Friday, 12 July 2019

Quibans 90: Goal sizes


This task was test-driven at the recent AMSP Core Maths days (as part of ‘Quibans Live’).  Thanks to those who made suggestions for improvements.

The story appeared in a number of places before the 2019 FIFA Women’s World Cup and there were discussions about whether women should play with smaller goals because women are, on average, shorter than men.  This version is taken from the Daily Telegraph.

'Reducing size of women's goals would not help fight for equality,' says England goalkeeper Karen Bardsley
England goalkeeper Karen Bardsley believes reducing the size of goals in the women's game would jeopardise the fight for equality in football.
Chelsea manager Emma Hayes has suggested the women’s game could be adapted to bridge the ‘physical differences’ between women and their male counterparts.
She suggested smaller goals could stem criticism often directed at female goalkeepers, who are generally of a smaller stature to their male counterparts.
Manchester City goalkeeper Bardsley, about to play in her third World Cup when she will add to her 77 international caps, does not believe narrowing the width between two posts is the way forward.
“My big thing is we need to change how people feel about goalkeeping, I don’t think there is enough respect for the position in the game, whether male or female.
“There is a stigma, that you have to be a certain size, or not very good with your feet, or you have had to go in goal as a last resort.”
How could we explore, using some Core Maths knowledge, whether the goals should be made smaller?

On one level it doesn’t matter.  Football is a relatively low-scoring game (compare it with American Football, basketball, rugby, cricket, etc, where the numbers of points (etc) are very much bigger).  If the goals are bigger than necessary and this results in more goals being scored then this isn’t a big deal.  The goals are the same size for both teams after all.

We could find data for the heights of male and of female goalkeepers and compare them. 
In fact, I think the size of the goal was set in the 1866 version of the laws of the game and according to the BBC the “average height of men has risen by almost 11cm since the mid-19th century”. 
Wikipedia has the current average height of UK men at 175cm and women at 162cm, so if both have increased by about 11cm since the size of the goals was fixed then women are now roughly the same height as men were back then.

Here’s my favourite way to explore this issue:  If the goalposts are too big for women then that presumably means more goals are being scored by women.  Let’s use the number of goals per game from the last two World Cups (Russia 2018 for men and France 2019 for women) and compare the data.

The Google Sheets file has the full data, but here are the relevant parts.  You might want to give the summary data to the class and ask them to work out averages and interpret them.


Before giving it out you might want to ask whether the average number of goals per game will be different in the group stage or the knock-out part of the tournament.

Is this a reasonable set of calculations to do?  Are there ways to improve this?

Those who followed the women’s competition will know that the eventual winners, the USA, beat Thailand 13-0.  This is clearly an outlier, so perhaps we should remove it from the women’s data?


At Quibans Live several people pointed out that if there was extra time in a match then there was more opportunity to score goals (the penalties that follow extra time have not been included here, though that is another avenue that could be explored).  A normal-length match is 90 mins, so extra-time of 30 mins is equivalent to another third of a match.

Here are the results with the 5 extra-time men’s matches and the three extra-time women’s matches included:


Very close indeed!  This data doesn’t suggest the goals should be smaller in the women’s game.

Final thought:
This is clearly relevant to the elite athletes at the last two world cups.  Does it still work for those playing at lower levels?   


https://docs.google.com/spreadsheets/d/1PupE-_c8zGfNsRBROBa8urFv48HQqk8oz42qu_lczUc/edit?usp=sharing

Friday, 17 May 2019

Quibans 89 – Men vs Women in the European elections 2019

I voted by post earlier in the week.  Some parties had more female candidates than others.  I wondered whether it was the case that across the country certain types of party have more male or female candidates.  Here’s what I did with my Year 12 Core Maths class.


Some background to the electoral system:

Each country in the EU has its own voting system for the European elections.  England is divided into 9 regions and Wales and Scotland are each a region (Northern Ireland has a different system, so I didn’t involve them).  In each region there are a number of seats, ranging from 10 seat for the South East region to 3 seats in the North East.

You don’t vote for an individual but for a party.  Each party has a ‘list’ of candidates for each region.  Last time around in the 7-seat East of England region most parties had seven candidates.  Having worked out that UKIP should get 3 seats, the Conservatives 3 seats and Labour 1 seat (using the D’Hondt method – but that’s not relevant here), the top three candidates from the UKIP list were elected as MEPs, the top three from the Conservative list and top Labour candidate.

