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.

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? !