Collaboration; even more encouraging news came in the form of a $12.7 million grant awarded by the National Science Foundation to upgrade the EHT's existing capabilities - might astronomers and astrophysicists therefore be able to expect higher resolution pictures of M87 and other supermassive black holes in the foreseeable future?

If only the late Richard Feynman could see this. (Image Credit: EHT Collaboration)

Equally laudable was NASA's New Horizons probe successfully completing a flyby of a double lobed 'snowman' (previously named Ultima Thule, now known as Arrokoth) 6.5 billion kilometers away in the Kuiper Belt on 1 January; more images are expected to be released when observational data collected by the probe is gradually downlinked in the next 2 years.

Back on earth, AI made yet another evolutionary leap of sorts when the Allen Institute for Artificial Intelligence's Aristo software passed an eighth-grade science test with a 90th percentile score. While the test attempted excluded examining one's ability to interpret diagram-based questions, nevertheless this particular achievement served to demonstrate how far AI has come since 2016, when all programs then competing for an $80,000 carrot dangled by the Allen AI Science Challenge flunked spectacularly.

Separately, in what could be construed as a medical breakthrough, the revolutionary CRISPR cas9 gene-splicing technique gave hope to millions suffering from sickle cell disease when researchers at the Sarah Cannon Research Institute in Nashville, Tennessee, announced they had used CRISPR to modify patient's bone marrow cells before reintroducing them back into the body; subsequent results appear promising given these modified cells begun producing healthy haemoglobin within months.

Might constructing an 'artifical sun' for the sole purpose of harvesting its energy strike you as frightfully ambitious? Apparently China did proceed with developing such a device - originally called the HL-2M nuclear fusion reactor. According to local news reports, it has already been completed, and is slated to commence operation sometime in 2020.

Damn thing looks ugly yet impressive, yeah oxymoron much. (Image Credit: Xinhua News)

Moving on, classroom affairs galore.

Legislation/curricula some schools introduced certainly raised eyebrows, such as Harry Potter books getting banned altogether by a catholic school in Nashville on account they could possibly conjure "evil spirits", male students being allowed to wear skirts at the New Taipei Municipal Banqiao Senior High School, a pornography themed module to be made available shortly to University of Exeter students and khat calligraphy planned for incorporation within the Standard 4 Bahasa Malaysia syllabus as far as Malaysian schools are concerned.

A few teachers also courted controversy by virtue of their opinions and actions; check out Valley View Elementary School's second grade educator Alyssa Rupp Bohenek who scribbled utterly discouraging remarks on a student's Math assignment, an unnamed member of Lakeland Senior High School's teaching staff inappropriately boasting about being capable of amassing a 1,000-person body count if he were a school shooter, Duke University's assistant professor Megan Neely losing her graduate programme directorship after a callous e-mail dispatched urging students to refrain from speaking Chinese and rookie Maths instructor Kelsie Schmidt at Beulah High School having no qualms transmitting X-rated images to a 17 year old student via Snapchat, even imploring him to play naked hide and seek with friends.

Smile Kelsie, you are now charged with luring minors by computer. (Image Credit: Facebook)

Then there was the FBI's "Varsity Blues" operation which laid bare that humongous college admissions bribery scandal, in the process ensnaring over thirty parents - amongst whom were prominent corporate head honchos and well known Hollywood actresses. Meanwhile in Hubei province, China, a mother almost literally died of a heart attack after her son couldn't solve a Maths problem. Oh please do take deep breaths and chill your tits mommy.

Bizarre sightings abound: student jumping into shark tank at South Bay aquarium on a dare, the remains of a Chinese teacher who went missing for more than 16 years finally being discovered buried under a school and one cow seen crashing a lecture at the Indian Institute of Technology (IIT-B) in Bombay to name a few.

Right here in Singapore, National University of Singapore (NUS) undergraduate Monica Baey caused quite a stir when she took to social media to openly chastise authorities over what she felt were excessively lenient punishments meted out to the man who filmed her in a hostel shower; her dogged pursuit of adequate justice for sex offenders sparked institutional change as universities scrambled to review disciplinary frameworks and reinforce victim support schemes. Another brouhaha arose with regards to the cancellation of a Yale-NUS module (titled “Dialogue and Dissent in Singapore”), consequently seeing playwright Alfian Sa’at engaged in a testy threeway 'he said, she said' exchange with Yale-NUS College and the education minister.

A screenshot of Monica Baey's Instagram Story (Image Credit: Monica Baey)

Not long after, the Ministry of Education (MOE) received brickbats for withholding a student's original PSLE results slip owing to $156 worth of outstanding school fees - despite one politically correct explanation proferred in the days that followed, many netizens evidently remained unappeased. (In any case a good samaritan helped clear the abovementioned arrears, and the child's original results slip has since been released to her.)

Dismay no doubt erupted when Singapore lost pole position at the 2018 OECD PISA rankings to China, but it was a rather unsettling survey metric citing more than 70 per cent of Singaporean teens being fearful of failure which had people opining we are possibly stuck in an unenviable one step forward, two steps back situation.

Initative-wise, I ain't charmed by the MOE's impending full subject-based banding rollout across secondary schools - IMHO this reeks of merely repackaging the old to pass off as new; regardless time will determine its overall efficacy. Thumbs up however for them plans to better support students with special educational needs, rejuvenate older junior college campuses plus provide more pathways for Insitute of Technical Education (ITE) students to upgrade their skillsets.

Once again I have reached the end, therefore bow out I must to rest up in anticipation of the new year. Thanks for reading; here's wishing y'all a splendid 2020. Peace.

(The above was also featured in the 8th February 2020 edition
of the Carnival of Mathematics )

noreply@blogger.com (Mr Koh) — Mon, 04 Nov 2019 15:05:00 +0000

Problem 30: Atypical Probability Related Question

noreply@blogger.com (Mr Koh) — Sat, 19 Oct 2019 15:48:00 +0000

A Level H2 Maths Question-Spotting Thoughts For AY 2019

The 2018 A Level H2 Maths examination while largely manageable still succeeded in delivering a fair bit of pain, as students were up in arms over an integration techniques problem calling for the computation of a curve's arc length (relevant to the F Maths syllabus), that colossal 13 mark context-based irritant which exacted tedious algebraic manipulation coupled with suitable use of differentiation techniques alongside one triple Venn diagram conundrum posed among others. Let's of course not simply dismiss the functions snafu erroneously articulated in paper 1 which didn't escape numerous keen eyes. Should folks therefore batten down the hatches this time round in anticipation of something even more severe occurring? I'd reckon not, but it certainly wouldn't hurt to shore up preparation efforts by getting acquainted with less routine musings - hence the thrust of my ~~debatably accurate~~ "prognostications". Here goes:

Graphing Techniques

Vertical asymptotes remain unaffected when the graph of y=f'(x) is premised on y=f(x), but can previously absent ones arise as a consequence of undergoing this particular transformation? The answer is actually yes, as long as points exist on the original curve such that the tangents to them are parallel to the y-axis.

In the upper half of a hyperbola example illustrated above, the tangents to both x-intercepts at (-11,0) and (1,0) are infinitely steep, thus implying that x=-11 and x=1 shall end up as vertical asymptotes happening in the y=f'(x) sketch subsequently produced. Also, no x-intercepts are expected when pursuing the graphical derivative of y=f(x) on account not a single stationary point transpired initially.

On an unrelated note, I cannot emphasize enough the importance of properly comprehending the asymptotic behaviour of graphs; simply relying on the graphic calculator output and thereafter applying visual approximates to equations of existing asymptotes is strongly discouraged. Considering y=[2/(e^x-1)]+1 for a moment, I had students mistaking curves on both sides of the y-axis to be simultaneously restricted by the x-axis, ie y=0 "appeared" as a common horizontal asymptote on the GC display. Well sorry to burst your bubble because it ain't so.

Pop quiz: can you prove that f(x)=[2/(e^x-1)]+1 is an odd function for x∈ℝ, x≠0 ?

Integration Techniques

Integrals featuring purely fractional exponential functions can on occasion confound, then again surveying them from a different perspective could enable surprising breakthroughs. Check out these two instances to better appreciate what I am driving at:

Vectors

Sometimes achieving the value of a scalar product via a deceptively simple triangular construct may require adept manipulation of information indirectly furnished - the following problem expounds this need unambiguously ( I have therefore included a pertinent hint to get you started off just in case).

Should you feel inclined to pursue the remaining parts of the question, they happen as such:

Functions

Discovering the domain and range of a typical composite function should be right up your alley, what then if you are confronted by three concatenated functions, such as hgf(x)? Should one be cognizant of how stuff are strung together as far as a two-tier process is concerned, then it is merely extrapolating that understanding once more to accommodate an additional plant. In any case I have detailed the required strategies accompanied by a box diagram below:

Statistics

I am not expecting anything devastatingly insurmountable to creep into the latter part of paper 2, that being said you may wish to take special note of the negative Binomial distribution (aka geometric distribution), where the nth success is assigned to a particular trial. Eg. Imagine tossing an unbiased die 10 times consecutively, what would be the probability alluding to the eighth throw displaying the third "6" ? (Food for thought: recognizing the first seven throws must yield exactly two "6"s, could you thus formulate a downsized Binomial distribution to adequately address affairs?)

You may or may not wish to have a go at another problem:

End here I must, given a final round of consolidation with my charges still awaits. As cautioned in previous editions the pointers offered herein are merely meant to supplement your revision efforts, nothing more. Stay frosty in the meanwhile for the upcoming As, 'em holidays shall beckon sooner than you think - trust me on this.

