Sheldon can't do probability

For the uninitiated, the people mentioned here are characters in the TV programme The Big Bang Theory.

In the episode The Russian Rocket Reaction Howard explains excitedly that some equipment his lab has been working on is going to the International Space Station and that someone needs to go there with it.  "Guess who that someone is?"

Sheldon's reply is "Mohammed Lee" (or other variants of the spelling).

When Howard asks "Who's Mohammed Lee?", Sheldon replies "Mohammed is the most common first name in the world, and Lee the most common surname. As I didn’t know the answer, I thought that gave me a mathematical edge."

This could be a way to introduce pupils to the idea of independent events.

First of all, why does Sheldon think this is a good answer?  If Mohammed is the most common first name in the world and Lee (or Li) is the most common surname then the most common full name is presumably the two of them combined.  Using independent probability, we can multiply p(firstname is Mohammed) by p(surname is Lee).

What is the problem with this reasoning?  The surname Lee/Li is one of the most common names in China and Korea.  The vast majority of people with this surname are likely to have family who originally came from China or Korea.

The boy's name Mohammed (which may be spelled in a number of different ways, including Muhammad and Mohamed), according to Wikipedia has its origins in the Arab World, and is particularly popular in the Middle East, north Africa, Pakistan, Bangladesh and India.

If names were distributed randomly then Sheldon would be right, but given the cultural, religious and geographic ties with names, the two events are not independent and it is likely that there are fewer people who have Mohammed as a forename and Lee as a surname than Sheldon expects.

Thought experiment:  What if we looked at the names of all the pupils who attend a school and found that the most common first name was "Jack" and the most common middle name was "Louise".  How many pupils would be named "Jack Louise"?

I suspect the answer is 'none', although Johnny Cash might know differently.

HS2 - Reams of Rubbish

The Daily Telegraph is carrying a story with this headline:

Two things immediately jump out here: we can work out how quickly we need to read each page, and we can work out the density of the paper involved.

The sub-heading says it “would weigh one tonne if it was (sic) printed out”.  This is because it was presented on a memory stick, which is significantly lighter, cheaper and easier than providing it in book form. 

Let's work out the density of the paper the campaigners intend to use.

One tonne = 1000kg = 1000000g

Divide by the number of pages to get 1000000g/50000pages = 20g per page.

Presumably it has been formatted to print on A4.  I know that A0 is 1m² of paper, so that means A1 is ½m²,  A2 is ¼m², A3 is one-eighth of a square metre and A4 is one-sixteenth of a square metre.

That gives the ‘weight’ of the paper that would be used to print this on as 20g x 16 = 320gsm.

The paper usually used in a photocopier is 80gsm, I think.  Card is 160gsm. 

320gsm seems rather heavy to me.

If we then revisit an earlier, unstated assumption, whereby I had assumed the document would be printed on a single side of the paper.  Surely some of the arguments against HS2 are environmental ones, so we would presumably want to print on both sides of the paper, meaning a 50,000-page document would need only (!) 25,000 pieces of paper. 

Let’s start a more sensible way around.  If the paper is 80gsm then a sheet of A4 (1/16 of a square metre) weighs 5g.  25,000 pieces of paper therefore weigh 125kg, which is sizeable but is a less impressive one-eighth of a tonne.

[The amount of reading is good.  Without any sleep I reckon you would need to read one page every 96 seconds to be able to get through it all.]

HS2 may well be A Bad Thing - I don’t know enough about it to be able to comment.  And giving people a 50,000-page document to read in under two months seems a little excessive (surely there is a summary?).   But overstating the size of the document by a factor of 8 doesn’t seem to help your cause.

Don’t do that.

The internet as soma

Over the next few days fall three 50th anniversaries.  Two of the events have become a major part of western culture and recent history, namely the assassination of President John F. Kennedy on 22nd November 1963 and the first episode of Doctor Who, broadcast a day later.  Here I focus on the third, the 50th anniversary of the death of Aldous Huxley, which also falls on 22nd November.

Huxley’s best-known work is Brave New World and for some reason this book is frequently compared to George Orwell’s Nineteen Eighty-Four.  While both books imagine a dystopian future and both predicted  a number of things that have come to pass, they are very different in their imagining of future society.

Nineteen Eighty-Four imagined a brutal world in the grip of perpetual war, where history is rewritten, the meaning and use of language is altered and surveillance is the norm.  There are many links here not only with the Soviet Union under Stalin but also with the level of surveillance carried out today, with CCTV cameras and the tapping of emails, text messages and phone calls being routine.

The society in Brave New World is less physically oppressive but rather focuses on the importance of consumerism, advances in medicine and on societal control.  The ideal sport is one that involves lots of equipment and to do your duty as a member of society you ought to spend your salary as you enjoy your leisure time.  Babies are not only created in a laboratory but are fully gestated there too, meaning that no-one has a parent and ‘mother’ has become a taboo word.  Children are conditioned, before birth with the use of chemicals and after birth using electric shocks and also by listening to audio while they sleep.  This conditioning not only prepares people physically for their future (with workers who will live in the tropics being given an immunity to malaria before they are born) and mentally (in terms of their capacity for learning) but also emotionally (a ‘Beta’ remarks that she is glad she isn’t an ‘Alpha’ because they have to work so hard but also glad she isn’t a member of one of the other classes because they wear horrible-coloured uniforms).

