Author Archives: Rick 1

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.

enWhen 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. develdeDM_468x336Think, 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…”.

5-4-3-2-1: song and the sequence

Paul jThe ever popular Manfred Mann song,
5-4-3-2-1 (a sixties hit and more recently part of a chocolate advert) surprisingly features references to the Charge of the Light Brigade (“onward rode the 600”) and Helen of Troy. Sung by Paul Jones, who joined the band after failing to complete his Oxford English degree, Paul was replaced by Mike d’Abo. Paul went on to have a varied solo career, currently still active as a music broadcaster and respected harmonica player, still playing the clubs. And looks hardly a day older even today!

The Mathemateer’s interest is not confined to the music, as the sequence “5-4-3-2-1” could one day feature – if not done already – as a GCSE question, set at around Grade 4 Level.

Namely, prove that the nth term of the sequence 5-4-3-2-1 is –n + 6.  (See below for the methods to prove this.)

Mm2Paul still plays in various bands, including the Manfreds and the Blues Band, and was interviewed on Breakfast TV with fellow original member Tom McGuiness (yes, that one). When a clip from “Do Wah Diddy Diddy” was played, Paul was asked whether he still did the famous knee-knocking dance style (think Mick Jagger, perhaps a precursor to Dad Dancing) and he replied “No, but I still play the maracas!”

There is a thriving club scene for bands – original and tributes – from the 60’s, 70’s and 80’s. For instance one of Paul’s other bands, the Blues Band, soon plays Blackheath Halls, which typically eatures comedians as well, such as Arthur Smith.

South West London features strongly in the history of rock’n’roll, and for instance the Half-Moon in Putney is still going strong, with Eddie and the Hotrods playing soon.  The Boom Boom Club in Sutton is soon host to Curved Air, and tribute bands like Alter Eagles (love that name) and Absolute Bowie. The Clapham Grand  will soon be featuring Like the Jam, with original member, that great bass player Bruce Foxton (As an aside, The Mathemateer went up to Somerset House recently to see the wonderful About the Young Idea retrospective of the Jam’s career. It reminded him that although the above websites are great, he misses the paper bill posters, showing for instance early Jam on the same bill as the Clash).

So, back to Maths questions: how on earth is 5,4,3,2,1 represented by “ – n + 6 ”?

This is intriguing because no less than three methods of solution are available in GCSE textbooks, summarised below. The first is the most often recommended, and the others are also of interest.

Method 1 The sequence reduces by 1 each time so there must be a “–n” in the answer. Then for n =1, what must be done to get to the first term of the actual sequence? To go from -1 to +5 you have to add 6.

 So answer is that an expression for the nth term is –n + 6. Check for n= 2, the answer should be -2 + 6, which as expected equals 4.

 Method 2 The formula for the nth term of a liner sequence like this is the nth term = dn + (a-d) where d is the difference in successive terms (-1) and a is the first term ( 5). So -1n + (5 –(-1))  implies the nth term is –n + 6, as before.

 Method 3. Form two simultaneous equations for the first two terms using a for the n part and b for the number. Using the first and second terms, 

For n = 1  a + b = 5
For n = 2  2a + b = 4

 Solving these, a = -1 and b = 6.,  So the answer as before is that the nth term (known as Un) = -n + 6

 Always double check the answer. For example, feed n= 5 (the 5th term) in. U (5) = -5 + 6 = 1

 It’s a strange but true answer. “5-4-3-2-1” expressed as –n + 6 ! But correct!

 And finally, Paul Jones sings “onward rode the 600”.  

 A supplementary foundation question might be: what is the 600th term? Answer of course is -600 + 6, namely -594.   

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.

From Free Schools to Benedict Cumberbatch

Recent news from the Government that a new (10th wave) of Free Schools is to be approved in England prompted me to look into the questions, what exactly is a Free School and do we have any in the TW area?

Free Schools are similar to Academies (like the excellent Waldegrave School) in that they are in the State system but not directly controlled by the Local Education Authority, so they are “free” in that sense, as well as making no charge to parents and having no academic selection criteria for admission.  But they differ in that normally they are new schools, sponsored or run by an education or learning charitable trust.

In the TW area, two Free Schools are opening as we speak, namely Turing House, run by the Russel Education Trust, and Twickenham Primary Academy at Heath Gate House Twickenham Green, run by the GEMS education trust, who are also next year opening a Primary Academy in Kingston. GEMS is a successful worldwide provider of education services, started in 1959.

The Turing House school is to open in a temporary site in Teddington, but is possibly later putting down roots in Whitton. This is strange for two reasons, first the idea was sold as filling a gap in secondary education in the Teddington area, and indeed there is some opposition in Whitton because of the effect upon traffic. You can argue that it is still within a couple of miles, but a trip across the A316 (either way) seems like a different place altogether. Second, the name Turing comes of course from Professor Alan Turing, associated with the NPL in Teddington, who many believe was the father of modern computers and artificial intelligence.

Turing, who graduated from Kings College Cambridge, was a talented marathon runner who regularly ran the 40 miles from Bletchley Park to London for meetings.  He lived at one point in Hampton, where you can see his Blue Plaque.

BeneHis life was recently portrayed memorably in the film The Imitation Game  (named after his “Turin Test” for artificial intelligence) by Benedict Cumberbatch, plotting his triumph in cracking the Enigma code through to his tragic death after undergoing treatment for his homosexuality, then illegal.

