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If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

Finding the volume of a rugby ball (prolate spheroid)

With the rugby union World Cup currently underway I thought I’d try and work out the volume of a rugby ball using some calculus.  This method works similarly for American football and Australian rules football.   The approach is to consider the rugby ball as an ellipse rotated 360 degrees around the x axis to create a volume of revolution.  We can find the equation of an ellipse centered at (0,0) by simply looking at the x and y intercepts.  An ellipse with y-intercept (0,b) and x intercept (a,0) will have equation:

Therefore for our rugby ball with a horizontal “radius” (vertex) of 14.2cm and a vertical “radius” (co-vertex) of 8.67cm will have equation:

We can see that when we plot this ellipse we get an equation which very closely resembles our rugby ball shape:

Therefore we can now find the volume of revolution by using the following formula:

But we can simplify matters by starting the rotation at x = 0 to find half the volume, before doubling our answer.  Therefore:

Rearranging our equation of the ellipse formula we get:

Therefore we have the following integration:

Therefore our rugby ball has a volume of around 4.5 litres.  We can compare this with the volume of a football (soccer ball) – which has a radius of around 10.5cm, therefore a volume of around 4800 cubic centimeters.

We can find the general volume of any rugby ball (mathematically defined as a prolate spheroid) by the following generalization:

We can see that this is very closely related to the formula for the volume of a sphere, which makes sense as the prolate spheroid behaves like a sphere deformed across its axes. Our prolate spheroid has “radii” b, b and a – therefore r cubed in the sphere formula becomes b squared a.

Prolate spheroids in nature

The image above [wiki image NASA] is of the Crab Nebula – a distant Supernova remnant around 6500 light years away.  The shape of Crab Nebula is described as a prolate spheroid.

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Essential Resources for IB Teachers

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

Simulating a Football Season

This is a nice example of how statistics are used in modeling – similar techniques are used when gambling companies are creating odds or when computer game designers are making football manager games.  We start with some statistics.  The soccer stats site has the data we need from the 2018-19 season, and we will use this to predict the outcome of the 2019-20 season (assuming teams stay at a similar level, and that no-one was relegated in 2018-19).

Attack and defense strength

For each team we need to calculate:

1. Home attack strength
2. Away attack strength
3. Home defense strength
4. Away defense strength.

For example for Liverpool (LFC)

LFC Home attack strength = (LFC home goals in 2018-19 season)/(average home goals in 2018-19 season)

LFC Away attack strength = (LFC away goals in 2018-19 season)/(average away goals in 2018-19 season)

LFC Home defense strength = (LFC home goals conceded in 2018-19 season)/(average home goals conceded in 2018-19 season)

LFC Away defense strength = (LFC away goals conceded in 2018-19 season)/(average away goals conceded in 2018-19 season)

Calculating lamda

We can then use a Poisson model to work out some probabilities.  First though we need to find our lamda value.  To make life easier we can also use the fact that the lamda value for a Poisson gives the mean value – and use this to give an approximate answer.

So, for example if Liverpool are playing at home to Arsenal we work out Liverpool’s lamda value as:

LFC home lamda = league average home goals per game x LFC home attack strength x Arsenal away defense strength.

We would work out Arsenal’s away lamda as:

Arsenal away lamda = league average away goals per game x Arsenal away attack strength x Liverpool home defense strength.

Putting in some values gives a home lamda for Liverpool as 3.38 and an away lamda for Arsenal as 0.69.  So we would expect Liverpool to win 3-1 (rounding to the nearest integer).

Using Excel

I then used an Excel spreadsheet to work out the home goals in each fixture in the league season (green column represents the home teams)

and then used the same method to work out the away goals in each fixture in the league (yellow column represents the away team)

I could then round these numbers to the nearest integer and fill in the scores for each match in the table:

Then I was able to work out the point totals to produce a predicted table:

Here we had both Liverpool and Manchester City on 104 points, but with Manchester City having a better goal difference, so winning the league again.

Using a Poisson model.

The poisson model allows us to calculate probabilities.  The mode is:

P(k goals) = (eλk)/k!

λ is the symbol lamda which we calculated before.

So, for example with Liverpool at home to Arsenal we calculate

Liverpool’s home lamda = league average home goals per game x LFC home attack strength x Arsenal away defense strength.

