Category Archives: Maths

Crocodile Maths problem goes viral

crocptintAnother maths exam problem has gone viral after the earlier “sweets in a bag” Twitter storm. This time a Scottish Highers Maths question about crocodiles and zebras (yes!) proved insurmountable. Over and above the technical solution (see below) there were a number of interesting aspects for us English in GCSE – land.

First,  the Scottish exam structure is completly different to England’s. There is no mention of GCSE or A-Level, so Higher in Scotland is roughly equivalent to A-Level in England, as it is described as a “pre-University qualification”.

Second, could such a question appear in English Maths GCSE ?  Very unikely for the reason above, and because the best solution involves calculus, which is still not in the new GCSE 9-1 syallabus. Calculus is in IGCSE, but even so the crocodile problem would swallow up time as a very tricky differentiation is involved. It is, however, still just possible that a problem like this could be in our GCSE 9-1 syallabus because an alternate solution for it is through “iteration”.  But solving it this way would surely eat up time, since perhaps 9 iterations might be needed with awkward square roots.

Third, it shows that quality control of questions is vital, especially when exam structures are changing. Ambiguity can be a killer. In this case many of the “descriptive” parts are not black and white (unlike the poor hunted zebra) .  For instance how important is the width of the river? This makes even the first two “easy” parts tricky as you spend time understanding the English meaning.  A shame – I feel the crocodile question writer (from Dundee?) crafted a potentially great question, but was let down at the end by the oversee process.

Fourth it shows there is a strong interest in Maths amongst the general public (I assume not crocodiles!) as the web post was No.1 in the charts for the BBC’s most read posts.  This is encouraging!

Finally it shows there is no place that examiners won’t go to make questions less purely numeric, and more “challenging”. Another question involved toads and frogs down a well – let’s not go there.

For the record, the techncial solution (in summary!)  is as follows – and it’s not a snappy answer!

(1) When x is 20, substituting in the equation for T gives T = 10.4 seconds
(2) When x is 0, substituting in the equation gives T =  11.0 seconds.

The minimum time T occurs when the dertivative (the differential) is equal to zero i.e. a turning point.
Differentiating the equation and solving, we find x = 8. Substituting back in the original equation, we find that when x = 8, T = 9.8 seconds.  We can prove that this turning point is a minimum by feeding in x values either side of 8 and showing that T is above 9.8 in both cases.

And that leads to the non-calculus “iteration” method, that in theory an English GCSE pupil could cope with. But you would have to start at x = 1, evaluate T, then use x = 2 and evaluate T again, and follow T down all the way down to x = 8, and find that T reduced to 9.8, Then for x = 9 find that T begins to increase again, i.e. a minimum had been reached at x = 8.

As I said, not snappy!

In conclusion, the question would not appear in GCSE south of the border due to content, and I don’t think it would make it to A-Level becasue of ambiguity – but its a salutory lesson for examiners.



England’s rugby demise – a GCSE lesson.

England’s rugby team failed the test as the Wales match approached its climax. This has lessons for how to approach exams. 

England exited the Rugby World Cup after losing to Australia, but for me the damage was done against Wales. I would argue that first loss was associated with “game management” and the parallels with “exam management” techniques are striking as we shall see.

A few weeks before the tournament, most critics would agree that the two teams were well matched. Then a few weeks before the match, Wales lost half their back line to injury, and during the match lost another half. Any small England selection errors were more than neutralised by Welsh misfortune. So duly, with 20 minutes to go, England were cruising, 10 points to the good, chances to extend.  And yet they lost. Why?

In my view, the following: some bad luck with events, but mostly game management.

Harold MacMillan, former U.K. Prime Minister, once famously answered a journalist who had asked what could blow Governments off course: “events, dear boy, events”. England could not cope with events that should have been surmountable.

When Lloyd Williams hopefully kicked cross field, the oval ball could have bounced anywhere but in fact bounced perfectly into the hands of Gareth Davies to score. Bad luck, but it is how you react to events, and England’s game fell apart from there.

Another penalty conceded – more inability to understand what the referee wanted. (Are England penalised more than others despite, or because of, complaining a lot?)  Then the fateful decision to go for the win instead of kicking for goal, not in itself illogical – the kick was missable, and risks sometimes are needed  –  but the decision to throw short at the line out, and risk being pushed into touch, was poor.  Then one final chance, possession lost.

Stuart Lancaster, England coach, is reportedly a fan of the book and philosophy “The Score Takes Care of Itself” , in which Bill Walsh describes his experience as an American NFL Coach, arguing that the preparation, the little things, make the difference in leadership.  Admirable, but no amount of preparation can overcome a coach or player’s inability to react to, or influence, events as they unfold.

