The mock exam grade thing is not quite as daft as it seems

What has happened?

33 ½ hours before the students are due to receive their A-level results the government announced that if their mock exam grade was higher than the grade they receive then they can choose to use the mock grade.

According to the Guardian:

Gavin Williamson, the education secretary, abruptly announced that pupils could substitute the grades they received in mock exams held by their schools earlier this year – so long as the mocks were held under exam conditions and could be “validated” by the school.

Why not use mock exam results to determine students’ grade this summer?

As I have written elsewhere, using mock exams to determine what grades to award every student is just not realistic.  Mocks are taken at different times in different schools (from Nov to March) and are not marked in the same way. 

Are mock results usually higher than the real exam?

In the vast majority of cases, in a normal year the mock exam results are lower than the actual result a student gets in their exams.  One would hope that an additional three to six months of study and revision would lead to an improvement!

So why mention them this year?

This year, a student ending up with a ‘moderated grade’ (as most people seem to be describing it) that is lower than their mock exam grade has almost certainly not been awarded the right grade and for those students to be allowed to have their mock grade instead does seem to be a reasonable way to right that injustice.

But what about ‘walking, talking mocks’?

In some schools the mocks are done with considerable support from the teachers – as a way of modelling exam technique and supporting students through the exam process.  The second half of the quote shows that those mocks won’t be able to be used: “so long as the mocks were held under exam conditions and could be “validated” by the school”.

How many students will this affect?

I would guess there won’t be many students whose mock exam results are higher than the grades that are received on Thursday.

Who will be disadvantaged by this?

In a way no-one will actively be disadvantaged.  This is a post-results change, so it isn’t the case that if one student’s grade goes up then someone else’s will go down.

There are some students who won’t be able to take advantage of this.  They include those whose schools had planned mock exams for after lockdown and those whose mock exams were not carried out under exam conditions.  That is a shame but isn’t a reason to allow other students’ clearly wrong grades to stand.

So what’s the problem?

·         After teachers have spent so much time trying to get the CAGs right ... 

·         After schools have explained the system and rules to students and parents ... 

·         After Ofqual have gone through multiple stages of proposal-consultation-plan ... 

·         After schools/teachers have read vast numbers of documents ... 

·         After teachers have had to upload data carefully ... 

·         After exam boards have had to change all of their systems and scramble to get the results out on time ... 

… the government has changed the rules only 33 ½ hours before the results will arrive at students’ email addresses.

Can we have a summary?

Pros:

This is an easy way to redress clearly unfair grades.

It should affect a small number of students.

Cons:

Some students won’t have valid (or any) mock results that they can use and won’t be able to take advantage of this.

The government is clearly very scared about negative backlash.

After teachers and schools have invested so much in this process, it is galling to see this last-minute change being rushed out.

 

Mathematically Distant

Here’s another article that tries to show some calculations but ends up being distant from the genuine mathematics.  And the source data is, in some ways, even funnier.

This blog starts with a filleted version of the article.  (The link is provided so you can see the whole thing, but none of the mathematical info has been removed from my version.)

https://www.theguardian.com/business/2020/jun/05/shops-reopen-england-social-distancing-rules-coronavirus

Plans to reopen shops in England 'in chaos' over social distancing rules

Study finds 10 sq metres of space is required per shopper to stick to government’s 2-metre guidelines

Fri 5 Jun 2020 14.19

High street retailers must give each shopper a minimum of 10 sq metres of space if they are to maintain social distancing rules, a new study has found, prompting businesses to complain that a lack of official guidance has left reopening plans in chaos.

With 10 days to go until most non-essential shops reopen in England, a research paper by Manchester Metropolitan and Cardiff univeristies has concluded that larger shops must allow substantially more than a 2-metre radius for each person so that they can move freely.

The study finds that people need 10 sq metres in smaller shops, 11 sq metres in larger shops, and 12 sq metres in outdoor spaces where they move more rapidly. That is up to three times as much space as the 2-metre radius required when people are stationary.

The report’s conclusions are based on the speed and unpredictability of people’s movements, which means they are much more likely to get near each other when shopping than when queuing or sitting at a desk – and therefore require a bigger “halo”.

