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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? !    


A banker GCSE science question – the environment; what students need to know.

There is much media talk about environmental issues like climate change, renewables, air quality and pollution. Penalties for diesel cars to discourage nitrogen dioxide and particulate emissions are being introduced, and, astonishingly, the Tesla electric car company’s market value at £40 billion has now overtaken Ford’s despite only achieving a fraction of Ford’s sales. What can we say about the inclusion of these topics in Science GCSE? Well, firstly, there are lots of examples, and “environment” is one of the few certain, banker questions in the whole of the GCSE syllabus. Secondly, examiners are looking for proof that students understand some of the technical language involved. Let’s take a look in more detail.

Questions about the environment in general have become so popular in Chemistry, Biology and Physics papers, in both core and triple science alike,  that I sometimes think you just have to mention “carbon dioxide” and you are half way to a pass ! Even if  you are a climate change skeptic, suspend that view until after the GCSE’s! Certainly the payback on a relatively small amount of revision on a not-too-difficult subject is high since one or more questions will almost certainly feature – which cannot be said about all science topics. After last year’s “drunken rat” controversy, students tweeted that they had learned their CGP Biology guide religiously, yet so little of the syllabus cropped up.

A common fault amongst pupils is to confuse climate change and pollution, so that’s a good place to start.   Students should understand the following three key points:

1. All three of the sciences begin this topic with fossil fuels, which are mainly oil and derivatives like petrol, also coal and gas.

2. There are two separate consequences of fuel combustion.  On the one hand, the generation of the greenhouse gas carbon dioxide as a natural but increasing product of combustion. Then on the other hand emissions of bi products such as sulphur dioxide and soot  which cause pollution and smog.

3. And to counter these problems is the emergence of renewable energy sources such as solar power which reduce dependence on fossil fuel.



Candidates should ensure they understand both the advantages of fossil fuels and derivatives (fairly cheap, easily available, engines designed for them) and disadvantages (may run out, greenhouse gas generation, pollution, scars on the landscape). Similar pro and con assessments should be learned for individual, different renewables.
The basic mechanics of global warming should be understood. Rays from the sun entering the earth’s atmosphere bounce off the surface, and we need this to happen to a certain extent to prevent overheating; but increased CO2 concentrations don’t allow enough infrared radiation to escape, leading to a small but significant warming of the earth. (This is the way greenhouse glass works).

Some effects of global warming should be learned such as polar ice-caps melting; sea levels rising and coral reefs deteriorating as ocean temperatures rise; and species migration patterns changing.


Students should be able to interpret graphs such as global temperatures rising on the y-axis – but note the typically narrowed scale – with time on the x-axis, especially since the industrial revolution. These show global average temperatures rising around 1 degree C to 14.6 degrees, which alongside CO2 atmospheric concentrations rising from 0.028 to 0.038 % seem to provide a link.

Now let us summarise what revision is needed in each of the three sciences in addition to the above, and the type of question likely to be asked. (More detail is available in my coaching card lesson plans). Each science begins with the basic assertions above, then develops different angles.


Pupils should understand how fractional distillation of crude oil works, including generating products such as petrol and diesel fuels as described in this BBC video about Grangemouth refinery where I visited many times in my work.


Students should learn the basic word equations associated with combustion of fuels which are generally alkane hydrocarbons .

Hydrocarbon +  oxygen –> carbon dioxide +  water + energy released

and one example, for natural gas combustion.

CH4     +            2O2             →         CO2          +        2H2O

And also the word equation for acid rain, which damages buildings and statues, especially limestone :

Sulphur Dioxide + Water -> Sulphuric Acid

Students should also know the formulae of Nitrogen Dioxides (NO and NO2) and also understand how incomplete combustion produces sooty carbon particles (turning bunsen burners yellow) and in extreme circumstances the poisonous carbon monoxide (CO). Methods of reducing emissions are important such as scrubbers at power stations and catalytic converters on cars.