Stuart is clearly a fantastic coach who oozes integrity, but before the tournament he said one thing which surprised me along the lines of, “my input to a match ceases just before it starts”. This refers of course to preparation, but did it betray an element of believing that events would follow the natural course, and so for instance substitutions would always follow at the preordained time?

You feel that New Zealand would also have taken the line out instead of the kick, but would have found a way to control it and win, borne of the confidence of winning late many times. They would have found a way to win.

The great sports people and teams keep their game management together as the pressure builds. Think, in contrast, of poor Jean van der Velde, the inexperienced French golfer who found himself only needing to avoid a triple bogey at the last to win the 1999 Open Championship at Carnoustie. In golf, we have “Course Management”, choosing the right clubs for distance, terrain, conditions. Unfortunately, Jean seemed to forget these guidelines, going via railings and rough to water. After removing socks and shoes he holed out for a 7 but lost the playoff,

England had one more chance, against Australia, but their confidence had gone. England lost the 1991 World Cup final against Australia because, by common consent, they listened to the critics and tried to play with flair instead of playing to their pack strength. Has the same thing happened recently?  England have focussed on addressing their perceived weakness – the attacking flair –  but judging by the way the Australian pack won scrum penalties, and had the edge at breakdown, it seems that England have let their forward advantage go.

And so England lost heavily to Australia. it probably would not have mattered had they beaten Wales. And that I would argue was due to Game Management.

Exam management

Think of the Maths exam as that rugby match. You are cruising through a twenty question paper, then just after half way you see a difficult question.  You get stuck, you take too long. You begin to panic, you answer a question on “direct proportion” but forget the principle of feeding back the answer to double check. It turns out to be a wrong answer.

Then you see an algebraic question which requires a quadratic equation to be solved. However much you try, you cannot get the factors. But you haven’t noticed the question says “answer to 3 decimal points” (if so you would realise there are no factors as such, you have to use the quadratic formula).

Then a question involving Pi asks you to leave the answer “exact”, but instead you insert many of its ongoing figures rather than leaving Pi itself in. More marks lost.  You fail to understand what the examiner wants, and what he will penalise you for.

You think you have an easy compound interest question. But you misread that it requires the final amount, not the interest paid. And you waste time doing the manual calculation as well, because you cannot find the “x to the n” button on the calculator.

A question on graphs  is on the next page. You think, “this used to be my strength, but now It is all about Real Life Graphs, with wordy problems about bike rides and punctures. It’s a weakness now, I will have to pass!”

The next one looks easier. But no, it’s on Transformations. I can remember Reflections, but not the one that also begins with “Tr”? It all seemed so easy on my “maths-to-go” and “maths R us” websites. For a moment you remember an old black and white video you saw, what was his name, Brain Clough? “We had a good team on paper. Unfortunately football is played on grass”. You muse that this exam is the reverse, I can do the questions on the computer, unfortunately exams are on paper”.

Then finally you come to the last question. The bell will sound in a few minutes. It looks difficult but features probability, your strength. Decision time. Should I go back and pick up some easy marks by finishing an earlier one, or go for the five marker? You go for the latter.

But what’s this, it starts with probability and bags of sweets, and ends with an algebraic proof of n² – n – 90 = 0.  “I have no clue how these things are connected! I give up”!

In conclusion

Could this happen, or is it just that nightmare where you dream you haven’t done your revision? Well consider this. It has happened and very recently. Thousands of students were approaching their Maths paper’s end – almost injury time so to speak – when they came across exactly that probability question above.  The complaints caused a Twitter storm. Read the story, it went viral,

In fact a reasonable student could have solved this, had they stayed calm at the vital moment andlinked two seperate methods.  Exam management, just like game and course management, can be the difference between achieving your goals and just missing out.  You still made you’re A* to C, but not the A*.  You have the abiility, but the sheer mechanics let you down

The week between the Australia match and the final, irrelevant match against Uruguay will be the longest week of the team’s lives. Plenty of time to reflect on what might have been, just like the Summer Holidays for a student who might think “if only…”.

Maths and the NPL Music Society

NPLConnections between Maths and Music are many and varied. Here is another, indirectly at least.  In Teddington the National Physical Laboratory and “Home of Measurement” plays host to the NPL Music Society, where small classical music lunchtime concerts are given in the Scientific Museum, Bushy House.  These concerts feature pianists, singers, small chamber groups and recently a harpsichordist who perform in a room overlooking Bushy Park, while surrounded by all manner of scientific measuring instruments. The next performance is Thursday October 22nd   2015, featuring Haydn and Granados.