Chris Turner, the chief executive of British BIDs – the representative group for Business Improvement Districts – said the report showed that maintaining a 2-metre distance was “hugely problematic”.

“There’s a real sense that the government is making it up as it goes along,” he said.

What’s the issue?

Have a look at the calculations on the diagrams.  Here is the first one.  What is going on here?

Each person is in a circle, radius 1 metre.  So the area is π m²

Suggesting 4-sq-metres is therefore a bit strange.

That is, however, the area of a square with each person at its centre.  The square would be of side 2m.  Maybe they assumed it was a square?  Maybe they have rounded up (despite pi being closer to 3 than to 4)?

OK – now let’s look at the next diagram:

There’s no explanation here as to why each person now needs to be in a circle with a radius of 1.75 metres.  This gives, apparently, an area of 11-sq-metres. 

Well a circle of radius 1.75m has an area of about 9.6m², which doesn’t round up to give 11, so that theory was wrong.  If this were a square for each person then it would have sides of 3.5m, which dives an area of 12.25m², which isn’t 11 either.

So are there two different errors here?  Or something else entirely?

The original report

I followed the link to the original report (https://www.highstreetstaskforce.org.uk/resources/details/?id=bc16b6bc-0ebb-4b7b-8df8-d8aa6b9a3a9f ) and found that some of it makes sense, some has been reported badly, and at least one part of it is hilariously mathematically ridiculous.

Starting with the sensible part:

The research paper points out that when people move around a shop they can’t stay exactly 2m away from everyone else.  We see this at traffic lights.  If you are four or five cars back when the lights turn green, you can’t immediately start moving forwards.  The first car moves, then the second driver reacts to that movement, then the third car responds a little later, and it is possible that the lights have turned red again before the tenth car has even started moving.  In a shop, if one person stops to browse, or moves towards a display, those around are all affected and need to respond.  To account for this the authors of the paper suggest a bigger circle is needed.  And they go for a circle of radius 1.75m (more on this later).

So that explains the bigger radius.  But what about the two dodgy areas?

The Areas

The paper points out that circles don’t tessellate, so they use a ‘square tessellation’. When arranged in this square formation, the total area divided by the total number of people gives an area of 4m² per person.

[Two diagrams (fig 4 and fig 6) from the paper.]

On the same diagrams I have shown the tessellations:

Later in the paper they mention the hexagonal tessellation.  Let’s try that out with a circle of radius 1.75m

The hexagon goes around the edge of the circle, so the ‘height’ of the shaded triangle is 1.75m.  The base is therefore 2 x 1.75 x tan30.  When we work out the area of this triangle and multiply by 6 (to find the area of the hexagon), we get an answer of 10.6088112 m².  Aha – that rounds off to 11 – and is presumably where that figure came from!

What a shame that the article doesn’t explain this, and what a shame their diagrams are so unclear.

But, I mentioned there are ridiculous things in the report as well.

The first maths error

Here is a quote from the report:

In square or rectangular tessellations, the density of the circles is 0.7854 (Williams, 1979, p. 49). In other words, 78.5% of the space can be utilised.

Based on a square tessellation, in fixed space each person will require a space of π(r)²/0.7894m² (3.9797m² when r = 1m) for the square or rectangular tessellation.

What is going on here?  The diagram shows the scenario, where a circle of radius 1 is inside a square, which clearly has sides of length 2.  The area of the circle is π, and the area of the square is 4, so the fraction of the area of the square that is taken up by circle is π ÷ 4 = 0.7853981634…

The book they are quoting uses a rounded version of this value (0.7854) , which is where the figure of 78.5% of the space being filled comes from.

Rather than just saying the area of the square is clearly 4, the report then tries to divide the area of the circle by 0.7854 to get the area of the square.  This would give 3.999990646, which is close to 4, but is a ridiculous way to do it!

Unfortunately, having written the correct rounded value of 0.7854 in the previous sentence, they then miscopy that in their calculations and divide the area of the circle by 0.7894 instead, which gives 0.979722135, and they have correctly rounded this off to 3.9797. 

