M a r / A p r 2 0 1 6 m e i . o r g . u k I s s u e 5 2

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Curriculum Update

GCSE and A level

Statistics

Subject content for GCSE Statistics and AS and A level Statistics for teaching from 2017

has been confirmed.

Have your say

Ofqual is seeking teacher views on how prepared they feel for changes to

qualifications.

Details are on the Association of School and College Leaders

(ASCL) website.

M4 is edited by Sue Owen, MEI’s Marketing Manager.

We’d love your feedback & suggestions!

Mathematical Modelling

From 2017, AS and A level Mathematics and Further Mathematics will have a greater emphasis on modelling, problem solving, reasoning and integration of technology, and statistics will have a new focus on interpretation of data. Math4teaching defines mathematical modelling as “the process of applying mathematics to a real world problem with a view of understanding the latter”, and uses the diagram below to show the key steps in the modelling process.

Howard Emmons, known as “the father of modern fire science”, said that the challenge in mathematical modelling is “...not to produce the most comprehensive descriptive model but to produce the simplest possible model that incorporates the major features of the phenomenon of interest.” In his 2001 paper Teaching Mathematical Modelling in Singapore Schools, Ang Keng Cheng (Associate Dean, National Institute of Education Singapore) describes mathematical modelling as “a process of representing real world problems in mathematical terms in an attempt to find solutions to

the problems.”

In this issue

Curriculum Update

This half term’s focus:

Mathematical Modelling

Climate Change, does it all add

up? Guest writer Chris Budd

OBE explains mathematical

models of weather and climate

Hugh’s Views: Guest writer Hugh

Hunt writes about Modelling and

the Climate

Site-seeing with... Paul

Chillingworth

KS4/5 Teaching Resource:

Modelling in mathematics

Cheng explores different examples of how the process of mathematical modelling may be introduced in the classroom using basic mathematical ideas, and how concepts are presented. He comments that a lack of ready resources and material may create a resistance towards teaching mathematical modelling. He suggests that teachers will need to be more resourceful in lesson preparation, but flags up the opportunities for cross-

curricular collaboration:

“Mathematical modelling also provides an excellent platform for studies and experiments of an inter-disciplinary nature. Problems may arise (and they usually do) from other disciplines. This provides the mathematics teacher with excellent opportunities to collaborate

with other teachers.”

(Click the image to view a larger version)

Earth Hour - what can we save?

“Coming out of a historic COP21*, Earth Hour 2016 will call upon its millions of supporters around the world to shine a light on climate action, to celebrate what we have achieved together and reiterate our collective commitment towards changing climate change. In 2016, coincidentally also the tenth lights out, Earth Hour will roll across the globe at 8:30pm local time on Saturday, 19 March.” (*see Jan/Feb

edition of M4 )

The Earth Hour website explains how to take part: “A simple event can be just turning off all non-essential lights from 8.30pm-9:30 pm. For one hour, focus on your commitment to our planet for the rest of this year. To celebrate, you can have a candle lit dinner, talk to your neighbours, stargaze, go camping, play board games, have a concert, screen an environmental documentary post the hour, create or join a community event - the possibilities are

endless.”

To calculate how much energy you would save for every hour each light bulb in your house is switched off, first check the watt rating printed on it. If the bulb is a 60-watt bulb and it is off for one hour, then you are saving .06 kilowatt hours. Although a single light doesn’t use much electricity (60-100W for a typical old-fashioned bulb), our homes can have dozens of them, so turning off all non-essential lights in a house adds up to quite a lot – around 18% of an average home’s electricity bill. uSwitch’s guide to kWh explores the difference between kWh and kW and gives you an idea of what a kWh actually represents to your

household energy consumption.

Cheng concludes:

“…mathematics is more than just about arithmetic – it is about problem solving. Teaching mathematical modelling involves high-order thinking skills in representation of the real world, as well as skills of problem solving. These are desirable outcomes that as important as getting the ‘right answers’ to ‘problem sums’.”

What if everyone were to switch lights off for an hour - how much energy would be saved? How could

this be calculated?

Earth Hour started in 2007 as a campaign backed by WWF Australia and the Sydney Morning Herald, asking all Sydney corporations, government departments, individuals and families to turn off their lights for one hour from 7:30pm to 8:30pm on March 31, 2007.

The standard Earth Hour '60' logo represents “the 60 minutes of Earth Hour where we focus on the impact we

are having on our planet and take positive action to address the

environmental issues we face”.

Earth Day Network

Earth Day Network is

a movement that

works with “tens of

thousands of partners

across 192 countries”

throughout the year to

defend the health of

our planet.

“Changing the world

starts by changing

your own little

corner of it.”

Earth Day 2016 takes

place on 22 April.

April 16-23 is

designated as

Climate Education

Week. The Climate

Education Toolkit for

K-12 (primary-

secondary) students

around the globe

includes a week’s

worth of cross-

curricular lesson

plans, activities and

contests. The CLEAN

Collection provides

other scientifically and

pedagogically

reviewed resources

on Climate Change.

Sydney Harbour Bridge and Sydney Opera

House during Earth hour 2007. By madradish

(Flickr) [CC BY-SA 2.0], via Wikimedia Commons.

Energy Modelling

money off people’s electricity bills, if you wanted to calculate how much money you would be saving by turning a light off for an hour, find your annual energy statement or look up current gas and electricity prices on the UK Power website, find out how much you are charged per kilowatt hour, and then multiply the price by the amount of kilowatt hours. For example, if your electricity rate is 10 pence per kilowatt hour (kWh unit price), then you are saving 0.6 pence for every hour that

one light bulb is turned off.

