Quibans 96: SATS Papers

This Quibans shows the power of some Core Maths techniques.

Here is part of the original version of an article in the education newspaper Schoolsweek.

Coronavirus: £2.7m cost of shredding cancelled SATs papers revealed
The government has shredded exam papers for this year’s cancelled SATs and phonics tests – writing off the £2.7m they cost to produce.

Something about this didn’t seem quite right, so I did some Core Maths to it.

First of all: it says there were 9 million ‘test papers’ and then looked at the height of a stack of paper 9 million pages tall.  My experience of exam papers is that they are usually more than one sheet. 

Here is the background knowledge I already had:

·         KS2 papers are taken by almost all Year 6 children in state schools in England (not the UK, just England).

·         England (not the UK as a whole) has a population of about 56 million people.

·         KS1 papers are taken by almost all Year 2 children in state schools in England.

·         The phonics screening test is taken by almost all Year 1 children in state schools in England.

·         There are 3 maths tests and 3 English tests at KS2, 2 maths tests and 2 English tests at KS1.

Core Maths-ing it

Here’s what I did with it:

·         Life expectancy in England is about 80 years.  Let’s assume that there are the same number of people at each age (might be a dodgy assumption).  56 ÷ 8 = 7, so 56 million ÷ 80 = 700,000.

·         That would give about 700,000 children in each year group – and I will assume they all take the relevant tests.

·         Next assumption is that each test booklet has about 20 pages, which means the equivalent of 10 sheets of A4 paper. 

·         Now it’s just multiplication and addition! 

·         The six papers that the Y6 children take (3 in maths and 3 in English) have a total of 60 sheets of paper per pupil.  That gives 60 × 700,000 sheets overall.

·         The four papers that Y2 children take have a total of 40 sheets per pupil: 40 × 700,000 sheets overall.

·         The phonics test might be 2 sheets per pupil: 2 × 700,000 sheets overall.

·         This gives a final total of 102 × 700,000 sheets, which I will happily round off to 100 × 700,000 and which gives 70 million sheets of paper.  This would have a height of about 7.3 km.

That’s nearly 8 times as much as in the article, which feels like quite a big difference!

Let’s check my assumptions.  Are any of them likely to be wildly out? 

·         Number of children in each year group:  If anything there are likely to be slightly more than I estimated/calculated because of population growth.

·         I didn’t account of children who didn’t take the test though (perhaps because they are in independent schools), so maybe this balances out the previous point?

·         I don’t know how many pages each test has, but the ones from previous years are stapled booklets and they last about an hour each.  Surely they can’t only be 2 or 3 sheets of paper each?

So I was fairly happy with my assumptions and with my answer of about 70 million sheets of paper. 

From the numbers in the article, it appeared that the author had assumed each of the 9 million papers was a single sheet of paper.  That seems like it might be a faulty assumption!

Looking things up

Next I looked up some figures to see how accurate my assumptions were.

This article says 600,000 children took the KS2 SATS last year.

This webpage allows you to download SATS papers (KS1 and KS2) from previous years.  Here are the figures for 2019:

KS2

English		Maths
Paper 1	32 pages	Paper 1	20 pages
Paper 2	4 pages	Paper 2	24 pages
Paper 3	12 pages (reading booklet) 20 pages (answer booklet)	Paper 3	24 pages
total	68 pages	total	68 pages

That gives 136 sides of paper, which is 68 sheets (because each piece of paper is double-sided!).

When I checked for 2018, the numbers were almost identical (the only difference was that the English answer booklet was 4 pages longer).

KS1

English		Maths
Paper 1	24 pages	Paper 1	20 pages
Paper 2	12 pages (reading booklet) 12 pages (answer booklet)	Paper 2	22 pages
Total	48 pages	Total	52 pages

That gives 100 sides of paper, which is 50 sheets.

Phonics screening test

It looks like (here) that the answer sheet is filled in by a teacher and that would mean each class only needs one copy of the question paper.  That’s a single sheet of paper for each pupil.

Comparing my estimate with the looked-up figures

This table shows my estimated calculations and shows a more accurate version using the figures I looked up:

	My estimate	Using the real figures
KS2 papers	60 sheets per pupil × 700,000 pupils	68 sheets per pupil × 600,000 pupils
KS1 papers	40 sheets per pupil × 700,000 pupils	50 sheets per pupil × 600,000 pupils
Phonics test	2 sheets per pupil × 700,000 pupils	1 sheets per pupil × 600,000 pupils
Total	102 × 700,000 sheets A height of about 7.3 km	119 × 600,000 sheets A height of about 7.4 km

I am happy with that!