A nice bit of maths (with another error)

Having done some bizarre things when working out the area of a square of side 2m, there is then some nice maths to decide that a circle of radius 1.75m is required. 

They use a study of walking speeds in different environments (roughly 1.5m/s), and a different study that looked at how long it takes someone to stop walking (0.5 seconds), worked out that this is therefore a stopping distance of 0.75m and that in order to stay 2m apart, that stopping distance of 0.75m should be added on to each person’s circle.  The original radius of 1m, plus the stopping distance of 0.75m gives 1.75m.

This assumes an instantaneous deceleration and that everyone walks and reacts in the same way. I think these are reasonable assumptions to make.  Unfortunately, the formula they use to calculate this extra amount is noted as this:

We set the value of x, the radius of the inner circle giving freedom of movement, as walking speed / stopping time.

It should be speed multiplied by stopping time.

This then gives them different areas required for different sizes of shop, because people walk at different speeds in smaller shops compared to larger ones and therefore the stopping distance is different.

What now with this?

Unfortunately, this is another reason not to trust the maths you see being used in the newspaper.  Rather disappointingly, there were basic errors in the news article (which didn’t explain the whole square and hexagon thing), and in the university report it was based on, which included multiple ridiculous things.

Perhaps worth using with a class to point out the folly of applying a formula when you don’t need to, and the danger of rounding off too early?

Given how much of this is just twaddle, it seems appropriate to end with the last line of the original article:
“There’s a real sense that the government is making it up as it goes along,” he said.

"It’s all OK – students can take the exam in autumn if they want"

I have written about my concerns about the fairness of the grades students will receive this summer. But it’s all OK – Ofqual have said this: “We recognise that there will be individuals who believe that their performance in the examinations, if they had taken place, would have resulted in them achieving a higher grade than the calculated grade they will receive this summer. This underlines the importance of the autumn examination series as an opportunity for students disappointed with their results to show what they can do.”

I am not reassured by this.

The consultation that is currently running about the autumn exam series suggests GCSEs will be taken in November and A-levels in October. Having had their last face-to-face lesson on or before 20 March, after receiving their results in mid to late August, students will have perhaps 2 months (or less) to prepare to sit full A-level exams and just over 2 months to prepare for GCSEs. (Most subjects are likely to be offered, so there will need to be an exam timetable as usual, and while it may not span the usual 6 weeks, it will need to last for several weeks.)

Let’s look at the logistics.

GCSE autumn exams

In those two months of time, the students will need to prepare for the exam. They will need to do everything they would have done in the 3 months between school closure and the end of GCSE exams.

They may still have had topics still to study, taught in class by the teacher.  They would have had practice papers to do, chased up by the teacher, marked and gone over carefully. They would have had other revision sessions and they would have been able to focus more on subjects like maths after they had finished their exams in other subjects. (One thing that might make this easier is if they are sitting exams in only one or two subjects.)

For those who are currently in Yr 11, how does this happen in Sept/Oct of Year 12?  Do they have timetabled time with a teacher?  Who funds that?  If they are taking the exam for GCSE maths, do they start A-level maths anyway, and do their GCSE work alongside that?  If social distancing needs to be in place in Sept then putting additional lessons on for certain students will make a crowded school even more packed.  If schools are not fully open in Sept, then all of the arguments as to why schools couldn’t take note of work done after 20 March will still apply, and it won’t be fair for those students to prepare for the exam remotely. What happens if a student moves to a different school or college for Yr 12; does the new place provide support for the GCSE exam, or the old one? Where do the exams take place?

Why might a student decide to take the autumn exam? In most cases that will be because they are unhappy with the grade they were awarded in August.  This might be because they have fallen below a grade 4 in English or maths, or because they haven’t got the grade they need to be allowed to take an A-level.  (The hassle involved in taking an exam means, I suspect, that very few students will take these exams unless it really is important to them.)

The results of the autumn GCSEs will be published in February. Let’s imagine a scenario whereby a sixth form college requires students to get a grade 6 at GCSE to be able to access an A-level in a particular subject. If the student is expecting a grade 6 but finds they have a grade 5, what will happen?  Will they need to take the GCSE exam and start their A-levels the following year? Or to run GCSE alongside A-level, awaiting their results in Feb?

