Tuesday, 28 July 2020

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

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 news...