The *Plus* teacher packages are designed to give teachers (and students) easy access to *Plus* content on a particular subject area. Most
*Plus* articles go far beyond the explicit
maths taught at school, while still being accessible to someone doing GCSE and A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world, find maths in unusual places,
and delve into mathematical history and philosophy.
We therefore hope that
our teacher packages provide an ideal resource for students
working on projects and teachers wanting to offer their students a deeper
insight into the world of maths.

### Mathematics in sport

This teacher package brings together all our articles that have to do with sport, from cricket to football and from the sport itself to sporting architecture and infrastructure. We have grouped our articles in the following categories:

- The physics of sport
- Sporting strategy
- Architecture and infrastructure
- Predicting results and sporting stats
- Scoring and ranking
- Betting and odds
- Doping

Don't forget that our sister site NRICH has hands-on problems, activities and articles on the mathematics of sport. We've also created a new website for our Maths and Sport: Countdown to the Games project celebrating the London 2012 Olympics, bringing together Plus articles and NRICH activities and problems, as well as some new project ideas, in one easily accessible place.

### The physics of sport

Spinning the perfect serve — A new mathematical analysis of how to hit a winning serve shows that spin is the thing. Perhaps it was maths that helped Andy Murray to gold in the Olympics?

The *Plus* Olympic calendar: Shuttlecocks — While medals were being awarded in badminton at London 2012, we started thinking about shuttlecocks. They are not like balls at all and this leads to some very strange physics.

The *Plus* Olympic calendar: Volleyball — As in tennis, spin's the thing when it comes to serving in volleyball. But why does putting a spin on a ball make such a difference to its path through the air?

When is a goal not a goal? — Remember Frank Lampard's disallowed goal in the 2010 World Cup match against Germany? The ball hit the crossbar, landed well behind the line but then bounced out again. And it all happened too quickly for the ref to spot it was a goal. But do how these kind of (non)-goals happen and what can we do about it?

Walk, trot, gallop — Horses, like all animals, have a number of different gaits. But how can they perform these complicated leg movements without having to stop and think? And why do they switch to a new gait when they want to go faster? Mathematics can shed some light on these questions.

Canoeing and kayaking — A simple question to ask about kayak races is whether having lots of paddlers helps or slows the boat down? The kayak with two paddlers has twice as many "engines" to power it but it also has twice as much weight to drag through the water. Which is the dominant factor?

By the skin of their suits — How did the controversial new swim suits, which have now been banned, lead to record breaking performances at the 2008 Olympics?

Rowing has its moments — Maths reveals new optimal rigs for rowing Fours and Eights. Will they be used at London 2012?

The answer to runnin' in the wind — Runners and cyclists can tolerate heat and cold but the thing they dislike most is wind. They know it produces slower times. Can we show them why?

A fly walks round a football — What makes a perfect football? Find out why the ball's surface is the most prized research goal in ball design.

Making gold for 2012 — Find out what sports engineers do to gain that extra hundredth of a second that makes the difference between gold and silver.

What goes up must come down — How large are the forces acting on a gymnast swinging on the high bar?

Making a racket: the science of tennis — Over recent decades new materials have made tennis rackets ever bigger, lighter and more powerful. So what kind of science goes into designing new rackets?

Supersonic bloodhound — In 1997 Andy Green was the first to break the sound barrier in his car Thrust SSC, which reached speeds of over 760mph. Now he and his team want to push things even further with a car called Bloodhound, designed to reach the dizzy heights of 1,000mph, about 1.3 times the speed of sound. This article explains how maths is used to build this car. You can also read our interview with Andy Green.

Career interview: aerodynamicist — *Plus* talks to Christine Hogan, programmer, sysadmin and author, who studied aerodynamics hoping to become a member of a Formula One team.

If you can't bend it, model it — Learn about the aerodynamics of footballs and perfect your free kick.

Eye on the ball — Why the 2006 World Cup ball caused controversy.

Outer space: Rugby and relativity — Relativity in action.

Outer space: Cushioning the blow — Where's the sweet spot on a pool ball and does it matter?

### Sporting strategy

Blast it like Beckham? — What tactics should a soccer player use when taking a penalty kick? And what can the goalkeeper do to foil his plans? This article uses game theory to find the answers.

Outer space: A question of tactics — Why randomness is a good strategy.

Formulaic football — When's the best time to make a substitution in a football match?