Good luck kids. Peace.

noreply@blogger.com (Mr Koh) — Sat, 27 Apr 2019 14:52:00 +0000

Problem 29: Interesting A Maths Sum And Product of Roots of Quadratic Equation Question

noreply@blogger.com (Mr Koh) — Sun, 23 Dec 2018 15:22:00 +0000

Final Look at 2018: School, Science and Education

The union of the year went to POTUS Donald Trump and North Korean dictator Kim Jong Un, a viciously squabbling pair who finally met for the first time in person at Capella Hotel, Singapore on 12 June 2018. Love at first sight it was apparently, as both struck up a bromantic relationship almost immediately upon setting eyes on each other; if anything it took the world's breath away, even Prince Harry and Meghan Markle's official coupling at St George’s Chapel played second fiddle in the attention department. After all the "I Do"s sworn by Trump and Kim endeared to the geopolitics of forging peaceful boundaries and borderlands, an ongoing international concern of behemoth proportions superseding even the most fairytale-esque of royal nuptials. (Sorry Mukesh Ambani, even with that $100 million splashed on your darling daughter's wedding, you'd still have to take a number - Beyonce ain't got shit on the 38th parallel either). For now Kim has promised to dismantle the collection of dangerous nuclear toys assembled in his backyard thus far in exchange for hitting the reset button on mutual ties, alongside US ceasing joint manoeuvres with South Korea. Still, it remains to be seen whether both fickle-minded leaders with attention spans shorter than goldfish will each keep up his end of the bargain. By the way, @Meghan Markle, did you actually make your sister-in-law bawl because of them overly tedious demands heaped over little Princess Charlotte’s bridesmaid dress?

Make love, not war.......until either one of us blinks? (Image Credit: VOX)

Trump can also give himself a real big pat on the shoulder knowing for sure he has succeeded in alienating the United States from almost the entire world by forging ahead with a tsunami of trade tariffs imposed against a total of 31 countries (and affecting some 12 billion bucks worth of imports) till date, not to mention a staggering number of investigations concerning anti-dumping/countervailing duties already being initiated by the US Department of Commerce (122 in all at last count, a majority targeting Chinese products). Thank goodness for a temporary ceasefire of sorts arrived at the eleventh hour during Trump's recent G20 dinner date with China's President Xi Jinping, where both countries agreed to halt fresh tit-for-tat sanctions for 90 days in a bid to surmount the ongoing stalemate.

Everyone in this room probably wants to punch him in the face, hell yeah (Image Credit: Angela Merkel's official Instagram account)

Might the ongoing trade wars be complicit in causing the Republicans to lose the House of Representatives in November's midterm elections? Most likely, the only question is to what extent. So while this loss didn't quite rattle Trump's grasp on power (who by the way remains stuck in denial mode it appears given his self-congratulatory tweet of tremendous success about the election results), nevertheless voters did give him a resounding slap across the face. Make that two, another courtesy of his former longtime legal 'pit bull' cum fixer Michael Cohen who has since bitten back, insisting he was "duty-bound to cover up the President's dirty deeds" after being handed a 3 year jail sentence by a New York federal court. Three actually, if the resignation of Defense Secretary James Mattis is to be taken into account.

Elsewhere, Taiwan President Tsai Ing-wen was also left licking her wounds, as her Democratic Progressive Party (DPP) suffered a walloping in the key mid-term polls and she accepted responsibility for the massive losses (even traditional strongholds such as Kaohsiung and Taichung fell to them China-friendly Kuomintang fellas) racked up by resigning as party chairperson. Looking forward, she is almost surely reduced to a lame duck leader for the remaining part of her term in office; with Taiwan's next presidential elections looming it isn't that difficult to realize she will be facing one helluva fight for her political life.

While some live to fight another day, others weren't so fortunate. Just look at how far Malaysia's 'Man of Steal' Najib Razak has fallen in a span of months after his ass was roundly trounced in the country's 14th General Elections......from despotic premier to despised felon slapped with more than two dozen charges of money laundering, power abuse and evidence tampering related to the billion dollar 1MDB scandal. His wife Rosmah Mansor (known to splurge on luxury items, expensive holidays and 1,200 ringgit haircuts) also faces a dozen charges related to tax evasion and corruption. Her cringeworthy comeback refuting every single allegation in court? She could only plead guilty to having stolen Najib's heart.

Do you believe this former no. 2 most beautiful First Lady in Asia is completely innocent? (Image Credit: Reddit Malaysia)

A moment of silence for all victims on board the downed Lion Air flight JT610 and one brutally murdered Saudi journalist Jamal Khashoggi, together with other remarkable individuals who left us - esteemed Physicist Stephen Hawking (a final book titled "Brief Answers to the Big Questions" was launched by his children on 15 October 2018 at the Science Museum in London), former US president George H. W. Bush and his wife Barbara Bush, outspoken political Maverick Senator John McCain, queen of soul Aretha Franklin, Italian auteur Bernardo Bertolucci and founding father of Marvel Comics Stan Lee among others.

Lest I forget, condolences to Germany for crashing out in the group stages of the 2018 FIFA World Cup Russia. Pun intended by the way.

One word: CHEERS. (Image Credit: Roger Cohen's official Twitter account)

Okay timeout from mucking around, here's the lowdown on science and education-related affairs which transpired this year.

Ever heard of CRISPR-Cas9? A piece of cutting edge genome editing technology which came on the scene fast and furious approximately 6 years ago, it enables geneticists and medical researchers to manipulate specific sections of the DNA, though much frustration has arisen thus far owing to the baffling nature of unpredictable outcomes observed (seemingly random mutations possibly occurring at targeted sites thereafter). It appears this might be a thing of the past, if scientists at the Francis Crick Institute are absolutely right about their discovery of a rather simple set of rules underpinning what exactly to expect post alteration. That being said, at the present moment ethical considerations have resulted in many countries banning intentional gene tweaking of human embryos. Yet this hasn't stopped one lone maverick (or nut job?) from disabling the CCR5 co-receptors of twin baby girls in a bid to make them HIV resistant, or so Shenzhen-based associate professor He Jiankui claimed. Explicit hacking the human genetic makeup aside, perhaps the non-invasive CRISPRa method developed by folks at UC San Francisco might serve as a reasonable alternative.

Another interesting breakthrough might just be around the corner, as Boston engineers are heavily invested in miniaturizing computer chips to eventually produce supercomputers no larger than fingernails which could mimic the computing abilities of the human brain (spanning some 100 trillion synapses established via 100 billion neurons). Equally cool is also written all over Intel’s new Vaunt smart glasses: information is projected onto the wearer's retina in an augmented reality environment literally wrapped around the eyes, then again observers on the side are non the wiser to the awesomeness that's actually happening. Now Iron Man fans are one step closer to digging Tony Stark proper.

Next up, to infinity and beyond. What exactly was the deal with SpaceX's top secret Zuma mission which launched on 7 January? For the uninformed, Elon Musk and gang went ahead with firing one U.S. spy satellite strapped onto their Falcon 9 rocket (by means of a payload adapter) into an orbit above earth. It seemed though the government's hush hush billion dollar toy didn't deploy correctly, with various news sources reporting that Zuma eventually came crashing back down into the Indian Ocean. Others like Netherlands scientist Dr Marco Langbroek opined the satellite itself did enter orbit, despite doing so it was already a mere piece of expensive dead junk. Meanwhile SpaceX has vehemently rejected allegations that sought to apportion blame to their equipment not performing up to the mark. An illuminating instance of "he said, she said"......outstanding.

The Zuma mission taking off from Cape Canaveral (Image Credit: SpaceX)

On a lighter note, NASA's InSight lander which successfully touched down on Mars in November did the most sensible thing expected of any exploratory spacecraft: taking its first ever selfie - this achieved through assembling 11 separate visuals snapped by a camera-fitted robotic arm. Looking pretty aside, many are counting on ya to thoroughly sniff out Martian territory and deliver some good news concerning the planet's habitability.

Thumbs up too for NASA's handling of things when misogynistic trolls from notoriously vile internet forum 4chan attempted to disrupt the Optimus Prime Spinoff Promotion and Research Challenge (OPSPARC); it suspended the public voting component of the competition following threats of a cyberattack to disparage an all-black, all-female team who made it to the finals. Eventually Mikayla Sharrieff, India Skinner, and Bria Snell from Banneker High School in Washington placed second with their project which focused on filtering poisonous lead impurities from school drinking fountains and were granted a special campus visit to MIT courtesy of an unofficial student group within its Department of Aeronautics and Astronautics.

Speaking of schools, hats off to New York University's School of Medicine in its extremely generous undertaking to make tuition free for all present and future enrolling students. In addition, the US states of Virginia and New York must be lauded for requiring schools within their jurisdictions to incorporate mandatory mental health education into standing curricula; meanwhile the pursuit of happiness is happening in full swing at Yale University thanks to psychology professor Laurie Santos' overwhelmingly popular Psyc 157 (Psychology and the Good Life) course. On the other hand, Harvard's ass was sued big time for discriminating against Asian-American applicants and Cornell University came under the spotlight as a consequence of one senior stripping down nearly all the way to express outright disapproval of her professor's supposedly sexist remarks. Add to these, police officers were deployed on Florida University's campus, reason being a first ever "White Racism" course was scheduled to be conducted by one Ted Thornhill. Not unexpectedly the assistant professor of Sociology received plenty of harassing emails and calls, thankfully day one of lessons kicked off uneventfully.