(In a sense I am pleased to see nitrogen and sulphur oxides (called colloquially “NOX and SOX”) being given priority once more. As a performance analyst in BP in the 1990’s I collated the emissions data from BP Chemicals’ factories including NOX and SOX. When CO2 was suddenly elevated to a much higher importance, I always worried that focus on these polluters might be lost).

The alternative fuels of particular interest in Chemistry are ethanol, bio-fuels like bio-diesel, and hydrogen along of course with their pros and cons.

The cracking of alkanes to alkenes and subsequent polymerisation also features in this context both for the fossil fuel origin and the non biodegradable nature of plastics.

Another branch of pollution features in metals extraction and mining, with heavy metal dis-colouration of rivers a possible disadvantage as seen in this article about the Colorado river.

Exam Questions have ranged from simple (what type of reaction is burning fuel in oxygen?);  to numeric (compare the parts per million figures for particulates, carbon dioxide, nitrogen dioxide for several types of fuel); to everyday experience (why do supermarkets charge for plastic bags?);  to involved (describe how a fractional distillation column works).


For Physics pupils need to know more detail about individual energy sources both conventional and renewable. The energy transfer steps for several of these should be understood. So for power stations running on fossil fuels, the transfer is from chemical (in the fuel), to thermal (burning it), to kinetic (turbines and generators); to electrical (the grid).

Nuclear power involves a plentiful supply of uranium and plutonium but they are finite resources so counted as non-renewable. And of course though they have the advantage of being green in a sense – no carbon dioxide emissions – the disadvantages include waste disposal and impact of major break downs (albeit rare) like Chernobyl.

The major renewable sources to learn are: solar panels and solar cells; wind turbines and wind farms; geothermal hot rocks; hydroelectric power; and tidal barriers. For each of these students should learn the energy transfer process, and advantages and disadvantages, perhaps two of each. For instance, for wind energy the transfer is from from kinetic wind energy to kinetic blade energy to electrical. The advantages include it’s renewable, and has zero carbon dioxide emissions and pollution; but it is not always available (when calm), the turbines can scar the landscape, and though costs are reducing they are expensive to build and maintain.

Typical questions have included: is global warming of 5 degrees C over the next 100 years a fact, a guess or a prediction?; why are copper pipes under a solar panel painted black?; calculate the cost of waste energy from a food processor and how it is manifested;  why do chemical salts used to store solar energy need a high specific heat capacity?; explain the difference in actual versus maximum electrical output percentages for a variety of energy sources; give 2 advantages and disadvantages of running gas fired versus nuclear power stations; why are transformers used between power stations and the national grid?

Finally, although electric cars are not specifically on-syllabus, that won’t stop AQA or Edxcel throwing in a wildcard question like “compare the advantages and disadvantages of electric cars versus conventional petrol or diesel engine cars”.  Answers should include reference to easy availability of petrol (difficult for electric chargers); petrol is from fossil fuel and so contributes to global warning (electric cars do not – though the charger itself has to be charged); petrol and especially diesel cause particulate, sulphur and nitrogen dioxide pollution whereas electric cars do not; and conventional cars currently have a higher mileage range than electric.(Note that a £300m electric taxi factory is opening in Coventry – truth is stranger than fiction as 3 years ago an A Level Business Studies question case study was built around just such a possibility, Even Business Studies is not immune from our topic ! )


Photosynthesis and the carbon cycle are highly relevant in this context.  This is a must for Biology exams, not just for the environment  question. The word equation for photosynthesis must be learned:

Carbon dioxide + water (with sunlight) –> glucose + oxygen
6CO2                  + 6H2O    light –>            C6 H12 O6  +      6O2

(The reverse equation for respiration of course also is important)

The carbon cycle includes the absorption of carbon dioxide through photosynthesis in leaves, and the production of carbon dioxide through respiration and also decay of dead animals, which eat vegetation.  This has been in balance until recently when from the industrial revolution onwards fossil fuel combustion is producing more carbon dioxide – only by a fraction but enough to mean an increase in the concentration of carbon dioxide in the atmosphere, which in turn links through to the so-called greenhouse effect and global warming.