Do note, that I have still made some estimates when I used the ‘real figures’. 

I have assumed there are the same number of children who take the tests in Y1 and Y2 as in Y6.  I also haven’t accounted for the class booklets for the phonics test, and for the teacher instructions for the other tests. (I think they provide teacher booklets in every pack of 10 papers, so this will add a couple of pages per pupil overall).

Finally, Wikipedia says the height of the Eiffel Tower is 300m.  So if we divide 7.4 km by 300 metres, we get that the height of the stack of paper would be about 25 times the height of the Eiffel Tower.

Are there better representations of the height?

(Another assumption: the paper doesn’t get compressed by the great pile of paper above it!)

The journalist who wrote the article was kind enough to confirm that they did assume that each test paper was a single sheet and added an extra line to the article.

UPDATE (29 July 2020)

Well this is exciting!  Since posting this Quibans the original article has been updated and now links here:

By the power of Core Maths!

Using this with a class

I clearly haven’t used this with a class yet.  Here is one way I might use it at the start of next term, stressing the importance of Core Maths techniques in realising when something was mis-stated.

Start by showing the excerpt from the article:

Coronavirus: £2.7m cost of shredding cancelled SATs papers revealed
The government has shredded exam papers for this year’s cancelled SATs and phonics tests – writing off the £2.7m they cost to produce.

Questions: What do you think?  Do these numbers seem feasible?  How could we tell?

Discussion about this.  Is it reasonable that 500 sheets of paper are 5.2 cm thick, are the calculations to get the height of a million sheets correct? (Yes to all of these.) 

Are there likely to have been 9 million pieces of paper in the tests (no), are the conversions to metres and Eiffel Towers right (yes).

Next question: There _is_ an error.  What is it? 

(The only thing it could be is the number of pieces of paper.)

Question: What info do you need to work out how many pieces of paper are involved?

In response to the students’ queries, I might give them some of the info from earlier in this post:

·         KS2 papers are taken by almost all Year 6 children in state schools in England (not the UK, just England).

·         England (not the UK as a whole) has a population of about 56 million people.

·         KS1 papers are taken by almost all Year 2 children in state schools in England.

·         The phonics screening test is taken by almost all Year 1 children in state schools in England.

·         There are 3 maths tests and 3 English tests at KS2, 2 maths tests and 2 English tests at KS1.

·         The Eiffel Tower is 300m tall.

They could then work out estimates for the number of children in each year group (again, as I did above), or I could tell them that about 600,000 children in each of the year-groups took the tests.

They could also estimate the number of pages in each test, perhaps comparing with the GCSE exams (or past papers if they didn’t get to sit the exam), or recalling their own SATS tests. 

Or I might give them this information:

KS2

English		Maths
Paper 1	32 pages	Paper 1	20 pages
Paper 2	4 pages	Paper 2	24 pages
Paper 3	12 pages (reading booklet) 20 pages (answer booklet)	Paper 3	24 pages
total	68 pages	total	68 pages

KS1

English		Maths
Paper 1	24 pages	Paper 1	20 pages
Paper 2	12 pages (reading booklet) 12 pages (answer booklet)	Paper 2	22 pages
Total	48 pages	Total	52 pages

Phonics screening test

One sheet of paper for each pupil.

It is worth checking they are clear that they need half as many sheets of paper as there are sides.

They might get calculations that look approximately like mine:

	My estimate	Using the real figures
KS2 papers	60 sheets per pupil × 700,000 pupils	68 sheets per pupil × 600,000 pupils
KS1 papers	40 sheets per pupil × 700,000 pupils	50 sheets per pupil × 600,000 pupils
Phonics test	2 sheets per pupil × 700,000 pupils	1 sheets per pupil × 600,000 pupils
Total	102 × 700,000 sheets A height of about 7.3 km	119 × 600,000 sheets A height of about 7.4 km

7.5 km divided by 300 metres = 25, so the stack of paper would be about 25 times the height of the Eiffel Tower.

Final question:  Can they think of better representations of the height?

Source: https://schoolsweek.co.uk/coronavirus-2-7m-cost-of-shredding-cancelled-sats-papers-revealed/

[NB: Some of the article appears in screen-shot form because I wanted to show it before it was updated.]

Quibans 95: The 50p at 50

From the Guardian.

50p at 50: how Britons' living costs have changed since 1969

The 50p coin entered circulation on 14 October 1969, in the run-up to D-Day on 15 February 1971, when the UK finally abandoned shillings and pence and moved to a decimal currency.