Leisure time revolves around the use of the drug soma, which is not only legal but the use of which is encouraged by the government.  The slogan: “a gramme is better than a damn”, refers to taking the drug.  The use of soma is linked with the state religion, ensures people are placid and that they spend their leisure time in a way that keeps them out of trouble.  If a character in the book displays any level of passion then another character is sure to suggest they take some soma.

In the same way that I earlier made links between Nineteen Eighty-Four and the present-day world, it is possible to draw comparisons between some of the themes of Brave New World and what we see around us, 80-odd years after it was written.  Examples include genetic modification and gene therapy, the acceptance and importance of consumerism and the way simple recreational pursuits now involve vast amounts of equipment (I used to put on a pair of shorts and a pair of plimsolls and go for a jog.  Nowadays there are shirts and shorts made of special fabrics, particular shoes to run in, hand-held computers, energy drinks, fluorescent tops, mobile phone apps, etc, all designed to help us run better.  All of them, coincidentally, cost rather a lot.)

What, then, is the 21st century equivalent of soma?  One could argue that it is a drug such as those used to combat ADHD, or a non-medical drug.  I want to make the case for it being the internet.

For me, born in the 1970s, TV was old hat.  It had existed before I was born and I grew up with it there.  I didn’t wonder how it worked but consumed what it produced and enjoyed it.  At that time there were three television channels and in the evening there would be ‘closedown’, when everything stopped (except perhaps on the evenings when the Open University would broadcast into the early hours).  For me “the internet” is still an amazing concept.  I watched the film War Games when it came out and marvelled at the idea that computers could be connected together.  I didn’t have an email address until after I had finished university and still have the habit of loading up lots of news webpages before beginning to read them from the days when my dial-up connection was so very slow.  I got excited when I learned how to use html and when I worked out how to create a basic website.

For children today the internet is old hat.  It has always been there and is only a means to access the content they want.  And ‘the internet’ is very different now from how it was only a few years ago.  Now almost everything children used to do is available on the internet, but with the only way of turning it off being the self-control of the user.  Instead of phoning up a friend and talking to them you can now interact with all of your friends using Facebook or Twitter.  Instead of watching TV you can see any TV programme or film by watching it on the internet.  Instead of playing physical games you can link up your games machine over the internet and play with others there.

Is this a bad thing?  Not in itself, and it is of course wonderful that we have so much choice, but it does bring to the fore the importance of self-control.  I used to be able to watch an episode of Doctor Who on the television on a Saturday night.  One episode.  If I missed it then I didn’t get to see it unless my friend who owned a video machine had recorded it.  Now it is possible to get hold of every episode with David Tennant as The Doctor online.  This might be through the BBC iPlayer, via Netflix, or on YouTube or an illegal file-sharing website.  When one episode finishes Netflix will immediately cue up the next one so you can continue to watch without distress!

It really does require a large amount of willpower to stop watching and would be very easy for a child to watch episode after episode.  The only problem with this comes for children who don’t have that willpower, or who aren’t aware that there are other things out there they could be doing instead. 

So how is the internet like soma?  Well, if you want to disengage from thought in Brave New World then you take soma.  Nowadays we can use the internet.  If you want to disengage from thought then there are scores (literally) of episodes of The Big Bang Theory you can watch, or an impossibly vast number of photos of cats looking cute, or videos of “the world’s biggest fails”, or of every goal Lionel Messi has scored this year, or … .

After spending several hours doing this you are as sedated as if you had taken a gramme of soma.  

Is this post a case of "when I were a lad ...", or "the youth of today ..." ?  Do I think that society has gone to hell in a handcart?  No, actually!  But it is important for those of us who remember a time Before the Internet to realise that children today can turn on, tune in and drop out in front of their computer.

So amongst the reviews of the life and times of JFK and the anniversary of Doctor Who, I would encourage everyone to read some Huxley and to enjoy the prescience of Brave New World in particular.

We’re going to need a bigger boat

Here are my initial reactions to the new GCSE subject content document published on 1 Nov 2013 by the DfE.

So far this document is all we have.  We don’t know what exams will look like, how the tiers will work, and how each statement will be interpreted (for example a “simple proof” will mean different things to different people), so much of what follows is necessarily speculative. 

At the moment it seems sensible to be aware of some of the implications (like the need to talk to SMT about an increase in mathematics teaching time) but not to do any detailed planning (like rewriting schemes of work) until we have more information.

New material

First of all, there are new topics.  Some of these are clear and obvious (such as finding the equation of a particular tangent to a circle centred on the origin), while others can be interpreted in a number of ways.  

An example of this is the way gradients of lines will be used.  The word “calculus” is not mentioned, but at the higher tier pupils are expected to “interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents)”. 

I hope this means that essentially we will need to teach the concepts behind calculus (that the gradient of the chord is an approximation to the gradient of the tangent and that as the length of the chord shrinks the approximation is likely to improve) rather than telling pupils the way to differentiate polynomials.  A potential exam question for this could give an unnamed graph and ask pupils to draw a tangent at a particular point and then find the gradient of this, with an appropriate level of variation of the position of the tangent line being permitted.

Other queries include whether function notation will be needed, whether estimating a square root is as straightforward as noting that sqrt(20) is between 4 and 5 (or whether an algorithm is required) and exactly what is meant by “find approximate solutions to equations numerically using iteration”.