In Biology, further emphases include the generation of another greenhouse gas methane through more intensive farming, and the reduction in CO2 adsorption through Amazon rain forest depletion, in tandem with the production of CO2 from burning those forests.

Fossil fuel pollution includes damage to leaves from acid rain because their waxy layer for mineral absorption is damaged, while health is affected by carbon monoxide because in red blood cells it binds more strongly to haemoglobin than oxygen.

Typical questions include; describe the main points of the carbon cycle and the role of photosynthesis; what can we do to slow global warming?; interpret a bar chart of billions of tonnes of carbon dioxide produced at each stage of the carbon cycle.


Environment as a subject is as near to a banker question as you can get, and one of the few where parents can easily help children, especially as GCSE age is just young enough for pupils to still accept parental advice! Further, you will hear almost daily on the news stories about this topic,  whose science may well feature in GCSE and so a round table discussion could follow at dinner. The key points are to start with are fossil fuels, but distinguish between carbon dioxide emissions – said to cause global warming;  and sulphur and nitrogen oxides, bi product polluters causing building damage and health issues. Then candidates should be able to explain the science and list some solutions for these problems. The examiners want balanced arguments, so be prepared to list both the pros and cons for conventional and renewable energy sources.

What to expect in the 11-plus

What to expect in 11 plus entry exams

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
 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
: 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!)


Maths English Non Verbal
Comments Sample Maths available
Tiffin Boys  Y Y  N 2 stages,  and stage 1 counts 10% each Maths and English, Stage 2 counts 40% each Maths and English  for entry. No
Tiffin Girls  Y Y N 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 No
Below are the Sutton Grammars taking  common SET
Nonsuch High for
 Y Y N 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
 Y Y  N 2 stage, 1st English and Maths common SET multi  choice then joint second stage Maths and English with NonSuch High School for Girls As above
Greenshaw High  Y Y  N 1 stage only Maths and English common SET multi choice. Pass for eligibility for  60 places. As above
Sutton Boys  Y Y  N 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 As above
Wallington County  Y Y N 1 stage only, Maths and English common SET, pass to be eligible for  place As above
Wilson’s Sutton  Y Y N 2 Stage , first is common SET Maths and English, second Maths and English. Count in ratio 2:4:4. As above
Maths English Non Verbal
Comments Sample Maths available
Hampton  Y  Y  N 1 stage, 3 exams : English, Words and Reasoning, Maths and an interview A few questions
Halliford Y Y Y 1 stage Maths, English, Verbal and Non Verbal reasoning No
Lady Eleanor Holllis Y Y Y 1 stage tests in Maths, English, VR, Non VR followed by Interview No
St Catherine’s Y Y N 1 Stage tests in Maths and English then interview No
Y Y N 1 Stage tests in Maths and English then interview. Note : it confirms some KS3 material will be tested. Yes full paper
Y Y N 1 stage tests in Maths and English Plus write a personal statement No
Kingston Grammar Y Y N 1 stage English Maths and verbal  reasoning followed by an interview Yes most of a sample paper
Reeds Y Y N 1 stage tests Maths English and Verbal Reasoning No
Y Y N 1 stage test Maths English and Verbal Reasoning Yes a full paper
St Pauls Y Y Y 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.

In conclusion

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.

Chinese Maths in English schools

News that Chinease Maths techniques are to be introduced to English schools needs some explanation and examination.


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.

ch1 ch2

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.

Measuring success

ch3In 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.

Drawing graphs – Top 180 pop records example

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. 



graph6As 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!

Brexit through lens of A Level Business Studies

The AQA Business Studies  A level Buss 4  exam is coming up, and Section B (a 50 minutes essay) is rumoured to perhaps contain a Brexit question.   Jim Riley, former businessman and consultant turned on line business academic, runs the excellent Tutor2U website with his twin brother. It covers business and humanities subjects and as well as briliantly assembling a knowledge bank offers sound advice on structuring exam question responses. He lists the various subjects covered already in the Section B past papers – margers, integration, planning, innovation etc. – and points out that European business is one of the few subjects not covered yet. It’s time may have come.