And then if they don’t get the required grade 6, what happens?  Do they get thrown off their A-level course?  Told they can only do AS-level?  Do they pick up a different subject at that point (having missed half a year of study) or do they only do 2 subjects?  Or will they then have to take a gap year before beginning Yr 12 again the following Sept?  Alternatively, if the school/college allows them to continue to study the A-level despite not getting a grade 6 in the GCSE exam, what was the point of sitting the autumn exam?

It seems as if the autumn GCSE exams will help very few students.

A-level autumn exams

For students currently in Yr 13 there are some similar issues, in that if they miss their university offer there won’t be time to take the autumn A-levels and get results before the university year begins. 

If they decide to sit the exams, there are similar issues as with GCSEs as to whether there is any teaching, revision, support, etc, who pays for this and how Yr 14 students can be accommodated in school/college.

I suspect that at many universities there won’t be a major issue if students miss their offer by up to a couple of grades, in that some students will be put off by scare stories of “lectures happening online” and will assume this means “all teaching will happen online” and will defer until the following year, and overseas students are much less likely to take up their places.  This will mean universities will be able to drop their offers a little (because they will still want to fill their places).  Some students will still miss out, though.  There will also be some who, even if they do get the university place they want, will feel hard done-by over their A-level grades and will worry there will be an enduring effect on their employability in the future.  Should they try to take the autumn exams alongside studying at university?  Where would they do the exams?

In conclusion

Whenever I have raised concerns about the fairness of the system that is being used to award grades this summer, I have been told that it’s OK because the students will be able to take exams in the autumn.  As I hope I have shown above, this will only be an option that is worth pursuing for a smallish number of students.

It will be interesting to see how many entries there are for those exams.

As in my previous post, this might well be the least-worst way of doing things. It still seems unfortunate though.

The worst way to give 2020 GCSE grades … apart from all the others

I have read several comments over the past week or so suggesting that the grades this summer will be rather accurate. They won’t. And I wish people would stop pretending they will.

My prediction (as I wrote about in a previous post) of the way grades would be awarded was largely accurate and I cannot see any other sensible way to generate such grades. In that post I raised some potential problems with this system and these problems appear to have been forgotten or ignored.

What is the problem?

This year’s grading system is (rightly) not going to be used to grade schools.

But the system will be closer to being accurate on a school level than it will on an individual pupil level.

If it’s not right on a pupil level then it’s unfair.

But we need something and this is the least-worst method, so what can we do about it?

In this post I am going to set out some of the reasons why this system is likely to be inaccurate and unfair. And something that I think will make it less bad (although not everyone will agree!).

Various issues

The grades a school gets overall will reflect the grades the school would have got from those pupils had they done as well as their counterparts at the same school over the previous couple of years. 

Problems with this:

1)  This might not be accurate as far as the school’s overall grades are concerned.  If a school is on an upward trajectory, or if a department has made changes (either recently, or to KS3 a few years ago) then this year’s students might be expected to do better.  Ofqual have said they will not take this into account because: “any statistical model is likely to be unacceptably unreliable in predicting trends in performance in 2020”.

2)  Subjects with small numbers of entries in some schools are likely to have a larger variance in their grades each year. This year that will lead to a larger risk of error in the grades that are assigned.

3)  Subjects being offered for the first time in a school will be difficult to manage fairly.

4)  New schools won’t have prior data to use.

Teachers are being asked to provide a Centre Assessment Grade (the grade they think each student would have got had they taken the exam) and a rank order of the students. The exam boards will then use the grades they think the school ‘should’ get and will move the grade boundaries as necessary for each school, maintaining the order of students submitted by the school.

It is worth noting that the grades which are provided are essentially irrelevant: the rank order is the only thing that is important here.  (It might be helpful to start with grades in order to help create the rank order, but the fact remains that the grades will not be used by the exam boards.)

It is also worth noting that this process could have been set up to happen the other way around. It would be possible for the exam boards to tell each school how many of each grade are available in each subject and for the schools then to distribute them amongst the students.

The key thing is getting the students in the _right order_.

