# The Maths in Greg Rutherford’s garden long-jump pit

Athletics has had a bad press recently, rightly so. But let’s celebrate one of Britain’s greats, Greg Rutherford, rightly nominated this week in the twelve for BBC Sports Personality of the Year

Greg Rutherford’s fantastic long jump win at the World Championships meant he joined the select band of Brits holding the four major athletics titles at once. It was all the more fascinating because he has built a long- jump training pit in his back garden, as you can see below.

And a genuine GCSE Physics or Maths Higher tier question might be this: end of paper “tricky”, but in line with the emphasis on “real world problem solving”.

Question: Greg builds a long-jump run up and pit in his back garden.  He typically accelerates evenly from 0 to 10 metres per second in 4 seconds, then runs for 2 more seconds at 10m/s before take off. The world record leap is then approximately 9 metres and he allows another 3 metres for landing.  What is the minimum length Greg’s garden must be, from beginning of run up to end of landing?

Answer: in the first phase the word “evenly” implies a straight line velocity versus time graph from 0 to 4 seconds, and the distance covered is the area under that graph, namely half the base (2 seconds) times the height (10 m/s) i.e. 20 m.

The second phase at constant speed is simply speed x time equals distance i.e. 2 seconds x 10 m/s equals 20m.

The sand pit must be 9+3 = 12 m so the total minimum length is 20+20+12 equals 52m.

Finally, back to those awards: why no cricketer?! (Joe Root, genuine personality, Ashes winner, record number of international runs in a year!)

Foornote May 2016: Greg has actually announced a world class competiition in hos own back garden using the afore-mentioned long jump pit!

# Diwali – great festival, great syllabus

One of many wonderful aspects of working with the Indian community – as people with IT experience like myself find – is the Diwali festival of lights. It is happening right now and in the office and at school it means sharing delicious highly coloured sweets specially made for the day – one of my favourite lunches of the year! In the streets it means bright lights and fireworks. The etymology behind Diwali is “rows of lighted lamps”.

You will find Diwali in the GCSE Religious Studies (RS) syllabus. For those not familiar, RS first covers broad topics like Matters of Life and Death, Belief in God, Marriage and Relationships and Community Cohesion.  Second, different Units cover different world religions and the sections are Beliefs and Values, Community, Worship and Celebration and Living the Religious Life. Diwali of course as a Hindu celebration is covered.

Diwali’s basis is the victory of good over evil, and has different emphases in different parts of the world. Two of the major focal points are the ending of the Ramayana story in which King Rama is reunited with his long lost wife Sita after fourteen years of exile; and also honouring Lakshmi the Goddess of wealth on day 3 of the 5 day festival. Customs include spring cleaning, making Melas including confectionary, drawing Rangoli or coloured patterns on the floor, sending cards to friends and relatives and it also marks the Hindu New Year and the start of the Business Year.

Diwali also features in the Unit on Sikhism because Sikhs celebrate the release of Guru Har Gobind and 52 Princes from imprisonment at Gwalior Fort. The Golden Temple in Amritsar is lit for the occasion.

The festival, also celebrated in the Jainism religion, occurs annually at the end of November or beginning of October depending on the new moon.  You can see that in India itself the lights are set up in many places such as next to railway tracks. It is highly photogenic as you can see in these examples.

I followed the RS GCSE syllabus with my son and it is truly fascinating. I am coming from a tradition where the subject matter was essentially Christianity only, with the most radical alternative being the Screwtape Letters by C.S.Lewis of Chronicles of Nania fame, in which a series of letters portray the various human temptations as viewed by the devil.

RS content varies from multi-ethnic and multi-racial societies, ethics of divorce, to theories on the origin of the world including creation, intelligent design, evolution and the Big Bang theory, which provides the link back to this science blog because Big Bang is covered also in Physics GCSE.  RS also provides an excellent lead into Philosophy and Ethics A-Level.

Finally, the design of the RS GCSE paper itself is interesting. There are two papers, one for general and another for specific religion. In both, the layout is that for each of the four headings described above, four questions are asked, with a choice. They test both the students’ ability to learn the meaning of key words and concepts, and also their reasoned opinion on specific topics.

So in the Worship and Celebration section of the 2014 Edexcel Unit 13 Hinduism GCSE paper, the following question occurs.

Do you think Diwali is the most important Hindu festival? Give two reasons for your point of view.

(Answers for the proposition include celebration of Rama and Vishnu, and the victory of good over evil. Answers against include other festivals such as Navaratri are more important).

RS GCSE – fascinating syllabus! For pupils and parents alike !

# Colorado yellow river pollution in GCSE?

The VW nitrogen oxide and now CO2 saga shows that pollution is still a concern, and reminds me of the recent story of heavy metal pollution in Colorado turning a river literally yellow: could it feature in GCSE Chemistry?