With that in mind I constructed – and answered myself – a possible Brexit question using Jim’s recommended layout of intro – 3 points (2 for 1 against) – conclusion. Here it is:

“In the event of a vote to leave the EU, would the risks of economic shock for the U.K. outweigh the benefits of controlling our borders to reduce immigration?”

In the upcoming Referendum the people of the U.K. will vote on whether to remain in, or leave, the European Union. The E.U. is a collection of 27 independent countries with a population of 500 million. Trade within the EU is governed by the Single Market, in which there are very few trade barriers or tariffs between members, and where trade agreements are negotiated on-bloc to the rest of the world.  Any reduction in Single Market access is likely to affect trade and cause uncertainty in financial markets, which may in turn diminish public finances.  In the EU there is free movement of goods and services, and also people for employment. This means that a dis-benefit of membership is reduced control over who settles in Britain, leading to an increase in net migration in recent years, which both changes the nature of the U.K. and itself puts pressure on public services.  In this essay we will examine some of the possible economic shocks of Brexit, and balance that against benefits of getting back control of our borders.

First let us examine trade barriers associated with Brexit and their effect on individual companies. The medium term economic risk of leaving the EU will be to reduce U.K. firms’ access to the Single Market. The impact will be to inhibit trade and increase trade barriers, in three respects. First financial tariffs, which U.K. firms may have to pay to export to EU countries; second the complexity and uncertainty of doing business with Europe would increase; and third inability for a prolonged period to strike new trade deals to replace EU deals as they unwound. If we examine Porter’s 5 competitive forces, new entrants to markets find it difficult to compete against existing companies if high barriers to entry exist. Brexit would create these barriers. This would reduce the ability of U.K. firms to compete against for instance German and French rivals, specifically new growth business would be difficult to secure, and existing business exports lost. Decisions on investment size and location would be at risk.

What evidence have we that trade might be more difficult? At a high level, the German finance minister has confirmed that Britain would be excluded from the Single Market, while President     Obama of the U.S.A. has said that Britain would be at the “back of the queue” in developing a U.K. independent agreement with the U.S.A. At a very local level, the former leader of the Republic of Ireland has said it is likely that trade tariffs would have to be introduced for products coming across the Northern Irish (UK) border into Eire. Tariffs vary – for instance 32% on wine, 4% on gas, and around 10% on cars and wheat, but typically they might average around 5% which would make UK products 5% more expensive when selling into Europe. Companies would either increase their prices accordingly to maintain profit margins – but this would risk losing sales volumes – or cut costs by 5%; for instance their labour costs.   An example of the increased complexity of doing business post Brexit is the risk to the “financial passport” which the UK financial services sector has by being in the EU.  This would put at risk the attractiveness of London as a financial centre for banks, insurances and currency trading.

So in summary the threat to individual companies of Brexit would come through trade barriers, tariffs and market access which would make individual UK firms un-competitive. Collectively, this would produce an economic shock to the UK.

If the economic case against Brexit is compelling, why as we approach Referendum Day is the Leave campaign pulling ahead in the polls? The answer lies in one word – “immigration” – and one phrase “get back control of our borders”.  Net immigration is running at about 300,000 each year. This means that the difference between gross immigration (600,000) and emigration i.e. people leaving, around 300,000, is about 300,000 net and has been increasing since the end of the 2008 recession as demand for labour has picked up. Free movement of labour is legally binding in the EU and so when David Cameron tried to win concessions from the EU before the vote, all could he achieve was a restriction on immigrants claiming benefits rather than a restriction of numbers per se. The attraction of the UK has increased to migrants as the minimum wage (the lowest wage employers must pay to employees) has risen. The rise of EU immigration has had economic, business and social effects. First, economically, more immigrants have increased pressure on social services like the NHS social housing and schools, at a time when budgets are already under pressure. Businesses need a supply of labour and are tempted to take the cheapest labour from Europe, and this in turn has a social effect. Namely communities especially on the East Coast are changing, with many different nationalities for instance Polish beginning settle and open their own supermarkets. The Leave campaign asserts that a way of addressing these issues is to “get back control of our borders” by leaving the EU, thus eliminating the requirement to accept immigrants without a proper assessment of their skills and suitability. This will be especially important as the EU expands in future, possibly taking in countries like Turkey and Albania which are nearer trouble spots in the Middle East.  This security issue makes it ever more important to control our borders.