Elsewwhere, winds of discontent blew strong as well - an alumni of Anglia Ruskin University in the UK took her alma mater to court on grounds that she graduated with a useless "mickey mouse" degree, Baptist University in Hong Kong had to resort to suspending two students involved in a bitter protest staged over the institution of a special test to ascertain one's proficiency in Mandarin while Taimei Elementary School in Tokyo, Japan had parents seeing red after new Armani uniforms costing more than £500 apiece were announced.

Remember Tommy Jordan, the awesome dude who pumped .45 slugs into his teenage daughter's laptop because she refused to perform household chores and whined like a self-entitled bitch? More than half a decade later, Matt Cox punished his little girl by making her walk 8 klicks in the freezing cold to school over a bullying episode. Utterly cruel or another contender for the dad of the year award? Apparently this incident has divided the online crowd down the middle. Three unambiguous cheers though for Texan mama Liza Martinez who spanked her son with a belt upon learning he took her BMW out for a joyride.

Lisa Martinez ain't no dominatrix but a real American hero. (Image Credit: CNN)

Next, a shout-out to educators who dared to make a difference despite the odds stacked against them. Meet Richard Appiah Akoto, who literally reproduced images of computer-related applications from memory on the blackboard for the benefit of underprivileged children attending school in Ghana. A collective of British teachers deserve high praise for going the extra mile to help children from poor families wash their clothes, even lending their parents money on occasion. Deyshia Hargrave from Rene Rost Middle School in Kaplan, Louisiana, couldn't in her conscience simply let the matter of her school district's superintendent enjoying an unwarranted salary raise slip quietly by, so she spoke up during a board meeting and ended up being led away in handcuffs. The Louisiana State Police subsequently stepped in to investigate, and the attorney general also intervened by suing members of the school board.

Over here in tiny Singapore, a few key happenings defined the year as well. For starters, there was a huge public outcry over the announcement made by the Ministry of Education that teachers had to pay up to S$960 a year for parking in schools come 1 August - a consequence of upholding the “clean wage policy” for all civil servants. Tackling inequality ranked high on the education minister's agenda, as the debate about elitism and social stratification rages on with no end in sight as of yet. Most unfortunately Catholic Junior College Vice Principal Mrs Yue-Chang Teck Hui's derogatory remarks about neighbourhood students only served to fan the flames.

While students will surely applaud a regime involving lesser examinations and assessments due to take effect next year, some parents remain on the fence regarding these changes, worried that an adequate measure of academic ability will no longer suffice. Then again in a world where practical skill sets become increasingly relevant compared to paper qualifications, steering away from an obsession with grades might not really be a bad thing. Besides numerous folks have shared Instagram stories on how their PSLE scores hardly correlate to the personal successes achieved later in life, they can't all be lying at once yes?

So we didn't do well in school back then as kids. Big fat deal. (Image Credit: Mothership)

And this concludes my exhausting but fun ramble. Thank you to those who endured reading till the end; I hope some of you have come away enlightened (or at least entertained). Here's wishing all a wonderful Christmas, and a happy 2019. Peace.

(The above was also featured in the 10th January 2019 edition
of the Carnival of Mathematics )

noreply@blogger.com (Mr Koh) — Wed, 24 Oct 2018 10:32:00 +0000

A Level H2 Maths Question-Spotting Thoughts For AY 2018

The first edition of the 9758 A Level H2 Maths examination which happened last year was quite uneventful, though I wouldn't venture as far as to label it a walk in the park in the absolute sense. In hindsight students typically tripped up because of carelessness committed during the calculation process and failure to consult the help sheet MF26 appropriately, rather than having encountered outright speed bumps as far as problem design was concerned. Therefore, while I wouldn't expect this year's papers to be genuinely insurmountable, based on a personal review of actual block tests, preliminary examinations as well as revision packages furnished by top and middle tier junior colleges in recent times, it appears teachers are somewhat leaning towards the nostalgic - which by implication could see 2018 favoring a disproportionately large quantity of "archaic" methodologies being poked and prodded once more. From the bottom of my heart I pray I am dead wrong, then again under the circumstances ain't it much safer to ~~"have a gun and not need it than to need a gun and not have it"~~ be overprepared than underprepared? (yikes Tarantino creeping from within....) Without further ado here's a summarized inventory of my forecasts which takes significant heed of the good ol' days:

Complex Numbers

The main deal here centres on deriving more complicated trigonometric identities via complex numbers theorems. I would like to share about two possible instances arising by citing actual problems. The first happens as follows:

Knowledge of Binomial series expansion is assumed* here, however you may wish to read about a primer on accomplishing the task in an efficient manner. That being said, with regards to solving the above problem posed, deriving the below generic association between a complex number and its conjugate would certainly be helpful:

The second type of problem involves obtaining an expression for tan(nθ):

Once again Binomial series expansion is involved and this is an intermediate result one must strive to achieve:

Thereafter can you satisfactorily reconcile both real and imaginary components to tie up loose ends?

Functions

Are you aware of what a floor function is?

And here's embedding the said concept within a problem concerning periodic piece-wise continuous functions:

The solution for (i) is offered below, and it shouldn't be all that difficult to figure out (ii). (Note though the integral requiring evaluation is not exactly about computing the area under the existing graph)

Inequalities Hybridized With Integration Techniques

The problem typically tasks the student with solving a mildly difficult inequality at the onset, subsequently posing an integral which references the context of the particular said inequality. The integral itself would naturally involve modulus expressions - care must therefore be exercised to treat them before invoking the actual integration process. A proper example would best articulate affairs:

Assuming you can crack the inequality wide open (and you should be able to), here's getting you started off with regards to tackling the integral:

How would you proceed to dismantle the modulus signs present?

Statistics

While I do not expect folks to run into any stiff headwinds here since mostly standard fare graphic calculator commands are being deployed to establish answers, it would still be prudent to remain cognizant of some really old school question templates. One always comes to mind: enacting a recurrence relation and thus solving for the mode of a Binomial distribution. An example is furnished below.

And that pretty much wraps things up; bear in mind the pointers offered herein are meant to supplement your revision efforts, not serve as an entirely standalone substitute. As I am drafting this you have slightly more than 2 weeks before Paper 1 (on 9 November) begins, so just hang in there for a wee bit longer........the A Levels shall end soon enough and then pop goes the proverbial champagne.

Good luck kids. Peace.

*Addendum: R-formula (Trigonometry), Remainder/Factor Theorem, partial fractions, differentiation from first principles and various other concepts previously acquired at the O Levels also fall under the ambit of assumed knowledge.

noreply@blogger.com (Mr Koh) — Sat, 26 May 2018 16:00:00 +0000

UNDERSTANDING MATTERS (12)

noreply@blogger.com (Mr Koh) — Fri, 06 Apr 2018 03:44:00 +0000

Problem 28: Interesting A Maths Indices Question

noreply@blogger.com (Mr Koh) — Wed, 28 Feb 2018 08:32:00 +0000

Random thoughts on the most dreaded mathematical concepts (6)

noreply@blogger.com (Mr Koh) — Wed, 27 Dec 2017 11:29:00 +0000

Final Look at 2017: School, Science and Education

2017 saw POTUS Donald J Trump attempting to be "sincerely" steadfast in sticking to his outrageous election promises made the year before, and things sure aren't turning out quite pretty. Beginning with the signing of an executive order endorsing a blanket travel ban targeting majority-Muslim countries in January (which thankfully was repealed by federal courts with common sense yet later being reinstated by the Supreme Court on a limited basis) and a rescinding of the Trans-Pacific Partnership, he subsequently thrust upon Congress with his thuggish demands of funding the construction of the great wall of Mexico (prototypes have since emerged and gasp, climbing tests conducted to ascertain the ease with which they can be breached). Then there was his proposal of a major tax reform bill where both corporate entities and individuals pay considerably reduced rates - this was successfully approved in a Republican controlled Congress mere days before the arrival of 2018.....congratulations on his first major legislative victory?

In the meanwhile, he also withdrew America from the Paris climate agreement and constantly verbalized threats of initiating a third world war with the non-compliant North Korea. Not forgetting his probably most disturbing decision exercised till date: recognizing Jerusalem as the capital of Israel. Besides obviously becoming a mortal enemy of both the Palestinians and Arabs, he also succeeded in getting CNN, the FBI, the democrats, the United Nations, almost the entire Europe plus one exasperated Attorney General Jeff Sessions to hate his 71 year old ass with a passion.

Do I look like give two hoots about what you or the other folks think about me?

This "fine" piece of work also remains largely trigger happy despite not having featured on The Apprentice for an eternity - in the first six months of taking office, he has fired an FBI director, a national security adviser and a holdover acting attorney general, while his White House press secretary, communications director, deputy chief of staff, deputy national security adviser and legal team spokesman have all resigned. When he isn't bawling and dismissing folks, he is roasting everyone else (from Hollywood celebrities, 'disrespectful' NFL players to North Korea's Little Rocket Man aka Kim Jong Un) on Twitter, so much so former undercover CIA agent Valerie Plame Wilson is seeking to buy the social media platform by means of crowdfunding in an attempt to ban Trump from it altogether. As far as getting fellow Earthlings all bent out of shape is concerned, he is surely doing a terrific job. Alas, his day of reckoning might just be round the corner, as the noose of a Russia probe into election tampering by the feds tightens around his excessively chunky billionaire neck. Michael Flynn acquiescing for starters, how about that? Fingers tightly crossed.