So far this is the DfE document.  The next stage is for the awarding bodies to create specifications and these will presumably include guidance that exemplifies topics like these one.

Beyond the headline new topics we will also need to ensure pupils are happy with using the kinematics formulae (which will be provided in exams) and that they have learned the other formulae they will need (because these, including the quadratic formula and the non-right-angled trig formulae, will no longer be provided).

Foundation will include harder topics

Aside from this there are a number of topics that will now fall into the Foundation tier.  This includes solving quadratic equations algebraically by factorising, and using the trigonometrical ratios.

Assessment Objectives

Assessment Objective 1 (AO1) is headed “use and apply standard techniques” and will be weighted at 40% of the higher and 50% of the foundation papers.  Much of the content in the document will fall under this heading.

The remaining marks on the papers will be equally split between AO2 (“reason, interpret and communicate mathematically”) and AO3 (“solve problems within mathematics and in other contexts”).  In the notes about AO2 and about AO3 there is the statement: “Where problems require candidates to ‘use and apply standard techniques’ … a proportion of those marks should be attributed to the corresponding Assessment Objective.”

I assume this means that a problem that involves reasoning within the context of Pythagoras’ Theorem will count as part of the AO1 marks (for Pythag) as well as part of the AO2 marks (for reasoning). 

This means that pupils will need to be comfortable with the problem solving and reasoning sections of the specification to be able to get all of the content marks.

What will the exams look like?

One of the key differences may well turn out to be in the structure of the exams.

A press release from Ofqual said this: “Exams only in the summer, apart from English language and maths, where there will also be exams in November for students who were at least 16 on the preceding 31st August.”  Finally, this gets rid of early entry completely.  

It appears that the highest attaining pupils will be stretched within the GCSE rather than needing to consider an extension course.  We don’t yet know what each grade will look like, so it is not clear whether the old G grade will map onto the new grade 1, up to grade 8 being the old A*.  I think this is unlikely because here is the only opportunity to recalibrate the system, so there will not be a direct link between old grades and new ones.

“Maths: Tiered with an improved overlapping tiers model. A foundation tier will cover grades 1-5 and the higher tier will cover grades 4-9. Assessed by external exam only, as now.”

If you just look at the tiers there doesn’t seem to be anything different about it.  Currently we have grades C-G (the lowest five grades) on the Foundation tier and the new version will cover the lowest five numerical grades.  There is currently an official overlap of two grades with the higher tier (with a compensatory E grade available for those who just miss a D) and the new system will have a two grade overlap. 

To describe the new tiering system as an “improved overlapping tiers model” must mean that model is improved.  Does this mean that the exam papers will be structured differently?  The document has some items in ‘standard’ type (for all pupils to know fully), some underlined (for all pupils to use and higher pupils to be more confident with) and some in bold (for higher pupils).  An early version of GCSE mathematics (when I took it in the first year it was available) had a common paper for all pupils and a lower paper and a higher paper (were they called Foundation and Higher back then?).  It will be interesting to see whether we return to a model like this.

If this is the case then we will need to have an increased focus on the higher tier topics.  At present the probability that any given different topic will crop up in a GCSE paper is incredibly small.  Half of the higher tier papers currently have to be questions at grades C and D, so A* questions topics can only be tested on about a sixth of the paper.  The new system might well give us more A and A* questions (sorry, grade 7/8/9 questions) so the pupils will need to study this higher content more thoroughly.  (As an aside, at the moment some students start AS-level mathematics not having previously been taught ‘completing the square’, or ‘rationalising denominators’ because they are unlikely to crop up in a GCSE exam.  An increased focus on the more difficult material in exams would make it less likely that this would happen.)

Teacher support

This will be important on a number of levels.  For those fully conversant with the mathematical topics involved there will be the need to flesh out exactly what is meant by each statement (such as the gradients referred to earlier).  For those who have taught mathematics in school up to GCSE level there may need to be some refreshing of subject knowledge for the new topics.  Teachers who are not maths specialists but who have taught GCSE (either at both tiers or only at foundation tier) may need to learn some new material. 

All mathematics teachers are likely to need to consider some new pedagogical ideas, because even the topics that currently appear at A-level will have a different emphasis in the new GCSE specifications.  (For example, we do not currently have long at A-level to delve into the meaning of the gradient and the new GCSE topic will afford us this opportunity.)

Teaching time for mathematics

In a written answer to the House of Commons, Michael Gove said:

“The new mathematics GCSE will be more demanding and we anticipate that schools will want to increase the time spent teaching mathematics. On average secondary schools in England spend only 116 hours per year teaching mathematics, which international studies show is far less time than that spent on this vital subject by our competitors. Just one extra lesson each week would put England closer to countries like Australia or Singapore who teach 143 and 138 hours a year of mathematics respectively. We announced on 14 October that mathematics, alongside English, will be double weighted in secondary school performance measures from 2016. This will also provide a strong incentive for schools to ensure that they are strengthening their mathematics provision.”

It will be important for mathematics departments to have the teaching time they need.  This will be important for the teachers, so they have sufficient time to cover the material (including AO2 and AO3), important for the pupils so they are able to achieve well on this new, more difficult course (if they don’t achieve a good enough grade the pupils will need to continue to study mathematics after Year 11), and important for schools because the maths grade will be double-weighted in performance tables.  (Incidentally, this double-weighting will start in 2016, before the new exams come into play.)