### Architecture and infrastructure

Leaning into 2012 — Rising, like a giant pringle from the Olympic Park construction site, the Velodrome is the first of the 2012 London Olympic venues to be completed. But why is the track inside tilted?

How the velodrome found its form — The Velodrome, with its striking curved shape, was the first venue to be completed in the London Olympic Park. We talked to two structural engineers who were part of the design team for the Velodrome about how mathematics helped create its iconic shape. You can also listen to the interview in our podcast.

Outer space: When errors snowball — Why small errors matter when it comes to breaking records.

Swimming in mathematics — One of the most impressive features of the 2008 Beijing Olympics was the beautiful aquatics venue, known as the *water cube*. Looking like it's been sliced from a giant bubble foam, the design was based on an unsolved maths problem and although the bubbles look completely random, the underlying structure is highly regular and buildable.

Career interview: IT project manager - Olympic Games — Travel, money, meeting new people, living in new cultures, and a whole lot of sport — that's where maths has lead Jamie Clarke, an IT project manager who specialises in international sport projects such as the recent Winter Olympics in Torino. Jamie tells *Plus* how he went from engineering to the Olympics.

Stadium maths: the podcast — Like tuning forks stadiums can resonate and if they resonate at the wrong frequency, disaster strikes. We talk to civil engineer Paul Shepherd about this and other things you need to keep in mind to make stadiums safe.

### Predicting results and sporting stats

Mapping the medals — Predicting the final Olympic medal count is a black art. But we like a challenge! So we came up with our very own predictions for the 2012 London Olympic medal count.

No limits for Usain — Usain Bolt, the "fastest man on the planet", aims to get his 100 metre world record of 9.58 seconds down to 9.40 seconds. What has mathematics got to say about this quest?

Anyone for tennis (and tennis and tennis...)? — What is the chance of another Isner-Mahut mega set at Wimbledon?

Understanding uncertainty: How psychic was Paul — How likely is it that Paul the octopus' accurate predictions for the 2010 World Cup were really due to psychic powers?

Mathematicians rival octopus in World Cup final prediction — Is maths better than Paul the psychic octopus?

The luck of the draw — How a mathematical improbability turned one day's football action into a once-in-a-lifetime quirk of fate.

On the ball — If your team scores first in a football match, how likely is it to win? And when is it worth committing a professional foul? This article shows us how to use probability to answer these and other questions, and explains the implications for the rules of the game.

Understanding uncertainty: Football crazy — Taking up a challenge from a BBC radio programme David Spiegelhalter and Yin-Lam Ng used their statistical finesse to predict the outcome of the last matches of the English Premier League — and they were 90% correct. Find out how they did it.

Power trip — How long do football managers' careers last and does talent matter? Surprisingly the answer may lie in mathematical power laws.

It's all cricket's fault — Is cricket to blame for the high price of oil? This is a good example to show that correlation (shown through plots of unrelated variables) does not necessarily imply causation.

El Niño and the Ashes — And do cricket results depend on the weather? This article uses statistical testing to test the significance of this rather surprising correlation.

The home advantage — Does the country hosting the Olympic Games fare better than others? This article gives several nice examples of simple data analysis.

The curse of the duck — How come that the most common score of cricket's best batsman is zero? This article explores different distributions for data.

### Scoring and ranking

The maths of gold medals: Four Olympic thoughts — It's not the winning, it's the taking part that counts. At least, that's what the Olympic creed would have us believe. But, like it or not, what the media and governments focus on is the tally of gold medals. This article explores some of the maths of gold.

Harder, better, faster, stronger — After every Olympics, there is speculation about which country performed best. Should we really be surprised when China, with its huge population, and the US, with its combination of high GDP and population, top the medal table? Can we take a look at the medal tables and see which countries did indeed perform better than expected?

Howzat! — This article discusses averages and explores a number of different ways to rank cricketers.

Understanding uncertainty: the Premiere League — Does a club's ranking in the Premiere League table reflect how good it really is? This article discusses the concepts of mean, variance, standard deviation and confidence intervals.

Deciding who's top — How normal distributions play a part in the ranking systems of chess and sumo.

### Betting and odds

Outer space: Racing certainties — Winning with horses.

### Doping

The logic of drug testing — When an athlete does fail a drug test can we really conclude that they are cheating? Not necessarily. Even with accurate tests the chance of guilt can be as low as 16%.