It isn’t quite as straightforward as that to calculate the total amount of money that could be saved by turning off lights through the world for an hour. Just as prices vary regionally in the UK, every country charges differently for electricity. OVO Energy has published a 2011 graph comparing average electricity prices around the world, as well as a graph showing the relative prices of electricity, taking into account the purchasing power of different currencies. Businesses are charged under different rates to domestic users – the Business Electricity Prices website details

typical kWh rates for UK businesses.

The savings by turning off lights for an hour might not appear very much, but the accumulative effects of turning off lights when not in use and employing other ways to ensure your home is as energy efficient as it can be will be significant, not only in financial terms for the homeowner, but also in terms of energy savings. For example, use energy efficient light bulbs, light sensors, avoid leaving appliances on standby, use smart heating controls, install energy efficient windows, insulate the home, use renewable energy, save water.

A 2014 Energy Research and Social Science study, ‘The electricity impacts of Earth Hour: An international comparative analysis of energy-saving behavior’, published by Science Direct, “compiled 274 measurements of observed changes in electricity demand caused by Earth Hour events in 10 countries, spanning six years. These events reduced electricity consumption an average of 4%, with a range of +2% (New Zealand) to −28% (Canada). While the goal of Earth Hour is not to achieve measurable electricity savings, the collective events illustrate how purposeful behavior can quantitatively affect regional

electricity demand.”

“With a new global climate change deal agreed at the end of last year, it’s never been more important to keep the momentum up,” it states on

the WWF Earth Hour website.

The main aim of Earth Hour is not to reduce energy or carbon during an hour

-long period, but rather:

“Earth Hour is an initiative to encourage individuals, businesses and governments around the world to take accountability for their ecological footprint and engage in dialogue and resource exchange that provides real solutions to our environmental challenges. Participation in Earth Hour symbolises a commitment to change

beyond the hour.”

How much money

could be saved?

Although Earth Hour isn’t intended as an exercise to save

Mathematical

problem solving

and modelling in

the new A levels

To support the

development of new

qualifications and

assessments

reflecting the new A

level content, Ofqual

convened an A level

mathematics working

group in March 2015

to provide expert

advice in the areas of

mathematical problem

solving, modelling

and the use of large

data sets in statistics.

The group produced a

report in December

2015, in which they

state that:

“Mathematical

problem solving is not

just for the highest-

achieving candidates:

it is a core part of

mathematics that can

and should be

accessible to the full

range of candidates.”

The group also

emphasised that

problem solving tasks

“must not become

formulaic or

predictable over time,

nor be reduced to a

learnt routine.”

Uses of Mathematical Modelling

MEI Conference 2016 sessions based on mathematical modelling See below and following pages for sessions relevant to this edition’s theme.

Modelling

population growth

Kevin Lord will model

and investigate

population growth

using matrices. This

will be a mainly

practical session

working through the

problems using

technology to help

with the investigation.

Statistical and

Financial Modelling

Core Maths is about

answering real world

questions using

maths. The modelling

cycle describes how

that happens. Keith

Proffitt will look at

some real world

statistical and

financial questions

and how teachers

and students might

answer them together

in the classroom. Is

the cost of going to

university worth it?

Other uses of mathematical

modelling

Financial modelling

Banks, insurers, and

other financial

institutions, as well

as organisations in

many other industries, increasingly use

mathematical models in the day-to-day

operations of their business to value

assets, liabilities and capital

requirements. These complex models

are used to evaluate capital and other

resource allocations, business

strategies, capital expenditure projects

and more, and to consider financing

options.

Stochastic modelling

In order to be solvent, a company has

to show that its assets exceeds its

liabilities, but in the insurance industry

assets and liabilities are unknown

quantities. They have to be estimated

using projections of what is expected to

happen. However, as Paul Fisher,

Deputy Head of the Prudential

Regulation Authority and Executive

Director, Insurance Supervision,

pointed out at the Westminster

Business Forum conference in

December 2015:

“...insurers play an important risk-

transfer role. In some instances,

individuals rely absolutely for their

future lifetime incomes on the continuity

of their insurance cover. To achieve an

appropriate level of policyholder

protection, insurers must be, and be

seen to be, safe and sound.”

Click the image to find out more about energy efficient lighting on the Energy Saving Trust’s

website.

Energy modelling: SAP

SAP (Standard Assessment Procedure) is the UK Government’s

recommended method for estimating the energy performance of residential dwellings: “SAP quantifies a dwelling’s performance in terms of: energy use per unit floor area, a fuel-cost-based energy efficiency rating (the SAP Rating) and emissions of CO2 (the Environmental Impact Rating). These indicators of performance are based on estimates of annual energy consumption for the provision of space heating, domestic hot water, lighting and ventilation.”

The SAP methodology is used to produce Energy Performance Certificates and is used in a number of other

government programmes to estimate the amount of energy typically used in a home. It is also the same methodology that the Energy Saving Trust uses to calculate the majority of their savings figures for upgrading insulation, draft proofing, glazing and heating systems

(including renewable space heating).

Uses of Mathematical Modelling

MEI Conference 2016 sessions based on mathematical modelling More sessions relevant to this

edition’s theme.

Resources and

Investigations: The

Normal Distribution

and Probability

Plots

Kate Richards will use

real data in context to

introduce the Normal

Distribution and show

how using this as a

statistical model can

be useful to solve

problems.

Statistical Modelling

in Football

Alun Owen will

consider the data

sources freely

available for many

sports, and describe

an application of

statistical modelling in

football that can be

used to illustrate how

computer games,

such as Football

Manager, rely on and

use statistical models

to ensure realism

during game play.

Stochastic modelling is used in the

insurance industry (‘stochastic’ meaning

having a random variable). This model

is used for estimating probability

distributions of potential outcomes by

allowing for random variation in one or

more inputs over time, also allowing for

volatility. For example, forthcoming

changes to the UK taxation system of

the ‘at retirement’ market will have a

significant effect on some insurers’

business models, warns Fisher.

“Changes to business models will

naturally lead to changes in risk

exposures which will need to be

carefully considered and managed.”

