UKMT is the United Kingdom Maths Trust which on foundation in 1996 brought together a trio of similar pre-existing Maths challenges at three different age groups, the Junior, Intermediate and Senior tests. Dr. Tony Gardiner is the name most associated with driving these competitions forward and now tens of thousands of pupils a year take part in the three competitions, which cover Years 8 and below, Year 11 and below, and Year 13 and below.
The format is very similar in each – 25 questions of increasing difficulty each with 5 multiple choice answer options. Clearly the standard of questions increases through the age groups but not in the way you might think. I have studied the Intermediate and Senior papers in particular and the syllabuses do not vary greatly, rather it is largely the same topics, but with more challenging questions. You will see in the appendix my research, that I have collated from past UKMT papers, the GCSE and A-Level individual topics that students need as a minimum to know to have a good chance of success in the UKMT competitions..
In order of most frequent first, Geometry, Number, Algebra, Trigonometry, Statistics, Probability (counting outcomes) and finally Ratio are the topics featured. You might think that calculus is included in the Senior challenge, but no, as I say, it sticks largely to GCSE topics but with more challenging questions, and set in a particular style which after a while becomes familiar. Both competitions challenge pupils in two ways especially; working at speed and doing problem solving, which is something the UKMT wishes to encourage.
The benefits to pupils are of the following kind. Most obviously, more exposure to subjects which will feature in their GCSE and A Level exams – for instance Pythagoras and Similarity feature extensively. Second, learning to work fast in terms of reading a problem, understanding what to do and executing the solution all within a few minutes (and without a calculator, thus developing mental arithmetic skills). Third, managing an exam – how should I proceed if my answer is not listed, when guessing a wrong answer may carry penalties, and which if any questions should be missed out? Fourth, the problems help you to think in a different way about Maths, imaginatively. out of the box if you like. And lastly, especially for those who are successful, it can add to your UCAS personal statement.
Just entering shows ambition, and there are further possible rewards for high scores, such as Gold,Silver,Bronze, Kangaroo (I’ll let you research that one!), and a national olympiad. There is also a team competition, which I am pleased to say my old school Newcastle Royal Grammar School won a few years back.
Do you have to be really good at school Maths for high scores? Well it helps of course, but the questions do encourage intuition and feeling for Maths rather than (just ) rewarding technical revision.
Shown below are some typical geometry questions, then number questions, from each of the three age group competitions, showing yes the progression and but also continuity. You will see that knowing some Maths formulae and definitions is a minimum essential – but that alone does not guarantee success, as intuition and for instance ability to quickly sketch, plan solutions, create equations or compile tables is needed too.
How do you enter? Well, the school usually helps with the administration. Remember the Junior, Intermediate and Senior competitions are typically in April, February and November respectively, with deadlines for entries a few weeks before each. A practice really is needed!
For tutors the benefit of helping pupils is that most tutors will have to stretch themselves to accurately and quickly answer the difficult questions; often there is more than one way to approach them and the challenge is to see the relative benefits of different methods; and learn to think in different ways about Maths problems beyond the confines of conventional exam requirements. My own approach to coaching UKMT is to start by going through topic by topic and set questions relevant to that subject from both UKMT and also GCSE past papers; and once any shortfalls are ironed out I begin to set full papers at first without time limits and finally within time constraints. Although model answers are available, they are sometimes a bit wordy, and I try to write out the solutions myself to force me to think through the problem and anticipate how a pupil may best understand an answer. I give tips on both managing the exam and also question-specific explanations and tips – for instance I point out that certain types of question such as circle/square combinations occur year after year.
In summary UKMT is a great initiative which encourages good Maths practices and techniques and I enjoy greatly helping pupils to become familiar with the challenging papers. More information is available through UKMT’s website
Appendix 1. List of Maths topics you need to know for Intermediate and Senior UKMT
The Extended Project (EPQ) which students can take in Years 12/13 is an opportunity to gain extra UCAS points, perhaps half a grade, and also to develop a whole new set of skills, both academic and future career related.
As a tutor I have been privileged to work recently on a fascinating EPQ and I hope played a good role in supervising and advising a student on a project related to the physics of rocket launch propulsion.
It is clear when you read the EPQ specification and marking system that approaching a half of the marks are awarded for the process of planning and executing the project, rather than purely marking the technical content. And so I bring some of my business and project management skills, as well as the academic aspect, into the mix for the student.
In choosing the title the student should do preliminary research on a topic that fascinates them and it is feasible to research and agree it with a nominated school supervisor. Begin to map out some objectives you wish to achieve and arrange them in the SMART form (specific, measurable, agreed, realistic, time-bound). The title may change a little as you go along don’t worry.
Set up a good document management system for articles you have found in the library or on the internet. Make sure you are always working on an up to date copy of your master not an out of date one. Keep notes of not just the technical content but also of the process steps you take such as how you take decisions about what to include or reject, how you avoid plagarism, how you are proceeding versus your objectives, and what you are learning; you have to complete Process Logs and these contribute to marks.
Use project management techniques such as Gant charts and stage gate control to ensure you plan out your work and use these to try and hit deadlines. Again useful to include in Process Logs.
Look at examples of projects to see how to establish a list of contents at the front, organise your paragraphs well and put a lot of work into the conclusion. A typical EPQ is 5,000 words and 25 pages. Keep structured references as you go along such as author name, article name, date. A good way of ensuring a validated paper is through Google Scholar.
You will find yourself on a technical project inevitably working way beyond A-Level syllabus. This is great!. It is introducing you to University level research and theory, and it will be a fantastic addition to your UCAS personal statement.
Do not worry if you realise that the more you uncover about your topic, the more questions emerge and you may feel your work is superficial – it is not! The writers of published Papers have years to do this, it is their job, and at the age of 17 you only have a few months on your project while focussing on A-Levels as a priority.
In summary you have to put in some extra work, but it may coincide with summer holidays anyway, and there are so many benefits ranging from UCAS points, through learning research and writing techniques in advance of a possible University dissertation, to expanding your academic and real-life knowledge.
November 14th. It is now time to end this particular blog or else it will go on forever!. Schools did indeed return in September and in my view teachers and their representatives, and pupils and parents have all done a great job in keeping the show on the road, at the time of writing, in difficult circumstances At this stage exams in England are going ahead, delayed a little to June or July, but it seems inevitable some changes such as reduced syllabus or exam questions options will be introduced. What is clear is that one aspect of education has changed forever, namely the use of on-line technology, which surely will be a permanent part of the mix even when things return to normal.