In summary the case for controlling our borders and hence immigration rests on difficulty in providing social services, a desire to preserve the UK’s ethnic mix, and worries about the future security of the UK. The Leave Campaign argues that the only way to address these issues is to leave the E.U. This is so important that any short term economic problems, if they arise, are insignificant especially if we no longer have to pay our EU membership fee of around £350 million a week, as advertised on the Leave Battle Bus.

Now let us use this membership fee to compare against macro-economic risks and assess whether a U.K.–wide economic shock would occur.  The fee amounts to £18 billion a year. A rebate of £5 billion occurs immediately leaving a net £13 billion payment. Our public expenditure is £750 billion a year so this represents 1.7% saving. Our Gross Domestic Product (GDP, the sum of all economic activity in the U.K.) is around £2000 billion so the saving would be 0.7%. Both of these savings are significant but how certain are the savings and would they be offset by economic risks of EU withdrawal? We have argued above that individual companies would find it more difficult to trade with the EU post Brexit. This in turn would reduce sales revenues and hence profits of U.K. companies, collectively meaning less corporation tax would be paid to the Government, and that jobs would be lost and unemployment would rise. This fall in Government revenues coupled with increased demand for unemployment benefit would lead to inability to balance the UK finances, which would require public national debt to rise, unless income tax rose or services like welfare or health care were cut.  The evidence to support this is that several independent economic forecasting bodies such as the Institute of Fiscal Studies (IFS), the International Monetary Fund, the Organisation of Economic Development and the Bank of England, have been warning of the economic effect of “Brexit”. There will be a short term shock and long term permanent shrinking of the economy. For instance the IFS have estimated that the negative effect of Brexit will create a £20 billion – £40billion fall in available public finances. This more than offsets the £13 billion gain. Taking a £30 billion midpoint, the net loss would be £17 billion, or 2% of public spending and 1% of GDP – indicating a recession. The Chancellor has claimed he will need to propose a short term emergency budget to increase taxes and lower public spending in reaction to these forecasts. He also has said that long term by 2030 the economy would be 6% smaller than otherwise would be the case, meaning a £4,300 loss of GDP per person. Other evidence for a shock very close to the date of the referendum is that share prices on the FTSE 100 index and value of the pound have fallen in correlation with the polls favouring Brexit, which leads to uncertainty in the markets.   Finally it is not certain how much of, or when, we would get the $13billion back, especially if some savings were maintained to preserve trade deals.

Summarising, the collective impact of companies’ difficulty in trading post Brexit would be both a short term shock and long term drag for the UK, which would offset the benefits of regaining the membership fee.

In conclusion I believe that on balance the high risks of economic shock following Brexit outweigh the benefits of reducing immigration for the following reasons. Leaving the EU would create barriers to trade for companies from a variety of factors like tariff increase and loss of trade agreements and this would in turn diminish competitiveness and restrict investments. The collective sum of these difficulties would mean a diminishing of the total public finances which the majority of economists agree would significantly outweigh the savings of the membership fee.  Forecasts are difficult and vary but the common factor of most is the sign – “negative”. The impact will be tax rises or spending cuts or increased debt – none desirable. However, this does not mean that staying in the EU should proceed without change. If immigration continues at its current level forever then clearly the U.K. would eventually, literally run out of space, and so I recommend that if the U.K votes to stay in the EU it uses its 2017 Presidency of the EU to continue the case for reform.  The most important argument for remaining is to avoid a recession and more austerity, caused by financial uncertainty and inability of companies to trade as they did before in Europe.