Liverpool’s home lamda = 1.57 x 1.84 x 1.17 = 3.38

Therefore

P(Liverpool score 0 goals) = (e-3.383.380)/0! = 0.034

P(Liverpool score 1 goal) = (e-3.383.381)/1! = 0.12

P(Liverpool score 2 goals) = (e-3.383.382)/2! = 0.19

P(Liverpool score 3 goals) = (e-3.383.383)/3! = 0.22

P(Liverpool score 4 goals) = (e-3.383.381)/1! = 0.19

P(Liverpool score 5 goals) = (e-3.383.385)/5! = 0.13 etc.

Arsenal’s away lamda = 1.25 x 1.30 x 0.42 = 0.68

P(Arsenal score 0 goals) = (e-0.680.680)/0! = 0.51

P(Arsenal score 1 goal) = (e-0.680.681)/1! = 0.34

P(Arsenal score 2 goals) = (e-0.680.682)/2! = 0.12

P(Arsenal score 3 goals) = (e-0.680.683)/3! = 0.03 etc.

Probability that Arsenal win

Arsenal can win if:

Liverpool score 0 goals and Arsenal score 1 or more

Liverpool score 1 goal and Arsenal score 2 or more

Liverpool score 2 goals and Arsenal score 3 or more etc.

i.e the approximate probability of Arsenal winning is:

0.034 x 0.49 + 0.12 x 0.15 + 0.19 x 0.03 = 0.04.

Using the same method we could work out the probability of a draw and a Liverpool win.  This is the sort of method that bookmakers will use to calculate the probabilities that ensure they make a profit when offering odds.

Essential Resources for IB Teachers

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

Statistics to win penalty shoot-outs

The last World Cup was a relatively rare one for England, with no heroic defeat on penalties, as normally seems to happen. England are in fact the worst country of any of the major footballing nations at taking penalties, having won only 1 out of 6 shoot-outs at the Euros and World Cup.   In fact of the 31 penalties taken in shoot-outs England have missed 10 – which is a miss rate of over 30%.  Germany by comparison have won 5 out of 7 – and have a miss rate of only 15%.

With the stakes in penalty shoot-outs so high there have been a number of studies to look at optimum strategies for players.

One study published in Psychological Science looked at all the penalties taken in penalty shoot-outs in the World Cup since 1982.  What they found was pretty incredible – goalkeepers have a subconscious bias for diving to the right when their team is behind.

As is clear from the graphic, this is not a small bias towards the right, but a very strong one.  When their team is behind the goalkeeper apparently favours his (likely) strong side 71% of the time.  The strikers’ shot meanwhile continues to be placed either left or right with roughly the same likelihood as in the other situations.  So, this built in bias makes the goalkeeper much less likely to help his team recover from a losing position in a shoot-out.

Shoot high

Analysis by Prozone looking at the data from the World Cups and European Championships between 1998 and 2010 compiled the following graphics:

The first graphic above shows the part of the goal that scoring penalties were aimed at.  With most strikers aiming bottom left and bottom right it’s no surprise to see that these were the most successful areas.

The second graphic which shows where penalties were saved shows a more complete picture – goalkeepers made nearly all their saves low down.  A striker who has the skill and control to lift the ball high makes it very unlikely that the goalkeeper will save his shot.

The last graphic also shows the risk involved in shooting high.  This data shows where all the missed penalties (which were off-target) were being aimed.  Unsurprisingly strikers who were aiming down the middle of the goal managed to hit the target!  Interestingly strikers aiming for the right corner (as the goalkeeper stands) were far more likely to drag their shot off target than those aiming for the left side.  Perhaps this is to do with them being predominantly right footed and the angle of their shooting arc?

Win the toss and go first

The Prozone data also showed the importance of winning the coin toss – 75% of the teams who went first went on to win.  Equally, missing the first penalty is disastrous to a team’s chances – they went on to lose 81% of the time.  The statistics also show a huge psychological role as well.  Players who needed to score to keep their teams in the competition only scored a miserable 14% of the time.  It would be interesting to see how these statistics are replicated over a larger data set.