August 17th. At this stage its is likely that schools will return in September but still not certain, with Case numbers creeping up. But the real story is A- Level results and the move to stick with Teacher grades. Comparing these to previous year actual outcomes versus predictions indicates significant grade inflation will therefore take place. The infamous algorithm actually did its’ job in bringing the broad sweep of grades back to where they should be. However: two problems. First, when applying correction factors, the algorithm produced some ridiculous individual results such as fails when no exam was taken. And second, it seemed to favour smaller class sizes, which are more common in private than state schools.
July 7th Various announcements have been made that schools will indeed go back full time in September for all Years which is good news. The emphasis will be on hygiene, from washing hands to cleaning surfaces, and minimising contact through staggered timetables, one way systems etc. Rather than a strict 2m rule throughout school, though avoiding 1m still seems required. This will be difficult, but the alternative of further virtual schooling may be worse. I think it will happen, but with nuances like cutting back on aspects of the syllabus content, shorter exams and perhaps still some virtual learning (after all, some of it has been very fruitful)
One aspect of the lockdown not much talked about is the loss for Year 11 and 13 of the “going into school to get results” day, and the leaving events like Proms, and so many end-of-school holiday trips have been cancelled. It is so sad for that generation.
June 19 Primary Schools have been back since June 1, years 1 and 6 at least. Years 10 and 12 have just begun to return, a few 2-hour lessons per week on face to face, mostly focussing on core subjects. Its is a slow start but we’re getting there. Some schools are really pushing on-line work rigorously, others less so. One school I am in touch with are setting exams at end of June for Year 10’s, not far off mock GCSE standard that’s good. I can see that the on-line novelty will wear off and we need to find a way of getting children back to school, safely of course but with an attitude of “we’re gonna do this”. If not for this school year then certainly in September. I think year 10 parents are the most worried the GCSE’s will be affected and why demand for Year 10 tuition remains very high.
For year 11’s (the forgotten year) two things are happening. First, yes we know their predicted grades will be formulated into actual grades in August. Some surveys have suggested they will be half a grade higher than last year. Perhaps the final examiners will bring them back down a touch but it seems reasonable. The issue for me is that children need four go’s at really learning a topic but Year 11’s missed out on the final pre exam revision push.
So that means that the if they take a topic forward to A Level they will have missed out on that final embedding of knowledge which forms the beginning of AS Level. Which is why – the second happening – it is a great thing that schools are beginning to use the June/July hiatus for Year 11’s to begin year 12 AS Level, even if its is with videos and on-line learning. (And why I am running Maths for A Level science courses for Year 11’s! )
Today we had the publication of plans for NTP the National Tutoring Programme and it certainly seems to have had a lot of thought put into it. The website is up and running and the aims and resources are clear. I think we should wish them well in trying to do the catch up of lost time, and maybe even at the other end of the programme providing a permanent means for disadvantaged pupils to keep up.
My tuition for International students continues about the same level but there’s just a hint that some are hesitating as to whether the British international schools will be open in September. We shall see.
May 11 The beginning of the end. Or the end of the beginning. The Prime Minister announced that some restrictions will be eased and said he hoped first and last year of primary schools could open from June, with secondary perhaps seeing some face to face teaching July. But I think it will take a lot to persuade parents and teachers alike to believe it is safe. I believe it is 50:50 whether any schools reopen before September – or at least more than they are now because we shouldn’t forget technically they are open to a small number of vulnerable pupils and those of front line workers.
May 8. Still full. I lost my first Chinese pupil whose parents understandably were hesitant to continue lessons in the uncertainty about resumption. But the place was quickly filled by an extra UK lesson. Zoom works well on Waiting Room but slightly annoyingly when 1 person is Waiting and 2 are in the lesson that counts as 3, which means maximum 40 minutes so you sometimes have to restart. I have found a way of helping with student’s school web tasks but feeding the questions back into a mix of past paper questions to check they can do them without help. I’m also extending Maths for A-Level Biology to Maths for A-Level Chemistry.
Still no sign of at-school restart : safety has to be guaranteed, so if not straight after half term, that would mean end of June earliest – and what would be the point for a few weeks. Are we into Alice Cooper territory? Schools Out for Summer. Schools Out Forever? The lyrics are eerily appropriate.
April 24 The first full week after Easter and it looks like all the pupils in my schedule have returned for on-line lessons. I have adjusted Zoom to include a password and the excellent waiting room feature. For GCSE students the Maths for A Level Biology programme seems to be working well; while continuing GCSE work is useful just in case resits are needed and to keep a learning focus, I’ve offered a programme which looks forward rather than back.
Still no sign of the plans for restart: these could vary for a phased resumption before half term on geographic and yeargroup basis, to a more widespread resumption immediately after half term, to a wait till September. My instinct is for the middle option, but we shall see. Years 10 and 12 will probably be a priority.
April 3 The second week complete and all my pupils have now used Zoom with me successfully , albeit I’ll adjust some settings during Easter. Some schools now looking forward rather than back, beginning A-Level introduction early for GCSE students rather than continuing GCSE work for which there’s no exam and its now become clear today that current work will not count towards GCSE because “schools have also been told not to set extra work to inform the predictions, because young people may not be able to do themselves justice if they are incapacitated by illness or have a difficult home environment”. Likewise with some of my GCSE students I will begin “Maths for A-Level Biology” early.
March 28 The first week of shutdown has completed and Zoom is working pretty well for my remote tuition. There is a boom in Zoom round the world it seems. Schools have been using Microsoft Teams, Google Classroom, Show My Homework, Hegarty Maths, Kerboodle among others to set on-line homework tasks which vary from watching videos to answering questions and entering answers. It looks like Year 13 A-Level students’ tasks do indeed still count towards final grade; with Year 11 GCSE it is a little less clear how important their continued diligence is.
March 20: schools have shut down. Some clarity received from Government that cancelled exams will NOT mean that GCSE s and A Levels are not awarded: rather that the criteria for allocating grades will be determined by predicted grades, mocks, and coursework which teachers will collate and inform examining boards of their recommendation. These grades will be awarded earlier than usual in July and so appeals may be received and possibly an optional Autumn term exam will be arranged. What is not quite clear is whether tasks submitted on line over the next few weeks will count towards grades. Until informed otherwise we have to assume they will.