What we did:

On a shared Excel file, I had each region on a separate sheet.  I divided these up around the class (some students had a sheet each, others worked as a pair).  First they sorted out whether each candidate was male or female.  This wasn’t always obvious, so they Googled if they weren’t sure.
Then they typed into another sheet the number of female and male candidates for each party.
They had to deal with some issues.  For example, some candidates are standing as ‘independents’, meaning they are not part of a party.  We decided to ignore them.  Some candidates have withdrawn from the election; we removed the ones we knew about.

These are the results, for the parties that are standing in every region:

11% of the UKIP candidates are female, through to Change UK, for whom 57% of the candidates are female.

Is it interesting that the more pro-Brexit parties tend to have more male candidates, while the more pro-Remain parties have roughly equal numbers of men and women?

Better methodology?

But hold on: maybe this doesn’t tell the whole story.  What if the LibDems have the same number of male and female candidates but the top few in each region are all men?  It is extremely unlikely that those further down each list will be elected, so this is important. 

Here is how I intend to deal with that.
The students recorded a list of Male and Female candidates for each party.  Here’s part of that list:

Since the lesson (we will talk about this next week) I have turned that into a set of binary numbers, where F becomes 1 and M becomes 0.

I then turned the binary into a decimal.  This means the first-ranked candidate in each list is worth double the one that follows, which is worth double the one after that, etc.  If there are 5 candidates in a list, the 5th one is worth 1 point, the 4th is worth 2 points, the 3rd gets 4 points, the second gets 8 points and the first is 16 points.  Add up the number of points for the female candidates and then divide by the total (which is 1 + 2 + 4 + 8 + 16 = 31).  That gives a fractional weighted-score for the female candidates in each region.
Then I found the average of these. 

The second column shows this new figure (with the first column showing the previous data):

Not much difference (apart from the Green Party)! 

Final thoughts:

It was good for the students to see a way to carry out this sort of analysis.  
It was good to divide up the work.  I couldn't find an easy way of getting hold of the gender of each candidate, so sharing the workload was a good thing.
We had to decide how to deal with problems (withdrawn candidates).
We had to think about how to do the analysis.

Here are links to my original spreadsheet and the one my class filled in.

If you want your class to use the original version then you will need to save it and share it with them (your IT people in College will be able to help if you haven't done this before).


Saturday, 4 May 2019

Quibans 88: Bottled water

I started by giving out just this paragraph from the Guardian article that follows:
Imagine laying out half-litre bottles on the pitch at Wembley Stadium. You could fit ### bottles on the grass, packed into a tight grid. Now imagine building up layers of bottles, covering the same area, to build a tower. To contain all the bottled water we buy each year [in the UK], you would end up with a ###-metre skyscraper.
There are some nice Fermi estimations required here.

What are the dimensions of a football pitch? What are the dimensions of a water bottle? How many bottles are used in this country in a year?

Then we need to convert some units and carry out the calculations, not forgetting to round off sensibly.

The students worked in groups. A typical estimate for the size of a football pitch was 110m by 60m. Some measured their water bottles (!) and found a diameter of about 6cm and a height of 20cm.

If working out the area covered by the pitch and by a bottle we need to be careful when converting units (10,000 cm2 = 1m2). We also need to decide what “packed into a tight grid means” (we took it to mean a rectangular grid). The easiest way is to work out how many bottles fit along each side of the pitch and then to multiply these. These figures gave us 1.8 million (pleasingly close to the 1.7 million in the article). Other estimates in the room varied between 1.6 million and 2.8 million.

Working out the height is more difficult. If everyone in the country has one bottle per week then that would give us about 1.7 billion bottles per year. That would need to be stacked 1000 rows high, which means a height of about 200m. In fact the article says it’s 2.2 billion _litres_ per year (each bottle holds 500ml), so we have underestimated the answer.

I then gave out a fuller version of the article (below) and asked what things they could work out.

If we care about plastic waste, why won’t we stop drinking bottled water?
Sun 28 Apr 2019

