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.