For year 10’s who are not yet taking exams the objective must be to take on- line tasks, teaching and tuition seriously and diligently to ensure the prolonged absence does not adversely affect their chances at GCSE next year
Today’s various announcements marked a Rubicon so from now I will be doing on-line tuition only till further notice, which some of my UK pupils have already started with me using Zoom. My Chinese students already do this and it works well.
March 19 : update: schools beginning to shut down and set up homework and revision material on the web systems. Some are timetabling the issue of new material to when their normal lesson times would be and some are planning to run live webinar lectures at lesson times. I am beginning to do on line tuition to UK students in the afternoon (already plenty of Chinese in the morning) and finding so far Zoom better than more well known Skype.
Still no word on decision of what might replace exams as a qualification.
March 18: update: announcement that all schools will close Friday and that exams will not take place in May/June. An announcement will be needed as to whether this means postponement till September, or waive through on Precited Grades. PM’s phrase “pupils will get qualifications” could indicate the latter. I am beginning to see how schools will keep their pupils busy: good on line portals like GCSE Pod or Show My Homework are places to set tasks.
A thought: one of the world’s most valuable Apps in moral terms is “Nextdoor” where you can find out what is happening locally, and who knows what its now worth in financial terms. Other Apps whose time has come include Zoom and Skype.
March 17 : update: Teddington has moved to closing most of the school but keeping Year 11/13 open. The reason is associated with shortage of staff, self isolating or on sickness.
Similarly Waldegrave is closing except for Year 7, 11 and 13 which remain open and Orleans Park is open for years 7,9,11,12 and 13 only.
This leaves keeps things moving for GCSE and A Level and leaves open the possibility of completing those exams but of course things are fast moving and may change.
Parents from year 10 are beginning to ask about possible extra tuition.
My personal opinion is that after this weekend the chances of UK schools having to close due to Coronavirus have moved from below 50% to over 50%. Whatever the science says, peer pressure may become irresistible. If closure happens, the length could be perhaps 4 weeks, 2 of which luckily are at Easter holiday; all the way up to 6 months including summer holidays.
With a short stop, perhaps pupils in Year 11/13 who would be most affected could receive remote schooling, reassemble for exams, and examiners might lower the grade boundaries. But for an extended outage, the question would then be, what about qualifications for 6th form and University, assuming that no exams would be possible in May unless on-line exams were mobilised quickly? I don’t believe that everyone repeating their year would be an option; firstly I do not believe pupils would want that, and second the capacity is not available unless you roll all the way back to nursery and delay the very first year of schooling.
Even a half way house of taking GCSE/A Level in September would be problematic as it would mean starting the next Year after Christmas, and requiring pupils to maintain “mental fitness” all over this summer. So an interesting alternative compromise is nearby Teddington’s plan to close the school except for Year 11/13, which at least keeps things moving.
If exams were to be cancelled altogether and yet pupils progress to the next level, that then implies that coursework and predicted grades at GCSE and A Level would come into play, as a means of determining 6th form and College admissions. But this is speculation. We shall see. Currently isolation for over 70’s seems to be the focus, but certainly schools are beginning to plan – for instance my school at Waldegrave is encouraging pupils to take more books and equipment home each day in case a sudden instruction comes.
As a tutor, whatever happens, I will offer options to parents of continuing as normal, or moving to on-line, or (and I hope not) stopping altogether. Note that better than Skype for on-line is a purpose built free programme called Zhumu, which I already use extensively with my morning Chinese students and remote Europeans and the tutoring works very well using this system. Needless to say we have already introduced handwashing.
The Biology of Coronavirus is interesting to say the least; at GCSE level we know that viruses, despite causing so much grief, are not actually living, as they do not have enough of the MRSGREN characteristics (more on that in future updates); they only live when a host is found, where they can rapidly replicate; and antibiotics do not work, instead a vaccine is needed to prevent infection rather than cure ; and at A Level you would know that the reason that soap and water is so effective is that the hydrophobic part of the soap can rupture the lipid membrane of the virus (see below)
On a lighter note
Regular readers will know that a pop song is never far away. Let’s hope the outcome is less of John Lennon’s “hold you in his armchair you can feel his disease” in Come Together, or Depeche Mode’s “you know how hard it is for me to shake the disease”; rather Paul McCartney’s “Its getting better all the time” (he always was more optimistic), a song which originated when Ringo fell ill in 1964, and was temporarily replaced with drummer Jimmy Nichol, who played five concerts before Ringo was well enough to return. During Nicol’s tenure John and Paul constantly asked him how he was coming along, to which he always replied, “It’s getting better,” In 1967 Paul made this into a song for Sergeant Pepper.
On July 20 1969 The Apollo 11 Lunar Module touched down on the surface of the moon and Neil Armstrong and Buzz Aldrin began their walk. Many (including me) judge this to be mankind’s greatest single-event achievement so far. Outlined below are the many aspects of this story which provide learning opportunities and potential exam questions across the three GCSE Sciences, particularly Physics.
Despite the enormous sound and visual fury of the launch, the fuel used by the Saturn rockets powering the mission was mainly not fossil fuel, rather it was a mixture of liquid oxygen and hydrogen. Normally gaseous, very low temperatures are required to liquefy them, -219 C and -253 C respectively. Being liquid rather than gas is safer, and occupies much less space because volume = mass / density and liquid density is higher. Saturn had sections which as fuel was used up were jettisoned to just leave the lunar and command modules. The enormous power was needed to enable the modules to reach the required speed to exit the earth’s atmosphere and escape the main gravitational pull.
The journey there
The distance from the earth to the moon is about 240,000 miles and the maximum speed was just over 24,000 miles per hour as it left earth’s orbit. So a “time = distance / speed “ calculation indicates a ten hour journey time and yet it took 3 days, so what happened to this slow-coach! Well, maximum speed does not mean average speed, and after the Saturn rockets were jettisoned, gravity slowed down the un-powered Module , as required, in order not to fly straight past the moon as it approached. Also, the journey included an orbit of the earth and several of the moon before descending to the moon so the distance was much higher.