The mathematics of queues

Queuing theory is a branch of

mathematics that studies and models

the act of waiting in queues.

It could be argued that intuition and

observation could be used to assess

which queue to join in a supermarket,

but it is necessary to predict queue

lengths and waiting times when making

business decisions about the resources

needed to provide the service for which

people are likely to queue.

The 6 minute video Queuing Models:

The Mathematics of Waiting Lines

provides a visual exploration of the

mathematics of waiting lines (queues).

In his Wolfram Blog post The

Mathematics of Queues, Devendra

Kapadia explains that it’s about more

than people waiting in line for a service:

“At a more abstract level, these waiting

lines, or queues, are also encountered

in computer and communication

systems.”

And for

people waiting

in one queue,

this will often

form part of a

chain or network of different queue.

“For example, passengers arriving at a

major airport will have to make their

way through a complicated network of

queues for checking in luggage,

security scans, and boarding flights to

different destinations. Thus, the study of

queueing networks...is of great

importance in applications.”

In his blog post Kapadia

describes a notation,

devised by mathematician

and statistician D. G.

Kendall, of the mathematical description

of queues. He explains the practical

applications of queuing theory along

with the mathematical models behind it.

For example, how to reduce bottlenecks

in traffic queues, and how to improve

the customer

experience when

phoning a busy call

centre.

A video of Kapadia’s presentation at the

Wolfram Technology Conference about

Queueing Networks provides an

introduction to queueing theory and

discusses the simulation and

performance analysis of single queues

and open or closed queueing networks.

Managing traffic queues

Many sat navs

can now receive

live traffic data

about congestion

on your route

and reroute you onto a less congested

and faster route. Traffic information

(including data about the location and

speed of movement of mobile phones

on certain networks, and traffic news

reports) is processed by the sat nav

companies’ computers and then

overlaid onto road maps. It is then

possible to work out which roads are

congested and which roads are not, by

comparing the information with the

same data taken at a different time.

Managed motorways with variable

speed limits have been in action in the

UK for some time now, designed to

slow traffic and help control the traffic

flow. Smart Motorways use sensors to

determine traffic flow and automatically

set the speed restrictions.

Reducing customer waiting times

Systems such as

QLess have been

developed for

businesses to reduce

waiting times and in so

doing, to transform the

customer experience by

“providing on-demand

status updates any time

a customer calls or

texts the system. These updates

include forecast wait times, and the

number of other customers ahead of

them in line. QLess also walks

customers through the initial process of

getting in line, ensuring that they’re

waiting in the right line to begin with.”

Phone apps have been

developed to help

consumers to avoid

queueing in shops and

bars. For example Q App was launched

in 2014 and acquired by Yoyo Wallet

in 2016 to be merged into ‘Jump’, with

some similar solutions having appeared

on the market, for example, HANGRY,

Starbucks, Westfield Dine on Time.

Customers can order and pay for food

and drink in busy bars and cafes using

an app downloaded onto their

smartphones. The developer of Q App,

Serge Taborin, says “there is a bigger

problem to be solved [the removal of

queuing] rather than simply replacing

the credit card with the mobile phone.”

Another Danish researcher, Kebin Zeng

created models to be used in

developing phone apps to tell users the

ideal time to go shopping if they want to

avoid long queues.

Uses of Mathematical Modelling

MEI Conference 2016 sessions based on mathematical modelling Another session relevant to this

edition’s theme.

Get set for

September 2017:

Mechanics and

modelling

From September

2017 all students

starting A level

Mathematics will learn

both Statistics and

Mechanics. This

applied content will

account for a third of

the qualification.

Simon Clay and

Sharon Tripconey will

explore the features

of the Mechanics

content including

changes in subject

content compared to

current M1

specifications, the

increased emphasis

on mathematical

modelling and the

connections to other

topics which can be

made while teaching

a linear course. Click image to view larger version.

Climate Change, does it all add up?

Chris Budd OBE,

Professor of Applied

Mathematics at the

University of Bath,

Professor of

Mathematics at the

Royal Institution, and

vice-president of the

Institute of

Mathematics and its

Applications, has

worked for the past

ten years in numerical

weather prediction

and data assimilation

in close collaboration

with the Met Office.

Chris also works on

climate modelling

using modern

mathematical and

computational

methods.

Chris is also co-

director of the

EPSRC/LWEC

CliMathNet network.

We are grateful to

Chris for writing this

article for M4

magazine.

Climate change, and its effects on our

weather, is important, controversial and

is possibly going to affect all of our

lives, especially those of school

students over the next fifty years. The

understanding and analysis of climate

change is an area where

mathematicians, scientists, policy

makers (such as members of

parliament) and anyone involved in

health, insurance or agriculture, meet to

try to interpret the current evidence and

to make predictions for the future. It is

truly an area where mathematics is

making a very big difference to the way

that people think about the future. But

why should we bother? Predictions

made by the Intergovernmental

Panel for Climate Change (IPCC) that

we might have a 3 degree rise in mean

temperature over the next 50 years or

so don’t on the face of it seem very

scary. However, we are seeing a lot of

extreme weather events at the moment,

such as the extensive flooding just

before Christmas. These cost billions of

pounds and can lead to great hardship

and even loss of life for those affected.

If these extreme events are just what

you would expect from random weather

variations then we can just about live

with them. If instead they are part of a

series of events due to climate change,

then we need to be worried indeed. It is

the job of the mathematician to help to

sort this out.

Evidence for climate change

There are at least five indicators that

make us think that climate change is

occurring. The first of these is the rise in

the Earth’s temperature.