On 1 December 2017, former National Security advisor Michael Thomas Flynn pleaded guilty to lying to the FBI about Russian interference in the 2016 presidential election; he lasted less than a month in the White House. (Image Credit: Chicago Tribune)

Moving from the spectacular mess spun by grandpa Trump in Yankeeland, enter super chubby North Korean dictator spotting the world's most absurd haircut - one Mr Kim Jong Un. He apparently takes great pride in antagonizing The United States, South Korea, Japan and China all at once with his ongoing pet nuclear weapons program, surfacing on occasion to officiate at the test firing of his ballistic missiles or to remind the global community of his reputation as planet Earth's number one warmonger courtesy of issuing ultimatum after ultimatum.

Check out the size of my........well, go figure. (Image Credit: Ross Hendrick)

Perhaps he isn't pure evil, at least he is pretty nice to his sister, Kim Yo Jong, whom he promoted straight to the country's all powerful politburo in early October. Oh wait, hang on a minute, then again his half brother Kim Jong Nam died gruesomely in Kuala Lumpur International Airport after having his face smeared with VX nerve agent....... surely marks have got to be deducted for this sneaky, ungentlemanly assassination attempt.

The svelte sister of Kim Jong Un; did she agree to donate her daily supper to him henceforth, thus the monster-sized promotion? (Image Credit: Bloomberg)

In another part of the world gone slightly amok, Catalonia apparently grew some balls and decided it wanted to divorce from the rest of Spain. Hence the holding of an independence referendum in October, and by virtue of a landslide win attained declared itself free from Spanish rule. The unilateral "binding" triumph was short-lived, as courts move quickly to denounce the whole affair as unconstitutional; the Catalan parliament was subsequently dissolved while its president Carles Puigdemont fled to Belgium along with four of his ministers after being charged with rebellion and sedition by the Spanish government. Snap elections for the region called on 21 December saw no clear majority won by a single party, however pro-independence camps managed to garner 70 seats in aggregate, which would therefore give them the ability to form a coalition government. Then again, can they successfully harmonize political differences on a range of other issues and march forth to forge a new republic? Would Madrid consequently resort to drastic measures in a strenuous bid to shatter their lofty dream of achieving full autonomy? Cautious optimism dancing uncomfortably with trepidation, that much we are sure of.

To be, or not to be a republic. (Image Credit: Al Jazeera)

Armed, angry and mentally unstable lone wolves continue to author fresh tragedies in blood anywhere and everywhere, the most shocking of which in recent times being the Las Vegas mass shooting incident, where a crazed gunman shattered the windows of his 32nd floor Mandalay Bay hotel suite and proceeded to unleash a massive hail of gunfire on some 22,000 people below who were then participating in a country music festival. After leaving at least 58 dead and over 500 injured, he turned the gun on himself. This invokes the memory of another horrifying episode of carnage which took place earlier in May, where a devastating explosion tore up the scene of an Ariana Grande pop concert at Manchester Arena, killing 22 people including children and injuring another 120. The suicide bomber was identified as one radicalised Salman Abedi who detonated an IED moments after the concert ended. Our collective hearts are bewildered, enraged, and most of all, thoroughly saddened. That said, the strength of humanity harnessed through faith and solidarity will prevail as history has clearly demonstrated time and again; darkness in corners shall be conquered by rays of righteousness and empathy. May the souls of those taken away too soon rest peacefully.

We danced, we laughed, we loved. (Image Credit: Jeff J Mitchell/Getty Images )

Two prominent musicians supposedly in the prime of their lives (Chester Bennington from rock band Linkin Park and Kim Jong Hyun from K-pop group SHINee) killed themselves almost 5 months apart, once again providing stoic reminders that depression is a very real illness that depletes and destroys; even famous folks can succumb to it. Souls mentally ravaged and stretched too thin to the point of sheer exhaustion see suicide as the only meaningful recourse to completely releasing themselves from the invisible pain gnawing at them. While precious lives can be saved and rehabilitated through timely medical intervention and treatment, many are still reluctant to press the panic button for help until its too late. A no-brainer: jobs, reputation and relationships are very much at stake. Which therefore begs the question - will the human race ever truly learn to dial down on being overly judgemental and better endear to its fellow members who are suffering terribly on the inside? Boy that sure is one depressing reservation to parse.

After inhaling much gloom and doom often typical of global current affairs, it's probably time to change down the gears and attempt a recapitulation of events that defined education and science in the 17th year of the 21st century yes? And no, I do not wish to talk about The Last Jedi, a monumental piece of garbage filmmaking featuring a plot so vapid it could induce seizures in a normally healthy mammal. George Lucas you were a fool for offloading your baby to Disney.

Star Wars: The Last Jedi SUCKS. End of story.

I begin by extending my heartiest congratulations to the trio of American scientists on winning this year's Nobel prize in Physics (Rainer Weiss, Kip Thorne and Barry Barish of the Ligo-Virgo observatories) for their groundbreaking work in detecting gravitational waves. A round of applause too for prolific writer Kazuo Ishiguro (best known for his enduring masterpiece "The Remains of the Day"), who walked away with this year's Nobel prize in Literature. For a man so truly dedicated to crafting emotionally penetrating novels over the decades, he certainly deserves to be honoured. To the likes of other equally outstanding folks including Margaret Atwood, Haruki Murakami, Salman Rushdie and Claudio Magris, better luck next time.

A special shout-out as well to the U.S. Department of Defense’s Defense Advanced Research Project Agency (DARPA) which engineered a non-invasive device to enhance cognitive abilities by 40%. However it must also be made clear the efficacy observed arises from tests conducted on monkeys, so it remains to be seen if human beings will respond in a similar fashion.

For the very first time, folks at NASA have discovered the presence of at least 7 Earth-sized planets happening in a tight orbit around an ultracool dwarf star, which thus increases the chances of success as far as locating an alternative habitable environment for the human species is concerned. Separately, after 5 years of probing the universe, NASA's Nuclear Spectroscopic Telescope Array (NuSTAR) endeavor has discovered the possible presence of a supermassive black hole masked by a thick column of gas. Adding to that, concepts for robotic missions to explore a comet and potential landing sites on Saturn’s largest moon, Titan, have been selected - these are scheduled for launch in the mid 2020s.

The Dragonfly is a dual-quadcopter lander that would be tasked to investigate Titan's organic chemistry, habitability and atmospheric conditions among other things. (Illustration Credit: NASA )

Besides seeking the smartest people on the planet to work for it, NASA also went searching to fill the "exotic" sounding planetary protection officer position, and guess who decided to send in an application? Jack Davis, an incredibly brave 9 year old from New Jersey who claims he is a bona fide alien (no thanks to his sister's consistently cruel teasing) and has watched almost all space/alien-related movies ever made save for Men In Black. Attaboy, he sure has a bright future ahead of him.

From one precocious kid to another, meet Reuben Paul - an 11-year-old “cyber ninja” from Austin, Texas, who properly schooled an audience of security experts when he effortlessly hacked into an innocent looking teddy bear connected to the cloud during an individual presentation at a cyber safety conference in the Hague, Netherlands. Hats off to this pint-sized genius, really.

Check out this kiddo's soft cuddly bear......as scary as the Annabelle doll from the Conjuring movies? (ImageCredit: AFP )

And then we have 12 year old Rahul from the London Borough of Barnet who has been found to possess an IQ of 162, thus putting him ahead of both Stephen Hawking and Albert Einstein in the intelligence department. As a contestant on Channel 4’s Child Genius, he made jaws drop during one particular recording when he answered every single question posed correctly. Among his feats: getting the scientific name for the apricot tree right (prunus armeniaca) and knowing that phoenix dactylifera refers to the date palm. In addition he succeeded in spelling words such as accouchement, hyponatraemia and garrulous. However, we also know the boy doesn't take too kindly to losing-he attempted to walk off the set after losing to a fellow contestant in a timed Maths test the very next day. As far as proper humility is concerned, he definitely still has a lot to learn.

While super smart kids dazzle on the international stage, super smart adults fret. Elon Musk, the real life Tony Stark whose companies SpaceX and Tesla Motors tinker with an inventory of cutting-edge toys science fiction geeks can only fantasize about, isn't quite a fan of Artificial Intelligence. In fact, he is quite adamant the rise of the machines will be complicit in the engineering of mankind's eventual doom if not adequately reined in. Can he be faulted for his supremely pessimistic forecast of things? Just ponder for a moment: "Autonomous armed robots, which can track and target people using facial recognition software, are just around the corner. Let loose, such machines would keep on killing until they ran out of targets or ammunition. This reminds us that AI has no social awareness, conscience, mercy, or remorse. It simply does what it’s been trained to do. (Steven Finlay, Fortune)." Still unpersuaded? On December 6 this year, Google's AI computer program AlphaZero taught itself to play chess in merely 4 hours and then proceeded to destroy Stockfish 8 (previously the highest rated AI chess engine) in a closed door 100-game chess marathon. If you require a breakdown of the actual results, here it is: 28 wins, 72 draws, and ZERO LOSSES. Surely that sends shivers down the spine?

It is probably time to start feeling genuinely frightened of Artificial Intelligence. (Image Credit: Now The End Begins )

Another self-inflicted doomsday scenario has been served up by prominent theoretical physicist Professor Stephen Hawking, where he envisions the Earth turning into a 'sizzling ball of fire' by the year 2600. The culprits responsible? Us human beings who siphon away too much and too quickly from the very planet we currently inhabit. The only plausible solution? Flee to another corner of the universe.