If the rockets were jettisoned, how did the modules get to the moon without their powerful fuel? Well, once the modules were propelled out of earth’s orbit at high speed, less force was acting upon them since air resistance was zero. There was still a backwards gravitational pull of earth but it became smaller and smaller. So Newton’s Law would suggest they just carry on in the direction they were pointing, namely towards the moon, even without Saturn rockets’s major fuel source, albeit gradually decelerating from initial 24,000 mph. Small amounts of fuel were needed for lighting, communication and landing/leaving the moon, and these were a mixture of conventional fuels and fuel cells developed in Cambridge University, which were the early versions of the fuel cells we learn about in Physics GCSE. Namely hydrogen plus oxygen combining through electrodes to produce water,and release energy as electricity. The maximum power was around 2000 Watts and the water was not wasted – it was drunk by the astronauts!
“In space, no one can hear you scream”
As the advert for the science fiction classic confirmed, sound cannot travel in space because the longitudinal sound waves, whose vibrations are parallel to the direction of travel, need particles such as air to vibrate – but there is no air in a vacuum. So how come we could hear the astronauts?
As David Bowie memorably told us in Space Oddity, a conversation was possible between Major Tom and Ground Control. Well, the answer is that communication was achieved by Radio waves, which are not sound waves but Electromagnetic waves which as transverse waves vibrate at right angles to the direction of travel. Just like other parts of the spectrum – like light waves from the sun – radio waves can travel through a vacuum at the speed of light namely 300 million meters per second. Since the 240,000 miles is around 360 million meters, then using time = distance / speed, the time for a radio signal to travel from the moon to the earth is only 1.2 seconds. Hence the only-slight delay between Houston asking a question and the astronauts answering.
Note however that Michael Collins, alone in the Command module while Armstrong and Aldrin walked on the moon, could not be contacted on the far side of the moon as radio contact was lost, as expected. Perhaps this was why Bowie’s Major Tom lost contact at the end of the record – “can you hear me Major Tom?”
The physics of an orbit
When the lunar module had jettisoned its rockets it performed an orbit of the earth before heading to the moon. How does this work? If the module is set in forward motion at just the right speed then the force at right angles to its motion – namely gravity – pulls it towards earth and the net result is a bisecting direction along the path of the orbit.
The speed of the orbit remains constant at 25,000 miles an hour but the velocity is constantly changing. How can this be? Well, it’s because velocity is a vector and speed is a scalar quantity and as Vector tells Gru in Despicable Me, a vector has magnitude as well as direction. So the velocity is constantly changing because the direction in a circular path is constantly changing. When a force creates a circular motion, this is a centripetal force. (Gravity is a non-contact force while other centripetal forces are contact forces – the friction when a motor bike turns, and the tension in the spokes of the London Eye)
The diameter of the earth is about 8000 miles and the Module initially orbited the earth at around 100 miles up. So the diameter of the orbit around the centre of the earth was 8200 miles, giving a circumference of approximately 25,000 miles using Pi. At almost 25,000 miles per hour, the initial orbit took 1 hour.
The Moon Landing
When Armstrong and Aldrin’s lunar module separated from Collins’s Command Module above the moon, it reduced its speed but slightly overshot the landing site in the Sea of Tranquility in order to avoid landing in a crater. Armstrong took over control from the Module computer to achieve this ( a computer with less processing power than an I Phone incidentally). Less than 30 seconds of fuel remained, so this was where both of the astronauts’ flying experience, including dog fights with Russian MIG’s in the Korean War, proved invaluable. They stayed impossibly cool, while Houston’s control centre personnel famously were so tense they almost “turned blue”.
Armstrong’s heart beat stayed normal at 70 beats per minute, almost until the “Eagle has landed” but even he succumbed at touchdown to the fight or flight adrenaline hormone at touch down, when his heartbeat reached 150.
After Armstrong stepped down off the ladder – “one small step for man, one giant leap for mankind” – Aldrin soon followed him and began, as the Police would later sing, while Walking on the Moon, to take “giant steps” with his “feet hardly touching the ground”. Why is this? Well ,gravity there is only a sixth of the earth’s gravity ( g is 1.6 rather than 10). So it was easy to hop around. And why is the gravitational force lower? Because the force is proportional to the mass of the two objects, and the moon is lighter than the earth, even if the man has the same mass. So a person of 50 kg faces a gravitational downward force of 500 N on earth but only 80 N on the moon.
They collected rocks and when later analysed they were found
to contain the chemical lelements silicon, iron, aluminum, calcium, magnesium, titanium and oxygen. No
carbon or nitrogen, so not enough ingredients for biological life. Years
later however , hydrogen and iced water were found at the moon’s poles and this
opens the possibility, with the presence of hydrogen and oxygen, of creating
fuel cells using electrolysis which could mean that the Moon could be used as
refuelling stop on the way to Mars.
The journey back
After taking off from the moon, the lunar module docked with the orbiting Command Module and together they returned to earth. Long before the mission, Aldrin had written a thesis on docking in space based on his experience as a scientist and Air Force pilot in Korea. As the Module approached the earth atmosphere the frictional force – this time a contact force – caused the heat shield to reach high temperatures and gradually melt – as planned.
A parachute slowed the Module down further, with air resistance offsetting the weight of the Module, which floated down at a leisurely terminal velocity to the sea.
The crew were kept in quarantine for several days in case they had caught viruses on the moon. A virus – unlike bacteria – is counted as non-living but nevertheless can contain DNA. It is worth recalling that DNA was discovered by Watson and Crick at Cambridge University only 16 years before the Apollo 11 mission.
All of the above science should be readily understandable by anyone taking Physics or Maths GCSE – if not it’s a definite revision topic! For those carrying on with Physics, the A Level and Physics Aptitude Test for Oxford will contain more advanced Space concepts like eclipses, Kepler’s Laws for orbits and what many consider to be one of the all-time great equations; namely Newton’s formula for the Force exerted by gravity on two objects, of mass m1 and m2: F = Gm1m2/r^2 where r is the distance between the masses and G is the universal gravitational constant.
Scientists are still not sure what Gravity truly is, yet in the 1700’s Newton could already quantify it, and in a sense invented the science behind Apollo.
I have just completed some Maths tutoring for two excellent students hoping to join a grammar or independent school in South West London. Their approach was exemplary, their Maths was already well in advance of Year 6, and they wanted to get even better, being prepared to work very hard in lessons and at home. One full practice paper was not enough for homework, they coped with two a week. Their parents hoped for a free or reduced fees place, but if not I have no doubt they would try to find a way to sacrifice to pay fees.
With the recent news about possible expansion of grammar schools, it made me think about what would happen if my two students did, or didn’t, make the grammar schools, and also how the various entry exams compared to each other, and to traditional year 6 SATs standards. In other words, what should pupils expect in their exam? Let’s start with this.