Meanwhile at Waldegrave School in Twickenham, a representative from the NPL recently gave a talk to the 6th Form Physics Group on the subject of standardised time zones and time measurement.   Before the advent of the railways in the mid-19th century there were no standard time zones in the UK, and time differences between cities could vary by as much as 20 minutes, as explained in this article.

The NPL is home to the first Atomic Clock developed 60 years ago this year. The Caesium atomic clock is accurate to 1 second in 158 million years.

Maths GCSE includes questions on converting ratios with different units into “1 to n” ratios. It is an extreme example, but in this case the accuracy would be 1 to 158, 000,000 times the number of seconds in a year, which is 31,556, 926 (you didn’t know this? Nor did I!). Making  :  4,982,688,000,000,000 in all, or about 1 in 5 million billion. 

If you find that mind boggling consider this: the next generation of atomic clock will make the above look piffling, and will be 100 times more accurate, making an accuracy of 1 second in the age of the universe. I cannot get my head around that! It presumably would enable us to figure out if the Big Bang was late in coming, but that is another story, although Big Bang is actually covered in GCSE Science and Physics and also in Religious Studies.  More on that another time.

Meanwhile back where we started, here is a link to an extensive review of a NPL concert from a couple of years ago and a more recent advert for a December 2015 concert featuring Natasha Hardy.

Philadelphia Soul helps GCSE Maths

Billy360 Degrees Of Billy Paul was one of the classic Philadelphia soul albums in the early 1970’s. It features the famous Gamble and Huff composition “Me and Mrs Jones”.  Billy went on to record “Let Em In”, one of the few occasions, like Joe Cocker with a Little Help from My Friends, where the cover is arguably better than the original by a Beatle.

To be pedantic, Billy’s face only appears to be rotating 180°, nonetheless it is a classic album cover, and 360° features of course throughout GCSE Maths, in “bearings” questions, circular geometry, symmetry and segment analysis.

A typical foundation level question might be:

In the shape above, where is the line of symmetry?  Answer is a line, drawn vertically down the middle.

Then a supplementary question about symmetry for higher level might be along the lines of:  

If we then assume there is fourth hidden face at the back, and it is a 3-dimensional model, and you look down on it from the top, how many lines of symmetry are there? Answer:  4

And what is the order of rotational symmetry? Answer: 4 because there are 4 points through a rotation of 360° where the shape would look identical.

Final;ly a typical mid-level higher tier geometry question featuring 360° would be:

A circle has a radius of 3cm and a sector is cut out with angle 60°. Find the exact area of the remaining shape, leaving pi in the answer.

Ans. The remaining shape must be a large sector of angle 360 less 60 = 300°.  It’s area must be
be  (300   x  pi   x  3²) / 360   = ( 5 pi  x  9)  / 6  =  15pi / 2  cm².

The Mathemateer is a very sad person who must get out more. Everywhere he gos he sees Maths questions!

Life of Pi – Maths makes you cool!

The Life of Pi – Maths makes you cool!

Watching The Life of Pi film again recently.  Most people (OK 99.9%) of people remember the tiger, but the Mathemateer was most struck by the scene in which young Piscine Patel, tired of mockery, jumps up and announces his nickname is Pi, and what’s more can recite it to many decimal places. He writes 3.14159 …etc to many hundreds of decimal places on the board and achieves instant stardom.

Pi features in many GCSE Maths questions in formulae and it is really important for pupils to know which formulae are given in the formula sheet (for instance the volumes of spheres and cones) and which are not (quite rightly the formulae for a circle’s area and perimeter are not).

A typical higher level question might ask this:

In a full, tightly packed golf ball box there are two golf balls. What % of the volume of the box is occupied by the golf balls?

At first you think, we are not given any dimensions, how on earth can we solve this? The trick as you will increasingly see in the new syllabus is to think about a problem laterally and say, “Ok let’s call the radius r, see what happens, and start doing some calculations”. You will soon find that the volume of the box is 16 r³, while the volume of the two spheres is 8 pi r³/3 and a quick division gives you an answer of 52.4% because the “r” terms cancel out.

And finally, one more thing to remember. pi is an irrational number, which means it cannot be expressed as a whole number nor even a fraction. In fact it goes on forever, which is why Piscine is such a hero! And why it is used in “express to 4 significant figures” questions! Or why, if a GCSE question’s answer involves pi, and says “ leave as an exact answer” the pupil has to simply leave pi in the answer rather than try to work out the never ending, inexact, result.  “Exam management” tips like this win points!