The 50p coin was worth a lot then – indeed, more than any coin in circulation now. In real terms it was worth the equivalent of just over £8. Fifty years on, especially after rampaging inflation in the 1970s, the 50p is a shadow of its former self – it has shrunk in size and weight, but above all in value.

But what do these figures mean in real terms? Inflation since 1969 has been an astonishing ## %, with the fast-rising “cost of living” already a major headache for the government at the end of the 1960s.

What is extraordinary about the wages data from the late 1960s is the huge gap between men and women. The ONS New Earnings Survey for 1970 shows that the average full-time male worker was paid £30 a week, while a female full-time worker earned just £16.30. Annually, that worked out at £ ## for men, which is equal to £24,050 today, but only £ ## for women, equal to £ ## today.

A look through the Guardian’s classified job adverts on the day the 50p was launched gives a snapshot of what real jobs paid. The Open University, which had just been established, was hiring professors at £3,780 a year (equal to £ ## today), lecturers on a range of £ ## to £ ## (£20,153-£47,147), and secretaries on £ ## -£1,068 (£12,506-£ ##).

The Open University is still hiring – and its adverts suggest that real pay for academics has barely edged ahead in 50 years. It currently has an advert for a professor of economics on a range of £67,700-£75,800, while it pays lecturers £33,000 to £49,000. But the secretary job pays lots more in today’s money, with the OU advertising a role at £23,000 to £26,000.


Questions:

Work out the numbers that have been replaced by ## symbols.

What is the real-terms increase of salary for Professors, Lecturers and Secretaries (using the top of the pay scale each time)?



Answers:

Inflation since 1969 has been an astonishing 1,554%

The ONS New Earnings Survey for 1970 shows that the average full-time male worker was paid £30 a week, while a female full-time worker earned just £16.30. Annually, that worked out at £1,560 for men, which is equal to £24,050 today, but only £847.60 for women, equal to £13,068 today.

The Open University, which had just been established, was hiring professors at £3,780 a year (equal to £62,532 today), lecturers on a range of £1,240 to £2,850 (£20,153-£47,147), and secretaries on £756-£1,068 (£12,506-£17,668).

• Professors: an increase of 21.2%
• Lecturers: increase of 3.9%
• Secretaries: increase of 47.2%

Source: https://www.theguardian.com/money/2019/sep/14/50p-britons-living-costs-1969-spending

Quibans 94: Transport in Luxembourg

This is a pre-coronavirus article about public transport in Luxembourg, from the BBC.

Free transport in Luxembourg, but what's the cost?

It has had months of hype and now finally Luxembourg's free public transport has begun.
With a population of only 614,000, it may be one of Europe's smallest countries and the idea is not unprecedented. But the "free mobility" drive has captured the imagination. Buses, trams and trains are now free to ride on and you don't need a ticket.

"The government wants Luxembourg to become a laboratory for mobility," says Mobility Minister François Bausch, who points to the grand duchy's fast-rising population, with a rise of 40% in 20 years.

Travelling on transport will be free for residents and visitors alike, except for first-class train passengers.

The price of the project will be the €41m (£35m; $44m) in lost ticket fares, but that will be shouldered by the taxpayer. "Of course, just because I call it free transport doesn't mean nobody pays," said Mr Bausch, who is part of Luxembourg's green party,

The total cost of running the service is more than €500m so the government sees the lost fare revenue as relatively small.

It was not exactly pricey before 29 February. A fare cost €2, and double for a day pass. Many workers have their annual travel pass subsidised in Luxembourg, so few people spend much on transport anyway.

Luxembourg spends more of its economic output on transport that most other European countries, with a reported €600 a year per person.

Questions:

1. What was the population of Luxembourg 20 years ago?
2. What is the exchange rate of the £ to the € , and the £ to the $ ? How can we use these exchange rates to work out the € to the $ ?
3. Using the cost of running the service and the population, work out how much it costs per person per year. What is strange about this?
4. Roughly how many public transport journeys are taken each year? How many per week?

Answers:

1. 614,000 divided by 1.4 = 438571, so the population was about 440,000
2. £1 = €1.1714 £1 = $1.2571 This means €1.1714 = $1.2571 , so €1 = $1.0732
3. €500m divided by 614,000 people is €814 each. Later in the article it says that it spends €600 per person per year on transport. (Presumably this includes roads as well. And airports?)
4. A ‘day pass’ used to cost €4 and brought in €41 million. Presumably most people made two journeys each day (there and back?). That’s about 10 million journeys. Dividing this by 50 (approx. number of weeks in a year) gives 200,000 journeys per week. If we were to assume that this is 100,000 people then only a small fraction of the population ever use public transport!

