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11 October 2015:

Of two maths questions from the UK.

Recently, a question set during a New Higher Maths exam by the Scottish Qualifications Authority was highlighted as being too tough for students to attempt. The pass mark for that exam was reduced to 34%.

I did some background check: SQA is the Scottish equivalent of SEAB in Singapore, and is in charge of setting exam papers for Scotland’s exams. For this particular exam, its standard was meant to match Singapore’s O levels, or Britain’s GCSE.

The question is:

A crocodile is stalking prey located 20 metres further upstream on the opposite bank of a river.

Crocodiles travel at different speeds on land and in water.

The time taken for the crocodile to reach its prey can be minimised if it swims to a particular point, P, x metres upstream on the other side of the river as shown in the diagram.

The time taken, T, measured in tenths of a second, is given by

(a)(i) Calculate the time taken if the crocodile does not travel on land. [1]

(a)(ii) Calculate the time taken if the crocodile swims the shortest distance possible. [1]

(b) Between these two extremes there is one value of x which minimises the time taken. Find this value of x and hence calculate the minimum possible time. [8]

~

Then, a few months ago, another maths question also made the headlines for stumping children. This was in Edexcel’s GCSE paper. (Edexcel is one of three exam providers in UK, unlike Singapore’s only one.)

There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow.

Hannah takes at random a sweet from the bag. She eats the sweet.

Hannah then takes at random another sweet from the bag. She eats the sweet.

The probability that Hannah eats two orange sweets is

.

(a) Show that

. [3]

(b) Solve to find the value of n. [3]

~

These two questions caused plenty of uproar in UK (and subsequently abroad) for their ‘excessively high standards’. However, honestly, I think the average Singaporean O level candidate will be able to answer both questions.

All along, Singaporean children and their parents have been complaining about the education system. I agree with some of the issues raised. Yet, I must say that I am thankful for the rigour of the system.

How so? When compared to other countries, I am confident knowing that for most subjects, Singapore pushes its children to learn more than what their peers are. This is especially so when comparing with US and UK.

How is this important? Simply put, in the future we may be facing global competition for jobs. Having this advantage, no matter how small, will put us in a greater stead, assuming all other factors remain constant.

If one complains about Singapore’s education, then they have probably not seen China’s. When I was Sec 2, I went to Yunnan for community service at a primary school. There was this P1 maths lesson, where they were already learning Singapore’s P3 stuff.

Just imagine these people competing for the same job as you in the future, or even university placings. Isn’t it scary?

So, whenever parents and children complain about how exam papers are getting too tough, I mostly roll my eyes. If MOE were to pander to their complaints and lower the standards, Singapore’s future generation would be gone – liào! (shout-out to LHL)

Also, parents and children need to understand that one difficult question in the exam paper doesn’t mean that the exam is doomed. If that’s their attitude in life (or what they teach their children), then I’d like to see how they react after one setback.

They must understand that such questions are meant to trip up most children, to sieve out the best of the best. (From there, y’know, meritocracy and stuff, and PAP member and stuff, y’know ...)

But I digress. Point being, unless every question feels like a torment, and unless everyone thinks so too, people should stop whining about how difficult exams are. Cos exams, like life, were never meant to be easy.

~

Now, to address the issue about maths problems. It seems like it is mainly maths that gets flak during exam season. My guess is that because maths is the most familiar topic to parents, since they have lost touch with the education system.

Hence, they extrapolate whatever maths they know / are using now, to compare it to school maths. Which is a very wrong thing to do. What students learn in school encompasses many disciplines of maths.

It is very wrong to use whatever arithmetic standards one has to judge questions on other topics, like trigonometry or calculus. Also, just because one has a maths degree doesn’t qualify one to judge.

Reason being, although yes maths degrees indicate that one has gone through much higher levels of maths, counter-intuitively sometimes primary-level questions are least solved by university graduates.

Having gone through uni maths, their brains are wired to think at that level. When faced with much more lower-level questions, most aren’t able to tune to primary level, hence find the problem tough.

That happened to me personally and my friends, and the gap is only between JC and secondary. What more after uni?

~

Another note with reference to the sweets question. UK students on Twitter have been using ‘that escalated quickly’ to comment. Which is very true, from sweets to equations.

This is, I find, one thing that stumps people in maths. They tend to draw very distinct lines between the real-world side and the theoretical side of it. And that is very dangerous when it comes to problem-solving.

Because maths has a very wide range of practical uses. From real-life problems, people assign variables to quantities and proceed to work out what they require. Algebra is about x and y, but they have to mean something too.

Too often schools are too focused on drilling how to find x, but do not teach students how to associate x with quantities in real life. Which is why, many were stumped with the sweets question.

Also, the question cleverly combines three concepts: algebraic fluency, probability and solving quadratic equations. Hence, students may feel overwhelmed as they subconsciously know the question is more than what it seems.

~

As for the crocodile question, I must say it tests a candidate’s language skills as well. However, for a diligent candidate who has attempted multiple past-year papers, this type of question, at least in Singapore, shouldn’t faze him / her.

