Quibans 100: Something Fishy

This is not a Quibans about politics (and when I use this with my class I won’t be taking about the politics), but it is about common mathematical/statistical errors that people make.

It is unusual in that it starts with a tweet.  I don’t usually create a Quibans based on a ‘random person tweeting’, but because this came from the account of a former MEP who has over 70,000 followers, it seems reasonable to do so.

With my class I will start by showing the following images.  Here’s the first tweet:

This image was part of that tweet:

Here’s part of an image from one of the other tweets in the thread:

I will ask the class:  What questions do you have for me?

I am anticipating some of the following:

1)      Where do the numbers come from?

2)      What do they mean?

3)      What is he working out?

4)      Why is he doing it ‘by hand’?

And here are my responses:

1)      This is from the ‘Annex’ to the Brexit Trade Deal that was agreed by the EU and the UK government on 24 December and which is due to be voted on by MPs before the end of 2020.  This is a printout of pages 893 and 894.

2)      It shows the percentage of each type of fish that is allowed to be caught in different areas of sea by UK and by EU fishing boats.

3)      The two orange highlighted columns show the percentage for UK boats in 2021 and in 2026.  He is working out the average of these for each page.

4)      He has presumably used a calculator to add them up and then to divide by the number of types of fish on the page.  Why hasn’t he just copied it into a spreadsheet?  The document it is in is a pdf, and the annex is an image, so the text can’t just be copied and pasted!  I have retyped those two columns (I think I have done it accurately).  They are available on this spreadsheet.  (The spreadsheet hasn’t been optimised for printing.  Pages 1 to 4 include a screenshot of the original document and my typed version of the relevant columns.  The final sheet includes all of the typed data.)

Tasks for the students:

A)      For page 1, calculate the total and the mean to check the figures he has worked out here.  How many different ways can you work out the mean?

B)      Now do the same on page 2.  What is going on here?  (Find the error!)

C)      Go to page 4.  If he makes the same mistake on page 4 as he did on page 2, what will his calculations for 2021 be?  What is the correct mean?

D)      Go to the final sheet.  What is the mean for all of the fish for 2021?  What is it for 2026?

E)      Why is it not reasonable to say there is a 2.32% increase?  Using his figures and his methodology, what would be a more sensible value to give for the increase?

F)       If UK boats catch 20% of a type of fish in one year and 30% the next, why might they not be landing more fish in the second year?  What information is missing here?

G)     Why can’t we find the mean of the 2021 figures and the mean of the 2026 figures and then calculate the change?  (This is a major issue!)

Some answers:

A)      This is a nice opportunity to use Excel in different ways.  The calculations for page 1 are correct.  (He has sensibly rounded the answers for the mean.)  We can use =SUM( : ) to work out the total and can then divide that answer by 24 (there are 24 types of fish stocks on the first page) or can do =AVERAGE( : )

B)      On page 2 the total for 2026 is correct.  But he has then divided by 27.  This should be 28.  If you use the ‘average’ command on Excel you get the correct answer and this is the same as dividing by 28.  Why has he got this wrong?  The second page starts with number 25 and goes up to number 52.  He has done 52 – 25 = 27.  This seems an obvious thing to do for lots of people.  Why is it wrong?  How many ways can they explain it? 

[If you count them you find there are 28 of them.  Or you could consider that on page 1 it goes from 1 to 24, but that is clearly 24 different fish – you don’t do 24 – 1.  Or: if you subtract 25 you are getting rid of fish number 25 – we need to remove fish number 24 (and the earlier ones), so it should be 52 – 24.]

C)      Page 4 includes fish number 77 to 87.  If he subtracts then he will assume there are 10 fish on that page, whereas it should be 11.  The total is 383.69.  He will divide by 10 to get 38.37% whereas it should be divided by 11 to give 34.88%.  (In fact he rounded off to 1dp this time and got 38.4%)  Note that you can use SUM and AVERAGE in Excel across the blank cells and cells with text in them – it just ignores them and still gives the correct answers.

D)      For 2021 the mean is 33.47% and for 2026 it is 35.93%.  These are close to the values he gives (of 33.6% and 35.92%) but not identical.  In fact, in the tweet he writes about 2020 and not 2021.  I tweeted to ask about this, but didn’t receive a reply.

E)      He seems to be saying that because the UK share (according to his calculations) is going from 33.6% to 35.92% then that is an increase of 2.32%.  We ought to refer to this a an increase of 2.32 percentage points.  From the perspective of the UK fishing industry, the percentage is going from 33.6 up to 35.92.  This is an increase of 2.32 over the original value of 33.6.  That’s actually an increase for the UK boats of 6.9%.