Biology GCSE drunken rats question

Three of my tutorial contacts talked to me today independently about the GCSE Biology exam that is going viral. I Googled “#aqa biology” and sure enough a torrent of links and Tweets popped up.

It seems that AQA included  questions in Biology GCSE biology about drunken rats, why boys drink beer and girls drink wine, including under aged drinking references, and a Business Studies question about what is an independent company.

Almost simultaneously a number of Scottish students complained that their Maths exam did not reflect the syllabus, and crucially “a common complaint was that the exam bore little or no resemblance to past papers and exemplar papers”.

What is going on? Let’s examine the issues, which raise important questions in general about the direction of exams. AQA are a terrific exam board, but have they got this one wrong?

First, this is not the same as last year’s most-tweeted GCSE Maths problem about “sweets in a bag: show that n²-n-90=0.” .  That was difficult, but on-syllabus, albeit requiring two rarely connected parts of the syllabus, probability and algebra. Not the same either as the “Scottish crocodile” question which was valid but ambiguously worded.

This year’s problems perhaps reflect a disturbing trend among exam boards. To appear to be “relevant”, “on-message”, “out of the box”, “contextual with society”, to focus on the “ethics of science” and “how science is applied” – rather than test simple scientific fact.  Also, a surely mistaken desire to be “cool with the kids”.  There is a whiff of Millenium Dome here – let’s make science exams more interesting and the kids will abandon their computer games and flock to science!

While some of these aspirations may be desirable their inclusion in vital exams is clumsy, unannounced and too dominant.  If such questions are included, it reduces room in the Hour test for basic questions about biology fundamentals. The implication is that examiners see science more as a matter of opinion, not fact.

Examiners, perhaps inadvertenly give the impression that they do not appreciate that teachers and children work very hard to learn the syllabus, practice on past papers, and despair when they open the exam paper and see a whole series of questions bearing little overt relation to the syllabus. They take the syllabus very seriously – more so than the examiners perhaps. It is like training all year to climb Ben Nevis, you reach the top exhausted, remove a prearranged stone from a cairn to claim your reward, only to find a message saying “Ha Ha, fooled you, you’ve climbed the wrong mountain!”  A teacher estimated that “only 25% of the course content” was covered in the Biology exam.

To paraphrase Donald Rumsfeld, known unknowns we can cope with, it’s the unknown unknowns that are the problem.

The question about “what is an independent company” was no doubt aimed at the idea of fair, not biased testing, and perhaps “controlled” and “independent variables”, but surely a better question would be, “why is an independent company used”? (for drug testing)?

biology2Some challenging examination of science experimentation and data analysis is fine – I am a data scientist and welcome the inclusion of graphs to represent data, and questions about interpretation.  Healthy living and drug testing are indeed in the Unit 1 specification. But when questions are sexist, appear to condone breaking the law, or are from another subject altogether, things have gone too far surely.

How many boxes can you tick in one question? Sexism, under-age drinking, animal testing, drug-taking, newspaper accuracy! One is tempted to ask, what were the examiners on when they wrote the question?! But seriously, didn’t the management have a quick sense check, and quietly suggest, ”I think you should have another look at this one, it is inappropriate”. Important social issues, but in a Biology exam? Better on Nicky Campbell’s excellent “The Big Question” on Sunday morning TV?

This raises questions about quality control at AQA, which along with Pearson/Edexcel and OCR is truly a great and professional organisation. (An independent company in fact, non-profit making)  I had assumed the Q stood for quality (actually it is for “qualifications” and AQA should remember that’s why pupils take the exam). One hopes this is a temporary blip. Questions like these are actually reducing standards not improving them.  If the majority of syllabus topics are no longer included in the exam, what’s the incentive to learn them?

I am detecting another general trend across exams. In their desire to become more challenging (good!) examiners are including more and more words in their questions, but not following through to ensure that the English is correct and unambiguous (bad!). An experienced tutor told me recently that sometimes a pupil often “has to guess” what the examiner wants.