For all the innovation and choice that define the food and drink industries, if you want to make money, you could do a lot worse than bung some water in a bottle and flog it. A litre of tap water, the stuff we have ingeniously piped into our homes, costs less than half a penny. A litre of bottled water can cost well over a pound, especially for something fancy that has been sucked through a mountain.
Yet the bottled water market is more buoyant than ever, defying the plastics backlash inspired by stricken albatrosses on the BBC’s Blue Planet, and a broader, growing sense that something has to change.
Sales in the UK were worth a record £558.4m in the year to last November, an increase of 7%, according to the latest figures from the market analyst Kantar. Separate data from the analysts Nielsen show that last year we guzzled more than 2.2bn litres of bottled water, including “take-home” and “on-the-go” products. That’s an annual rise in volume of 8.5%.
Imagine laying out half-litre bottles on the pitch at Wembley Stadium. You could fit 1.7m bottles on the grass, packed into a tight grid. Now imagine building up layers of bottles, covering the same area, to build a tower. To contain all the bottled water we buy each year, you would end up with a 514-metre skyscraper – 200 metres taller than the Shard.
Hope is not entirely out of reach. That plastic skyscraper conceals attempts in the bottled water industry to change. If nothing else, the rate of growth has begun to ease (sales were up 7% in the year to November 2018, compared with 8% the previous year).
But even if large numbers of us are quitting bottled water because of care for the environment, others are taking it up. The introduction of the “sugar tax” on juices and fizzy drinks has pushed more people to bottled water, while health awareness has boosted its desirability. Kantar says tap water consumption is growing at roughly the same pace (we still drink almost three times as much tap water as bottled water).
So the plastic tide only creeps higher. The industry is quick to point out that all its bottles are recyclable. “But collection rates are, at the most generous estimates, 56%, so the actual recycling rate will be lower than that,” Chetan-Walsh says. And while bottles may be recyclable, very few are made of recycled plastic. Highland Spring launched recycled half-litre “eco” bottles alongside its standard bottles in January; Evian has vowed to use only recycled plastic across its range by 2025.
Kinvara Carey, general manager of the Natural Hydration Council, an association of the biggest bottled water manufacturers, cites a survey in which people were asked what they would do if bottled water were not available. “Forty-four per cent would buy another drink, which is not great, 14% would go without and 4.5% said they would find a fountain,” she says. “The choice is important.”

We can work out things like:

1) The cost of a litre of bottled water
2) How the price has changed in the past year
3) What the amounts were last year
4) How many bottles are recycled
5) The volume of landfill taken up by bottles
6) Why the percentages in the final paragraph don’t add up to 100%
7) How much water each person drinks each day

Friday, 26 April 2019

Quibans 87: McDonald’s straws


I gave the article below (from the BBC website) to my Yr 12 class and asked them what they could work out.  They had some great ideas (which are shown below the article), largely because we have been working on this sort of thing throughout the year.  If you wanted to use this with a class that isn’t used to working in this way then the ideas my class came up with could be turned into questions to ask.

McDonald's plastic straw petition: Call to ditch paper straws

24 April 2019

McDonald's is being called on to stop its roll out of paper straws in the UK and Ireland, amid claims that they "dissolve" in drinks.

The fast food giant is switching from plastic to paper straws at their 1,361 restaurants after customer pressure. An online petition calling for a return to plastic straws has so far garnered more than 35,000 signatures.

McDonald's say they are "doing the right thing" while a supplier said customers need to "compromise". The restaurant chain supplies 1.8 million straws to its four million UK customers each day.


Things my class worked out:
  • 35,000 people have signed the petition.  That’s less than 1% of the people who visit McDonald’s in one day.
  • About 6% of the population of the country go to McDonald’s each day.
  • On average each restaurant uses 1323 straws per day.  [We talked about rounding this off more appropriately – and decided that 1300 is probably better.]
  • Just under half (45%) of customers use a straw.  This might be because some people don’t have a drink, or have a hot drink, or have juice without a straw.
  • There are 1.46 billion visits to McDonald’s in the UK and Ireland each year.  [Better as 1.5 billion?]  We know this can’t be 1.5 billion different people because that would be about a fifth of the population of the planet!  On average, that’s 22 visits per year per person, which is equivalent to about two visits per month.  Maybe half the population go once a week and the other half never go? 
  • A straw weighs about 0.42g.  That means 756kg of straws per day.  Over the course of a year that’s the weight of 3600 adults.




Saturday, 15 December 2018

Quibans 86: Recycling Ferrero Rocher


This is interesting because it contains an error, a lack of clarity and something misleading.  Perfect for a Quibans!

Article from the Daily Telegraph:
Ferrero Rocher shamed as Christmas chocolates with the least recyclable packaging 
12 DECEMBER 2018 
Ferrero Rocher has been named and shamed as the Christmas chocolate box with the least recyclable packaging. 
The vast majority (89 per cent) of the packaging in a 359g Ferrero Rocher Collection box may be destined for landfill as it is made from non-recyclable plastic and foil, a study by consumer group Which? found.
By contrast the most recyclable chocolate boxes were Waitrose Christmas Chocolate Favourites 240g which is 96.3 per cent recyclable) Cadbury's Milk Tray, which is 92.5 per cent easily recyclable. The second least recyclable packaging was for Marks & Spencer's The Big One Selection 600g, which is 71.4 per cent recyclable. 
In addition Which? analysed chocolate boxes for their packaging to chocolate weight ratio, and found boxes of Ferrero Rocher Collections had by far the highest. 
Nearly half of the weight of a 359g Ferrero Rocher Collection box is made up of packaging (42.4 per cent), with the rest constituting (67.6 per cent) the chocolates themselves, it found.  
The boxes with the highest proportion of chocolate weight were Marks & Spencer's The Big One Selection 600g (just 8.5 per cent packaging), and Lindt Lindor Assorted Mix 384g (just 11.5 per cent packaging), the study showed.  