And so the world shall eventually burn, says one man.

Looking once again to the present year, numerous schools all over the world have courted controversy with high profile incidents happening in their backyards: one Ryan International School in India (a seven year old student was sexually assaulted and murdered), a nursery run by RYB Education in China (horrifying allegations of child abuse being levied at it), an elementary school of Moritomo Gakuen in Japan (Prime Minister Shinzo Abe's wife dragged into the fray and accused of cronyism in a purported shady land deal) and even Harvard University in the United States (US Justice Department threatening legal action over its Asian-American admissions practices). Other notably disturbing observations include pervasive sexual abuse in Pakistan's Islamic schools, nearly a fifth of Indonesian high school and university students supporting the establishment of a caliphate within the country and curricula circulated by the Maharashtra State Board authority which aims to teach students about the essence of air for breathing by means of a kitten killing experiment.

Just when you reckon things can't get any crazier, check out the antics of these few individuals who think they are still living in the wild wild west: a student who blew £850,000 in 73 days because the annual food grant wired to her account had erroneously included a whole bunch of additional zeroes, a high school teacher caught on video snorting narcotics in an empty classroom plus one high school principal exposed by student reporters to be a complete fraud.

Samantha Cox is a 24 year old teacher who loves teaching...........and doing drugs. (Image Credit: Lake County Sheriff )

On home soil, the Singapore Education Ministry announced in April its intention to merge 28 schools due to falling enrollment, among which include 8 middle and lower-tiered junior colleges; this resulted in countless students, teachers and alumni of affected institutions seeing red. The main reason for the huge disquiet on the ground: elite brands were spared, while the axe fell on lesser mortals. In November, private pre-school operators were on edge after it was made known children from Ministry of Education (MOE) kindergartens would be accorded priority admission in the Primary 1 registration exercise. So much for fair and square competition.

Yet all these couldn't rival the spectacular failure of public rail operator SMRT, which experienced traveling delays due to a train point fault on the very first day of the Primary Six Leaving Examinations (PSLE). Frantic candidates were either left stuck on board fearing the worst, or scrambling for alternate modes of transportation to reach their respective exam centres in time. This FUBAR was to be exceeded shortly after in both severity and scale when a stretch of tunnels along the North-South Line became flooded as a consequence of faulty water pumps, resulting in a massive disruption of train services for an unprecedented 20 hours. Oh wait, then there was also a train collision incident which left 36 passengers injured..........bravo SMRT.

And so let the curtains fall yet again I must, for it is time that I retreat for a bit and catch my breath. Many thanks for reading, and wishing everyone a splendid, joyous 2018. Peace.

My only wish for the coming new year: no more water sports pretty please SMRT. (Image Credit: sqfeed )

noreply@blogger.com (Mr Koh) — Wed, 22 Nov 2017 12:06:00 +0000

Guest Post: Ladders Go Both Ways

This guest post by Mr Kartik Chandra first appeared here on White Group Mathematics on 18 November 2017

Earlier this month, Frederick Koh, a Singaporean math tutor, invited me to write a guest post for this website of his. Of course I accepted --- blogging is, after all, a conversation.

But what could I write about?

I spent the month of November on the lookout for an idea worth sharing, and last week I found one. I follow a lot of math-educator blogs (blogging is, after all, a conversation!), and one of my favorites is that of Dan Meyers. Recently, Dan posted a rather provocatively-titled piece, "Dismantling the Privilege of the Mathematical 1%". He makes the case that those with a mathematical education --- those who are mathematically privileged --- those who make up the mathematical 1% --- they are the ones with the responsibility to define what mathematics is. Dan says, eloquently as always, that "through our action or inaction we create systems that preserve our status as the knowers and doers of mathematics."

Dan's post made me think deeply about where I myself fit into his spectrum: I'm hardly one of the 1% --- and certainly won't graduate with a math degree --- but at the same time, I love math, both as an activity and as a set of truths.

And I realized that this gives me a privileged position, one where I can comment on mathematical topics from a somewhat neutral perspective. I am neither the person who barely scraped through high school calculus, nor the person who skipped high school calculus because it was too easy. So, trapped between "math people" and "not a math person people", I wish to use this post to explore what exactly creates this divide --- or, rather, explore it in a way slightly different from what you have probably heard a dozen times already.

It is almost universally acknowledged that "education is good." More education, says society, will lead to a happier, more prosperous world. Educate everyone, says society.

I agree, of course.

And yet, paradoxically, we are so attached to the notion that education --- college, in my case, grad school or high school for others --- is a means to distinguish oneself from one's peers. A degree, we argue, makes us stand out in both the workplace and in society. The more competitive the institution, the more valuable the degree.

Less enthusiastically, I agree with this as well: reality forces me to concede the point. Why else would college admissions be so competitive? Why else would classes be graded on curves? Why else would we even have grades in the first place? As disturbed as I am by it, society cares very much about my academic performance, especially in relation to others.

And therein lies the rub. Reader, are these not contradictory notions? If the purpose of an education for an individual is to separate him or her from the general populace, then what follows is the absurd notion that universal education is self-defeating: that the more people we educate, the less an education is worth to the individual.

How does one reconcile this? Can education in the limit benefit both the individual and the society? As an optimist, I wish it could. And in fact I believe it can, but only if we rethink what education means to the individual.

Here is what I think. I believe that too often, we conceive education to be a ladder that lifts us --- above others, if we're fast enough --- rung by rung. The more you climb, the higher you get.

But too often, we forget that you can also climb down a ladder. That an education can also lower the arrogant to humility and place them alongside the less privileged, on common ground. That education builds capacity for empathy and communication, empowering the individual but also society at large to have a dialogue. That this view resolves the paradox of the previous section, because both individuals and society benefit from the capacity for having that dialogue.

What do I mean by this? Let's return to mathematics for a moment.

If you are reading my blog, you most likely have been at a gathering of mathematicians at some point in your lives. It is quite a marvellous thing to behold: a congregation of brilliant minds sharing ideas. Mathematics as a community has its own folklore, its own in-jokes, and its own language. You need only to glance at sites like The Art of Problem Solving or mathmo.org to see this community in all its glory.

I love it. There is charm to the way mathematicians celebrate their field, unlike any other profession I have come across.

And yet, imagine being an outsider for a moment. Imagine being part of the mathematical 99%. How would you feel if someone responded to a question of yours with a grin, saying "left as an exercise to the reader"? Or if someone made you use this weird software package with lots of backslashes to write up your homework? Or if someone went off on a tangent about why their coffee mug said "donut" on it? Or if someone makes an arithmetic mistake and then says "it's true in base 12" before you even notice the error? Or if someone claimed a very unobvious-to-you solution was "trivial"?

These phrases aren't meant to be exclusionary. Some are common inside jokes. Others are part of the mathematical vocabulary. The word "trivial," after all, has a very specific mathematical meaning --- think of the "trivial group" with one element, for example. I use it all the time.

Yet to an outsider, they make the mathematical community seem simply impenetrable. How am I ever going to understand all this? I can't think that fast!

"Math people" tend to be remarkably self-selecting, and I believe this is one of the reasons why: there is a divide between the initiated and the uninitiated, and far too few resources for the latter.

Education, I believe, should be tasked with bridging this divide --- rather than exacerbating it as it does now. Education should give the 99% the opportunity to join the magical world of mathematics, but education should also show the 1% how to open up the world to new members. It should teach students to write about mathematics, finding a middle ground between dense manuscripts weighed down by Greek-letter jargon, and airy puff-pieces that contain nothing of substance. Where will future Ian Stewarts, Martin Gardners, and Brian Hayes come from? I myself would almost certainly be very firmly a "not a math person" person were it not for an Ian Stewart book in my dad's bookshelf that taught me about Fermat's Last Theorem and the Mandelbrot Set when I was very young.

Education should also encourage the next generation of professors to move away from lessons that consist of the copying of lecture notes onto a chalkboard, and provide them with the tools they need to create engaging, interactive lessons that appeal not only to those whom we believe are preordained to be mathematicians, but to artists, musicians, writers, and athletes. It should teach students about the bigger picture of where their mathematics fits into society, about who produces and who consumes mathematics, and why.

At the same time, it should explain to the mathematical 99% what exactly those math people are on about all the time. It should teach them the mathematical canon, help them learn the language, and help them discover the beauty that the 1% have already found.

Yes, such an education will produce a more diverse generation of mathematicians, not only in terms of demographics, but also in terms of ways of thinking. It will produce a generation that writes not only more effective grant proposals, but also clearer papers. That's what the individuals get out of it.

But it will also inspire the mathematical community to rise up against the tyranny of Alice-gives-Bob-three-bananas standardized tests, to use their passion for mathematics and their perception of its beauty to guide the development of curricula and lesson plans. Not to reject non-math-people as heretics who refuse to see the light, but rather to see them as evidence of a broken education system that failed to convey the beauty of mathematics.

To redefine mathematics to be the way they see it, because, whether or not they realize it, the way the mathematical 1% sees mathematics --- as the pursuit of beautiful truth --- is far from what the mathematical 99% sees it as --- the painfully rushed manipulation of symbols on a midterm. To care.

That's what society gets out of it.