Truth is stranger than fiction. The theme of the Simpson’s movie was that the USA Environmental Protection Agency (E.P.A.) turned into bad guys and erected a giant dome around Springfield in an attempt to contain the water pollution that Homer had started.

In August however the tables were turned when the real E.P.A, accidentally pumped polluted water into a Colorado river while clearing up a mine. The pollution spread and extended over the border to New Mexico and the river turned yellow, becoming contaminated by heavy metals including lead, iron, zinc, copper, mercury and arsenic. Read more in the BBC’s account and CNN broadcast

Could pollution and heavy metals feature in GCSE chemistry? Well, it is at the margins, but yes they could. Heavy metals are amongst the transition metals of groups 2 and 3 of the periodic table, of which students must know the layout.

Pollution as a whole is often featured these days in GCSE chemistry as a supplementary question, sometimes around acid rain. For instance one specific sample question was to “list the advantages and disadvantages of mining metal ores” to which a good answer might include “they can cause difficulty in clearing up once closed down, as shown in a recent incident in America”. It is worth students talking to their Chemistry teacher about the incident.

Water purity sometimes crops up as well in GCSE, with one of the purposes being to remove heavy metals.

It remains to be seen whether the American Government will fine itself several billion dollars, as they did to BP after the Gulf of Mexico oil spill (I declare an interest as a BP shareholder and former employee !) The affected Narajavo Nation for instance is already threatening to sue. It reminds us in Europe of the sad decline in colour of the Blue Danube.

To finish on a lighter note, while Yellow River certainly was not a Heavy Metal record (it was by UK Group Christie) it is worth mentioning the possible origin of the term Heavy Metal. In Chemistry it is because the metals mentioned have high relative atomic mass. In music it is probably from the phrase “heavy metal thunder” in Steppenwolf’s “Born to be Wild” which featured in the biking film Easy Rider (above) , or from the title of Iron Butterfly’s 1968 album “Heavy”. Many songs claim to be the first truly heavy metal song, the most famous of which are the Kinks “You Really Got Me” and  the Beatles’ “Helter Skelter” at the end of which Ringo, after drumming so loudly, famously agonises “I’ve got blisters on my fingers”!

GCSE-taking teenagers (OK, of the boy variety) will probably associate  heavy metal most closely with AC/DC, and their soundtracks from the films Iron Man 2 (Robert Downey Junior) and Battleship ( Liam Neeson, Rhianna), or from the video games like Rock Band and Mad Max. Parents may be interested to know that AC/DC’s Back in Black is the second highest selling album ever behind Michael Jackson’s Thriller, and ahead of Pink Floyd’s Dark Side of the Moon, Whitney Houston’s Bodyguard, and Meatloaf’s Bat Out of Hell. Could there be such a diverse set of albums?

# Origin of the word Google – it’s Maths!

The Economist this week speculates that we are running out of combinations of letters for company names, and mentions the best and worst examples of made up names. One of the best is Google, which lead me to research its origin.

The good news is, there is a Maths angle.

The word Google comes from the googol,  namely 10 to the power of 100, or 1 followed by one hundred zeros.

The founders of the company used the googol to represent the search engine idea of identifying an extremely large number of options.  But the story goes that googol was
mis-spelled as google and the rest is history.

A nice GCSE question, in the new mode of “challenging”, might be:

A googol is 10 to the power 100

(a) What is a googol divided by ten to the power 98
(b) Write in standard form 15 googols

These could be seen as frightening, yet easy at the same time:

(a)  answer = 10² = 100
1.5 x ten to the power 101

The word googol itself was invented by a nine year old (why am I not surprised?) in the 1920’s.  The nephew of American mathematician Edward Kasner.  To get an idea of what a googol “looks like” it is similar to the ratio of the mass of an electron to the mass of the whole visible universe.

The word google in fact was mentioned before the company invention by an unlikely author, Enid Blyton. Not in “A very large number of people go the smuggler’s top” but in the term “Google Bun” in Faraway Magic Tree. Also (much more likely)  Douglas Adams used the term Googleplex in the Hitchhiker’s Guide to the Galaxy, while Google itself uses “Googleplex” as the name for it’s HQ.

Googleplex is in fact the term for 10 to the power googol ( ten to the ten to the 100)  which is a very large number indeed, perhaps to infinity and beyond. The mind boogles. I mean boggols. I mean boggles. in “Back to the Future 3″ the Doc says about future wife Clara ” She’s one in a billion. One in a Googleplex!”

The word googol surfaced again when it was the £1 million question in 2001 in Who Wants to Be a Millionaire?, the one where Charles Ingram was revealed to have used an accomplice.