Questions:
1)      What is the error?
2)      The lack of clarity is that ‘359g’ is the weight of the chocolate (and nuts, etc) in the box.  The packaging weighs more on top of that.  42.4% of the total is the packaging, so what is the weight of packaging?
3)      The misleading part is that an implication of the two mentions of M&S is not followed up.  What is going on there?

(Answers are below.)

Further info:
Here are screenshots of the tables from the Which? report:



Here it is as a single table (to be pasted into Excel?)
Chocolate
Official weight (g)
Weight in packaging (g)
Packaging weight (g)
Percentage packaging 
Percentage easily recyclable
Cadbury's Heroes
660
808
144
17.80%
88.90%
Cadbury's Milk Tray
360
470
106
22.60%
92.50%
Cadbury's Roses
660
810
144
17.80%
90.30%
Chocolate Treats By Sainsburys
650
806
154
19.10%
88.30%
Ferrero Rocher Collection
359
642
272
42.40%
11%
Lindt Lindor Assorted Mix
337
384
44
11.50%
90.90%
Marks & Spencer The Big One Selection
600
662
56
8.50%
71.40%
Mars Celebrations
650
812
150
18.50%
90.70%
Mars Malteaser Teasers
275
364
70
19.20%
88.60%
Morrisons Mega Mix
1050
1216
162
13.30%
81.50%
Nestle Quality Street
720
846
130
15.40%
83.10%
Thorntons Continental Selection
284
408
120
29.40%
86.70%
Waitrose Christmas Chocolate Favourites
240
356
108
30.30%
96.30%

Additional questions:

4) What should happen when you add up the Official Weight and the Packaging Weight columns?
5) Do any of them mis-state the amount of chocolate they contain?
6) Work out the percentage of the Total Weight for each product that cannot be recycled. How does M&S compare?


Answers:

1)      The error is that 42.4% plus 67.6% doesn’t equal 100% (and it should).  The former figure is correct (as seen in the Which? tables).
2)      The chocolate in the Rochers are 57.6%, so when we divide 359g by 0.576 we get the total weight (623g) and this leads to a packaging weight of 264g.  This doesn’t agree with the figures from Which? because there is slightly over 359g of choc.
3)      M&S may have the second-least recyclable packaging, but there is much less of it, so maybe they have cut down on the packaging overall (which seems like a good thing: just because something is ‘recyclable’ doesn’t mean everyone will recycle it!).
4)      You should get the ‘Weight in packaging’ column. 
5)      Most of them give us slightly more chocolate than advertised, with only Nestle falling under (but that is presumably only on the single box that Which? tested).
6)      I used Excel to work this out.  38% of the weight of the in-package Ferrero Rocher is non-recyclable, but only 2% of the M&S.  The median was 2%.



Tuesday, 10 April 2018

Quibans 85: Crime and Police figures

From the Cambridge News:


Violent crime in Cambridge has nearly doubled in a decade as police numbers drop
9 APR 2018
Bottom of Form
Violent crime is 90 per cent higher than nine years ago, while police officer numbers have dropped by nearly a tenth.

In Cambridgeshire, there were 15,975 violence against the person crimes reported in the year to September 2017, the latest figures, a ## per cent rise from 8,364 reports in the year ending September 2009. 
However, over the same period, the full-time equivalent number of police officers at the force has fallen by ## per cent, from 1,438 in September 2009 to ## in September 2017, a loss of 106 officers. 
In the past year alone, Cambridgeshire has seen a 30 per cent rise in violent crime, up from 12,316 cases, while police officer numbers have dropped by 1 per cent or seven officers.
A police spokesperson said: “Some of the increases in violent crime can be attributed to improving recording standards and increased reporting of domestic abuse. Most violent crime happens behind closed doors rather than on the streets and the force is working hard to reduce all violent crime.”
Across England and Wales, there was a 20 per cent rise in violent crime, made up of violence against the person, both with and without injury, and homicide, in a year.
In the year ending September 2016, 1.08m violent crimes were reported, rising to 1.29m in the year ending September 2017. 
Compared to the year ending September 2009, when there were 706,859 reports, numbers have more than doubled, a 114 per cent rise, according to figures from the Office for National Statistics, with most forces seeing low in the number of reports in 2012 or 2013.
Over the same period, police forces across England and Wales have lost 22,155 officers, a 15 per cent drop in numbers between September 2009 and September 2017, according to Home Office figures. 
Numbers have dropped by 1 per cent in the past year, from 126,252 full-time equivalent officers in September 2016 to 125,364 in September 2017, a loss of 888 officers.