In Figure 1 we show the measured

temperature (relative to a reference

temperature) over the last 150 years. In

this figure you can see that the average

temperature is showing a steady rise

over this time, with 2015 looking like

being the warmest year on record. On

top of this rise is what appears to be a

random variation. This variation causes

a lot of discussion in the climate

science debate. The second indicator is

the loss of the Arctic Sea ice. It is an

undisputable fact that the amount of this

ice has been decreasing dramatically in

recent years, with an annual loss in

area of about the size of Scotland.

In Figure 2 (next page) we show the

measured values. If you fit a straight

line to this data using the statistical

methods taught in A level maths, then

the prediction is that all of the Arctic Ice

will have vanished by the end of this

century. (Interestingly the amount of

Antarctic Sea ice is currently increasing

slowly, again leading to many

discussions. However the evidence is

that the Antarctic Land ice is also

decreasing).

Figure 1: The changes in temperature over the

last 150 years. Note the overall upwards trend

with a random variability on top.

long term trends can be hard to

determine. Secondly, the equations for

the climate are ’nonlinear’. This means

that they have the potential of having

solutions which are ’chaotic’, showing a

lot of variability and therefore are hard

to predict.

We can see this in the weather, which

is basically impossible to predict with

any accuracy much more than a week

into the future. It is often argued that if

we can’t predict the weather what hope

have we of predicting climate 100 years

ahead. This is not really the case as

climate is much more about finding

general trends rather than day to day

variations (climate is what you expect

and weather is what you get). Thirdly,

the climate is genuinely very complex

indeed, involving the atmosphere, the

oceans, the sun, vegetation and ice, not

to mention human activity. Whilst this

does not prevent us from predicting it

(indeed the weather is also very

complex involving over 109 different

variables, but we can still predict

tomorrow’s weather with a good

accuracy), it does make the job much

harder.

Finally, a very real problem in climate

science is distinguishing between cause

and effect (for example does a rise in

Carbon Dioxide cause a rise in

temperature or is it the other way

round), and distinguishing between

human made change and natural

variations.

The three other main indicators are: the

increase in mean sea level over the last

100 years, the increase in the number

of extreme rainfall events, and the year

on year rise in the level of Carbon

Dioxide in the atmosphere, with

measured values now above 400 parts

per million (twice the level before the

industrial revolution). Later on I will

show the (mathematical) link between

Carbon Dioxide levels and temperature,

which is a cause of concern.

The climate change debate

Most scientists (and this includes

mathematicians) believe that climate

change is occurring, but this is certainly

not a universally held opinion. One

reason for this is that predicting the

climate is genuinely hard. As the great

scientist Niels Bohr famously remarked,

“It is difficult to predict anything,

especially about the future.” There are a

number of (mathematical) reasons why

this is the case for climate. Firstly, as

we have seen from the temperature

measurements, there is a lot of

statistical variation and uncertainty in

the data that is being measured, so

Climate Change, does it all add up?

About CliMathNet

“CliMathNet is a

network which aims to

bring together Climate

Scientists,

Mathematicians and

Statisticians to

answer the key

questions around

Climate modelling (in

particular

understanding and

reducing uncertainties

in observation and

prediction). This is an

area of science that

ranges from

numerical weather

prediction to the

science underpinning

the Intergovernmental

Panel for Climate

Change (IPCC).”

The website links to

its own set of teaching

resources

mathMETics, as well

as to resources on

external sites,

including an earlier

edition of MEI’s

Monthly Maths.

The Fourth Annual

CliMathNet

Conference will be

held at the University

of Exeter, from 5th -

8th July 2016.

Figure 2: The changes in the Arctic Sea ice cover

over the last 35 years, showing a decline. A best fit

line is added to allow us to predict future trends.

Mathematical models of weather and climate:

the full and glory details Mathematicians can help a great deal to clarify the issues in this debate, using and extending the mathematics taught at A level. Firstly, they can look at past variations in climate (such as the sequence of ice ages in the last

million years) and find mathematical models which explain these. Then they can use these (and other) models, combined with a lot of statistics and probability to make sense of the data that we are currently measuring about the weather and climate, so that we can distinguish between cause and effect. Finally, they can combine all of this knowledge to produce models which can predict what the climate might do in the next 100 years or so. These results are used to inform policy makers such as the IPCC. It is very important to say that these models, and the data which informs them, are far from perfect. A vital part of all of this analysis is identifying and then quantifying the level of uncertainty in all of these predictions. In short, never trust any prediction unless you can estimate how uncertain it is!

Mathematicians around the world are

heavily involved with constructing,

studying and solving, models for the

future climate. Many of these work in

climate centres, such as the Hadley

Centre which is part of the Met Office in

Exeter, UK. The basis of all of these

models are mathematical equations.

These take Newton’s laws of motion

(which are covered in A level) applied

to the pressure and movement of the air

and the oceans, combined with the laws

of Thermodynamics, which were

discovered by Lord Kelvin and which

describe how heat is transported around

and how water is turned into vapour and

then back into rain.

Many other great mathematicians have

contributed to these equations including

Euler, Navier and Stokes who

discovered the laws of fluid motion,

which are also used to predict the

weather and even to design aircraft. We

also need to add in the effects of the

rotation of the Earth (called the Coriolis

terms), and for climate predictions need

to include the effects of ice, Carbon

Dioxide (and other greenhouse gases),

vegetation, volcanoes, solar variation

and (as best as we can predict), human

activity. The result is a set of partial

differential equations which explain how

the various quantities involved in the

weather and climate, change in time t

and in space x. Here are the splendid

partial differential equations for the

weather (brace yourselves).

Here u is the velocity of the air, T is its

temperature, p its pressure, ρ its

density, q its moisture content and Sh is

the solar heating.

To simulate the climate starts with

these and adds more equations for all

of the other effects. These partial

differential equations are too hard to

solve by hand, and instead we find

approximate solutions on a (super-)

computer. To do this the computer then

has to solve billions of different

problems, as well as incorporating as

much data about the system as

possible, such as measurements of the

air and ocean temperatures and

velocities.