Don’t dive

A different study which looked at 286 penalties from both domestic leagues and international competitions found that goalkeepers are actually best advised to stay in the centre of the goal rather than diving to one side.  This had quite a significant affect on their ability to save the penalties – increasing the likelihood from around 13% to 33%.  So, why don’t more goalkeepers stay still?  Well, again this might come down to psychology – a diving save looks more dramatic and showcases the goalkeepers skill more than standing stationary in the centre.

Test yourself

You can test your penalty taking skills with this online game from the Open University – choose which players are best suited to the pressure, decide what advice they need and aim your shot in the best position.

If you liked this post you might also like:

Championship Wages Predict League Position? A look at how statistics can predict where teams finish in the league.

Premier League Wages Predict League Positions? A similar analysis of Premier League teams.

IB Revision

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

Statistics to win penalty shoot-outs

With the World Cup nearly upon us we can look forward to another heroic defeat on penalties by England. England are in fact the worst country of any of the major footballing nations at taking penalties, having won only 1 out of 7 shoot-outs at the Euros and World Cup. In fact of the 35 penalties taken in shoot-outs England have missed 12 – which is a miss rate of over 30%. Germany by comparison have won 5 out of 7 – and have a miss rate of only 15%.

With the stakes in penalty shoot-outs so high there have been a number of studies to look at optimum strategies for players.

One study published in Psychological Science looked at all the penalties taken in penalty shoot-outs in the World Cup since 1982. What they found was pretty incredible – goalkeepers have a subconscious bias for diving to the right when their team is behind.

As is clear from the graphic, this is not a small bias towards the right, but a very strong one. When their team is behind the goalkeeper apparently favours his (likely) strong side 71% of the time. The strikers’ shot meanwhile continues to be placed either left or right with roughly the same likelihood as in the other situations. So, this built in bias makes the goalkeeper much less likely to help his team recover from a losing position in a shoot-out.

Shoot high

Analysis by Prozone looking at the data from the World Cups and European Championships between 1998 and 2010 compiled the following graphics:

The first graphic above shows the part of the goal that scoring penalties were aimed at. With most strikers aiming bottom left and bottom right it’s no surprise to see that these were the most successful areas.

The second graphic which shows where penalties were saved shows a more complete picture – goalkeepers made nearly all their saves low down. A striker who has the skill and control to lift the ball high makes it very unlikely that the goalkeeper will save his shot.

The last graphic also shows the risk involved in shooting high. This data shows where all the missed penalties (which were off-target) were being aimed. Unsurprisingly strikers who were aiming down the middle of the goal managed to hit the target! Interestingly strikers aiming for the right corner (as the goalkeeper stands) were far more likely to drag their shot off target than those aiming for the left side. Perhaps this is to do with them being predominantly right footed and the angle of their shooting arc?

Win the toss and go first

The Prozone data also showed the importance of winning the coin toss – 75% of the teams who went first went on to win. Equally, missing the first penalty is disastrous to a team’s chances – they went on to lose 81% of the time. The statistics also show a huge psychological role as well. Players who needed to score to keep their teams in the competition only scored a miserable 14% of the time. It would be interesting to see how these statistics are replicated over a larger data set.

Don’t dive

A different study which looked at 286 penalties from both domestic leagues and international competitions found that goalkeepers are actually best advised to stay in the centre of the goal rather than diving to one side. This had quite a significant affect on their ability to save the penalties – increasing the likelihood from around 13% to 33%. So, why don’t more goalkeepers stay still? Well, again this might come down to psychology – a diving save looks more dramatic and showcases the goalkeeper’s skill more than standing stationary in the centre.

So, why do England always lose on penalties?

There are some interesting psychological studies which suggest that England suffer more than other teams because English players are inhibited by their high public status (in other words, there is more pressure on them to perform – and hence that pressure is harder to deal with).  One such study noted that the best penalty takers are the ones who compose themselves prior to the penalty.  England’s players start to run to the ball only 0.2 seconds after the referee has blown – making them much less composed than other teams.

However, I think you can put too much analysis on psychology – the answer is probably simpler – that other teams beat England because they have technically better players.  English footballing culture revolves much less around technical skill than elsewhere in Europe and South America – and when it comes to the penalty shoot-outs this has a dramatic effect.

As we can see from the statistics, players who are technically gifted enough to lift their shots into the top corners give the goalkeepers virtually no chance of saving them.  England’s less technically gifted players have to rely on hitting it hard and low to the corner – which gives the goalkeeper a much higher percentage chance of saving them.