Where will this lesson that Mr Gove suggests we need come from?  Which subjects will lose out?  I am afraid I don’t know, but what I do know is that it will be important for the mathematics teaching to be done well and the extra time will be necessary for this.

It is also worth bearing in mind that it appears this extra period per week should happen in every year group at secondary school and not just at KS4.

Implications

These changes are for first teaching Sept 2015, so the current Year 8 will be the first year group to take the new exams.  Presumably it would be ideal for next year’s Yr 7, 8 and 9 to have an extra lesson per week. 

Are there enough mathematics teachers for the country to be able to staff this?  Probably not at the moment, so schools will want to appoint new teachers fairly early next term.

Scheme of Work

Much of the language in the KS4 document echoes language in the new KS3 national curriculum.  We will want and need to update our scheme of work to ensure it will still be appropriate not only for Sept 2015, but also to get the current Yr 7 and 8 pupils to 2015 with the requisite background knowledge and skills. 

If feels sensible to have an updated SoW for KS3 in place for Sept 2014, but then to wait until the final GCSE specifications and associated document (exemplification, sample papers, etc) are published and approved by Ofqual before making major changes to the KS4 SoW. 

Summary

So, we are going to need to start thinking about some of the implications (such as timetabling for next year), and will need to consider our schemes of work, etc.  We know we will need a bigger mathematical boat, but will also need to be patient and wait for further information before we can do all of our planning for the new GCSE.

Finally, I haven't yet mentioned that I like many of the new features in the document.  For example, being able to explore the background to calculus will be important and interesting.  It is good that AO2 and AO3 will maintain the focus on reasoning and problem solving.  Having more time to teach mathematics well will be welcome, as will be the increased importance of the subject.   

Football Transfer Record

The Football Transfer Window is about to shut (where 'football' = 'soccer' and 'the transfer window' is the period during which players may move from one football club to another, subject to certain conditions, the chief one involving vast sums of money).

Why does the Transfer Window close at 11pm?  There might be all sorts of reasons but I suspect this is because it will be midnight in continental/central Europe.

The BBC Sport website has a nice article about the way the record for the highest transfer has changed over time.

They then use a fairly rubbish graph to display the data!

The original is in the article, but here is a copy.  What is wrong with it?

There are lots of things worthy of comment:

·         At a glance it appears to be exponential.  The y-axis helps to show this.

·         The y-axis is useless, though, if you are particularly interested in the left-hand half of the graph.  The first 11 data points all appear to be of identical height - despite increasing from £1000 to half a million.   

·         The x-axis isn’t used properly, with huge gaps between some of the dates and small ones at other times.

·         Inflation hasn’t been considered.  I used thisismoney.co.uk to check out what Alf Common’s £1000 fee would be equivalent to in today’s money.  It is £100,000.  [Which means that Gareth Bale could buy an Alf Commons every 56 hours or so.]

·         Currency conversion needs to be considered.  Bale cost €100million, which currently equates to about £86million, but in the future this will change.  When compiling a league table of big transfers, which currency should be considered?

·         There is some missing data.  Amusingly, the Daily Mail version of thisarticle includes screenshots from Wikipedia.  If we accept the Wiki figures we can see that there are some other record amounts that haven’t been included.  Why not? [This also gives the intriguing fact that at one stage the world’s most expensive player was owned by Falkirk!]

BBC News - the cost of a CD

The BBC News website has an article called  If CDs cost £8 where does the money go?

The article starts with this:

Despite digital's complete dominance of the singles market, most albums are still bought on CD. The average cost is £8 but where does the money go? Of 100 million albums sold in the UK last year, 70% were CDs.

It also includes this graphic:

Later in the article the percentages for each part are given.  I am going to blank out the percentages so a class could work them out.

Here is the blanked out version:
About ##% goes to the artists, while ##% goes to the label, with a ##% cut going to the government in the form of VAT (applied at 20% and therefore 1/6 of purchase price). About ##% goes to the retailer, while the rest goes to manufacturers (##%), distributors (##%) and the spend on administering copyright (#%).

And here is the original version:
About 13% goes to the artists, while 30% goes to the label, with a 17% cut going to the government in the form of VAT (applied at 20% and therefore 1/6 of purchase price). About 17% goes to the retailer, while the rest goes to manufacturers (9%), distributors (8%) and the spend on administering copyright (6%).

Further thoughts:
The graphic refers to the 'Record Company' while the text calls it 'the label'.
Why is VAT applied at 20% only 1/6 of the purchase price?
Is it easier to draw a pie chart from the original data or from the percentages?

Formula One - winning by a mile?

This afternoon (Sunday 25 August 2013) the Formula One Belgian Grand Prix was held at Circuit de Spa-Francorchamps.  I listened to the start of the race on Radio Five Live (while driving) and heard one of the commentators point out that Sebastian Vettel was gaining 0.7 seconds per lap over Lewis Hamilton (then in second place) “which doesn’t sound like much but is an absolute mile”. 

I think he was using the phrase “absolute mile” to mean “a long way”, but would an advantage of 0.7 seconds in every lap work out as being a mile? 