A 2015 report by Cambridge Assessment compared predicted grades for the 2014 OCR GCSE exams with the actual grades achieved. The report found that overall 44% of the grades were ‘accurate’, while 42% were ‘optimistic’ and 14% were ‘pessimistic’.  These exams in 2014 were based on the old specifications that, in many cases, had been used in schools for many years, unlike the fairly new exams we have now.

The report found that overall 13% of the results were more than one grade different from the predicted grade. 

5)  One in eight exams results were two or more grades away from the predicted grade. Given that it was impossible to be more than one grade too optimistic when considering a grade of A or A*, or more than one grade too pessimistic when considering a grade G, this suggests that grades in the middle are less easy to assign accurately.

I have long-argued that it is irrelevant to talk of predicted grades as being ‘right’ or ‘wrong’, because between the predicted grades being produced in March and the exam being taken in May/June a lot can change.  This year, schools were told not to collect any additional data after lockdown began, so centre assessed grades, while being submitted in early June, are based on data from prior to 20 March, just like the previous predicted grades were.

6)  This is a problem because it is extremely difficult to tell how much a pupil will change between their predicted grade and the actual exam. Should we try to factor that in when making our Centre Assessment Grades?  Or should we only use the mock exams and other work that we have records for?  This seems to risk disadvantaging those students who would have worked hard after 20 March.

This then is the major problem for me:

7)  While the grades awarded to the school may largely be ‘right’ (with the caveats above), we are stupendously bad as a profession at giving them to the correct students.  This leads to the ludicrous situation whereby: 

a.       The school results are ‘right’ overall – but the school won’t be judged on these.

b.       The individual results are incredibly iffy – and the students _will_ be judged on these.

That is plain unfair for the students.

Surely the opportunity to take the exams in the autumn means this is OK?

No it isn’t.  A further post on this will appear over the next couple of days.

A footnote to the results

Football fans will know that Liverpool were on the cusp of winning their first league title in several decades. They have dominated the league this year and have won 27 out of the 29 leagues games played so far. They are so far ahead of the rest that they only need two wins in the final 9 matches to guarantee they win the league, and if their nearest rivals lose matches even that won’t be required.

In the early days of lockdown a frequent topic of conversation on the radio was that it would be unfortunate were the season not to be completed and for Liverpool to have an asterisk or a footnote next to their league title for this year.

Similarly, I have seen it argued that it would be unfair on the class of 2020 were their grades to be treated differently from those of other years.  In fact, I think they _should_ be treated differently, for the benefit of those students.

We just don’t know that they will be accurate and it would be wrong to pretend they will be.  A student getting a grade 5 this summer in maths might have been someone who would have got a grade 5 in the exam, or might have got a grade 4 or a grade 6 or might have got a grade 3 or a grade 7 (or further away from grade 5). We cannot tell, yet by giving _a_ grade we will be saying we think we do know.

In an ‘ordinary’ year the grade a student gets might not be an accurate reflection of their ability in that subject.  Perhaps they misunderstand the way a question is phrased and lose marks when they do understand the subject content. Perhaps the vagaries of a mark scheme affect their results (see the bizarre marking of the KS2 ‘semi-colon’ question), or perhaps they happen to revise a topic over lunchtime that then appeared as question 1 in the afternoon paper, or perhaps the marking was done incorrectly (this certainly happens because post-exam appeals are sometimes successful).

We know this, though.  We know that an exam result is just that: a result on an exam. While we might use a GCSE grade in a subject as a proxy for how ‘good’ a student is at that subject, all we really know is the grade they got in that exam.  This year everything is different.  Different schools will interpret the guidance in different ways, the number of high grades available to schools might not be ‘fair’ (see earlier in this post) and the students might not get the grade they ‘should’.

I would love for us to be able to put confidence intervals on grades – in normal years as well as this year.  This would be too complicated to understand, however, and the middle value would inevitably be used as ‘the grade’ the student got. 