The entrance exam
My focus was upon my local South West London schools, 10 fee paying private independent schools and 8 free, state, selective grammar schools. I drew broad conclusions about the latest exam processes, likely to be reasonably applicable outside London too. The first thing to say is that in these 18 Schools, it is very difficult to find free sample papers or even sample questions on their websites. This is to avoid advantaged children “buying” their entrance through expensive “teaching to the test” tuition. However, for some of the Surrey schools typical common entrance papers can be purchased, some schools just outside this area do publish sample papers, and of course national publishers like CGP and Bond make practice papers available.
So you can piece together what the typical test will look like. Maths rather than English is my speciality so here are some of features of the typical Maths entrance paper.
The number of questions will be between 25 and 50, students have 45 minutes to 75 minutes to complete, so at 1.5 to 2 minutes each these are short sharp questions. But the complexity varies significantly from beginning to end, so you should expect to spend 30 seconds on the easy ones and perhaps 3 minutes on the difficult ones. The ability to work fast is almost as important as the ability to answer the question. The paper typically divides, in order of questions, into what I’ll call the four quartiles of difficulty. Remember that the higher the reputation of the school, the higher the demand for places, the higher proportion of questions in quartiles 3 and 4, as follows:
1st quartile – simple KS2 topics Number : Addition, subtraction, multiplication, division (always without calculator)
Fractions, percent and decimals, number lines 2nd quartile – tricky KS2 topics Number and measurement: clock times, square and prime numbers, ratios, units of measure Algebra: graph coordinates, sequences, simple algebra expressions, Geometry:, Angles along straight lines, at a point and in triangles, areas and perimeters of regular shapes, recognise 2D and 3D shapes, simple translation and reflections. Data : Mean (average),Tables, Pictograms, Bar Charts, Pie Charts, Line graphs Problems: Inverse Logic problems such as “what number did I start with” 3rd quartile – still KS2 but highly developed problems
Number: Factor pairs, place list of fractions and decimals in ascending order Algebra: Solving linear equations, Create equations from areas and perimeters, including odd shapes; substitution of numbers in equations Geometry: Combination of angles rules in one problem, Nets, angles round a clock-face circle Rotations, Symmetry, Mirror (e.g. what would “WINTER SALE” be on a window’s other side Problems: Speed x times = distance problems, Number reasoning, Railway timetables, Time-zones
4th quartile – Beyond KS2 to KS3 and KS4 GCSE, and Puzzles Number : Exchange rate conversions, Fibonacci sequence, Prime factor trees, Ratio problems such as cake recipe; HCF and LCM; powers. Algebra: simultaneous equations created from e.g. prices of burgers and soft drinks, Multiply double brackets using grid or FOIL Geometry Parallel line angles, enlargements and scale factors, 3-D cuboids Data: Venn diagrams, Probability, Mode, Range and Median Problems: Sudoku-like magic number puzzles, Shapes representing operations, number machines Shortest route problems such as through the streets of New York; full page multi-paragraph problems featuring combination of numeric and verbal reason logic culminating in for example, which of five children got a present, which of five animal lives on which island?
This last, 4th quartile frequently goes well beyond KS2 in two respects. Firstly, what I’ll call “puzzles” – which ironically will never resurface in secondary exams. Secondly, school syllabus content stretching well into KS3 Year 8 and 9 even in rare cases up to KS4 GCSE level (yes!). This last quartile contains the differentiator questions, the ones you have to be able to do to be really confident of gaining entry. When tutoring an 11 plus pupil followed by a GCSE pupil I sometimes find myself using the same sample questions.
Most of the school websites say, to be politically correct, that the questions should be suitable for any KS2 student (only one admitted that some questions may stretch to KS3). Parents should not be fooled. With demand outstripping supply by 4 or more to 1, the higher reputation schools do throw in the puzzles and year 7-11 level questions to identify the brightest pupils.
How many exams?
The table below shows, for the 8 grammar schools sampled in SW London, most have 2 stages, although the Sutton set start with the common SET test. The 10 independents all have just one stage except St Pauls, which starts with the common ISBE test, and most have an interview to confirm selection. Most schools feature Maths, English and either a separate verbal reasoning test or similar questions within English. What is noticeable is that Non Verbal Reasoning is becoming quite rare now (thank goodness – awfully difficult to teach!)
Sample Maths available
2 stages, and stage 1 counts 10% each Maths and English, Stage 2 counts 40% each Maths and English for entry.
2 stages, and stage 1 is Maths and English OMR multi choice, passing gets you to Stage 2 Maths and English which alone determines entry
Below are the Sutton Grammars taking common SET
Nonsuch High for
2 stage, 1st English and Maths common SET multi choice, then joint second stage Maths and English with Wallington High School for Girls
No but SET samples can be ordered
High for Girls
2 stage, 1st English and Maths common SET multi choice then joint second stage Maths and English with NonSuch High School for Girls
1 stage only Maths and English common SET multi choice. Pass for eligibility for 60 places.
2 stages, first is common SET English and Maths multi choice , to get you to second stage Sutton specific English and Maths. 1st and 2nd stage tests all affect final entry, ratio is 2:2:3:3
1 stage only, Maths and English common SET, pass to be eligible for place
2 Stage , first is common SET Maths and English, second Maths and English. Count in ratio 2:4:4.
Sample Maths available
1 stage, 3 exams : English, Words and Reasoning, Maths and an interview
A few questions
1 stage Maths, English, Verbal and Non Verbal reasoning
Lady Eleanor Holllis
1 stage tests in Maths, English, VR, Non VR followed by Interview
1 Stage tests in Maths and English then interview
1 Stage tests in Maths and English then interview. Note : it confirms some KS3 material will be tested.
Yes full paper
1 stage tests in Maths and English Plus write a personal statement
1 stage English Maths and verbal reasoning followed by an interview
Yes most of a sample paper
1 stage tests Maths English and Verbal Reasoning
1 stage test Maths English and Verbal Reasoning
Yes a full paper
1st stage ISBE / GL Multiple Choice in English, Maths, Verbal and non Verbal reasoning. 2nd stage is English and Maths and interview
No but ISBE sample papers can be ordered
Grammars – the pros and cons.