Questions:
1)      Work out the missing numbers (shown by ## in the text above).
2)      Is the subheading reasonable?  (Violent crime is 90 per cent higher than nine years ago, while police officer numbers have dropped by nearly a tenth.)
3)      Are the other figures in the article consistent?  (Hint: they aren’t!)
4)      How does Cambridgeshire compare to the national situation?




Answers:
Q1)  From the article:
In Cambridgeshire, there were 15,975 violence against the person crimes reported in the year to September 2017, the latest figures, a 91 per cent rise from 8,364 reports in the year ending September 2009. 
However, over the same period, the full-time equivalent number of police officers at the force has fallen by 7 per cent, from 1,438 in September 2009 to 1,332 in September 2017, a loss of 106 officers. 
Checking this:  15975 ÷ 8364 = 1.90997, which is a 90.997% increase.  91% is accurate.
1332 ÷ 1438 = 0.92629, which is a 7.4% decrease.  7% is accurate.

Q2)  The subheading says:
Violent crime is 90 per cent higher than nine years ago, while police officer numbers have dropped by nearly a tenth.
91% has been rounded to 90%, which seems fine.  7% from the article has become “nearly a tenth” in the subhead.  That seems a little out given that it is a way away from 10%.  If both numbers have been rounded to the nearest 10% then …
2009 is eight years before 2017.  The heading says “nine years ago”.  Perhaps they are referring to 2009 being eight years ago from now, but that seems a bit odd, given that we don’t know what the current figures are for 2018.

Q3)  From the article:
In the past year alone, Cambridgeshire has seen a 30 per cent rise in violent crime, up from 12,316 cases, while police officer numbers have dropped by 1 per cent or seven officers.
We have, from earlier in the article, 15,975 violent crimes.  15975 ÷ 12316 = 1.297, which is a 29.7% increase.  30% is fine.  There are now 1332 police officers.  The percentage is 1332 ÷ 1339 = 0.99477, which is a drop of 0.52%.  This has been rounded (unfairly?) to 1% 
A police spokesperson said: “Some of the increases in violent crime can be attributed to improving recording standards and increased reporting of domestic abuse. Most violent crime happens behind closed doors rather than on the streets and the force is working hard to reduce all violent crime.”
This is important.  The article refers to ‘recorded crimes’, so those that aren’t recorded aren’t included and it could be the case that there aren’t in fact more crimes being committed, but rather more crimes being reported.  It would also be worth comparing the increase in crimes to the increase in population (but only for particular age-groups?).
Across England and Wales, there was a 20 per cent rise in violent crime, made up of violence against the person, both with and without injury, and homicide, in a year.
In the year ending September 2016, 1.08m violent crimes were reported, rising to 1.29m in the year ending September 2017. 
1.29million ÷ 1.08million = 1.1944, which is an increase of 19%.  This has been shown as 20%, but it is possible that the accurate value is in fact 20% and rounding issues have intervened.
Compared to the year ending September 2009, when there were 706,859 reports, numbers have more than doubled, a 114 per cent rise, according to figures from the Office for National Statistics, with most forces seeing low in the number of reports in 2012 or 2013.
I can’t make this one work at all.  There are now 1.29 million reports and there used to be 706,859.  Clearly this isn’t more than double.
Over the same period, police forces across England and Wales have lost 22,155 officers, a 15 per cent drop in numbers between September 2009 and September 2017, according to Home Office figures. 
Numbers have dropped by 1 per cent in the past year, from 126,252 full-time equivalent officers in September 2016 to 125,364 in September 2017, a loss of 888 officers.
125364 ÷ (125364 + 22155) = 0.8498, which is a drop of 15%
125364 ÷ 126252 = 0.992966, which is a drop of 0.7%

Q4)  I’ll leave it to you to compare the numbers …




Quibans 90: Goal sizes

This task was test-driven at the recent AMSP Core Maths days (as part of ‘Quibans Live’).  Thanks to those who made suggestions for improv...