F)       The document talks about the percentage of fish that can be caught by the two fishing fleets.  It doesn’t confirm that the total amount is the same each year.  If 10 tonnes can be caught one year and 1 tonne the next, then 20% of 10 tonnes is 2 tonnes (2000kg) whereas 30% of 1 tonne is 300kg.  (I couldn’t find any figures about the total quotas, so I have no idea whether the amounts are the same each year.)

G)     There is a massive issue here, which means all of the information in the original tweet (and everything that has happened so far in the earlier tasks) is just nonsense.  We cannot find the average of the percentages in the way the former-MEP did, because they are likely to be percentages of different amounts.  Here’s a simplified example.  If the UK fleet gets 10% of Fish A and 20% of Fish B, it doesn’t necessarily get 15% of all of the fish.  Suppose 100 tonnes of Fish A is landed in total and 1 tonne of Fish B is landed then 10% of A is 10 tonnes and 20% of B is 0.2 tonnes.  Altogether the UK boats get 10.2 tonnes out of 101 tonnes, which is 10.1%.  Or if the UK fleet has 100% of Fish A and 0% of Fish B then it gets 100/101 = 99% of the fish and not the average of 100% and 0%.

Any time you see averages of averages it is worth asking whether it is fair/reasonable to do this!

Sources: https://twitter.com/MartinDaubney/status/1343147201112010753

https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/948104/EU-UK_Trade_and_Cooperation_Agreement_24.12.2020.pdf

Quibans 99: Bus Gate Fines

‘Bus Gates’ seem to be the new way to restrict access to certain roads.  In the past in Cambridge we had rising bollards that would disappear into the road to allow buses and emergency vehicles to pass through.  Bus Gates are ordinary roads, with lots of signs and cameras.  A new Bus Gate was installed on what had always previously been a clear section of road.  The story below is from the Cambridge News.

Thousands of fines issued to people driving through Mill Road bridge bus gate

Nearly 5,000 fines have been issued to people driving over Mill Road bridge.

From August 28 to October 16, 4,840 fines have been issued to drivers. The number of drivers themselves being issued with fines will be fewer, since one driver can receive several fines.

Since the summer, Cambridgeshire County Council has closed the bridge to all traffic except buses, cyclists, and pedestrians.

Prior to fines being issued, there was a grace period of more than two weeks when 1,630 notices were sent out to warn drivers that they were committing an offence and should they do so again, then they would face a fine.

The fine is £60, but is reduced to £30 if the motorist pays within two weeks.

More than 96 per cent of fines paid were just £30.

This means the council has made an estimated £151,008 from the partial closure of Mill Road bridge.

The council said last year money from fines was used for frontline highways maintenance.

 

You might like to present this to a class and to ask what is wrong with it.  Alternatively, the following questions could lead the students:

Q1)  Check every number.  Which ones are exact?  Which ones have been estimated by the journalist? 

Q2)  How did they do these estimations?

Q3)  Which number is definitely wrong?  Explain!

Q4)  Produce a better version of that number.

Q5)  Compare the number of fines issued to the number of notices sent out during the grace period.  What can we conclude?

 

Some possible answers follow.

A1)  “Nearly 5,000 fines” – has been rounded. 

“4,840  fines” and “1,630 notices”.  Is it suspicious that the number of fines (4,840) and notices (1,630) are both multiples of 10?  Maybe they have been rounded off?  (For both to end in a zero by chance the probability would be 1/100.)  It is possible that they are exact, though.

The numbers for the fines are accurate.

“More than 96 per cent” is an estimate, as is “an estimated £151,008”.

A2)  4,840 rounded to the nearest thousand is 5,000.

96% of the fines were £30.  Work out 96% of 4,840 and multiply that by £30.  Work out 4% of 4,840 and multiply by £60.  Add them together.  We get £151,008.

A3)  £151,008.  You can’t possibly get a number that ends in an ‘8’ by adding integer multiples of 60 and 30. 

A4)  A better answer/method would be to say: there are about 5,000 fines and almost all were £30, so that makes a total of 5,000 x £30 = £150,000.  The total is therefore an estimated £150,000 in fines.  I think this is the best estimate to give.  If you wanted to go more deeply into this we need to decide what the language means.  Can we assume “more than 96%” means it’s up to 96.5%?  (suspecting that were to be over 96.5% then it would be written as “nearly 97%?).

If so, then a lower bound would be to take 96.5% of the fines as £30 and the rest as £60.

That gives the newspaper figure as an upper bound (but it would need to be a multiple of £30) and 150,282 as the lower bound (but again – needs to be a multiple of 30).  The bounds would therefore be 150,300 and 150,990.  If we assume the 4840 has been rounded to the nearest 10 then that broadens those bounds just a little.  (To be clear, I don’t think any of this is worth doing: £150,000 is a perfectly sensible estimate here!)

A5)  The ‘grace period of more than two weeks’ was presumably there to allow those who use the road regularly to realise that they would be fined and to warn them of that.  I would expect lots of warnings during the grace period (because it was a new thing) and fewer fines when it was actually rolled out.