About The Author

Kartik is a student who studies and writes about mathematics. He cares deeply about education and the communication of powerful ideas. You can learn more about him at his personal blog, Comfortably Numbered.

noreply@blogger.com (Mr Koh) — Fri, 03 Nov 2017 11:39:00 +0000

151st Carnival of Mathematics

Probably the most "famous" association with this particular number is the highly alcoholic rum Bacardi 151, so at least we are assured Mr One Hundred And Fifty-One here isn't a bore by any account. Then again there is so much more to what it represents on the mathematical front. 151 is a palindrome prime (a prime number which remains exactly the same when its digits are enunciated in reverse order) and get this, a lucky number. Say that again?

All eliminated numbers are highlighted in orange. Smile if you are still alive.

In the context of number theory, this basically means it manages to endure a 3-stage sieving process of a list comprising all positive integers:

Stage 1 - eliminating every second number from the original list;

Stage 2 - eliminating every third number from the remaining list retrieved from the first stage;

Stage 3 - further eliminating every seventh number in the remaining list retrieved from the second stage.

Evidently our dearest 151 is very much a survivor. Outstanding.

Oh so very lucky bastard aside, 151 also has the distinction of being known as a centered decagonal number. Ascribed to paper, it visually represents an expanding series of concentric decagonal layers (of dots) surrounding a single core dot. Such numbers are governed by the basic recurrence relation where the (n+1)th centered decagonal number is equivalent to the nth centered decagonal number plus 10n:

And a set of accompanying illustrations to clarify matters if all these still somewhat sound gibberish to you.

Source: Wikipedia

On a completely irrelevant note, did you know that Singapore (where yours sincerely is based in) most unfortunately ranks a measly 151st on the 2017 World Press Freedom Index? Talk about room for improvement, obviously there's plenty.

Oh yes, extremely tragic that the Singapore government still endeavors to stifle the ordinary citizen's voice in this day and age where information is beamed instantaneously all over the internet with an effortless click of the mouse. Had to get it off my chest, now I feel much better. Or do I really?

Okie-doke, with the customary introduction of things out of the way, lets get down to featuring the awesome content created by online Maths bloggers in recent times.

Tony's Maths Blog pays tribute to the late Monty Hall who died on the last day of September this year. The legendary game show host behind the culturally phenomenal "Let’s Make a Deal" was clearly the inspiration for a popular probability-based teaser which arose thereafter bearing his namesake.

What exactly were the origins of Mathematics? Mary Bellis attempts to provide a coherent, reconciled narration of affairs over at the Thought Co website: who invented zero, when the "=" sign was first used, who invented differential and integral calculus (hint: it wasn't Isaac Newton, though this is still being debated till the present day) and many other bite-sized nuggets of trivia. Meanwhile, Kevin Houston also gets all curious with why the Matrix (no, not in reference to Neo and gang) is named as such. Ed Southall of Solve My Maths on the other hand however feels a little perplexed (perhaps distressed) about being labeled a 'Mathematician'. He bemoans the possibly unrealistic expectations thrust upon such an individual:

"All mathematicians know EVERYTHING about ALL elements of maths. You can give them an exam paper from Cambridge on an obscure module about astrophysics and they’ll soon crack it. It’s just maths right? You’re a mathematician, you can do it. We’re all our own accountants, and we could do an actuary’s job for them if we wanted, after all, it’s what we do. Number stuff. All of it."

Source: THE NIB

So asserts one Olivia Walch at THE NIB, where she proceeds to reinforce this seemingly outrageous hypothesis by means of employing Mathematical and statistical considerations in a light-hearted, animated manner.

Ever encountered an arithmetic question that got you so badly stumped at that moment in time all you want to do is grow a pair of wings and simply fly away? Well, Math With Bad Drawings has aggregated an entire list of possible ways to somewhat achieve this in an absolutely not subtle but rather hilarious post titled "Fake an Answer and Run Away". Wings not included of course.

For the extremely mathematically inclined folks, Mark Dominus of The Universe of Discourse blog shares two recently drafted pieces, the first being "Counting increasing sequences with Burnside's lemma" (a description of some interesting combinatorial relationships between Stirling numbers, polynomials, and increasing finite sequences), and the second titled "The Blind Spot and the cut rule" (a technical discussion of The Blind Spot by Jean-Yves Girard).

Ever heard of the Art Gallery Problem? Fundamentally it involves deploying and positioning a minimal number of guards within an art gallery such that they can altogether visually monitor every single corner of its premises. Anthony Bonato who helms The Intrepid Mathematician blog chooses to apply this Mathematical conundrum to The Royal Ontario Museum in Toronto. He reasons:

"The Royal Ontario Museum in Toronto, with its striking Frank Gehry design, is a great introduction to the art gallery problem. The problem is to place the fewest number of guards so they can see everywhere inside the museum. The blog post introduces the problem in a fun and simple way, and recasts it as a problem in networks. A proof of the art gallery theorem is given. Applications are discussed in the fields of computational geometry, computer vision, and driverless cars."

And it's time once again to wrap things up in this edition. It is humbly hoped you will derive considerable joy from ingesting the stuff presented herein. The upcoming 152th Carnival of Mathematics will be hosted at the Chalkdust Magazine, so stay tuned. Cheers and stay sober if you can help it.

Peace.

(PS: I would like to accord a sincere thank-you to Katie Steckles for giving me the opportunity to contribute to this blossoming math blogging community. )

3 November 2017

noreply@blogger.com (Mr Koh) — Tue, 17 Oct 2017 11:54:00 +0000

A Level H2 Maths Question-Spotting Thoughts For AY 2017

After exactly a decade of offering the 9740 H2 Maths curricula, JC2 Mathematics students this year shall constitute the first batch of A Level candidates being examined under the somewhat truncated 9758 H2 Maths syllabus which covers considerably lesser topics compared to its predecessor. Does that mean things will turn out to be much easier? Possible, though I ain't going to wager on that. This is because the actual A Level papers themselves in the past were already largely doable, and it is therefore unlikely a further downgrade in overall difficulty (of questions set) will come to pass given the education ministry's obvious intentions to adequately orientate youngsters in the rigorously demanding wilderness of STEM learning. A reduction in scope is compensated for by an enlargement in both depth and application of concepts taught, so if one is to expect anything, it would be that the problems might get a wee bit more intense. Cue: more critical thinking and improvisation involved. In this vein, allow me to present my main list of forecasts.

Complex Numbers

Hooray, the bulk dealing with De Moivre's Theorem and loci construction has been banished (to the Further Maths syllabus), as a consequence the student is left with very little content to internalize. That's the good news. The not so good news, is that he/she might perhaps be required to be more nimble in formulating solutions. To articulate this in a more concrete sense, examine the problem posed below-can you solve this set of simultaneous equations? Hint: consider multiplying both equations together, and then using the z=x+iy assignment.

Integration Techniques

Of late it has been observed that this particular integral (possessing both a linear factor in its numerator and square root of a quadratic polynomial in its denominator) is sort of making a resurgence:

Pretty please, with a cherry on top, make sure you know how to properly sort this out-till this day I still witness some of my students fumbling around on occasion despite my repeated attempts to communicate the proper manner of resolution to them. So here goes once again just in case:

Vectors

I seriously pray this is an anomaly, however apparently problems on calculating the shortest distance between two skew lines (technically not examinable within current syllabus) have been starting to surface in revision packages distributed to students in top-tier JCs, so it might be useful to thoroughly appreciate the solving process (you may wish to consult a detailed treatment I wrote HERE.) In any case, should such a problem arise, it must be packaged with various assisting pointers to nudge you in the right direction; that said it's better to be safe than sorry, so take the initiative to perform some reading up on your own first.

Functions

Officially the tiny segment concerning the restriction of a domain to enable the proper existence of a composite function isn't examined, but I am not absolutely convinced. I have encountered modified instances of this still terrorizing folks out there, so consider yourselves warned. Being also able to discern whether a function is even, ie f(-x)= f(x) or odd, ie f(-x) = -f(x) is climbing the popularity charts these days with a possible tie-in with calculating areas under graphs.....I think you get the drift.

By the way, since you are equipped with the knowledge to discover the range of a composite function fg(x), might you be able to extrapolate the logic employed in such a circumstance to similarly ascertain the range of a larger-sized composite function say fgh(x)? Food for thought.

Statistics

In all, a significant chunk of the standing syllabus has been removed (including Poisson distribution, approximation of Binomial/Poisson distribution via the Central Limit Theorem pathway etc) ; since the only "newcomers" to the party are general random discrete variables, I'd say it is almost guaranteed this section will feature in a standalone problem. Naturally the ability to compute both the expectation and variance of a given probability distribution function is assumed, so kindly don't muck those up either. Beyond that, I wouldn't be too petrified (or nervous) about the non-pure Maths section of the paper.

And here we go, a reasonably detailed account of stuff to possibly anticipate what could come your way courtesy of yours sincerely. As I have always emphasized in previous years, this is merely a tabulation of personal predictions and should in no way serve as a wholesome replacement for your revision efforts. Leave nothing to chance, be doubly (best still, triply) prepared. At this point in time while I am drafting this, the H2 Maths papers are scheduled to happen in less than a month, hopefully you are almost already to do battle. God made you cool, so be cool. Remember a well deserved break awaits your weary soul soon enough.