First off, what information do we need to work this out?  They had already told me that the race was over 44 laps.  44 x 0.7 = 30.8 seconds, so if they drive at an average speed of 120mph then this would be about a mile.  Is an average speed of 120mph reasonable?  It feels like it to me, because I know they can drive faster than that but also drive more slowly through some of the corners.

I am happy with that as an approximate solution, but let’s work out a more accurate answer anyway.

According to Wikipedia, the length of the track is 4.352 miles.  Over 44 laps that gives a total race length of 4.352 x 44 = 191.488 miles.

Sebastian Vettel (the eventual winner) took 1 hour 23mins and 42 seconds.  

This gives an average speed over the race of 137.3 mph.  30.8 seconds at that speed equates to 1.17 miles.

As it happens, at the end of the race Vettel beat Fernando Alonso by 16.9 seconds, with Hamilton in third place, 27.7 seconds adrift of Vettel.  This means that Hamilton finished 1.05 miles behind Vettel.  Not bad!

Now the Daily Telegraph could do better too

Oh dear, Daily Telegraph: you are too easily led.  Just because The Guardian published a rubbish article in which they turned 'A-level' mathematics questions into a multiple-choice quiz it doesn't mean you need to copy them by making a GCSE vs O-level quiz of your own.  It's not big and it's certainly not clever.

Here's the headline from the Telegraph article (linked to from the front page of their website on GCSE results day (22 Aug 2013).

This doesn't bode well already.  Surely they can't pick a handful of GCSE paper questions and a handful of O-level questions and then claim that this can help us tell which exam is more difficult?  Can they?

Here's the test rubric:

Ah, apparently they can.

In the article that follows the quiz it says this:

Perhaps unsurprisingly, at least some of the people who did the quiz immediately zoomed down to add their comments and did not read the article.  I know this because of the number of comments that mention how ridiculous it is to have a multiple-choice GCSE exam.  Here is an example:

Just in case anyone is uncertain on this point, neither the current GCSE exam, nor the O-level questions that were used were originally multiple-choice!

But what does it say at the end of the article?  "GCSE questions are from AQA 'GCSE November 2012 Sample Questions'."  These questions don't appear to be freely available online.  When the new exam specifications were put forward for approval by Ofqual, the awarding bodies were also asked to submit 'sample assessment materials' so people could see what they intended the exams to look like.  These sample materials were not 'live' exam questions and were never used as such.  I strongly suspect that if they had been considered to be n accurate reflection of the real papers then they would have been made available on the AQA website.  I don't know if these are the source for the Telegraph questions.

Something I applaud is that the author of the article responded to some of the early comments under the article.  This leads nicely into the obvious concern which is that 5 GCSE and 5 O-level questions have been chosen to be representative of their respective exams.

Someone had pointed out that Simultaneous equations appear at GCSE too so they could have been included.

So what they are really comparing is whether the 'easier' GCSE questions from a sample paper are considered easier than the 'easier' O-level questions from an actual paper.

There are lots of 'interesting' comments.  My favourite is this one:

To answer point (1) it says clearly that "NO CALCULATORS were allowed for either set of questions" [capitals in the Telegraph].  How the use of a device that hadn't been invented would have resulted in disqualification is left to the imagination.  Calculators are not permitted in one of the papers in each exam.

(2) Does the author think there are mutiple-choice questions in the current GCSE?  There aren't.  It mentions that in the article.

(3) They have obviously never heard of QWC questions, where the process is vital for gaining all of the marks.

As for: "just impose these conditions on GCSE maths exams alone" - they do!

The final statement is also lovely: "I would doubt whether a single person taking GCSE maths would be able to even recognise or understand a slide rule let alone use it."  Let's imagine two groups of people: one group is taught how to use a slide rule and the other isn't.  What is amazing is that the second group of people don't know how to do it!  Stunning!

[I know that the quality of the comments is not the fault of the Telegraph, but it does show how mis-conceived articles like this one add to the perception that exams can be compared directly.]

The Questions

These have, as mentioned above, been cherry-picked, but there are still some interesting things that crop up.  The Q-level simultaneous equation question is made significantly easier by having multiple-choice answers provided because you can just try them out (and the first option happens to be the answer!).

 I do like Q6:

I think it is nice that this could be bashed through by hand (no calculator, slide-rule or tables), but that there is a nice way to make it easier.  Anyone who can grind through calculations can get the answer, but to avoid wasting time you need to know a bit more and this advantages those with a more flexible understanding of maths.  The bracket is the difference of two squares so the expression can be factorised to give:

3.14 (5.3+4.7)(5.3-4.7), which is helpful because the first bracket equals 10.

3.14 x 10 x 0.6, which is the same as 3.14 x 6, is much easier than carrying out four separate calculations.

There is also the obligatory (it appears) error.

A little disappointingly the seemingly irrelevant information is in fact useful for the following part (given as Q10).  The correct answer does not appear (because of a rounding error).

To sum up: it is good to have been introduced to Q6, but the rest of the quiz was sloppily put together.

Guardian Newspaper - could do better

Oh dear, Guardian.  You posted a set of ten questions under the heading "A-levels: test your maths" leading people to assume that this is representative of A-level mathematics. Unfortunately you failed to give the background to this, failed to explain how you cherry-picked the questions you gave, failed to explain that mathematics A-level modules are not multiple-choice and included errors in the questions/answers too.  See me after school.