Through the lens of my two students, if they started Year 7 even in the best of the local state schools, they would be so far ahead that they would, to be honest, be bored and held back. Like many bright children they need the challenge. The supply of free grammar schools is limited. At many of our local grammars the ratio of applicants to places is 4 to 1 and at some even higher, where queues around the block form at the start of exam day. (Some now phase exams through the day to avoid this). In business, if supply is limited and demand is high, you increase prices or create more capacity – in this case by creating more grammar schools, because prices are fixed at zero.
However the downside is of course that if the brightest pupils are creamed off from state schools, the overall standard must surely fall. This is detrimental to the remaining pupils, who lose the chance to learn from the approach and abilities of high achieving pupils, and dispiriting for teachers who enjoy challenging them and getting a positive can-do response. Some teachers would surely jump ship. Some Headteachers have said this would recreate “secondary moderns”.
One compromise – which one of our local state schools already employs – is to offer a limited number of exam-selective places, while mainly offering free places for local pupils. The question then is, do you sprinkle the selected pupils among the classes, or “set” from the start. The problem with the first approach is that schools are constrained by the national curriculum which prescribes certain content for certain years, so the brighter pupils would be constrained by the pace of the slowest. The alternative is to “Set” from year 7 and effectively teach the top set Year 8 or 9 level content from age 11, and take all GCSE’s (not just Maths) a year early. This “grammar stream” approach is advocated by former UCAS Chief Executive Mary Curnock Cook Or go further (as my old school used to do) and identify the brightest year 7 pupils and to remove them at year end from Year 8 and place them straight into Year 9 (we were called “removes”).
Is tutoring needed? As noted above, the questions definitely stretch beyond standard KS2 (whatever schools say). The question is, how do you get access to, and practice these. In theory, purchase of Bond or CGP practice books can do the trick, but the risk is that the pupil will miss the personal explanation and without homework being set, may not practice enough, and even these excellent publications don’t include the outrageously tricky questions which do crop up. Note also that while common entrance papers like SET the Selective Eligibility Test can be purchased, frequently these are only for Stage 1 permission to sit the really challenging Stage 2 papers which are not formally available. So structured learning, and exam tips are needed over and above school provision. Parents might provide this but many would struggle with the vital end of paper questions. Extra tutoring is your insurance policy (but not a guarantee) and this can come in several forms, including private one to one, or exam centre cramming.
What is tutoring providing?
What you are trying to do is this: First make sure the basics of KS2 are in place. Second, introduce the pupil to a selection of KS3 topics which may crop up. Third, help the pupil work at speed. Fourth, teach exam techniques. Finally set a sufficient quantity and quality of challenging tasks from which gradual improvement instils confidence – the “more I practice the luckier I get”. What is difficult to teach is the natural mathematical abilities such as puzzle solving and spatial awareness, and my guess is that is why such puzzles are included – there may be disadvantaged pupils who cannot afford tutoring yet have that innate mathematical ability which money can’t buy.
The 11 plus is highly challenging. A good KS2 performance – an 11 plus “pass” – will probably not be enough to get through. There are many pupils and parents willing to take up that challenge, to achieve that extra level of excellence. Schools, the State and Tutors all have a part to play in meeting that demand.
Initially to be piloted in around half of our primary schools, the technique involves learning techniques more by rote, asking one child to answer a question, then asking the remainder of children to repeat the answer. The class does not move on until all the class has “got it”. The brighter children avoid being held back because they have a role in leading the other children with the first answer. There are some similarities with Kumon, namely keep practising by repetition until “mastery” of a topic is achieved to an advanced level, but differences too: Chinese Maths emphasises the role of the respected teacher at the front of class, Kumon relies more on self learning through worksheets.
Chinsese children themselves are believed to be 2-3 years ahead by the time they move to Year 11; so 16 year olds in China are already at the same level of maths as an 18 year old A level student in the UK.
There is a view that culturally some British pupils are not ready for this and our cultural diversity and child centred participation doesn’t sit easily with chanting and learning by rote which is common and part of the educational ethos in China. The benefits are not at all questioned.
Chinese Maths versus English real world approach
More important I believe is this. The direction in Maths and Science in England is to introduce more “real world” relevance to exam questions, not just at GCSE KS4 but also at earlier KS3 and KS2 as well.
So while introducing a “back to basics” learning approach in Maths is very good, not least because we are slipping down the international educational league tables, I wonder if joined up thinking is taking place in Government in terms of the following two factors:
If teaching methods move in the direction of focusing upon purely numerical excellence, and yet examiners insist on setting real world applied questions where the maths technique is merely a small means to an end, do we risk the recent Biology GCSE “drunken rat” exam problem ? By this I mean that the children aim to learn the syllabus and technical methods to the best of their ability, and they put a lot of effort into mastering the knowledge and technique required in the syllabus, but meanwhile the examiners smother the questions in “real-world” unfathomable words and situations. So the child learns the techniques but can’t do the exam questions because they haven’t been schooled in the methods of deciphering them, or applying the technical knwledge they have acquired.
An example in Maths itself is the 2015 GCSE question that went viral. The question involved two techniques rarely seen together: algebra, and probability. One can imagine pupils achieving high levels in these two topics individually using Chinese techniques of practising lots of examples, but being unbable to piece together the required jigsaw which requires a different sort of skill altogether.
Mile long and centimeter deep
One other phrase associated with Chinese Maths is interesting: their criticism of the British Maths syallabus is that it is a “mile long but a centimetre deep”. There is something in this. For GCSE Maths there are five basic topics such as Number and Algebra but within those there are many sub-topics making around 80 in all. One wonders if all of these are necessary, for instance frequency density histograms are beloved by specification setters but in practice are never used by businesses. Could some topics be left out allowing time for in-depth understanding of the core?
But we are where we are: my philosophy as a tutor is to “teach to the test”, whether GCSE exams or earlier end-term tests. Because that’s what parents want. And the last thing a child wants is to open an exam paper and find there are topics they don’t even recognise. So you have to teach the whole syallabus, not just the mathematic principles but the ability to understand and answer increasingly inscrutable questions.
In summary there will almost certainly be benefits and we need somehow to catch up on global competitors. An intangible benefit may be a cultural change, to make Maths excellence expected rather than optional. But ultimately, the acid test is this: will the programme lead to better GCSE results, either higher marks, or the same marks at a younger age? This may depend on whether the new techniques are compatible with the direction of Maths exam question designers. Sound learning of fundamentals is essential and surely must be improved – so we have to start somewhere; but it may be only the first base-camp stage in achieving the summit of maths mastery. We may not be able to judge success for half a decade.