Good luck kids. Peace.

noreply@blogger.com (Mr Koh) — Tue, 25 Jul 2017 10:16:00 +0000

Eigenvectors and Eigenvalues

Worked Problems

noreply@blogger.com (Mr Koh) — Sat, 08 Jul 2017 16:50:00 +0000

The Chi-Square Test

Worked Problems

noreply@blogger.com (Mr Koh) — Sat, 17 Jun 2017 15:33:00 +0000

2-Sample Hypothesis Testing of Difference in Population Means

Worked Problems

noreply@blogger.com (Mr Koh) — Fri, 05 May 2017 08:23:00 +0000

UNDERSTANDING MATTERS (11)

noreply@blogger.com (Mr Koh) — Sun, 02 Apr 2017 04:35:00 +0000

144th Carnival of Mathematics

The most obvious thing about 144 is probably that it is a perfect square, ie 12×12=144. But it isn't just any square number, it is in fact the largest one to exist in the Fibonacci series:

The aesthetically pleasing Fibonacci spiral constructed by means of the Golden Ratio

And it's rather eerie current US president Donald Trump possesses a side profile that resembles the same spiral:

A truly sad,sad day for Mathematics to be associated with this particularly vile, obnoxious Homo Sapien? Hmm.

Apart from being a Fibonacci number, 144 is also unofficially recognized as a Harshad number in recreational Mathematics. But what exactly is a Harshad number? Well, a clean definition goes as such: it is an integer (in base 10 format aka the conventional number system) that is divisible by the sum of its digits. In this instance, the sum of the digits constituting 144 = 1+4+4=9, and 144 is completely divisible by 9 as 144 ÷ 9 =16, so there you go.

A third and final fun fact about 144: it is the smallest number whose fifth power is a sum of four (smaller) fifth powers. Discovered in 1966 by L. J. Lander and T. R. Parkin, it is mathematically articulated as such:

Now it's time to get down to serious business of showcasing them works of blogging Maths folks. Welcome to the 144th Carnival of Mathematics.

Manan Shah offers some problems he personally feels teachers should add to their repertoire when teaching the topic on parabolas. He further explained:

"These problems are important, yet are often being overlooked. They are designed to get students thinking a little bit more about the structure of parabolas, quadratics, and the quadratic formula as well as how the study of these objects leads to other areas of Mathematics."

Over at The Maths Mentors, a "Crack My Treasure Chest" game has been fashioned to help students thoroughly explore the Greatest Common Factor (GCF) concept. Educators seeking inspiration for fresh classroom activities may wish to check this one out.

Do you know what Havercosine or Excosecant is? The Scientific American website has Evelyn Lamb serving up a list of 10 trigonometric functions that have, erm, somewhat gone extinct in the present world order of Mathematics, which may or may not have been 'sincerely' motivated by The Onion's outrageous false piece titled "Nation’s Math Teachers Introduce 27 New Trig Functions".

Former engineer turned passionate Mathematics teacher Yana Mohanty received a 360 camera on her birthday approximately 6 months ago, and it dawned on her that it can be used to help people see the transformations that topologists call homeomorphisms- roughly stretching and bending a shape without tearing. Hence her blog post "Bagels, Pretzels... Cubical Frames", which seeks to demonstrate how a cubical frame can be stretched to a pretzel with 5 holes. A side remark in her submission to this edition of the carnival mentioned that the post was inspired by the prominent role topology played in the research of last year's Physics Nobel Prize winners (David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz).

In "Fractivities!", Jennifer Schuetz from the Fractal Foundation helps kids get up close and personal with the mathematical premise of fractals in a fun-filled hands on session; younger kids were seen happily drawing Sierpinski triangles on guided worksheets, while the older ones got to physically construct them using wooden sticks and glitter glue.

As far as pedagogy is concerned, Lucy Rycroft-Smith recommends a series of posts Maths educators can consult at the Cambridge Mathematics portal: "The Unreasonable Man" (a lovely blog about proof, reasoning and the history of Maths), "Mental Imagery" (an incredibly thought-provoking blog about what mental imagery (and other senses) might be and how it can help learners of maths at all levels) and "What's In The Pot?" (about probability and how to teach it using 'urn models', in which the author gently provokes thought about the basic models behind early probability teaching). Denise Gaskins over at Let's Play Math generously dishes out a wholesomely curated buffet of links to various Maths forums, blogs, journals and e-book downloads which will certainly have both parents and teachers feeling most delighted and grateful for.

Jason Merrill endeavors to relate algebraic and pictorial ways of understanding the Pythagorean theorem and the Law of Cosines, which he largely succeeds as evidenced in his "Geometry, Algebra, and Intuition" piece. He expands upon the rationale for doing so, stating that such an attempt "shows a lesser-used way of visualizing the dot-product that makes its main properties (bilinearity, symmetry, relationship to squared length) clearer than the "scaled projection" picture."

Finally, meet Tony. He is a Mathematics lecturer at a University in London who rides on public buses daily, and he recently contemplated the implementation of a possibly more efficient transport solution by means of employing number theory. Talk about making really good use of time even when one is on the move.

And things draw to a close once more in this edition. It is humbly hoped you will have as much fun reading the content showcased as I did assembling it all together. The upcoming 145th Carnival of Mathematics will be hosted at the Aperiodical blog, so stay tuned.

Peace.

(PS: I would like to accord a sincere thank-you to Katie Steckles for giving me the opportunity to contribute to this blossoming math blogging community. )

2 April 2017

noreply@blogger.com (Mr Koh) — Fri, 23 Dec 2016 08:15:00 +0000

Final Look at 2016: School, Science and Education

The biggest shocker on the world stage without a doubt came in the form of one Mr Donald J. Trump, who bulldozed over one very, very ambitious Hillary Rodham Clinton in the November United States Presidential Election by gobbling up the lion's share of electoral votes-this despite her securing nearly 3 million more in the popular vote territory. A final desperate attempt to sway majority of the electoral college into voting for the former Secretary of State obviously proved senselessly futile, as even more electors ran away from Clinton as compared to Trump on Monday, 19 December 2016. What lies in store for everyone as the Trump administration takes charge next January? Will he deliver on his isolationist threats in a bid to put the great back in America once more? Or will he take a step away from all the pompous election rhetoric and strike a more conciliatory stance with the rest of the world? What exactly will Time magazine's person of the year (anointed "President of the Divided States of America") do as the new incoming Commander-In-Chief? It's anyone's guess. But if anything is certain, it sure as hell will be that Mr Trump isn't always going to play by the rules.

The New incoming US President. Imaginary devil's horns or not, suck it up. (Image Credit: Time Magazine)

While grandma Hillary is sobbing away loudly, at least she can take comfort in the fact the FBI have decided to close the investigation into the e-mail scandal without pressing any criminal charges. Understandably, her supporters are not taking things lying down, demanding for heads to roll (in particular FBI director James Comey's) on grounds her campaign was somewhat derailed no thanks to last minute baseless assertions raised by the Feds. Still, she is free to roam and ain't gotten her ass thrown into jail - isn't that cause for cheer? Maybe she should consider the much more unfortunate circumstance of Korea's President Park Geun-hye and count her blessings. President Park has fallen into a shithole so deep as a consequence of getting way too chummy and trusting with a Choi Soon Sil, to the extent numerous accusations have been volleyed at her for being tediously manipulated by Choi, both mind and soul. Her political career is now pretty much in tatters after majority of standing MPs voted to have her impeached. Hundreds of public apologies aside, her nightmares have only just begun. She will need a lot more than good luck and a steady stream of prayers to tide her over.

Clearly female leaders aren't having it easy in 2016. Another case in point: German Chancellor Angela Merkel. Her Christian Democratic Union (CDU) faltered big time in the September elections, culminating in her losing governmental control of Berlin. Then only just very recently, the deadly truck attack at a Christmas market led Germany's far-right leaders to pin the blame squarely on her seemingly over liberal migrant policy.

On a lighter, completely irrelevant note, Darth Vader returns to the big screen in a Star Wars flick reasoned to have taken place between episodes III and IV. He didn't disappoint as James Earl Jones voiced him once more with the menace and cruelty expected of a longtime cult villain. And that one scene where he showed up to effortlessly slaughter them rebels sure makes it more than worth the ticket admission price. Beyond that, congratulations to martial arts superstar Donnie Yen for becoming the first Asian to have a Star Wars Hot Toys action figurine produced in his likeness. Step aside Jackie Chan and Jet Li.

I'm one with the Force. The Force is with me. So is Hot Toys. (Image Credit: Hot Toys)

Still on the topic of rock-stars and movie gods, Bob Dylan was quite literally rendered "speechless" after it was made known that he had won this year's Nobel Prize in Literature. For two whole weeks following the official announcement by the Swedish Academy, he kept completely mum and made no acknowledgement of his win whatsoever, leading many to chastise him as being rude and arrogant. In any case, he has also confirmed that he will be a no-show for the actual presentation ceremony in Stockholm, Sweden due to pre-existing commitments. Being a prick once more? You decide.

Chanel's legendary creative director Karl Lagerfeld literally digitized the world of fashion when he unveiled a gorgeous visual setup featuring meticulously conceived mock-up data centers and models fronting robot gear in his October Spring ready-to-wear presentation at Paris’s Grand Palais. Bits and bytes never looked so good till now, honestly.

Fashion re-imagined. In a discrete sense that is. (Image Credit: Associated Press)

2016 also marked the departure of numerous famed celebrities and well loved country leaders from this world, including actors Carrie Fisher (of the Star Wars movies) and Alan Rickman (of the Die Hard and Harry potter movies), musicians David Bowie, Prince and George Michael, as well as the highly revered Thailand King Bhumibol Adulyadej. Such is the frailty of life, and the eventuality of death which ensnares each and everyone of us. May their souls rest in peace.