Another question in the Biology exam is about Malaria. OK so far, it is in the syllabus. But the question is shrouded in “extras” – for instance needing to know Maths GCSE standard form “power of 10” notation. A core of scientific experimentation is to change and test one independent variable at a time – but here the examiners themselves are simultaneously testing biology, and beyond-basic maths.

AQA have responded and are standing by their exam, saying they do not want it to be “predictable”. I have taught and trained many pupils and adults and what I have found is this: provided they are given clear instructions with no surprises they will pursue a difficult task to completion, otherwise many become confused and disheartened. The danger for AQA is that in their efforts to make science more “interesting” and “challenging” they will discourage interest, especially as these questions were in the basic science Unit 1 paper. They seem more appropriate for advanced students

The fact that AQA felt the need to explain the question on social media at all suggests it wasn’t very clear in the first place. The fact they defended it suggests we should expect more of the same, we should “expect the unexpected”.  This will mean more teaching time is dedicated to predicting and practicing for these flowery questions, and less time for the fundamentals of biology.  Examiners may be surprised, but optimising your grade really does matter.  Appliance of science is of course important, but not at the expense of simply knowing the fundamentals?

At the risk of sounding like a “grumpy old man”, and another thing: core Unit 1 science GCSE contains no questions about electrical circuits or electrical safety but generally contains questions in Biology, Chemistry and Physics about the evils of fossil fuels,  carbon dioxide and global warming. On-message indeed!

The other issue emerging is the annual use of Twitter and other social media by pupils to vent frustrations with exams. While I am not a great fan of “trial by social media” I think this method of scrutiny is here to stay and exam boards must expect more in the future.

In summary, my beef is this. Though “application of science” is directionally right, and AQA are a fine organisation, the quality control on questions needs to be stepped up.  We need less social posturing in science exams. The syllabus content may be reasonable, but the questions do not sufficiently or overtly reflect the syllabus.  And when they do, they are shrouded in unnecessary, periphery extras, obscuring the basic facts around the subject.

If an 11-year old were to learn the subjunctive

If an 11-year old were to learn the subjunctive…..then so what?  Would they say, “if only I could use the subjunctive mood in my conversation, how cool would I look!” Perhaps not!

It seems ridiculous that such young children should learn the finer points of grammar like the subjunctive, or be tested on the difference between a preposition and a subordinating conjunction (the distinction which I heard the Schools Minister Nick Gibbs get wrong live on air), or that 6 year olds should face national tests on punctuation, tenses, nouns, adverbs, statements and verbs.

On the other hand, consider this. The U.K. is falling down the international league tables in English education. Yet we invented the language. Shouldn’t we be nearer the top?  Whereas we might soon be behind the Marx Brothers’ imaginary countries like Freedonia  or Moldova (OK that turned out to be a real one).

Also I rather like the idea of 11 year olds knowing more than me about English grammar. Not difficult I admit.  I was never really taught it at school until I did Latin, taught by our wonderful North of the Border Latin master whom we nicknamed “Scotch Mist”.  “The having been marched up the hill Romans won the battle”, or “he was so stupid he was fed to the lions”.

As ever I turn to pop music. All of these songs use the subjunctive mood to a degree. “If I were a rich man” (Fiddler on the Roof).  Correct. “If I was” (Midge Ure); incorrect – should be “were”. ”If I was a boy” – Beyonce: incorrect – likewise. “If I were the only girl in the world and you were the only boy” (Dean Martin – correct), and the classic Tim Hardin/Four Tops “If I were a carpenter”; correct.

I guess that if you are going to raise standards you have got to start somewhere.  Possibly these tests are a step too far, but on balance it is better to aim too high than low, and they are a counter to the seemingly unstoppable rise of Twitter-like abbreviations.  If the building blocks of Maths like fractions are being taught at an early age, why not do the same for English?  Maybe the ability of children to pick up grammar, if taught, is greater than we think.  We shall find out in due course.

