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.]

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/