It is remarkable that we can solve these

equations at all, given their complexity,

but this is done every six hours when

forecasting the weather. And weather

forecasting (at least for the next few

days) is now pretty accurate.

Met Office supercomputer

higher when the Earth is covered in

ice). The heat energy radiated back into

space is given by

σeT 4

where σ = 5.67 * 10^{-8} is Boltzman’s

constant, and e is the emissivity, which

is a measure of how transparent the

atmosphere is. On the moon, with

almost no atmosphere, we have e = 1.

Currently on the Earth we have e =0.55.

To find the Earth’s temperature we

balance these two expressions so that

σeT 4 = (1 − a)S,

and then we solve this for T to give

which you can evaluate on a calculator.

Isn’t that nice! Try it with the values

above to find the current mean

temperature of the Earth. Now take e =

1 to find the temperature of the Moon.

The power of this expression is that we

can perform what if experiments to see

what can happen to the climate in the

future. For example, if the ice melts

then the albedo a decreases, which

means that (1 − a) and hence T

increases. Similarly if the emissivity e

decreases then the temperature T

increases. This is a worrying prediction

as it is well known that increasing the

amount of greenhouse gases, such as

Carbon Dioxide, in the atmosphere

leads to a decrease in e. Thus there is a

direct cause and effect link between an

increase in Carbon Dioxide (which is of

course what we are seeing) and a rise

in the predicted mean temperature of

the Earth.

Weather forecasting is basically an

honest process, in that every day you

are confronted with the results of your

calculations and compare whether you

predicted the weather correctly or not.

You then get into trouble if you get it

wrong!

Forecasting the climate using these

complex models is less easy to test as

we can’t test our predictions for the next

100 years against what will happen

then. What is usually done is to

compare the predictions of past climate

with what is observed. This is a useful

check but is far from perfect as a

means of testing the climate models as

their sheer complexity makes it hard to

run lots of simulations over long times,

which is necessary for a realistic test.

One way to make progress is to look at

much simpler models which incorporate

significant features of climate and which

can be more easily tested. One of the

most useful of these, called the Energy

Balance Model uses ideas from A level

mathematics. In this we assume that

the Earth is heated by the radiation

from the Sun and that it has an average

(absolute) temperature T . Some of this

heat energy is absorbed and the rest is

radiated back into space. We then

reach equilibrium when these two

balance. Now the heat energy from the

Sun is given by

(1 − a)S

where S is the incoming power from the

Sun (which is around 342Wm−2 on

average, and a is the albedo of the

Earth which measures how much of this

energy is reflected back. The current

value of a = 0.32 . (The albedo is

Simpler models of the climate

CliMathNet has

developed a set of

teaching resources

under the

name mathMETics.

“MathMETics allows

pupils to gain an

insight into how

mathematics is

applied to understand

the climate and

predict the

weather. Pupils can

have a go at

collecting weather

data and running a

climate model for

themselves and think

about how the

information can be

used.

The mathMETics

website provides

resources developed

by a team of

mathematicians at the

Universities of Exeter

and Bath in

collaboration with the

Met Office. The

resources explain

how to collect and

record data, how to

verify collected data

against Met Office

forecasts and gives

an insight into the use

of mathematics and

statistics in weather

and climate

forecasting.”

Conclusions: What can a

mathematician do next?

There are many ways that a

mathematician can help in the climate

debate, from making and analysing

climate models, from better

understanding of data, and to a more

informed presentation to policy makers

of the nature of the issues involved.

But, the moral of this article, is that you

should always use your mathematical

judgement to test whatever is said, in

the media and otherwise, about

weather and climate.

In Figure 3 we show the predictions of

the future mean temperature from

various climate centres around the

world. These are made using the

sophisticated climate models described

earlier, but give the same predictions as

the simpler model on the previous

page.

Note that these predictions are not all

the same. This is because the models

make different assumptions about the

level of Carbon Dioxide and other

factors. However, they are all predicting

a significant temperature rise by the

end of the 21st Century.

Figure 3: Predictions of future temperatures from various climate centres around the world.

Simpler models of the climate

Mathematics of Planet Earth (MPE) was launched by a group of mathematical sciences research institutes “to promote awareness of the ways in which the mathematical sciences are used in modelling the earth and its systems both natural and manmade. MPE aims to increase the contributions of the mathematical sciences community to protecting our planet by: strengthening connections with other disciplines; involving a broader community of mathematical scientists in related applications; and educating students and the general population about the relevance of the mathematical sciences. MPE’s mission is to increase engagement of mathematical scientists researchers, teachers, and students in issues affecting the earth and its future.”

In 1943 Barnes Wallis worked out how to make bouncing bombs to blow up dams. He did experiments so that he could have confidence in his models. He had to scale up from dams that were a few metres tall to one that was 30 metres tall. He worked out that the explosive energy needed is proportional to the fourth power of the height of the dam. Why “fourth power”? Well, look at the units. Energy E is in Joules which is kg m2 s-2, height H is in metres, density

(of water and concrete) are kg m-3 and

gravity g = 9.81m s-2

The way a dam stays up is by gravity – blocks one on another held in place by their weight. If we assume that the strength of cement and mortar doesn’t matter then the only way an equation can be assembled based on gravity alone, one that has the right units, is

that E/( g H4) must be dimensionless.

Mathematical models can be pretty simple. It’s 180 miles to Newcastle and I’m averaging 60mph on the A1(M). For constant velocity u the distance s = u * t, so I’ll be with you in about 3 hours. Curiously, this is rocket science. The Apollo missions and more recently the gorgeous Philae Lander and the exciting New Horizon probe all used

Newton’s mathematical laws of motion.