First of all, is this really a problem?  Surely people realise that a set of multiple-choice questions on a website is only a bit of fun and shouldn't be taken seriously.  Unfortunately this has been taken seriously as the comments beneath the article attest.

The lack of context is a genuine concern.  We are used to smiling cynically when a footballer or politician claims that something he said has been "taken out of context" but here the background to the exam is rather important, particularly to respond to those who contrast the 10/10 they achieved on this test with the D-grade they gained back in 1979.  Nowadays A-level mathematics is modular.  The first three modular exams are generally taken during Yr 12 (the Lower Sixth) and are AS-level standard.  At the end of that year a student can 'cash in' their three modules and gain an AS-level in mathematics.  This is not an A-level.  In Yr 13 (the Upper Sixth) three further modules are studied and they are more challenging than those taken in Yr 12.  The six exams together form the full A-level.

The paper the Guardian article used is from module C1.  This is the very first module from AS mathematics, includes more difficult applications of some GCSE topics and some of the core mathematics required for other topics and modules.  The full paper contributes one-sixth of the marks for the final A-level grade.  The material is covers is arguably the easiest part of AS-level, which is the easier year of the A-level course, so the mathematical demand here is less than one-sixth of that required for A-level mathematics.

Some of the comments recognise this:

The next problem is that multiple-choice answers have been given.  I appreciate that it is difficult to mark algebraic answers via computer, but it should have been made clear to readers that A-level mathematics is not a multiple-choice exam!  [Or perhaps the fact that some people clearly believe that A-level mathematics is multiple-choice shows the degree of scepticism amongst the general public about A-level standards.]

Even after other comments have pointed out that A-level maths is not multiple-choice other commentators persist in assuming it is.  The comments below are given in chronological order, so if some of the later commentators had read the earlier ones before posting then they could have saved themselves the trouble.  [I have excerpted these from the comments threads - other comments intervened between some of these.]

And this raises one of the issues about making it multiple choice.  You can not only just blindly guess (and on average you should get 5/10 by doing this) but can also in many cases dismiss an obviously wrong answer, or in some cases can work backwards from the two given answers and find out which is correct.

The comment above clears this one up and also points out that this paper is not representative of A-level mathematics.  More than an hour after this comment was posted someone else asked whether A-level mathematics is multiple-choice or not.

Another problem with deciding to have multiple-choice answers is that they cannot use any questions that need explanation, which means some of the more challenging questions cannot be used.

As other comments point out, the quiz has cherry-picked its questions and it often includes just the easiest part of a multi-part question.

Here are further comments about the individual questions:
Question 1:

Because this has two answers to choose from, it is really asking "do you know the difference between integration and differentiation", which is significantly easier than asking someone to carry out the integration for themselves.

Questions 2&3:

One problem here is that Q3 relies on the answer to Q2, so it would have made sense to have allowed people still to be able to see the answer to Q2.  Unfortunately neither answer is correct for Q2.  As has been noted in the comments, the first answer is closer but is missing a power of (1/3), which should appear next to the second x.

Question 4:

Oh dear.  Several problems here.  The misspelling in the question is by the Guardian and was rendered correctly in the original exam paper.  Commentators have pointed out that this is rather straight-forward.  They are correct, but haven't been made aware that this is the initial part of a 4-part question.
The big error here, though, is in the first answer, which is the one the Guardian considers to be correct, but which has errors in the units that are provided.  The answer is £0.80, which can also be written as 80p.  As given it is meaningless.

Questions 5 & 6:

These have been cherry-picked.  They are each worth one mark.  The rest of the question, which hasn't been included here, is worth 6 marks.

Question 7:

Cherry-picked.  This is the first part of a 5-part question.  It is worth one mark out of the 15 available for the complete question.  Again, the answers that are given are helpful.

Question 8:

The typo is the Guardian's.  The two answers are so far apart and the first clearly must be wrong because it is so large.  Had two closer answers been given then this would be more challenging.

Question 9:

I think it is worth seeing the original question from which this has been taken.

The typo in the Guardian question doesn't appear in the original.  It is also Question 2 from the original exam paper (which the Guardian quiz links to) and should therefore be expected to be easier than some of those that have preceded it.  Giving multiple-choice answers makes it easier, and part (b) of the original question, which is harder, has been omitted here.

Question 10:

The typo in this question isn't just a misspelling and makes the question meaningless.  Also, because only one of the two answers includes 'c' the way the Guardian has done it really boils down to whether you understand what "write an expression in terms of c" means.

Here is the original question from the exam paper:

Once again, the first part of the question has been cherry-picked, giving only one mark out of the 7 available for the full question.

Tennis Tension

It’s a week since Wimbledon finished and I have just got back from playing tennis (we went last week, immediately after Murray’s victory to find that both of our village tennis courts were in use when we got there; this is the first time this has ever happened!).  I like to think this means we weren’t just caught up in Murray-mania.

I hadn’t noticed before that my racquet includes information about string tension, printed in the V-shape where the head meets the handle.

A couple of things are a little strange.  Have a look.

First of all, the tension, which is a force, is given in kg (a unit of mass).

Secondly, the two ranges don’t appear to match up.  The KGS range is 16% of the lower figure.  If you increase the lower end of the LBS by 16% you get to 63.8 lb

So let’s look at the conversion factor for pounds and kg.  1kg is equivalent to 2.205lb (to 4sf).