Google (the word) is often in the news. It was the subject of an imaginary merger of the future with Amazon and subsequent war with Microsoft in (the Epic 2014 Googlezon wars).

It has officially become a verb (to Google, to search). Ironically Google the company doesn’t like this use, because it has come to mean “to search the whole web”, not just using their search engine, although most people do actually use Google as their primary search tool.

Google has been translated for instance into Chinese

After a financial reorganisation, Google the company name, has technically become “Alphabet” (a combination of word search and alpha-bet, the best algorithm choices). Personally I don’t think “Alphabet” will stick – the word will never catch on!

Finally, the Economist rated Google one of the best company names (becoming a verb clinched it). The worst? A large consultancy expensively renamed itself “Monday”, a name judged so bad that it did not last to the Friday, when it was taken over.

# Headmaster suspended for letting pupil take exam early

A headmaster in Wolverhampton has been suspended, and then reinstated after an enquiry, for allowing a pupil to take a GCSE English exam a day early. The reason seemed a little lax, namely to allow the pupil to go on holiday with their parents.

One assumes the enquiry involved checking his phone records and those of his friends in the few hours after the exam!

It reminds me of another “exam made easier” story from June when the answer to one GCSE question was helpfully supplied in another question, in the same paper. An AQA Chemistry paper contained the following:

2a. Fill in the blank. Limestone is mostly calcium ————
5b Limestone is made mostly of calcium carbonate…

In terms of making exams easier, let’s finish on a more serious note, well slightly more serious; allowing computers in exams.

The head of the OCR exam board suggests that Google be allowed in exams. The responses have varied from “ridiculous” and “rubbish” to “it would test resourcefulness and initiative rather than just your memory”.

Another proponent of the use of computers in exams is Dr. Sugata Mitra who conducted the famous experiment to place a computer in a hole in the wall adjacent to an Indian slum and found 7- year old children very quickly picked up skills with no assistance. It is a topic that won’t go away. But that is for another blog!

# One Direction’s Maths Song – it’s good!

One Direction were in the news yesterday for postponing a concert, but the Mathemateer is more interested in the group’s Maths song.  Yes there is one, and it’s good !
As described below – can you do the mental Maths? !  Hear it here.

One Direction were in the news earlier as they are rumoured to be taking a break in 2016. Recalling the excellent parody of their own song “What Makes You Beautiful” on Radio 1, the band wrote and recorded “Maths Song” whose main chorus is “Your Maths Skills = Terrible”

It features a series of quick, simple, mental maths tasks whose eventual answer is 130.  At GCSE level this would not constitute a genuine question but on the other hand the lost art of mental maths should not be underestimated. As a warm up exercise before a test you could do worse than follow this through.  Well done 1D as I think I should call them!

# Could you wear Pink Floyd in Physics exam?

I, like many aging rockers, proudly wear my Pink Floyd Dark Side of the Moon T shirt around the house.

So the question is, could a pupil wear this in a GCSE physics exam, and would it be of any use?

The answers are possibly, and yes.

Most pupils have to wear school uniform in GCSE exams, but it is possible some don’t (think re-takers or adult education).  But it is likely they would have to change or cover up, as “notes that would help” are precluded.  If however all those barriers were crossed, would it actually be of use? The answer is definitely yes. Useful both to you – and here’ s the catch – everyone else!

A very typical Physics GCSE question might be to predict and explain the path of white light entering a prism, and what would the positions of red and violet light be?

The T-shirt goes a long way to answering the question.

White light disperses as it enters a prism because different wavelengths of light refract by different amounts. Unlike a rectangular block, the boundaries of a prism are not parallel so the different colours of different wavelengths do not recombine.

But why is red at the top of the spectrum and violet at the bottom, and how do you remember which way round it is? Well, red has the longest wavelength of the visible part of the electromagnetic spectrum and is refracted i.e. bent the least, whereas violet has the shortest wavelength and is bent the most.

How could you remember this? For the exam you certainly need to learn the key parts of the whole of the electromagnetic spectrum from radio waves down to gamma waves; and within that, the order of the visible light colours – but how to do that?

Well, you could wear the T shirt and be asked to leave the exam room. or use a technique close to the Mathemeteer’s heart – the mnemonic (always wondered how to spell that!)

Richard of York Gave Battle in Vain       Red Orange Yellow Green Blue Indigo Violet!

Pink Floyd’s chorus children famously sang “We don’t need no educashion”. Oh but you do!

# Crocodile Maths problem goes viral

Another 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…”.

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

The 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.)

Paul 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

Connections 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.

His 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

360 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!

# 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.

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!)

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.

Note 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

The 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.

Mostly 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? !