Test yourself

You can test your penalty taking skills with this online game from the Open University – choose which players are best suited to the pressure, decide what advice they need and aim your shot in the best position.

If you liked this post you might also like:

Championship Wages Predict League Position? A look at how statistics can predict where teams finish in the league.

Premier League Wages Predict League Positions? A similar analysis of Premier League teams.

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

Following on from the data released a couple of weeks ago about Premier League clubs’ financial data, the data from the Championship (England’s second tier) have just been published by the Guardian.  These are from 12 months ago (the most recent data available).  The Championship is famously very competitive – so it will be interesting to see if the same wages and league position correlation that we see in the Premier League also holds here.

Wage Bill 2012-13 Season (millions of pounds)

Using an online scatter plot program we can get the following graph:

Here the league position is on the x axis and the wage bill is on the y axis. In this case if there was a correlation between wages paid and league position we would expect the slope to be negative (as greater wages would lead to a lower league position).  From the graph we can see a pretty weak correlation:

Correlation coefficient (r): -0.3451690473979

Regression line equation: y=24.59-0.43x

The correlation coefficient shows that there is a weak negative correlation.

If we compare this to the scatter graph for the Premier League for the same period (this time the wages are plotted on the y axis – though this will not affect the calculations!)

We can see a stark difference.  This time the correlation coefficient is -0.73, which shows a pretty strong negative correlation.  So what does this show?  Well, it confirms what many people already think about the 2 leagues – the Premier League is overall quite predictable – just by looking at the relative wage bills you can get a pretty good idea about league positions.  The Championship on the other hand is really pretty unpredictable – wages seem to have only a weak correlation with league position.  Indeed Wolves had one of the highest wage bills and yet finished 2nd from bottom.

Championship Debt Timebomb

This remarkable graphic shows the terrible state of the finances for most Championship clubs.  It shows wages as a percentage of turnover.  Spending 50% of turnover on wages is generally considered a sustainable model for football clubs – yet every club in the table is above this – and the vast majority are spending 95% or more of their turnover just on wages.  Bristol City (who were relegated) were spending a staggering 190% of turnover on wages – i.e nearly twice their total turnover!

This is again quite a contrast to the Premier League, where clubs seem much better run:

Whilst most clubs are spending more than 50% of their turnover, there are only 3 clubs spending more than 90% – and only QPR (who were relegated) spending more than their turnover.

So, the Championship is a much more unpredictable league – but also a financial basketcase.

If you liked this post you might also like:

Premier League Wages Predict League Positions? A look at the Premier League data.

Does Sacking a Manager Improve Results? How an improvement in team results is often just down to a statistical result – regression to the mean.

Maths Studies IA Exploration Topics – A large number of examples of statistics investigations to explore.

Premier League Finances – Debt and Wages

This is a great article from the Guardian DataBlog analysing the finances for last season’s Premier League clubs. As the Guardian says, “More than two thirds of the Premier League’s record £2.4bn income in 2011-12 was paid out in wages, according to the most recently published accounts of all 20 clubs. The Guardian’s annual special report of Premier League clubs’ finances shows they spent £1.6bn on wages last season, most of it going to players.”

The first graph (above) shows the net debt levels for different clubs.

The second graph shows the total turnover:

The third graph shows wages as a proportion of turnover:

and the last one is particularly interesting – as it ranks clubs on their wage bills and their league position. This would be an interesting piece of data to test for the strength of correlation:

I’ve used an online scatter plot to calculate both the regression line and the correlation coefficient:

Which clearly shows a strong positive correlation.  This would be an interesting exercise for both IGCSE or  IB students (especially Maths Studies).

For even more data, a club by club full breakdown is also provided by the Guardian here.  I have also made the data above into a word document to be used as a some A4 posters – and you can download that here: Premier League Debt

If you enjoyed this post you might also like:

Which Times Tables do Students Find Difficult? An Investigation.

Why Study Maths? Careers Inspiration

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

### New website for International teachers

I’ve just launched a brand new maths site for international schools – over 2000 pdf pages of resources to support IB teachers.  If you are an IB teacher this could save you 200+ hours of preparation time.

Explore here!

### Free HL Paper 3 Questions

P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.