In GCSE Science and Maths you are often asked to draw or interpret graphs – representing and visualising data are the technical terms. Often it depends on whether the data is discreet or continuous.
Continuous data can be almost anything – a temperature measurement for instance – and line graphs are generaly used – whereas discreet tends to be categories that can only have certain values and bar charts are best. As an example here is an assessement I did for my hobby – assembled the top 3 pop singles each year for the last 60 years. I used a bar chart to show which artists had appeared more than twice. Not suprisingly the Beatles, Elvis and Michael Jackson were at the top. If you are pop rock and soul fan you can see the full list and how they they were chosen in this link.
As part of my tuition I run through each of the types of graphs you can see here including scatter, line, Pie, box plot, bar, cumulative frequency, histogram. These are becoming ever more important to understand with the new GCSE’s coming next year with Maths.
Another favourite with the examiners expecially with science is the concept of independent and dependent variables. Independent variables are the things you change deliberately e.g. the size of the pellets in a chemcial reaction, and these normally go on the x-axis. These “cause” a change in the dependent variables which are the “effect” i.e.they tend to be continuous, could be the reaction rate, and are usually on the y axis of a graph. Finally the “control variable” is something you keep the same to be fair, such as as room temperature or weight of pellets.
There is often a cross over between Maths and Physics so if you learn about Distance Time graphs in Maths you will also see efectively the same graph in Physics.
And often you will be asked to interpret a graph about which you know nothing such as the drunkne rat biology question – the key is not to panic and instead apply the pronciples you have learned about graph interpretaton.
And don’t of course forget algebra graphs , classic y against x, straight lines or quadratic curves, measuring gradients, shading inequalities for instance.
All in all graphs pop up everywhere in Maths and Science GCSE not to mention Business Studies!
News today that the Department for Education inadvertently but helpfully posted a SATS test on a practice paper website some time before the real thing (not a good week for the department, the National Audit Office found holes in their accounts). This made me think, how much would it really help to see an exam before?
The answer is, if you didn’t know it was going to be the real thing, then it wouldnt help that much. It would be easy to forget the solutions, especially if time passed.
However, if you did know it was going to be the exam, you would take extra care to remember the methods and solutions.
In practice the chances of this happening are almost zero. Or are they? In the following sense this does happen.
Certain questions occur time after time in pretty much the same form – just with different numbers. Actually this tends to happen more in A-Level than GCSE, but consider these examples:
June 2015 : Expand and simplify (t +2)(t + 4) November 2014 Expand and simplify (2x + 3) (x – 8) June 2013 Expand and simplify (m + 3) (m + 10)
They are in effect the same question, same technique, but with different numbers.
Will esentially the same question occur in 2016? We shall find out soon. You could look at it in two ways. Either, it occurs so often it’s time for a break: or it’s a staple question, it will occur agan. Second guessing the examiner’s mind is impossible in terms of exact questions, but broadly you can predict the type of question.
What’s clear is that this type of algebra, whether “expand the brackets”, or perhaps the reverse – “factorise”, introduce the brackets, and solve the quadratic – is likely to crop up.
Therefore if you have done your past paper practice, and it does reappear, then in effect you have seen the question before. At least the method, which you have practiced and mastered. If you turn over the paper and see this type of question, you think “joy, I know how to do this”.
Of course not every question is a “repeat” question, but broadly quite a large proportion have similarities. As a back up to learning the methods, past paper practice, with access to worked answers, is so incredibly useful ! And why my tutoring homework always includes some real (on paper, not PC ) past paper examples.
June 2015 : Expand and simplify (t +2)(t + 4) t² + 6t + 8 November 2014 Expand and simplify (2x + 3) (x – 8) 2x² – 13x – 24 June 2013 Expand ans simplify (m + 3) (m + 10) m² + 13m + 30
The start of the new English cricket season reminds me of that moment a few weeks ago when England were about to win the T20 World Cup. West Indies batsmen had a mountain to climb from their last over and Ben Stokes, England’s expert “death” bowler, was ready (Stokes had won the game for England in the semi final in similar circumstances). However (and this is a personal opinion) I think Stokes was already imagining his celebration, particularly to his nemesis Marlon Samuels, and his concentration wavered.
Carlos Brathwaite, West Indian batsman, had other ideas and England’s hopes disappeared in a blaze of sixes. His achievement lends itself to a form of GCSE Maths question proving popular with examiners. Namely the “reverse mean” question where pupils have to calculate not the average of a given set of numbers, rather the number needed to change an old mean to a new mean.
England cricket team had an average of 7.75 runs from their 20 overs. West Indies after 19 overs had averaged 7.21 runs per over. How many runs did they need from the last over to win the match ? (i.e. exceed England’s total by 1 run) (See below for answer)
Although West Indian cricket has struggled of late, the win was eventually going to happen as they have a tremendous competitive spirit. As can be see in this fascinating BBC article about the use of gamification in Jamacan classes. A small company, Edufocal, has set up computer aided classrooms for core subjects like Maths which reward the children for scoring right answers. The company is growing and results are improving. Sponsorship from Virgin’s Branson Centre of Entrepreurship is helping. Some of my pupils use CGP Mathsbuster which has a similar philosophy – bronze, silver, gold trophies are awarded as pupils move through the questions. But in Edufocal’s case the prizes are real – cinema tickets etc ( funded by subscription).
And just to keep the Jamaican theme going, one of my favourite artists and songs is Bob Marley’s One Love, the video here being not in Jamaica but London, with a yound Suggs and Paul McCartney.
So, returning to the cricket, the final was featured in the “Cricinfo” website which specialises in statistical coverage of cricket matches and includes graphs which, while not exactly the same as in GCSE, do show the power of using visual techniques to bring numbers to life.
Carlos Brathwaite perhaps wasn’t thinking of solving a GCSE puzzle as he awaited Stokes’s first ball (a glance at the scoreboard may have been easier!) But if he was. here is what he would be calculating:
England’s average (mean) of 7.75 runs per over from 20 overs meant they scored 7.75 x 20 =155 runs in total. So West Indies needed to score 156 to win. But after 19 overs, their average was only 7.21 per over so they had scored 7.21 x 19 = 137 runs. So from the last over they needed 156 minus 137 equals 19 runs to win. (In fact they scored 3 sixes from 3 balls, and another from the next for good measure, to win with 2 balls left. Bravo!)