My suggestion then is to put a symbol next the grades to show they have been created in a different way and are not exam results.  Maybe ~B could refer to an A-level B grade that was arrived at by this year’s system.  ~4 could be a GCSE grade from this year.  This wouldn’t be to devalue the grades, but rather to point out that they are just less likely to be accurate.  If a student currently in Yr 11 needs a grade 7 to be allowed to take A-level maths, for example, the sixth form could look carefully at those with a grade ~6, rather than that automatically disbarring the student.  If a Yr 13 student needs three A grades to meet their university offer, the university could consider carefully those who get A,A,~B

To sum up:

This year’s system is not good for the very people it is supposed to be there for: the students.  There isn’t a realistic alternative, however, so we need to find a way to make this work to serve the needs of those students and those who would ordinarily make use of exam grades.

Geometric Social Distancing

I have another blog on which I post what I call “Quibans” – Questions Inspired By A News Story.  Those tasks are aimed at Yr 12/13 Core Maths classes.  While this post is inspired by a news story, it isn’t really Core Maths appropriate, so I’m putting it here instead.  It might be of interest to Yr 10 and Yr 11 Higher Tier GCSE classes, or as some trig revision at the start of a Yr 12 A-level course.

The article has been edited to remove material that is irrelevant (more on this in a future post); nothing has been altered or misrepresented.

Here is the article, taken from the Daily Telegraph:

Parallelogram, hexagon or pentagon? The safest way for six people to picnic revealed 

A basic knowledge of geometry could come in handy for people who wish to meet when the coronavirus lockdown rules are relaxed next week 

29 May 2020 • 6:51pm  

Groups should picnic in a parallelogram, hexagon or pentagon and if they want to meet together while still maintaining social distancing and good hygiene, experts have said.

On Monday, lockdown restriction will ease so that six people can gather together for the first time since measures were imposed on March 23rd. 

Maths specialist Bobby Seagull who presents the BBC Two series, Monkman & Seagull's Genius Adventures, has suggested that the most efficient way to keep people the correct distance apart is to form a series of triangles, with each person sitting at an apex. When six people do this it forms a parallelogram. 

Mr Seagull said: “My solution is not perfect but means anyone could leave without upsetting others' social distancing. we think that this problem is trickier than at first sight.” 

However mathematicians at Oxford University believe that sitting in a hexagon or pentagon would be a more efficient solution, taking up less space and allowing minimum distance between people. 

Jason Lotay, Professor of Pure Mathematics at Oxford University said that putting six people in a hexagon, each two metres apart, would require just 10.39 square metres. 

Forming a hexagon means the maximum distance between two people is four metres at the extremities, while the average distance that one person has to another person is around 2.29m.  

Alternatively, if one person is placed in the middle so that the other five form a petagram then the maximum distance between any two people is just 3.80m.However, although the pentagram requires a smaller area - 9.51 square metres - the average distance that one person has between another person rises to 2.72m.   

“Of course, things are much better for the person in the middle, who is 2m from everyone, but worse for everyone else, who now have an average of 2.86m to anyone else,” added Prof Lotay. 

“I would probably say that efficiency is given by the total area required and minimising the maximum distance between any two people, in which case the pentagram model is better.   

“However, as I say, it means that one person gets preferential treatment, and it is worse for everyone else.”

There are so many issues with this, which would make it fun to do with a class!

First of all, the article is a bit of fluff, it’s not meant to be taken seriously and the errors are really not a big deal. It may appear as if what follows is being ridiculously pedantic (and of course it is!), but the bullet points at the end make it clear that none of this is really serious.  

Let’s look at some of the figures involved.

Which of these two configurations looks like it has the smaller area?

Which one looks as if the “average distance that one person has to another” is smaller?

To me it appears as if the pentagon has the smaller area and the smaller average distance between the people.

The article says:

However, although the pentagram requires a smaller area - 9.51 square metres - the average distance that one person has between another person rises to 2.72m.  

Ignoring the typo of “pentagram” (which isn’t the same as a pentagon or, as it’s written elsewhere in the article “petagram”), this suggests that my intuition is partly wrong, in that the area of the pentagon is smaller but the average distance for the pentagon is bigger.  Hmm.  Let’s check that out.

The Hexagon

Here’s the regular hexagon, with 2m along each side.  There are lots of ways to find the area.  We could use Pythagoras to find the height of one of the triangles:

The height is 

The area is therefore √3 and the area of the hexagon is 6√3

(An alternative would be to use the area formula   which isn’t as bad an idea as it looks, because this is  

 (we know the value of sin 60 degrees) and this is clearly also the square root of 3.)