Source: https://www.bbc.co.uk/news/world-europe-51657085

Quibans 93 – Royal Families

Quibans 93 – Royal Families

from CNN.

Here's how much Europe's royal families really cost

Question 1:

The UK monarchy is twice as expensive as every other one.  What is potentially misleading about this?

Answer 1:

The population of the UK is massively bigger than that of Monaco!

Question 2:

If we work out the cost per person in each country, where do you think the UK would be?

Answer 2:

Here is the next part of the article:

Question 3:

If we combine these two images, what can we work out?  (Then work it out!)

Answer 3:

We can work out the population of each country.

Here are my workings (with upper and lower bounds included too, perhaps as a talking point):

Question 4:

What else might we need to consider when working out how expensive a Royal Family is?

Answer 4:

There might be lots of other issues.  If palaces are being renovated then that will have a one-off cost.  The number of ‘working royals’ might be different in different countries, so there are different numbers of people being ‘paid’.  More working royals will require more admin support and staff. 

This doesn’t factor in the benefits to a nation of having a monarchy.   This might include the tourism that is attracted.  In the UK system the Queen is the head of state.  If we didn’t have a monarchy then we would need a different person doing that role (such as a president) who would need offices, staff, etc, so some of these costs would still be necessary.

Source: https://edition.cnn.com/2020/03/14/europe/european-royal-families-intl/index.html

Quibans 92: Hoarding toilet paper

Quibans 92 – Hoarding toilet paper

From the Daily Telegraph, near the start of the coronavirus pandemic:

Why the UK won't run out of toilet roll

13 March 2020
 As panic over the coronavirus outbreak continues to spread, empty shelves where toilet roll once would have been has become a common sight in many supermarkets. Consumers have been stockpiling the bathroom staple, sparking fears that supermarkets could soon be caught short. However,  there might not be any need to panic.

The UK exported $129m (£103m) of toilet roll in 2019 and exports more than it imports.

“As the world’s 11th-largest exporter of toilet roll, at least if our supply routes shut down, we will be able to keep calm, stop exporting, and carry on,” Dr Rebecca Harding of data firm Coriolis Technologies said.

Ireland accounted for $90.4m or ##% of all UK toilet roll exports in 2019 and is dependent on the UK for 80pc of all toilet roll. Dr Harding said the Irish reliance on the UK for toilet roll is evidence of the importance of borderless trade on the island of Ireland.

Questions:

1. What is the exchange rate? 
2. What percentage of UK exports of toilet roll go to Ireland?
3. How much does Ireland spend on toilet roll per year?
4. The population of Ireland is 4.83 million.  How much does each person spend on toilet roll per week?
5. Four sheets of toilet paper cost about 1p.  How many sheets is that per person per day?

Answers:

1. £1 = $1.2524
2. 70%
3. 90.4 million divided by 0.8 = $113 million
4. 45 cents per week
5. 45 cents is about 36p per week.  Divide by 7 to get the cost per day and multiply by 4 to find the number.  20.5 sheets per day.

Source: https://www.telegraph.co.uk/business/2020/03/13/uk-wont-run-toilet-roll/

Quibans 91: SPOTY and the Christmas Number 1

'SPOTY' is the annual BBC 'Sports Personality of the Year'. 

This Quibans takes an article from a year ago.  It appears that it is updated each year (and is nonsense every time!).  The versionfor 2019 has been done as a video (but includes the same graphics as in 2018).

Here I present the article a section at a time, with a question after each.  My thoughts and comments appear together at the end.  You may want to give students the address of the webpage and ask them to go through it for themselves.

BBC Sports Personality of the Year: How do you win the main award?

14 December 2018
The BBC Sports Personality of the Year 2018 will be crowned on Sunday but what are the key factors in the make-up of previous winners? From birthplace to facial attributes, we've been analysing the ideal candidate...
The award was first handed out in 1954 and there have been 60 different winners, with Andy Murray the only person to win it on three occasions.

Question 1: This image appears at the end of the article.  It suggests that to win SPOTY you ideally need to do athletics, to have been born in London, to have blue eyes, to be 29 years old and to have had your achievement in August.  Any comments at this stage?

Here is the first image:

Q2)  Why might it not be surprising that there are more Athletics winners than there are from other sports?

============================================= 

Q3)  Why is this unsurprising?