At one point Cumberbatch’s character says “There are 159 million, million, million possible Enigma settings…it is 20 million years to check each of the settings (manually)”

This links to a potentially typical GCSE foundation question which might be:

Express 159 million, million, million in Standard Form. Ans. Each million has 6 zeroes i.e. 106, and when you multiply such numbers you add the powers.
So it is 159 x 1018  and finally in standard form 1.59 x 1020

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.

King the tiger as Richard Parker, with Suraj Sharma as Pi, in Ang Lee's Life of Pi.

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!

GCSE – new 9-1 Grade structure

GCSE grade structure is changing – is it getting more difficult?. Well yes, and one of the first signs of this was even before 9-1 came in: the first GCSE Maths question that went viral on Twitter was in 2015 even before the grades had changed. On this BBC video you can see that a probability question was deemed to be impossible because it mixed in algebra unexpectedly. But this type of challenge will become typical. Let’s examine grade structure and difficulty in more detail.

Media coverage of GCSE results of course shows mainly groups of girls jumping for joy (why girls? – partly because girls’ results tend to be better than boys’ and because boys think acknowledging success is uncool). Results are broadly stable year on year – typically 69% achieved A* to C  while 6% achieve A*

But one of the main points often missed in the coverage is that from September 2015 the new GCSE syllabus and grading scheme started for Maths and English (first results in 2017) while the remaining subjects start a year later.

What will this mean for pupils and parents? First of all the grading system is changing from A* down to G, to 9 down to 1, with 9 being the best. You will see from the chart below that A*/A now cover 9,8 and 7, while the definition of the minimum level for a “good pass” – namely C grade, changes to grade 4.GCSE structure

Reasonably straight forward. But parents will have to get used to conversations around “your child is on National Curriculum Level 7c but is expected to get a grade 8”.

But that is not the main change in the Mathemateer’s opinion, certainly not in Maths which we will look at in more detail.

There is now a foundation and higher paper in Maths, and Foundation will cover grades 1 up to 5, and Higher grades 4 up to 9 (so a slight overlap). But really it is content we should be looking at. It’s not a case of “meet the new boss, same as the old boss”.

Having benchmarked international results (a good thing, from the Mathemateer’s business experience) the Government now wishes to raise standards by making exams, well, more “challenging” – that’s harder to you and me. So the following changes will take place:

More difficult topics.  The diagram below summarises a Pearson guide, showing the cascading of some more difficult topics to the levels below. So “quadratic sequences” is among eight topics formerly just A-Level now in GCSE higher, and long division is among fourteen new topics in KS2 (and hooray it’s not needed manually in GCSE!)lev222

More formulae must be learned. Back in the day you had to learn all formulae, then it was relaxed, now only a few will be given. Even the dreaded Quadratic Formula has to be learned.

More questions of a “problem solving” nature. So a basic question like “divide 100 in the ratio 3 parts to 2 parts” will become something like “Brad and Angelina went to a film and shared a 100 gm box of popcorn in the ratio 3 to 2. How many grams did they each get? And what film did they watch? (OK the last bit is a joke but you get the picture, there is just more to read and understand, more to misinterpret).
And that is not all. Remember the “amazing Twitter rant”, as Al Murray would say, last June straight after the GCSE Maths exam. The unfair Maths problem that went viral, (see BBC or Guardian links) involving choosing coloured sweets and ending with a request to prove n² – n – 90 = 0.  At first you think, what on earth is the connection? But the point of the question is to reward those who realise there is a connection.  The individual steps are not unreasonable (probability diagram, create a formula, multiply brackets, rearrange) but putting these together proved too much for most.

viNote there was a rumour that the timing of the Twitter storm actually was clocked as starting before the end of the exam. If this is correct then some students either had mobile phones in the exam, or walked out early and the first thing they did was take to Twitter. If so, it confirms my despair!

This tricky question was before the syllabus change. This type of question – where techniques across the piece have to be linked – will only become more common.

Foundation or Higher?

A great presentation comparing grades is attached. It predicts that to get a C 4 in Higher you need around 40% but if you take Foundation you need 70% to get C 4. Is Foundation that much easier? I’m not convinced it is. You could argue that unless you are worried about failing altogether (by getting less than 17% in Higher) you should stick to Higher. The actual boundaries when issued should prove interesting.

Finally, more exams: 3 times 1.5 hours exams instead of 2 exams covering 3.5 hours.

Some slight pieces of good news for students. A small element of multiple choice will be introduced (always easier in the Mathemateer’s opinion) and the syllabus still does not include Calculus (wish it would, it’s not that hard and would bring GCSEs and IGCSE’s pretty much together).

But on the whole the majority of factors above will mean that standards will be raised – that’s good – but more preparation will be needed for a more challenging set of Maths exams.

 

The Mathemateer Blog

BLOGThe Mathemateer blog has the hallmarks of the pupeteer – it presents an entertaining show by modeling well known characters and events from film, media, sport, music, and TV in order to illuminate GCSE questions and education matters.  It also rsembles the musketeer and mutineer – it fights conventional wisdom about what an education blog should be.

GREAT EQUATIONS BlogjpegMostly for Maths, with a soupcon of Science, with the aim of helping parents to understand the type of question their children are faced with, and perhaps to risk a dinner table conversation around “could you answer this question?” Great expectations indeed – or should that be Great Equations? !