We need to find the rate for both numbers.  August 28 to October 16: That’s 4 days in August, 30 days in Sept and 16 days in Oct: a total of 50 days.  4,840 fines divided by 50 days gives an average of 96.8 fines per day.  (5000 divided by 50 is perhaps more sensible – giving 100 per day.)

The grace period was “more than two weeks”.  If that means 15 days then it is was 108.7 per day, if 16 days then 101.9 per day and if 17 days then 95.9 per day.

All of these values are very close to the 96.8 per day that were actually fined.  The number of people being fined each day is the same as the number of people who drove through during the grace period.  Does that mean the bus gate just isn’t working?  Or that the fines are not a deterrent?  Or that the grace period was in August, during the holiday times when the traffic was lower anyway, and that the massive increase in traffic on the roads in Sept means a smaller percentage are bursting through the bus gate? 

 

Source: https://www.cambridge-news.co.uk/news/cambridge-news/thousands-fines-issued-people-driving-19173924

Quibans 98: US Election night 2020

In this Quibans you may want to copy the images so you can project them one at a time.  The questions can then be posed verbally.

In the USA’s presidential election, because of the time differences across the country, the polls close on the east coast earlier than elsewhere and they start counting the votes and releasing the results while voting is still taking place in other states.

I expected the states in New England (the north-east of the US) to support Joe Biden, so I was very surprised in the early hours of the following morning to see, on the election webpage of the Wall Street Journal, this (as a ticker running across the screen):

Q1) What does this mean?  And what doesn’t it mean?

A1) It would be usual to think of this as “Trump has 23.1% more than Biden”.  But we need to be careful.  If Biden has 100,000 votes, does that mean Trump has 123,100 votes?  Or does it mean that Trump’s lead divided by the total number of votes cast is 23.1% ?  We would usually refer to the latter as a lead of “23.1 percentage points” to avoid confusion.

After my surprise at this huge lead, I clicked on the image of the state of New Hampshire to get further information:

Q2) What is going on?  What do you notice?  What is strange?  What can you work out?

A2) So far, 26 votes have been counted!  Just 26.  And 16 of those went to Trump, while only 10 were for Biden. 

The difference between 61.5% and 38.5% is 23.0% - so there must be some rounding involved.

16/26 = 0.61538…, and 10/26 = 0.384615…, so Trump’s figure has been rounded down and Biden’s rounded up, and when we subtract we get 0.230769…, which rounds to 23.1

An alternative way to get this value is to do 6/26 (where the numerator is the difference between 16 and 10).

It says that 1.0% of the expected total vote has been reported.  If it’s exactly 1% then we would expect 2600 votes to be cast in the state of New Hampshire altogether.  Using the upper and lower bounds of 1.0% we get:

26/0.0095 = 2736.8 – so the upper bound for the total number of votes is 2736 (mustn’t round up!)

26/0.0105 = 2476.1 – so the lower bound for the total number of votes is 2477 (must round up!)

That seems like a small number!

 

Here is the current state of play in New Hampshire:

Q3) What is surprising?

Q4) How many percentage points is Biden leading by?

Q5) How many votes are still uncounted?

Q6) Biden won all 4 of the electoral college votes from New Hampshire.  Why might that be considered unfair?

A3) They have counted 803,831 votes so far – which is a long way above our upper bound of 2736 ! (And Biden managed to overturn Trump’s 6-vote lead!)

A4) Biden is leading Trump by 59,275 votes.  Dividing this by the 803,831 votes counted gives us 0.07374…, which is a lead of 7.37 percentage points.

A5) I am deeply suspicious of the figure for the percentage of votes that have been counted, but if we take it as correct we get that 830,831 is between 98.95% and 99.05%

830,831 / 0.9895 = 812,360 (rounding down)

830,831 / 0.9905 = 811,541 (rounding up)

Hence, there are between 7710 and 8529 still to count.

A6) The way the US Presidential election system works, each state votes and then (for 48 of the 50 states) the electoral college votes for that state all go to the winner of the state.  In the election in 2000 the state of Florida was won by a margin of only 500 votes.  That gave all of the electoral college votes for Florida (25 of them – because Florida has a greater population than New Hampshire) to George W. Bush and resulted in him winning the election. 

In New Hampshire, Biden benefited (he got 53% of the votes but 100% of the electoral college votes), whereas in other states Trump benefited from the system.  In the 2016 election Hillary Clinton won more votes than Donald Trump, but ended up with fewer electoral college votes, so Trump was the winner.

Here is the final map from 2020:

Q7) Is there more red or blue?

A7) I think there appears to be more red.  But because the very big red states in the north have small populations, the number of electoral votes for the blue states (Biden) significantly exceeds the red states (Trump).