============================================= 

Q4)  Any comments?

============================================= 

Recall the line of text near the start of the article:

The award was first handed out in 1954 and there have been 60 different winners, with Andy Murray the only person to win it on three occasions.

Q6)  Looking at the previous image, what can we work out?

 ============================================= 

Q7)  Why is this not a surprise?

Q8) So let’s look back at the diagram we started with.  Any further comments?

Here are my answers:

Q2)  There are lots more athletes than there are motor racing drivers and there are lots of sports included under the heading ‘Athletics’.  There are really only 11 England cricketers who might be in with a chance and footballers suffer because people will only vote for national team players and not those from clubs they don’t support, 

Q3)  London is bigger than anywhere else, so it has a larger proportion of the population.

Q4)  What has this got to do with success in sport or with popularity?  Does having green eyes make you less popular?  No.  Does it make you less successful?  No.  And the facial hair is presumably partly to do with fashion.

Q5) If you have blonde hair you are more likely to have blue eyes!  So these two things are linked.

Q6)  There are 65 winners here, so some people have won it twice.

Q7)  Many awards will have been because of events that take place in the summer, such as the Olympic Games (July/August) or Wimbledon (June).  There are most winners from athletics and that is overwhelmingly a summer sport.  Is it also likely that those who win things in Jan/Feb/March are forgotten by the time you get to December and the voting for SPOTY?

Q8) We have said already that athletics and August are likely to be linked and that London is the biggest city in the UK, so is likely to have more people who are born there.  The blue eye colour is interesting: for the men it was equal, but by including the female winners you get blue being the mode.  Clearly, if you are male then this isn’t relevant!  But hold on: vastly more men have won than women, so should that be mentioned here?

Here is a list of the Athletics winners of SPOTY (from Wikipedia):

Of the Athletes, 5 were born in London (Christopher Chataway, Seb Coe, Daley Thompson, Fatima Whitbread and Jonathan Edwards). 

The final four of those appear to have brown eyes, leaving only Christopher Chataway.  He was 23  years old when he won and his big achievement (breaking the 5000m world record) took place in October. 

Chataway meets only the first 3 of the 5 criteria.  Are there any who meet 4 of them?  Jonathan Edwards is the only one of the athletes who makes 4 out of 5.  He was a Triple-Jumper, born in London, was 29 and his world championships win was in August. 

None of the other winners (from any sport) get 4 of the 5. 

So this is an example of where appealing to the averages at every stage doesn’t help us in predicting the winner.

Extension:  Number 1 song at Christmas

This used to be one of the biggest songs of the year.  Many different newspapers carried exactly the same story.  This version comes from the Independent:

Music experts find formula for ‘perfect’ Christmas number one

Music researchers have crunched the numbers on Christmas songs to find the perfect formula for a festive hit.

Experts at a UK music label looked at every Christmas number one from the last 50 years to see what they have in common and determined that the Pet Shop Boys’ 'Always On My Mind' came closest to being the quintessential tune for the holidays.

“I think we’re a long way from an algorithmically-generated Christmas number one,” said Howard Murphy, founder of Ostereo which conducted the research, in a press release accompanying the data.

“But certain characteristics do make a song more likely to resonate with audiences at Christmas,” he added.

For a song to hit the top spot on Christmas Day it needs to be three minutes and 57 seconds, in the key of G major, played at 114 beats-per-minute and performed by a 27-year-old solo artist, according to the research.

Other key trends among the songs analysed were that the majority were ballads, nearly half were cover versions and nearly all were about something other than Christmas.

'Always On My Mind' hits the ideal length and key exactly, as well as being a cover of Elvis’ 1972 hit, but is slightly faster than the formula suggests at 125 bpm and performed by a duo who had an average age of 31.5 when the song hit number one at Christmas 1988.

'Mary’s Boy' by Boney M came a close second, researchers said, with a tempo of 113 bpm, length of four minutes two seconds and a key of F, two semitones lower than G.

Clearly this isn’t a ‘formula’ – they have just averaged the attributes of all the Christmas number one songs over the years.  What types of average have they used for each attribute?  How is this story related to the SPOTY one?

Sources:

https://www.bbc.co.uk/sport/sports-personality/42367914

https://www.bbc.co.uk/sport/av/sports-personality/50752816

https://en.wikipedia.org/wiki/BBC_Sports_Personality_of_the_Year_Award

https://www.indy100.com/article/christmas-number-one-how-to-guide-perfect-formula-pet-shop-boys-boney-m-elvis-8674946