It will be the 50th anniversary of the Apollo 11 moon landing in 2019 and the maths hasn’t changed at all. What is different is that the mathematicians back then had to do everything. They programmed up their own code and they took responsibility for their mistakes and pride in their successes. To be the mathematician at the end of the phone when Neil Armstrong was having problems landing on the moon

must have been quite a thrill.

The thrill hasn’t gone – no way! We get a buzz out of making things work not by accident, not by trial and error but by

mathematical modelling.

Not long ago I was asked to recreate the classic world-war II Dambusters

mission.

Dr Hugh Hunt is a

Senior Lecturer in the

Department of

Engineering at

Cambridge University

and a Fellow of Trinity

College.

In the previous edition

of M4 magazine, Hugh

wrote about problem

solving with CO2

emissions. We

received positive

feedback from

teachers regarding

this article; it’s good

news to know that our

writers are inspiring

you with their ideas for

the classroom! M4 is

very grateful to our

guest writers for their

contributions and we

welcome feedback

(see page 1 for how to

contact the editor)

Just a reminder that

the COP21 website

has 10 videos to help

viewers understand

climate change.

View Hugh’s

videos on

his YouTube channel:

spinfun

Follow Hugh on

Twitter:

@hughhunt

Hugh’s Views

Modelling and the Climate

By NASA (The Project Apollo Image

Gallery) Public domain], via Wikimedia

Commons

By NASA (The Project Apollo Image

Gallery) Public domain], via Wikimedia

Commons

expand our horizons when it comes to

phasing out fossil fuels.

How far can we go? We are seeing now that the Arctic is warming rapidly and that the Greenland ice shelf is melting. As the permafrost thaws it will probably release a great deal of methane, which is a far more potent greenhouse gas than CO2. How much trouble are we in? Sea-level rise, floods, drought, crop failure, storm surges – well, we just don’t know. Ought we to err on the side of caution? Barnes Wallis might be interested in the new field of ‘geoengineering’ – man-made intervention into the climate system. Can we refreeze the arctic? Complete madness, maybe, but so was the

bouncing bomb.

Mathematical models don’t need to be complicated, at least not at first. We know for instance that the eruption of Mt Pinatubo in 1991 caused a global cooling of about 0.5oC for a year or so. Perhaps we can simulate a volcanic eruption? Indeed climate scientists have put a great deal of effort into studying Pinatubo and it seems that a modest injection 300kg per second of sulphate aerosol into the stratosphere at a height of 20km would cause a global cooling of 2oC. Would we ever dare do this? Would we even dare trust ourselves to do experiments that might lead us in this direction? In fact we know the answer to this because some very benign experiments have already been cancelled. We have to rely then on computer models. Are they

accurate? Are they reliable?

We know the answer to that – Philae, New Horizon, Apollo – none would have succeeded without mathematical models. And who wrote them?

Mathematicians, of course.

That is where the fourth power comes from. Magnificent! So to blow up the Möhne dam at 30 metres high needs 81 times as much explosive as a 10 metre dam. The dam we built was about 10m high so we knew how much explosive to use. It worked perfectly!

What might Barnes Wallis be putting his mind to now, in 2016? My guess is he’d be very concerned about climate change and he’d be wondering if we could engineer our way out of trouble. He’d need to make good use of models. For instance, if we know that the sun delivers 1200 watts of energy per square metre to the Earth’s surface then we can work out if there is any way that energy from the sun can be a substitute for our dependence on fossil

fuels.

Globally we consume something like 500 billion gigajoules of energy per year – which sounds a lot but is actually a tiny fraction (around 0.01%) of the total power the Earth gets from the sun. As mathematicians, scientists, engineers we ought to be thinking imaginatively, using the full might of our models to

Hugh’s Views

In 2011 Dr Hugh Hunt and Windfall Films won The Royal Television Society (RTS) Programme Award for best history programme for their documentary, Dambusters: Building The Bouncing Bomb, screened in the UK on Channel 4 2 May 2011. The Windfall film (length 1:33:42) is available to view on Hugh’s web page. Other media coverage about the mission and film can be access from Hugh’s web page.

Images of Hugh’s recreation of the ‘bouncing bomb’ have been used in this article, with Hugh’s permission, from his web page: Dambusters: Building the Bouncing Bomb. You can view more videos and still photos on this page.

Modelling and the Climate

Click image to view video

Keele University’s Mathematics Department

developed a new first year module on problem solving and mathematical modelling, which aimed to develop these skills and use innovative methods that allow students to express their creativity. The materials from this project (which include Group Round questions from the United Kingdom

Mathematics Trust’s (UKMT) Senior Team Mathematics Challenge (STMC)) will be available as part of the HE STEM program and can be freely used at other institutions. In their report Problem Solving and Mathematical Modelling: Applicable Mathematics, Dr Martyn Parker and Dr David Bedford (Department of Mathematics, Keele University), ask how students can best develop their problem solving skills during their mathematics education, with particular relevance to students in transition from school to higher education and employment. They explored the issue of school students preferring the ‘safety’ of exam-style questions with a familiar format, but finding it challenging to solve unstructured problems posed within the world of work or university. The authors explain: “We sought to address these issues by introducing problem solving with contextual problems, then progressing on to problems that require a qualitative rather than quantitative analysis, before finally developing the students’

modelling skills.”

(David Bedford will be delivering a session at the MEI Conference 2016

on Transcendental Numbers.)

Mathematical modelling and problem solving

For the HE STEM project (see opposite), thirteen STEM departments across eight universities (Leeds, Manchester, Keele, West of England, Loughborough, Swansea, Portsmouth and Bradford) collaborated to ensure students possess the skills to develop mathematical models, apply mathematics, and find solutions to real problems. The project aim was “to provide mathematics, physics and engineering undergraduates with the skills and abilities to develop mathematical models and apply mathematics to analyse and solve problems in science and engineering.” A poster was produced for a regional dissemination event in April 2012, providing a summary of the activities to date to enhance the mathematical modelling and problem solving skills of undergraduate STEM students. Four Universities (the original Project Partners) are engaged in outreach work with local schools and colleges, helping sixth formers studying A Level mathematics to develop their mathematical modelling and problem solving skills and helping them and their teachers to understand the importance

of these skills in STEM degree courses.

