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## Teacher package: Mathematical Modelling

This is the second installment of a new feature in *Plus*: the teacher package. Every issue contains a package bringing together all *Plus* articles about a particular subject from the UK National Curriculum. Whether you're a student studying the subject, or a teacher teaching it, all relevant *Plus* articles are available to you at a glance.

## What do you think?

This is the second package in a new series for *Plus*, and we'd be very pleased to hear what our readers think. So if you are teacher, a student or any other interested *Plus* reader with thoughts on this new series, then please get in touch.

Thank you!

*Plus* articles go far beyond the explicit maths taught at school, while still being accessible to someone doing 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 in the world of maths.

## Mathematical modelling

Mathematics is often called "the language of the universe". With mathematics, we can describe and make predictions about the behaviour of things around us. The results are often better than ever expected — in fact, one mathematician even wrote an essay about what he called the "unreasonable effectiveness" of mathematics in solving physical problems! This power to describe a complicated system in a simple way is useful in many different branches of study, from analysing Rubik's cube to choosing a national defence strategy.

The *Plus* articles listed below all deal with mathematical modelling. We have divided them into three categories. *Explicit maths* articles contain explicit examples or proofs that can be worked directly into classroom activities and discussions. Articles in the *middle ground* category also contain explicit maths, but require the reader to fill in the details — possible
material for student projects. The *bigger picture* category contains articles that focus on concepts and give an overview of an area, making for eye-opening background reading. In addition to the *Plus* articles, the *try it yourself* section provides links to related problems on our sister site NRICH.

- Explicit maths — get your hands dirty with some real maths.
- The middle ground — enter the wonderful world of modelling and get a glimpse of the equations too.
- The bigger picture — go beyond what you can do in the classroom.
- Try it yourself! — if you've enjoyed reading about how other people solve problems in mathematical modelling, why not have a go? From our sister website NRICH, these problems are graded by school level and challenge difficulty, so that you can find investigations suitable for yourself or your students.

## Explicit maths

### Mathematical modelling in medicine and nature

Classroom activity: Build your own disease — Explore how to model the spread of an infectious disease.

Maths and climate change: the melting Arctic — The Arctic ice cap is melting fast and the consequences are grim. Mathematical modelling is key to predicting how much longer the ice will be around and assessing the impact of an ice free Arctic on the rest of the planet.

Sex, evolution and parasitic wasps — Some things are so familiar to us that they are simply expected, and we may forget to wonder why they should be that way in the first place. Sex ratios are a good example of this: the number of men and women in the world is roughly equal, but why should this be the case? A mathematical model provides an answer.

Pools of blood — A biologist has developed a blood test for detecting a certain minor abnormality in infants. Obviously if you have blood samples from 100 children, you could find out which children are affected by running 100 separate tests. But mathematicians are never satisfied by the obvious answer. Keith Ball uses information theory to explain how to cut down the number of tests significantly, by pooling samples of blood.

Matrix: Simulating the world, Part I — If you've ever watched a flock of birds flying at dusk, or a school of fish reacting to a predator, you'll have been amazed by their perfectly choreographed moves. Yet, complex as this behaviour may seem, it's not all that hard to model it on a computer. Lewis Dartnell presents a hands-on guide for creating your own simulations — no previous experience necessary.

Modelling, step by step — Why can't human beings walk as fast as they run? And why do we prefer to break into a run rather than walk above a certain speed? Using mathematical modelling, R. McNeill Alexander finds some answers.

Guilt counts — Guilt, so some people have suggested, is what makes us nice. When we do someone a favour or choose not to exploit someone vulnerable, we do it because we fear the guilt we'd feel otherwise. A team of scientists have recently month produced some new results in this area, using a model from psychological game theory.

### Mathematical modelling in economics, politics and human interaction

Game theory and the Cuban missile crisis — Steven J. Brams uses the Cuban missile crisis to illustrate the Theory of Moves, which is not just an abstract mathematical model but one that mirrors the real-life choices, and underlying thinking, of flesh-and-blood decision makers.

A risky business: how to price derivatives — This article takes us on the road to the famous Black-Scholes equation of financial mathematics. There's an explicit derivation of the formula for a simplified case, and an appendix with more advanced material for the brave.

Kissing the frog: A mathematician's guide to mating — What's your strategy for love? Hold out for The Only One? Simply try and avoid the really bad ones? How long should you wait before cutting your losses and settling down with the next best who comes along? John Billingham models the problem and saves the national grid in the process.