25kg is equivalent to 55.125lb

29kg is equivalent to 63.945lb

Hmm - it looks as if the imperial set of figures is wrong.

But instead, let’s start with the imperial numbers:

55lb is equivalent to 24.943kg

65lb is equivalent to 29.478kg

Ah - now it works.  Presumably my tennis racquet was strung in imperial units which were then converted to metric.

I will try this with a class next week to see if they can work out what is going on. 

And then, if they manage to work it out I will ask them to help me with the tensions on my son’s racquet (which don’t work either way around!).

Brilliant Bounds!

In a previous post I wrote about fractions that appear on roadsigns. 

I had gone with an initial assumption that quarters and halves were the fractions that appear on UK road-signs, but then we saw one that featured one-third!   

My new assumption is that these fractions appear on UK road signs:

 

This is brilliant if we want to think about upper/lower bounds.

Assuming that the bounds are halfway between one measurement and the next, this gives:

If we put them all over 24 then the upper/lower bounds are:

 

There is something rather pleasing about these!

Least accurate measurement?

According to the Department for Transport: “A sign may indicate the distance to a destination in miles. Fractions of a mile may be shown for distances less than 3 miles.”  This means that distances less than 3 miles can be given rather accurately because they can use the fractions shown above. 

A final question: which exact measurement (whether greater or smaller than 3 miles) has the largest percentage error?

As I was going to St Ives 1: See the signs

No - not the old riddle, even though it appears in DieHard: with a Vengeance and would link with an earlier blog (and it probably isn’t that St Ives anyway - I was in Cambridgeshire).

[Incidentally, I never quite understood why I couldn’t be going in the same direction as the polygamist - surely all those “kits, cats, sacks, wives” would slow a man down so I could easily catch him up?  2802 seems a perfectly sensible answer to me (although it assumes that I am not carrying sacks, cats, etc too).]

I took these photographs on a bike ride close to St Ives.

What maths can we do with a class with these?  Lots, it turns out.  Let’s start with just the first picture (but before you do you might want to ask pupils what is strange about the way fractions are written on road signs - most of them will never have spotted that they are written without a line):

• How far is St Ives from The River?  
• What problems might there be with that answer?   There might be a more direct route from St Ives to The River that is shorter than 2¼ miles.
• What about the accuracy of the measurements?  Presumably they have been rounded
• What degree of accuracy might have been used?  
• So: what is the maximum distance of The River from St Ives?  From this sign the most precise measurement involves quarters, so maybe everything has been rounded to the nearest quarter.  In this case the biggest River distance would be 5/8 and the biggest St Ives distance would be 1 7/8, giving 2½ miles as the maximum.

Now let’s put picture 2 alongside picture 1:

What can we do with this one?  I would let pupils decide what questions they can answer.  Ideas include:

• How far from St Ives to Fenstanton?
• How far from St Ives to Hilton?
• How far from Hilton to Fenstanton?
• Why might these answers not be accurate?
• What if we consider the rounding that has taken place? 
• What happens if a lorry needs to go to Fenstanton?
• Combining it with the first picture: what is the maximum distance of Hilton from The River? 
• What is the minimum distance?

Finally, let’s add picture 3:

We could repeat similar sorts of questions with this one, including: “How far did I cycle between taking pictures 2 and 3?”. 

But hold on: on this one it says that Fenstanton is 2 1/3 miles away.  Our previous assumption about rounding to the nearest quarter is incorrect! 

So, what is the upper bound of signpost measurement given as ¼ ?

More on this in a future post!

Binary trading is bad

Some of my mechanics-studying sixth-formers were discussing Binary Options yesterday.  This worried me a little because I have always taken it for granted that Mechanics was a superior discipline to Statistics and their scary lack of statistical knowledge has made me start to reconsider this!

When I described it as ‘gambling’ they corrected me: “it’s investing”.

The way it works is that you bet (sorry, ‘invest’) that over a particular time-frame (that might be only 5 minutes) a particular share or basket of shares will rise or fall in value.  That’s it.  Now, using your superior knowledge of the markets to decide that over a period of several days/weeks/months/years a particular company will rise or fall in value is one thing, but to get through the ‘noise’ of minute-by-minute share-dealing to say that a share will rise/fall immediately is very unlikely (unless Sir Alex has given you the tip that he is about to announce his resignation!).

Forbes business magazine puts it like this: http://www.forbes.com/sites/investor/2010/07/27/dont-gamble-on-binary-options/

Quoting that article:  One particular binary option website

pays $71 for each successful $100 “trade.” If you lose, you get back $15. Let’s say you make 1,000 “trades” and win 545 of them. Your profit is $38,695. But your 455 losses will cost you $38,675. In other words, you must win 54.5% of the time just to break even.

How did Forbes know that it would be 54.5% of the time?  I reckon my non-stats sixth-formers (and Higher tier GCSE pupils) can work this out for themselves.

Let’s call the probability you win p.

In the long run you will win 71xp dollars per ‘trade’.

The probability you will lose is (1 - p) and in the long run you will lose 85 x (1 - p) dollars.

To break even your wins and losses must be equal, so we have the equation:

71p = 85(1 - p)

Solving this gives p = 85/156

Which is about 0.54487

How to present this to a class?

Slide 1:

There is a particular type of gambling that is close to being random.  You pay $100 and if you win you get your stake back and also $71 in winnings.  If you lose you get $15 back.  Should you play?