… putting six people in a hexagon, each two metres apart, would require just 10.39 square metres. 

Forming a hexagon means the maximum distance between two people is four metres at the extremities, while the average distance that one person has to another person is around 2.29m. 

6√3 = 10.39 (2dp), so we have worked out the area required and this agrees with the article.

The maximum distance between any two people is indeed 4m.  (F and C on the diagram are 4 metres apart, for example.)

How do we work out the ‘average distance’ between people? There is nothing special or different about any of the positions, so if we work out the average distance that A is from all the others, this will be the same as the average distance of B from everyone else, etc.

AB is easy – this is 2m.

AD = 4m and AF = 2m.

AE could be worked out using the cosine rule, but we could look back at the previous diagram and see that AE is the height of two of the triangles, so it is 2√3.

AC is the same as AE.

That gives the average distance of the other 5 points from A as:

This is 2.98564 …  and not 2.29 as in the article.

It is also greater than the values given in the article for the pentagon.

The Pentagon

I think the pentagon values are accurate.  (A class could be asked to work these out, using whatever method they want.)

Here, briefly, are my workings:

Area of a triangle = 
 so the area of the pentagon is (to 2dp)

Now we work out the lengths on the diagram:

Distance AF = 2.

We can work out AB using the cosine rule in the earlier diagram:

AE = AB = 2.3511…

To work out AC, I used the triangle ACF and the cosine rule. 

AD = AC = 3.8042…

This time we need to be careful when working out the average distance, because F is an average distance of 2m away from everyone else, whereas A (and B, C, D, E) are an average of

(This is a useful reminder to use the unrounded versions of the answers …)

To work out the average distance for all of the six people in this arrangement, I did

These figures agree with the article.

The Parallelogram

Bobby Seagull tentatively suggested a parallelogram seating arrangement based on equilateral triangles:

The area and the average distances aren’t included in the article, so our class could work those out!

Here is my brief working:

Area is 4 lots of the triangle we worked out earlier, so it’s 4√3

The distance AD is the height of two of the triangles, so this is the same as AE was on the hexagon: 2√3

AD is equal to CF and BE.

AB = AC = BC = BD = CD = CE = DE = DF = EF = 2

That just leaves AF to work out. I used the cosine rule to get √28 = 2√7

AF is the furthest distance in this configuration, which is 5.29m (2dp).

The average distance is, I think, 2.78m (2dp) which is better than the hexagon but worse than the pentagon.

Real-life implications

• There are clearly a few mathematical simplifications going on here, but there are some other issues too.
• People are being treated as points (I think that is reasonable in that the ‘danger area’ is presumably the mouth/nose which, when looking vertically downwards, are not particularly large.)
• How good are people at estimating 2m? I would guess that no-one is exactly accurate, so much of this is moot anyway.
• Given that the measurements won’t be perfect, it is a bit odd to give answers to 2dp, which suggests that they are accurate to the nearest cm !
• A practical drawback with the parallelogram is that C and D are the only people who can see everyone else (B and F can’t see each other because D is in the way).
• Bobby points out that the parallelogram is good because anyone can leave of their own volition. This is true for the hexagon as well. In the pentagon arrangement, for the one in the middle to be able to escape, a minimum of two other people (adjacent to each other) have to move.
• Finally, the article suggests that the person in the middle of the pentagon has an unfair advantage because they are closer to everyone else.  It will be an uncomfortable conversation for them, though, because they will need to twist their head like an owl to be able to look at everyone!

Postscript 1:

The Daily Mail has an article that starts in a very similar way. Not sure whether they both got it from the same source or whether someone has been copying their homework!  They include the error in the calculation too.

https://www.dailymail.co.uk/news/article-8372009/The-way-picnic-Scientists-say-groups-sit-hexagon-pentagon-parallelogram.html

Postscript 2:

I have written above that the article is clearly a bit of fun.  The comments under both the Telegraph and the Mail articles from members of the public suggest that not many of their readers realised this …