In fact, apart from mark allocation, I think this question is a decent O level question.

~

Just in case anyone wants it, I attempt to present solutions for the two questions.

(a)(i) [thought process: ‘does not travel on land’ would mean that the crocodile swims from the very start and gets out of the river only when he reaches the zebra. Hence, point P would be at the zebra, then x = 20.]

$\textup{When }x=20, \newline T=5\sqrt{36+20^2}+4(20-20) \newline {\color{White} T}=5\sqrt{36+400} \newline {\color{White} T}=5\sqrt{436} \newline {\color{White} T}=5\sqrt{436} \newline {\color{White} T}=10\sqrt{109} \newline {\color{White} T}=104.403065089 \newline {\color{White} T}=10.4 \textup{ s (3 s.f.)}$

(a)(ii) [thought process: ‘shortest distance’ would mean that the crocodile immediately swims across perpendicular to the river. Hence, \[x=0\].]

$\textup{When }x=0, \newline T=5\sqrt{36+0^2}+4(20-0) \newline {\color{White} T}=5\sqrt{36} + 4(20) \newline {\color{White} T}=5\cdot 6+80 \newline {\color{White} T}=30 + 80 \newline {\color{White} T}=110 \newline {\color{White} T}=11 \textup{ s}$

(b) [thought process: for an O level candidate, such a phrasing would trigger one response: differentiate!]

$T(x)=5\sqrt{36+x^2}+4(20-x) \newline{\color{White} T(x)}\; \; \; \; \; =5(36+x^2)^\frac{1}{2}+80 -4x \newline T'(x)=\frac{1}{2}\cdot 5(36+x^2)^{-\frac{1}{2}}(2x)-4 \newline {\color{White} T'(x)}=5x(36+x^2)^{-\frac{1}{2}}-4 \newline {\color{White} T'(x)}=\frac{5x}{\sqrt{36+x^2}}-4$

[thought process: now to find the value of x for which T is a minimum!]

$\textup{At min. }T, T'(x)=0. \newline \Rightarrow \frac{5x}{\sqrt{36+x^2}}-4=0 \newline 5x=4\sqrt{36+x^2} \newline \frac{5}{4}x=\sqrt{36+x^2} \newline \frac{25}{16}x^2=36+x^2 \newline \frac{9}{16}x^2=36 \newline x^2=64 \newline \therefore x=8 \, (\because x\geq 0)$

[thought process: finally, find the minimum T!]

$\newline \therefore \textup{min. }T=T(8) \newline {\color{White} \therefore \textup{min. }T}=5\sqrt{36+8^2}+4(20-8) \newline {\color{White} \therefore \textup{min. }T}=5\sqrt{36+64}+4(12) \newline {\color{White} \therefore \textup{min. }T}=5\sqrt{100}+48 \newline {\color{White} \therefore \textup{min. }T}=5\cdot 10+48 \newline {\color{White} \therefore \textup{min. }T}=50+48 \newline {\color{White} \therefore \textup{min. }T}=98 \newline {\color{White} \therefore \textup{min. }T}=9.8 \textup{ s}$

~

(a)

$\newline\textup{no. of orange sweets}= 6 \newline\textup{no. of yellow sweets}= n-6 \newline\newline P(\textup{both sweets eaten are orange})=\frac{6}{n}\cdot \frac{5}{n-1} \newline {\color{White} P(\textup{both sweets eaten are orange})}=\frac{30}{n(n-1)}=\frac{1}{3} \newline\Rightarrow n(n-1)=90 \newline n^2-n-90=0\textup{ (shown)}$

(b)

$\newline n^2-n-90=0 \newline (n-10)(n+9)=0 \newline \therefore n=10\textup{ or }n=-9\textup{ (rej.}\because n\geq 0)$

TTFN.

P.S. For the SQA paper, it was typeset in Trebuchet MS. I don’t know who did it, but Trebuchet MS is a big no-no for maths typography. If one were to see the original paper, they had to switch to Times New Roman for numbers and formulae, which was disastrous. I wonder why no one thought of using Times New Roman all the way, or at least a respectable serif font.

3 October 2015:

Quick point: two lives.

On the way back from camp, I heard a radio programme which interviewed two TV producers that went to ground zero of the Indonesian forest fires, which has been causing the much-hated haze. Their recounts were terrifying.

One of them was talking about how Indonesian schoolchildren were without masks in PSI 1500 environment. When asked, they replied, "so what (else can be done)?" Such is the life that these children go through.

Meanwhile, when I opened my Twitter feed, I see someone complaining about having university exams on a Saturday. Sure, I get the logic. This would eat into your weekend, resulting in a loss of time to cultivate your social life.

But hey, if the haze worsens, these children may even lose their lives.

TTFN.

aboutme.

From Singapore. 20 years of age. Blogs as and when inspiration comes, in British English (and Singlish), Traditional Chinese and (hopefully) Russian. Not a lifestyle blogger, expect posts to be serious, dull or even obscure. I enjoy comedy, in particular British humour.

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