Big data is a term used increasingly to describe the use of large amounts of data gathered electronically to determine insights otherwise lost in the detail. It is often characterised as the 3 V’s, Volume ( e.g. terrabytes); Variety (e.g. social media insights as well as surveys) and Velocity (fast data transfer and processing). A famous early example was use of location-specific Google searches on flu medicine to predict and track the spread of a flu virus through America quicker than conventional methids.
How can this principle help revision? Well, a subset of Big Data is simply “more data than usual” – big data light to coin a phrase – and I have done this with Maths past papers. Not just answers and methods are available on line for all past papers – that is well known – but also examiners’ comments are available question by question.
Initially I have looked in detail at 4 past papers, 100 questions, and captured the comments from each, 130 in all, for which the examiners highlighted common causes of lost marks from tens of thousands of entries. Then I grouped them and tallied them GCSE style in a frequency table and chart as seen left. This sheds light on general areas for revision, with (lack of) “basic maths skills” ferquently bemoaned by examiners, as well as subtle tips such as “read the question very carefully and make sure you show working”.
Then, further I picked out twenty very specific examples of errors that seemed to occur – this time syallabus technical content rather than functional categories above – and wrote and illustrated a list of “20 things examiners do and don’t like to see”. A typical one is shown. I recently took some examiner marking training and I can assure you this is true. If a question asks you to “express a number as a product of its prime factors” then merely listing them, with commas, will lose you a mark, if the “times” sign is missed out. Even if the numbers are riight as above.
I have put these and other tips together into an Easter/Summer Term special lesson covering:
– reminder of key basic maths skills, especially the ones that get overlooked
– exam technique : start, during and end of exam
– problem solving techniques for difficult, wordy, end of paper high mark questions
– active revision methods
– the twenty things examiners do and don’t want to see.
With half time biscuits covering the Jack Black “school of rock” Math video.
Another phrase used in the Big Data field is “wisdom of the crowds”. This is being applied superbly by the excellent on-line Maths tutor “MrBartonMaths” (he is also a real teacher). One of the blog pages he runs is “Diagnostic Questions – Guess the Misconception” where students are invited on-line to answer a multiple choice question and give their reason (a typical weekly question is shown). Typically around a thousand students vote (hence the “crowd”) and reveal what errors are often being made (in the example A is correct of course, C was the most common error). The misconceptions are both the students’ errors, and tutors’ sometimes incorrect expectations of what errors might be most frequent.
A lot of data is available out there on-line – the key is to process and present it in the best way to understand and hence help students.
News that Psy’s worldwide hit “Gangnam Style” has exceeded 2.5 billion video views is astonishing. That’s two thousand five hundred million (like when my football team loses 8-0, the teleprinter helpfully adds “eight”). A Maths GCSE question could be:
Write 2,500,000,000 in standard form : Ans. 2.5 x 10 to the 9th
But where or what is Gangnam? Well, the Economist reported recently from Gangnam itself in Seoul, South Korea, where just to get into the best private tuition after-school study groups, children have to pass exams; the children are cramming for crammers. These are the Hagwon schools and the best are called Sekki (cub) – most of them in fashionable Daechi-dong in stylish Gangnam (yes that one). Students work at a level up to 5 years ahead of their age group syllabus and often arrive home tired and late after a double day in education. A law is now being proposed to ban children from studying in private tuition after 10 pm.
Children also spend their free periods at school doing extra homework for Hagwon. Parents spend 0.8 % of GDP (or a tenth of all household income) on private education, which puts South Korea top on the Private Tuition World League (Britain is 8th with 0.4% of GDP). But few parents actually admit to enrolling.
But this is what we in the West are up against – huge achievement in South East Asia. Demand for tuition is so high (sigh!) in Seoul, South Korea that no advertising is needed.
But does this have a measurable impact upon results? Well, yes. according to the latest PISA study (not the leaning tower, rather the international education benchmark for 15 year olds in 72 countries). Korea is in the top ten for Maths and reading and 11th for Science (Singapore as ever dominates). While the U.K. has climbed to 15th in Science it has dropped to 27th in Maths. A sobering thought. Should the U.K. strive to match SE Asia by copying their “learning by wrote” mastery techniques, or push on with our strategy of “real world” syllabus questions perhaps more relevant to the workplace. That’s for a future blog!
Gangnam is a fashionable district of Seoul in South Korea described as affluent and the equivalent of Beverley Hills or Chelsea. Psy wrote “Gangnam Style” as a slightly ironic social comment on Gangnam residents lifestyle.
News perhaps lost over Christmas was that national tests are to be introduced by the Goverment for times tables. up to times 12 by age 11. Momentous not so much for the fact that “3R’s back to basics” are being tested – it seems to makes sense to do so – but for the first time ever a national test is to be conducted on-line with results available immediately. It is another test for teachers to organise, so more workload, but hopefully the automation minimises administration and marking (provided the iT works !)
No doubt someone will beaver away analysing where the hotspots and coldspots are ( will x7 prove the most difficult, except in Sevenoaks? Will x2 prove the easiest, especially in Twice Brewed?). A benefit of “Big Data” analysis is that it reveals “Wisdom of the Crowds”, or “Bulk Crime” as Police would call it, where when you are able to easily consolidate data, patterns emerge, which can lead to actions being addressed.
We are all getting used to using on-line Maths coaching and testing, there are scores of websites. My own favourites are CGP Mathsbuster, BBC Bitesize, AQA AllAboutMaths, http://www.cimt.plymouth.ac.uk/ .
And last but not least http://www.mrbartonmaths.com/ where he uses “Essential Skills” diagnostic Maths quizzes with the ingenious requirement to add a few lines on “why you believe the answer is correct”, which on compilation reveals the top reasons why pupils get a particular question wrong e.g in BIDMAS. And so “Wisdom of the Crowds” helps tutors and teachers identify problem areas with the certain knowledge that a large number of other pupils also find a topic difficult.
In conclusion, how relevent is the story for GCSE? Well, the national on line test is another step on the road to automation (how far will it go?) and while Times tables will clearly not be asked directly in GCSE, many steps in GCSE questions do require a thorough knowledge of the basics, especially the non-calculator exam, otherwise slow or incorrect answers will result.
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. 20m.
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!
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
(b) answer 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.
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 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!
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 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.
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”!
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…”.
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.
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!