Graphical methods I: Slug wars, Graphical Methods II: The return of the slime and Graphical methods III: the slugs bounce back — Phil Wilson models the fall and rise of slug world, looking at arms proliferation and slug economy.

## The middle ground

### Mathematical modelling in medicine and nature

Counting calories — Struggling with that new year's resolution to lose a few pounds? Weight not dropping off as fast as you'd expected? A new mathematical model has some good news and some bad news for you. Which would you like to hear first?

What can birds tell us about flying through ash clouds? — Why does a financial mathematician think about birds when trying to understand the grounding of aeroplanes after the Icelandic volcano eruption?

Baby robots feel the love — Researchers have unveiled the first prototypes of robots that can develop emotions and express them too. But how do you get emotions into machines that only understand the language of maths?

Modelling cell suicide — This article sheds light on suicidal cells and a mathematical model that could help fight cancer.

Uncoiling the Spiral: Maths and Hallucinations — Think drug-induced hallucinations, and the whirly, spirally, tunnel-vision-like patterns of psychedelic imagery immediately spring to mind. So what can these patterns tell us about the structure of our brains?

Eat, drink and be merry: making it go down well — This article takes a dive into the rather smelly business of digesting food, and how a crazy application of chaos theory shows the best way to digest a medicinal drug.

Shaping our bones — We know that applying a force to a bone during its development can influence its growth and shape. But can we use our understanding of how developing bone reacts to mechanical forces to help people suffering from diseases that lead to bone deformities?

The mathematics of diseases — Over the past one hundred years, mathematics has been used to understand and predict the spread of diseases, relating important public-health questions to basic infection parameters. Matthew Keeling describes some of the mathematical developments that have improved our understanding and predictive ability. See also this related activity on our sister website NRICH.

How the leopard got its spots — How does the uniform ball of cells that make up an embryo differentiate to create the dramatic patterns of a zebra or leopard? How come there are spotty animals with stripy tails, but no stripy animals with spotty tails? Lewis Dartnell solves these, and other, puzzles of animal patterning.

### Mathematical modelling in games and sport

Harder, better, faster, stronger — Modelling Olympic success.

The *Plus* sports page: The curse of the duck — Why do the best cricketers in the world keep scoring zero?

The *Plus* sports page: Power trip — This article looks at the tenure length of football managers and fits a model to the data.

If you can't bend it, model it! — David Beckham and his fellow players may intuitively know how to bend a football's flight as they wish, but for the rest of us Ken Bray explains the aerodynamics of footballs and the secret to the perfect free kick. For a football-related modelling problem see our sister site NRICH.

Games people play — Combinatorial Game Theory is a powerful tool for analysing mathematical games. Lewis Dartnell explains how the technique can be used to analyse games such as Twentyone and Nim, and even some chess endgames.

## The bigger picture

Model behaviour — A quick introduction to modelling, this article shows how mathematicians model complex systems by describing the most crucial elements in the simplest possible way.

### Mathematical modelling in medicine and nature

Met Office in for another roasting? — How should the Met Office communicate the predictions of its meteorological models?

Creating a virtual cancer — A mathematical cancer model may lead to personalised treatment.

Protecting the nation — Vaccination is an emotive business. To make sure it works, we need to model how diseases spread.

The speed of climate change — Scientists model how fast temperature changes sweep the Earth.

Feeling tense about healing wounds? — Squeamish about cuts and scrapes? Mathematical modelling can help you feel better.

And now, the weather... — Met Office forecasters use giant numerical models of global climate to produce the weather forecasts each day - but the complexity of the simulation means that they can't always be right!

Modelling catastrophes — Hardly six months go by without a natural disaster striking some part of the globe. While it's next to impossible to predict these catastrophes, let alone prevent them, mathematical modelling gives a way to prepare for their impact.

Chaos on the brain — Saying that someone is a chaotic thinker might seem like an insult — but, according to Lewis Dartnell, it could be that the mathematical phenomenon of chaos is a crucial part of what makes our brains work.

When will they blow? — How mathematical models are used to predict volcanic eruptions and earthquakes, and issue volcano alerts that could save thousands of lives.

Quantum geometry — One of the many strange ideas from quantum mechanics is that space isn't continuous but consists of tiny chunks. Ordinary geometry is useless when it comes to dealing with such a space, but algebra makes it possible to come up with a model of spacetime that might do the trick. And it can all be tested by a satellite.
Shahn Majid met up with *Plus* to explain.