Let us not forget in this “Shakespeare year” that the Bard, or rather Macbeth, whispered one of the greatest subjunctives of all. “If it were done when ’tis done, then ’twere well
It were done quickly

So to finish here are two sample tests, first  KS2 English for 11 year olds, in which you will find the controversial “subjunctive” topic in question  41, and also for KS1 6 year olds   Can you manage them?  (I didn’t even score 100% in the KS1 grammar test! But then I am a scientist and mathematician so there is something reassuring about that!)

Would seeing an exam beforehand really help?

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)

(Answers below)

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

West Indian cricket wins Maths question

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.

cricketCarlos 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)

edufocalsAlthough 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 helps Maths GCSE revision

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.

erInitially 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.

misAnother 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.


A GCSE Chemistry substance found on planet Mercury

mercuryNews today from the NASA Mercury probe that an unexpected substance has been found on Mercury, in fact making up the outer crust of planet Mercury. The substance is graphite, which will be familiar to GCSE Science and Chemistry students, not only because it is used in pencil lead due to its slippery nature and black colour, but also because its unusual structure is frequently the subject of GCSE questions.

graphiteGraphite is made of carbon atoms, arranged in layers in a giant covalent structure. While the bonds between adjacent carbon atoms are strong covalent bonds, the bonds between layers are weaker enabling slippage to occur. Further, the structure includes “free electrons” enabling graphite to conduct electricity. This is highly unusual for a covalent, non-ionic structure, as is its high melting point. Graphite is also used as part of nanotubes in tennis rackets.

 In some senses graphite is like diamond because it is also a giant covalent structure of carbon atoms, but in diamond there are no layers, just a continuously strong bonding arrnangement making diamond much harder. And of course you’d look a bit silly with a pencil for an ear-ring.

What other GCSE substances are there on planets in our solar system? Well, on many planets we can find an iron and nickel crust, and on Mars recent photogrpahs indicated the possible presence of both water and methane, with the water creating channels that are still changing in appearance.  This indicates that the water is still moving and not completly frozen, perhaps because of hydrated salts, which lower the melting point.

BowieThis, together with the presence of methane (carbon with four hydrogens – natural gas – and another GCSE bonding question) indicate that Life on Mars is possible. And so I need no excuse to include a video from the sadly missed David Bowie’s Life on Mars, one of the great rock records.

So it may not get you any more marks, but in a GCSE question it would surely look cool to your examiner that you knew that graphite had been found on Mercury and methane and water on Mars !

As a footnote, a wonderful website called PeriodicTable.com contains picture links to all the elements, including carbon of course – try it!

Gangnam Style Exam Cramming

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 

Gangnam_Style_Official_CoverBut 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.

First ever on-line national Maths test

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 !)

timesNo 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.





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

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

greg3Greg Rutherford’s fantastic long jump win at the World Championships meant he joined the select band of Brits holding the four major athletics titles at once. It was all the more fascinating because he has built a long- jump training pit in his back garden, as you can see below.


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

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

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

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

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

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


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

Diwali – great festival, great syllabus

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

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

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

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

The festival, also ceDiwali 1lebrated in the Jainism religion, occurs annually at the end of November or beginning of October depending on the new moon.  You can see that in India itself the lights are set up in many places such as next to railway tracks. It is highly photogenic as you can see in these examples. 


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

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

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

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

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

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

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

Colorado yellow river pollution in GCSE?

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

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

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

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

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

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

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

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

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

Origin of the word Google – it’s Maths!

The Economist googlemathsthis 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.

back-to-the-future-part-iii-2Googleplex is in fact the term for 10 to the power googol ( ten to the ten to the 100)  which is a very large number indeed, perhaps to infinity and beyond. The mind boogles. I mean boggols. I mean boggles. in “Back to the Future 3″ the Doc says about future wife Clara ” She’s one in a billion. One in a Googleplex!”

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

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

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

Google has been translated for instance into Chinese


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

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

Headmaster suspended for letting pupil take exam early

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

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

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

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

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

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

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

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

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


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

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

Could you wear Pink Floyd in Physics exam?


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

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

The answers are possibly, and yes.

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

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

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

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

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

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

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

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

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