Now that we are sufficiently warmed up, I reckon it is an opportune moment to begin examining events that have taken place during the course of the year and impacted the scientific and educational communities significantly. Shall we?

For the very first time in the history of Physics, gravitational waves were actually detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO) in Louisiana and Washington, confirming longstanding rumours and opening up new exciting frontiers in the science of Black Holes amongst other stuff in the Astronomy realm. A definite candidate for next year's Nobel Prize? Fingers crossed for now.

Speaking of the Nobel Prize, Mathematics won in a big way given that Topology featured extensively in the research efforts of the three physicists who shared this year's top honours in Physics. Heartiest congratulations to David Thouless of the University of Washington in Seattle, J. Michael Kosterlitz of Brown University and Duncan Haldane of Princeton University.

Then again not everyone in academia enjoy happy outcomes. Some get thoroughly inconvenienced by their field of work, like the next chap's rather unpleasant personal experience I am about to share.

What happens when one can't tell the difference between terrorist codes and Differential Equations? A plane gets grounded for an extended period while an Economics professor of Italian descent was interviewed by security personnel. All because an ignorant woman reported him for scribbling away somewhat furiously and also looking a tad too Middle Eastern for comfort.

Professor Guido Menzio gets racially profiled and accused all because he was seen scribbling away. Plus he looked Middle Eastern. (Image Credit: Washington Post)

Artifical Intelligence gave us humans a drubbing of sorts when Google DeepMind’s artificial-intelligence program, AlphaGo won 4 out of 5 rounds against the world's top GO player Lee Sedol at the Four Seasons Hotel earlier in March this year. Is Director James Cameron's scary Skynet hypothesis in the Terminator movies therefore becoming an accurate prediction of the future? Opponents of smart gadgets/devices will certainly imagine the worst and conjure up scenarios of mankind being completely outsmarted and eventually enslaved by technology.

Meanwhile, the Mechanical Engineering folks at Purdue University are seeking to empower non-technical folks with the ability to design and create stuff with the development of MakerPad, a system that allows for the scanning of 3 dimensional objects by means of depth sensing cameras and subsequent storage of data for a physical recreation/modification of the objects themselves. IMHO, super cool stuff.

On the education front, the US College Board found itself mired in controversy when it proceeded forth to administer SATs around the world despite possessing intimate knowledge of security breaches which involved leaking of test contents. The reputation of American college education was further thrown into doubt as Reuters uncovered possible instances of Chinese students blatantly cheating by using ringers in varsity examinations.

How should the country India be construed in textbooks? In the state of California, attempts to revise and update the history curricula for sixth and seventh graders have hit a serious snag, as scores of educators and students have come forward to protest the writing committee's interpretation of Indians as South Asians. Judging by the gargantuan amount of impassioned pleas and online petitions, this simmering debate looks like it will last for quite a while.

In 2012, a dad was cheered by the internet community when he blasted 9 rounds into his daughter's laptop with a .45 in the name of teaching her proper decorum and manners. This year, a badass mum put up her 15 year old girl's truck for sale on Craigslist after the kiddo skipped school and turned all rotten. World's meanest mum she self proclaims. I say best parent of the year.

In a much sincere effort to deliver the gift of education to refugee students, The Platform for Education in Emergencies Response (PEER) was established.. While current endeavors are mainly focused on helping those trapped in the Syrian crisis, the longer term aim is to eventually being able to reach out to the needy in every corner of the world. Kudos to the brains behind this lofty vision.

Striking a decent balance between religious demands and sentiment of the common folk is no mean feat, as is evident from the heated debate in Germany on whether Muslim girls should take part in swimming lessons alongside boys. Stemming from a court case involving an 11-year-old Muslim girl of Moroccan descent who was accorded an “unsatisfactory” grade due to her refusal to participate in school swimming lessons on religious grounds, the country's constitutional court eventually ruled against her.

Over here in Singapore, there is much to celebrate. Young swimming ace Joseph Schooling won the nation's first Olympic gold medal ever when he pipped American Olympic legend Michael Phelps to finish first in the 100m Butterfly event at this year's Rio Olympic games. Paralympians Yip Pin Xiu and Theresa Goh subsequently brought an equal measure of pride to the little red dot when they netted two golds and a bronze respectively at the Rio Paralympics shortly after.

Singapore's proud son: Joseph Schooling (Image Credit: USA Today)

In academics, our 15 year-olds also did us proud by securing pole positions in all three test categories (reading, Mathematics and Science) in the latest Programme for International Student Assessment (PISA). While we cheer this monumental achievement, does that mean we are doing everything right by our students? In all honesty, I believe it is still pretty much work in progress. We should take a moment of pause and seriously ponder whether such success was achieved at the expense of the well being of our kids. Just look at the alarming number of teen suicides which happened in 2015 (The Straits Times: "Last year's total of 27 was twice as many as the year before and the highest in 15 years.")

The annual Primary School Leaving Examination (PSLE) is perhaps the clearest demonstration of Singapore's never ceasing obsession with grades. Each year anxious parents ready their 12 year olds in a year long campaign (read: tuition, tuition and more tuition day in and out) for the first major national assessment of their lives, scrambling once more to compare scores and discuss secondary school placement strategies when the results are released. This year isn't any different, except one particular mother (who happened to run a popular local education forum) chose to deny her boy a Nintendo DS for not meeting a previously agreed upon target, and apparently said some rather unkind stuff to him in the aftermath. Consequently the online community ripped into her, and she did herself a great disservice by flip-flopping when giving her side of the tale.

And as they say, that's all folks. I have served up as much as I possibly can in one single breath, so it is only fair I am allowed to retire a bit for now. As always, I hope you have taken something away from all the ramblings here. Wishing everyone out there a blessed Christmas, and a happy 2017. Peace.

noreply@blogger.com (Mr Koh) — Fri, 14 Oct 2016 04:58:00 +0000

A Level H2 Maths Question-Spotting Thoughts For AY 2016

2016 marks the final (official) year where students get to offer the 9740 H2 Maths examination format; come 2017 things will transition to a watered down 9758 syllabus, where parts of the standing curriculum are removed and migrated to the newly reinstated 9649 Further Maths variant. Which naturally means the odds of examination questions this year (Papers 1 and Paper 2 fall on 10 and 14 November respectively) testing those sections currently poised for deletion are well...... greater than great. And this will be the main thrust of my theorizing what would actually be laid bare out on the dance floor.

Recurrence Relations

Definitely the stuff of nightmares for too many a student, might this show up for one final act? To which I say, why wouldn't it happen? Ensure you are well versed in computing the value of convergence for a given series, and on top of that, be able to retrieve the exact description for the nth term if you are presented with the recurrence relation itself. Give the following problem a try, in particular part (i):

Complex Numbers (Loci)

In 2017 these will be completely wiped clean from existing lesson plans (jealous of your juniors I reckon? Then again life's never fair), so bring along your compass, protractor and ruler, 'cause you will probably be called upon to sketch a geometrical construct or two (circle, perpendicular bisector or half-line) in an Argand diagram, and subsequently discern meaningful calculations based on the interactions between whatever it is you have thus drawn (eg points of intersection, max/min value of |z-a| and/or max/min value of arg(z-a) where a is a specified fixed point etc). Probably routine work which shouldn't cause any jitters, however never let your guard down. Adhere strictly to the "be absolutely meticulous cum careful" policy here, and I can't emphasize this enough, because I have witnessed some cartoon characters who actually misread things right from the beginning and created circles in wrong quadrants, totally misplaced perpendicular bisectors so on and so forth.

Poisson Distribution

Truth be told, I am unsure why this super minuscule segment in statistics is also being done away with, but because it shall never be encountered again come next year, get ready to jam with it for one last time. In particular, know how to solve for the range of values of an unknown Poisson parameter λ by hand when posed with the inequality P(X≥1)=k (underlying distribution being phrased as X~Po(λ) ), where a constant value will be explicitly assigned to k in the problem itself. And of course, hone your ability to tackle supply and demand related questions. Examine the following question, see if you can get it properly sorted out:

Correlation And Regression

What here exactly vanishes in 2017 and beyond? That lesser known equation r^2 =b*d, where r denotes the product moment correlation coefficient and b, d are in reference to the separate gradient values of the regression lines of both y on x as well as x on y. This typically becomes highly necessary for use when one isn't presented with data tables, and yet is still required to do the Math sans the graphic calculator. If you are somewhat unclear about what I just articulated, examine part (ii) of the problem presented below to see what I am getting at:

Functions

In honesty nothing gets axed here, though I'd like to share a personal hunch that has been eating at me in the past 2 months. Can you discover an unknown function g(x) from, say, a given known function f(x) and composite function fg(x)? Well, if you don't, spend some time mulling over this short write-up I produced HERE (see no.4 under Pure Mathematics on the page).

That pretty much sums up my forecasts, which are mostly no-brainers this time round. Whoever gets elected as POTUS, life goes on, and so will every other major examination on the planet thereafter. While you are burning the midnight oil during this season, get an abundance of nutrition within your system and keep a lid on them stress levels. Stay calm, and remember there is much celebration to look forward to after the As.

Good luck kids. Peace.

noreply@blogger.com (Mr Koh) — Fri, 30 Sep 2016 06:43:00 +0000

UNDERSTANDING MATTERS (10)