The Wall Street Journal provides this as an alternative version of the map:

Q8) What is going on here?

A8) The map has been scaled to show one square for each electoral college vote.  (You may want to flick back and forth between the real map and this one.  Montana is a particular casualty!).  Now there is more blue. 

Source: https://www.wsj.com/election-results-2020/live-coverage.html

 

 

 

Quibans 97: Curious coronavirus cases

This Quibans uses the first part of an article from the Bristol Post, published on Weds 21 October 2020. I intend to use this a bit at a time, asking questions and revealing the next part of the article after each answer. My answers are at the end.



Part 1:

Bristol University coronavirus cases nearing 1,100 as more staff members test positive

The number of staff members to have tested positive for coronavirus at the University of Bristol has more than doubled in a week, latest figures show.

Question 1) The focus here is on the number of members of staff who have tested positive. What percentage of the 1,100 do you guess are staff and what percentage are students?




Part 2:

A total of 1,050 students have tested positive for covid-19 at the University of Bristol 

Q2) Now you have got some additional information, calculate the percentage of the cases that are staff and that are students.



Part 3:

A total of 1,050 students have tested positive for covid-19 at the University of Bristol now, together with 19 members of staff, taking the total of confirmed cases at the university to 1,079 - a daily rise of 67 positive tests.

Q3) Any comments?

Q4) What is the actual percentage of cases that are staff and students?



Part 4:

In recent days, the number of new positive tests among students seems to have slowed down and Tuesday saw the biggest rise in six days (83) as the University hadn't confirmed that many positive tests among its students since last Wednesday, October 14, when 126 cases were confirmed. However, the number of positive tests among staff members now seems to be rising quickly, having more than doubled since last Wednesday when eight coronavirus cases were confirmed among university staff.

Q5) What is the percentage increase for staff over the week?

Q6) If the staff cases continue to increase at the same rate, in which week will the number of staff cases reach 1000?

Q7) Why is this unlikely to happen?



Part 5:

Belowis a graph from the article.

Q8) How does it work?

Q9) Work out the numbers for some of the orange bars and blue dots.

Q10) What criticisms do you have of the graph?

Q11) Why is Oct 19 lower?

 





Answers:

1) It doesn’t really matter what the class suggest; the reason for doing this is that they will very likely get a percentage for the staff that is vastly too low.

2) Presumably that means 50 staff and 1050 students, so 4.5% are staff and 95.5% are students.

3) The numbers don’t add up! 1050 students plus 19 members of staff equals 1069 and not 1079. And the headline says “nearing 1,100”, which is a rounded version of 1069.

4) 19/1069 gives a staff percentage of 1.8%, with 98.2% of the cases being amongst students. This focus on staff seems less than important!

5) Last week it was 8 and now it’s 19, so the multiplier is 19/8 = 2.375, so the increase is 1.375, which is a 137.5% increase.

6) Keep multiplying the number of staff cases by 2.375

After 1 week: 19 x 2.375 = 45.125 (It might appear that we should round this to 45, but it’s not going to be an exact increase, so I am not particularly bothered about using 45.125)

After 2 weeks: 45.125 x 2.375 = 107.171875

After 3 weeks: 107.171875 x 2.375 = 255

After 4 weeks: 605

After 5 weeks: 1436

We could have done this by using 2.3753, etc. 

7) 1000 staff would be a considerable fraction of the staff of the university, whereas 1050 students is tiny compared to the number of students. It seems unlikely that this figure will be reached for staff. The student figure is declining after only a few days, so a sustained increase over 5 weeks seems unlikely.

8) It appears that the scale for the yellow bars is on the left hand side, while the scale for the blue dots is on the right hand side. The horizontal lines work for the left hand scale; you have to look very carefully at the tiny marks next to the numbers on the right hand scale.

The first day has 119 cases (the blue dot) and after that the height of the yellow bar is added to the previous day’s blue dot to get the next blue dot.



Q9) If you go to the graph on the website you can hover over the bars and dots to see the actual numbers.



Q10) Plenty of criticisms!

Why doesn’t the right hand scale start at zero? Starting at 100 doesn’t save very much space!

What happened to Oct 8, 10 and 11 ? Given that the blue dots (for the total number of cases) matches up with the yellow bars that have been added on, does that mean about half of the cases shown for Oct 9 were really from Oct 8? How would the graph look different if Oct 8, 10, 11 were included too?


Q11) It seems odd to have Oct 19 being so much lower and the day after much bigger again. Oct 19 was a Monday. The graph shows when cases were confirmed, so presumably this is 24 or 48 hours after the students were given a covid test. It might be that fewer people were tests on 17 or 18 Oct (because it was a weekend).

Source: https://www.bristolpost.co.uk/news/bristol-news/bristol-university-coronavirus-cases-nearing-4627979

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.]