This work is facilitated through strong links with MEI (see pp 13-19 of Integrating Mathematical Problem Solving Applying Mathematics and Statistics across the curriculum at level 3 End of Project Report) and the Further Mathematics Support

Programme.

Higher level skills for HE STEM students

Led by Professor

Mike Savage

(University of Leeds),

a project: Higher level

skills for HE STEM

students:

mathematical

modelling and

problem solving was

set up by through the

National HE STEM

Programme, initially

with four project

partners: Leeds,

Manchester, Keele

and West of England.

This interim 2011

report describes why

problem solving using

mathematics

suddenly emerged as

a problem in Higher

Education. The

approach taken by the

four universities to

address this problem

involved “introducing

the two modelling

skills ‘setting up a

model’ and ‘multi-

stage modelling’ into

the university

curriculum for those

STEM

undergraduates who

need them, in a way

that is most suitable

to their needs.”

Site seeing with… Paul Chillingworth

Paul is a Central Coordinator for the Further Mathematics Support Programme, which is managed by MEI. Paul coordinates the Senior Team Maths Challenge and the Live Online Tuition programme. He liaises with HEIs over entry requirements and is responsible for the development of resources to help prepare students for STEM degrees. After graduating in mathematics at Cambridge University, Paul qualified as a teacher. He has had experience in a variety of school based roles including coordinating professional development and curriculum leadership. He was Deputy Head of an international school for 8 years and has considerable experience of different mathematics curricula worldwide.

Modelling

The assessment objectives for GCSE

require students to translate problems

in non-mathematical contexts into a

process or a series of mathematical

processes and to evaluate solutions to

identify how they may have been

affected by assumptions made. At A

Level, in Mechanics and Statistics, we

have long made use of the modelling

cycle:

Modelling is an excellent way to show

the usefulness and power of

mathematics, to promote interest and to

learn new concepts or apply those

already learnt, so it would be good to

provide more opportunities for students

to undertake this throughout their

mathematics education.

Nrich has a

collection of

both short and

longer modelling

problems.

One particular favourite of mine is the

‘Where to Land’ Problem. Chris is

swimming in a lake, 50m from the

shore. Her family are 100m along the

shore. She'd like to get back to her

family as quickly as possible.

If she can swim

at 3 m/sec and

run at 7m/sec,

how far along

the shore

should she land

in order to get back as quickly as

possible?

Some of the shorter problems might

make good starter activities.

The Bowland Maths Project provides

some modelling tasks originally

designed to help assess pupils'

progression against the Key Processes

defined in the Key Stage 3 National

Curriculum. These tasks provide some

rich ideas for problems that allow

students to improve their reasoning

skills.

The first task ‘110

Years On’ shows the

picture of a girl taken

110 years ago.

Now, 110 years later,

all this girl’s

descendants are

meeting for a family

party. How many descendants would

you expect there to be altogether?

A key part of the modelling process is

the discussion of the assumptions

made which might include birth rates,

average age of giving birth and at what

age people die. These factors will have

changed over time!

Modelling in Mathematics What do we mean by ‘Modelling’?

Using mathematics to represent something in the

real world to make something simpler to work with,

or so that we understand it better, draw some kind of

conclusion or make a prediction.

Modelling can be quite simple or very complex.

Often, we have to make assumptions about

something, or make an educated guess about

missing information.

Modelling in Mathematics Example: simple

When we think about the Earth in

Mathematics and Science, we

model it as a sphere.

It’s not really a sphere, it’s

slightly flattened at the top and

bottom and there are lots of

lumps and bumps on its surface

– but for most purposes, a

sphere is close enough.

Modelling in Mathematics Example: complex

Weather forecasters use complex models of

weather systems to predict what is likely to happen

in the next few days.

Challenge 1 By the age of 15, what percentage of their life has a

person spent at school?

Challenge 1: assumptions What assumptions did you make?

You will probably have had to make assumptions

about some or all of the following:

• Regular attendance

• Length of school day

• Age to start school

• Number of weeks of the year spent at school

• Number of days in a school week

Challenge 2 Four minute mile

In the late 19th and early 20th Centuries, people

speculated whether or not it would be possible for a

man to run a mile in under 4 minutes.

If your school has a running track, that would be four

times round the track in 4 minutes.

Look at the data on the next slide and predict if and

when it might have happened.

Challenge 2 Four minute mile data* (in seconds)

Year Time 1893 257.8

1861 286 1895 255.6

1862 273 1911 255.4

1868 268.8 1913 254.4

1873 268.6 1915 252.6

1874 266 1923 250.4

1875 264.25 1931 249.2

1880 263.2 1933 247.6

1882 259.4 1934 246.8

1884 258.4 1937 246.4

*IAAF data from 1913, amateur data pre-1913; only final record of any one year cited.

Challenge 2: Graph

240

245

250

255

260

265

270

275

280

285

290

1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980

Tim

e (

s)

Year

Men’s world record in seconds

Challenge 2 Four minute mile

It actually happened in 1954. How close were you?

Did you envisage a straight line or something else?

Why?

Bringing this up to date, on the next slide is a graph

of the World Records since 1861.

Do you think a man will ever run a mile in under 3

minutes?

Challenge 2: Graph 2

180

200

220

240

260

280

300

1860 1880 1900 1920 1940 1960 1980 2000 2020

Tim

e (

s)

Year

Men's world record in seconds

The Modelling Cycle This is a model of how we model in mathematics.

There are many different versions in existence, the

one on the next slide is a mixture of two classroom

ones.

It just captures what it is we’re doing when we’re

modelling.