Slide 2:   (Scaffolding)

Let’s call the probability that you will win each time p. 

How much will you win each game on average?

What is the probability you will lose?

How much will you lose each game on average?

What probability do you need so you will break even in the long run?

Slide 3:  Why do you get $15 back even if you lose?  What is the point of starting with $100 in that case?

I will try it out with a KS4 Higher class in a couple of week’s time.

Everyone learns!

At school we have been trying out some of the ‘three act’ lessons by Dan Meyer.  There has been a real buzz from them and, wonderfully, everyone (teachers and pupils) appears to have learned something from them!

A colleague, who hadn’t come across these lessons before learned what a great resource they can be.  He used Best Circle with his Yr 11 group.

The group learned more about circles, specifically that a circle is the shape that bounds the maximum area for a given perimeter.

One pupil in the class is a bit of a programmer: he took things further.  That afternoon he created a webpage (written in javascript) where you can draw a circle and it will tell you how accurate you are.  Brilliantly, this gives a percentage and features sad/happy kittens to show you how impressed the feline world is with your effort.  My colleague put it up on his website.  The page is here.

A couple of weeks on he turned it into an iPad app, which can be downloaded for free here.  

He learned that maths can be rather cool!

And me?  Well I saw yet another example of why sharing is so important, whether it is ideas, experiences, materials or iPad apps!

How do the programs work? 

Well, first of all they join up the line that is drawn so it forms a closed shape.  Then it measures the perimeter and the area of the shape. 

From this it calculates the radius you would get if the perimeter were the circumference of a circle and the radius you would get if the area were the area of a circle.  Then it divides the latter radius by the former one and gives the percentage.

Questions:

1]  Can you get close to 100% ?

2]  Can you get close to 0% ?

3]  What happens to the area if you draw a shape that crosses itself (such as a figure of 8) ?

Here are two alternative methods for giving a percentage score for the accuracy of a circle:

Method A]  Assume the perimeter drawn is that of a circle and work out the radius of the circle.  Use this to calculate the area of the perfect circle.  Compare that to the actual area drawn.

Method B]  Assume the area drawn is that of a circle and work out the radius of the circle.  Use this to calculate the perimeter of the perfect circle.  Compare that to the actual perimeter drawn.

4]  What is the link between the answers the two methods give?

5]  Which one is equivalent to the method used in the app?

6]  Which method is better and why?

Black Ops 2

After the blog entry about Sky Sports, this is the second in an occasional series about changes brought about by technology.

“Technology is changing at a rapid rate.”  I suspect this sentence could have been written in Europe at any time since the renaissance.  Alongside this is the idea that advances in technology mean that society is changing … almost certainly for the worse.

The printing press ruffled feathers when Guttenburg introduced it (making literature available to everyone for a modest fee).  Ned Ludd didn’t like automated weaving looms (hence the word ‘luddite’).  Television was believed to herald the death of theatre and movies.

One of the things I am interested in at the moment is the way technology is developing and then the ways we change in response to this.  I think it is dangerous to start drawing conclusions about effects on society or culture right now (not least because my local cinemas appear to be thriving even though every house has a TV).

A couple of quick examples of changes I have seen recently:

1]  Fewer pupils seem to wear a watch - instead many people use their phones to tell the time.  On local commercial radio on Sunday morning the presenter talked about not knowing whether her phone had updated the time to BST automatically or whether she would need to do it manually.  Presumably she doesn’t own a watch or any other clock to compare the time against.  Is this a problem?  No.  I am happy wearing a watch because I am used to it.  Would I want to use a fob and chain?  No.

2]  We have a ‘phone chain’ of people to call if school is ever closed (it has only been used in cases of severe snow).  Older colleagues have two phone numbers listed (home and mobile).  Younger colleagues only have a mobile number.  Maybe I don’t need a landline any more? 

Computer Games

In the 1980s computer games were mostly mind-numbingly repetitious.  If you were unsuccessful at Pacman or Space Invaders or Chuckie Egg or Manic Minor, etc then you restarted from the very beginning and had to replay every single moment of the game.  If you were successful for a while (#) you just did the same thing on every level, but with it getting slightly more difficult or faster.

(# In all of the listed games apart from Manic Minor you play until you fail.  Manic Minor did have an end.  I never got to it.)

Nowadays the games where you control a character have an author.  They don’t just look cinematic but they also have a story much like a film.  I haven’t played computer games like this in years, but have recently played Call of Duty: Black Ops 2.  New skills are taught as they are needed and if you fail you restart at almost exactly the same place.  The idea of repeating the same tasks over and over is gone: this is a world away from Pacman.  Despite this there must be huge amounts of repetition going on.  I find I have to stop and think about which button (or combination) to press, while my son and his friends just seem to know which controls to use without thinking about it. 

Another difference that I find strange is that I am not always clear when I am in control of a character or whether the games console is doing it for me.  This is not just true in Black Ops, where it sometimes takes over to move the story forwards, but also in FIFA 13, where my goalkeeper makes astounding saves without my input, and in car-racing games where the system moves me to the starting point of a race.   

Implications

I don’t know what the implications of this massive change will ultimately be.  We won’t find this out for a decade or so.  What I do know is that if you play games like this you are getting used to the idea that you have enormous amounts of control  and little repetition of story, while not getting frustrated when the computer takes over.