Career Interview: biomechanical engineer — Jose Munoz explains how engineering can allow you to explore the unknown, from understanding how mechanical structures bend to investigating the way genes affect the shape of embryos.

Career Interview: fluid mechanics researcher — André Léger studies the fluid mechanics of food travelling through the intestines for consumer goods giant Unilever.

From restaurants to climate change — This article explains how linear models are used to explain diners' tipping behaviour, global warming and genes.

### Mathematical modelling in games, sport and art

Making gold for 2012 — Recently leading researchers in sports technology met at the Royal Academy of Engineering in London to demonstrate just how far their field has come over recent years. The changes they make to athletes' equipment and clothes may only make a tiny difference to their performance, but once they're added up they can mean the difference between gold and silver.

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 modelling is used to build this car.

How to measure a million — You're sweating in the spotlights and squirming in your chair as the whole nation is watching you. One more question. Four possible answers. If you get it right, you're a millionaire. If not, you go home with a mere £32,000. Should you go for it? Or pack up and settle for what you've got?

Rubik success in twenty-six steps — A simple toy, but a fiendishly complicated mathematical model is needed to prove that any scrambled cube can be solved in a maximum of twenty-six steps.

Career Interview: games developer — In the real world, balls bounce and water splashes because of the laws of physics. In computer games, a physics engine ensures the virtual world behaves realistically. Mathematician and computer programmer Nick Gray tells us about playing God in a virtual world.

Career Interview: furniture design — Two designers tell us how they took the long way round to design, and how the maths and science they took in on the way helps them with their work today.

Find out how to use computers to solve a fiendishly difficult jigsaw puzzle on our sister site NRICH.

### Mathematical modelling in technology, economics, politics and human interaction.

Is maths to blame? — Are mathematical models to blame for the financial crisis?

Trust me, I've evolved — Game theorists model the evolution of trust and trustworthiness.

Paying the price — Can a scientific approach to risk in finance avoid the next financial crisis?

Career Interview: defence analyst — Helen is a defence analyst with the MoD, using her maths skills to help defend the nation. *Plus* finds out about her career path.

Call routing in telephone networks — Find out how modern telephone networks use mathematics to make it possible for a person to dial a friend in another country just as easily as if they were in the same street, or to read web pages that are on a computer in another continent.

Model Trains — As customers will tell you, overcrowding is a problem on trains. Fortunately, mathematical modelling techniques can help to analyse the changing demands on services through the day. Tim Gent explains.

I'm not paying that! — It's not that long ago that all you needed to run an airline was a few planes and some competent pilots. But now, with more of us zipping around the globe every year and the advent of no frills airlines, keeping an airline competitive has become a complicated business. Christine Currie explains how your airfare is calculated. Our sister website NRICH has a problem involving another airline conundrum: overbooking.

The crystal ball — If you had a crystal ball that allowed you to see your future, what would you arrange differently about your finances? *Plus* talks to the Government Actuary, Chris Daykin about the pensions crisis, and how actuaries use statistical and modelling techniques to plan for all our futures.

Network news — Why do rich and popular people tend to get richer and more popular, and why do train companies choose to build more and more lines to the biggest cities? A new mathematical theory of networks can be used to simulate real-life social, biological, physical and technical connections.

Career Interview: financial modelling — David Spaughton and Anton Merlushkin work for Credit Suisse First Boston, where they provide traders in the hectic dealing room with software based on complicated mathematical models of the financial markets. *Plus* interviewed them at their offices in Canary Wharf in London.

## Now try it yourself on our sister site NRICH!

Epidemic modelling — Join in this ongoing investigation. Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

Over-booking — Airlines know that any given passenger has a probability of just not turning up at check-in, leaving an empty seat, but they want to maximise their profits by filling as many seats as possible. So they routinely overbook flights. But how should they do this, in order to make the most profit without incurring the wrath of too many disgruntled passengers who have to be turned away from a double-booked seat?

Escalator — At Holborn underground station there is a very long escalator. Two people are in a hurry. . . .

FA Cup — In the four years 2001 to 2004 Arsenal were drawn against Chelsea in the FA cup and beat Chelsea every time. What was the probability of this? Lots of fractions in the calculations!

Where to land? — Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

Impuzzable — One for enthusiastic computer programmers only! This is about a fiendishly difficult jigsaw and how to solve it using a computer program. (Uses BASIC).