You might find it helpful if you get stuck or are not

quite sure what to do next.

(A version of) the Modelling Cycle

Choose a challenge On the next slide are some challenges.

Remember, you may have to make a sensible

estimate of something – don’t look it up!

Information:

There are approximately 65 000 000 people living in

the UK, about 55 000 000 of whom live in England.

Choose one of the following problems to work on with

a partner. Show your working to communicate your

solution to others.

• How long is a line of a million dots?

• How heavy is the food that a person eats in a lifetime?

• How many pets are there in the UK?

• How many people are there on the Isle of Wight? (or

your county). You may use a map of England for this

one.

Choose a challenge

You may have to think more carefully about the

information you will need to have or to estimate for

these problems.

Some information is given which may be helpful, if

you need it, but you might like to estimate it first.

Longer challenges

Imagine working at a large theme park

To help customers plan their time, information

boards need to be placed to let them know how

long they can expect to wait for a rollercoaster.

Problem

Where should

you place signs to

indicate a waiting

time of 30 minutes?

Theme Park Queue

Millennium Force Ride at Cedar Point in Ohio

Height 310ft

Drop 300ft

Length 6595ft

Max Speed 93 mph

Duration 2.00 minutes

3 trains with 9 cars (riders are arranged 2

across in 2 rows per car)

Train leaves loading station every 1 minute

40 seconds

Theme Park Queue: Facts and Figures

Charity marathons usually have a mass start.

Several thousand runners assemble behind the

start line.

Problem

How long would it take for all the competitors of a

marathon to cross the start line?

Starting a Marathon

The London Marathon Number of entrants: 40 000 Starting points: 3 Width of start lines: 10 to 20 metres

Wheelchair and paralympic athletes set off about

an hour ahead of most of the rest of the field.

Elite athletes lead the way at the main start time.

Starting a Marathon: Facts and Figures

Each year Earth day raises awareness of

environmental issues.

Some things to work out:

• How much water do you use each day?

• Putting a brick in a toilet’s cistern saves 1.5

litres per flush. How much water would your

school save a year if there was a brick in each

cistern?

Earth Day: April 22nd

Teacher notes: Modelling in Mathematics

This issue looks at modelling in mathematics. Using real contexts can

often act as a motivator for young people and help them to understand

how mathematics is useful in a range of situations.

With modelling, the emphasis is on the processes, reasoning and

justification students give rather than on the answer, however, some

guides to answers have been given as knowing the right magnitude of

an answer is often helpful in the classroom.

Teacher notes: Modelling in Mathematics

» Students should have the opportunity to discuss this

with a partner or in a small group

» Students should sketch or calculate (as appropriate)

Challenge 1 What percentage of a person’s life is spent at school by age 15?

Assumptions:

• Start school at exactly age 5

• Attend every day

• 39 school weeks a year

• School day from 8:30 to 3:30

• At exactly age 15.

Time at school: 10 x 39 x 5 x 7 = 13 650 hours

Hours alive: 15 x 365 x 24 = 131 400 hours

Approximately 10% (10.38%)

Challenge 2 Four minute mile

The data for this problem is subject to dispute as there are different

records available. Additionally, timing is more accurate in recent years.

However, the data do give a sense that time is decreasing.

Question: how come people are running faster now?

It could be improved technology of running shoes, people are taller,

people train harder, have a better understanding of nutrition etc.

Question: Should we use a straight line or something different?

Can’t be a straight line to extrapolate, otherwise at some point in the

future it will take zero time or even negative time to run a mile. This

would suggest that a curve such as an exponential decay curve might

be helpful.

Choose a challenge How long is a line of a million dots?

It all depends on the size of the dots and the spacing.

If the dots are close together and created with a sharp pencil then one

dot per mm should be achievable.

1 000 000mm = 1000m

How heavy is the food that a person eats in a lifetime?

http://wiki.answers.com/Q/How_many_lbs_of_food_does_a_person_ea

t_in_a_lifetime#page2

suggests that we eat 30 600 pounds/ 14 000 kg of food in a lifetime

How many pets are there in the UK?

Approximately 67 million according to the Pet Food Manufacturers

Association, including:

• 8 million dogs

• 8 million cats

• 20-25 million indoor fish

• 20-25 million outdoor fish

• 1 million rabbits

• …and 100,000 pigeons!

Choose a challenge

How many people live on the Isle of Wight (or in your county)?

• 2011 census: 133 713

• Increasing at approximately 0.7% per year (UK population growth

rate)

• 138 459 expected in 2016

Choose a challenge

Where to place a 30 minute sign

In the example given, a train with a maximum of 36 people leaves

every 100 seconds. If we assume that it’s not always full, there are

perhaps 30 riders each time.

30 minutes is 1800 seconds, so 18 cars leave every 30 minutes.

18 x 30 = 540 riders.

Students will need to decide how long a queue of 540 people is. This

will depend on how wide the queue is and how close people stand.

People tend to not like standing too close to the people in front of them,

so a metre per row of people would seem a reasonable estimate.

If they were in threes (on average) then it would be 180m.

Theme Park Queue

How long to start a Marathon?

Assumptions:

• The start line is 10 to 20 m wide and in the picture shown there are

approximately 30 runners crossing the start line.

• Most runners are in the main body of competitors, perhaps 36 000 of

them.

• It takes 2 seconds to cross the start line – so 30 rows a minute

• The runners cross in a steady flow.

This would mean:

36 000÷ 30 = 1200 ‘rows’ of runners

1200 ÷ 30 = 40 minutes

If runners only take 1 second to cross, then it will take 20 minutes.

London Marathon

Acknowledgements

https://en.wikipedia.org/wiki/Four-minute_mile

http://www.flemings-mayfair.co.uk/blog/2015/04/17/spectators-guide-to-

the-best-spots-at-london-marathon-2015/