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.
m e i . o r g . u k
I s s u e
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.
M a r / A p r
5 2
2 0 1 6
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 crosscurricular 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)
In this issue
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.”

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
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.”

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
Click here for the MEI
Maths Item of the Month
M4 is edited by Sue Owen, MEI’s Marketing Manager.
We’d love your feedback & suggestions!
Disclaimer: This magazine provides links to other Internet sites for the convenience of users. MEI is not responsible for the availability or content of these
external sites, nor does MEI endorse or guarantee the products, services, or information described or offered at these other Internet sites.
Earth Hour what can we save?
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.”
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.
Earth Day 2016 takes
place on 22 April.
April 16-23 is
designated as
Climate Education
Week. The Climate
Education Toolkit for
K-12 (primarysecondary) students
around the globe
includes a week’s
worth of crosscurricular lesson
Sydney Harbour Bridge and Sydney Opera
House during Earth hour 2007. By madradish
plans, activities and
contests. The CLEAN (Flickr) [CC BY-SA 2.0], via Wikimedia Commons.
Collection provides
The standard Earth Hour
'60' logo represents “the 60
other scientifically and
minutes of Earth Hour where
pedagogically
we focus on the impact we
reviewed resources
are having on our planet and take
on Climate Change.
positive action to address the
environmental issues we face”.
“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.
Energy Modelling
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 highestachieving 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.”
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
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.
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?
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).
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 risktransfer 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.”
Uses of Mathematical
Modelling
MEI Conference
2016 sessions
based on
mathematical
modelling
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 sessions
more inputs over time, also allowing for
relevant to this
volatility. For example, forthcoming
edition’s theme.
changes to the UK taxation system of
the ‘at retirement’ market will have a
Resources and
Investigations: The significant effect on some insurers’
Normal Distribution business models, warns Fisher.
“Changes to business models will
and Probability
naturally lead to changes in risk
Plots
exposures which will need to be
Kate Richards will use carefully considered and managed.”
real data in context to
introduce the Normal The mathematics of queues
Distribution and show
how using this as a
statistical model can
be useful to solve
Queuing theory is a branch of
problems.
mathematics that studies and models
Statistical Modelling the act of waiting in queues.
in Football
It could be argued that intuition and
observation could be used to assess
Alun Owen will
which queue to join in a supermarket,
consider the data
but it is necessary to predict queue
sources freely
lengths and waiting times when making
available for many
business decisions about the resources
sports, and describe
needed to provide the service for which
an application of
people are likely to queue.
statistical modelling in
football that can be
The 6 minute video Queuing Models:
used to illustrate how The Mathematics of Waiting Lines
computer games,
provides a visual exploration of the
such as Football
mathematics of waiting lines (queues).
Manager, rely on and In his Wolfram Blog post The
use statistical models Mathematics of Queues, Devendra
to ensure realism
Kapadia explains that it’s about more
during game play.
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.
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.
Reducing customer waiting times
Managing traffic queues
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.”
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.
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.
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.
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 codirector 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.
Figure 1: The changes in temperature over the
last 150 years. Note the overall upwards trend
with a random variability on top.
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).
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.
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
Figure 2: The changes in the Arctic Sea ice cover into the future. It is often argued that if
over the last 35 years, showing a decline. A best fit
we can’t predict the weather what hope
line is added to allow us to predict future trends.
have we of predicting climate 100 years
The three other main indicators are: the ahead. This is not really the case as
increase in mean sea level over the last climate is much more about finding
general trends rather than day to day
100 years, the increase in the number
variations (climate is what you expect
of extreme rainfall events, and the year
and weather is what you get). Thirdly,
on year rise in the level of Carbon
the climate is genuinely very complex
Dioxide in the atmosphere, with
indeed, involving the atmosphere, the
measured values now above 400 parts
oceans, the sun, vegetation and ice, not
per million (twice the level before the
to mention human activity. Whilst this
industrial revolution). Later on I will
does not prevent us from predicting it
show the (mathematical) link between
Carbon Dioxide levels and temperature, (indeed the weather is also very
complex involving over 109 different
which is a cause of concern.
variables, but we can still predict
The climate change debate
tomorrow’s weather with a good
Most scientists (and this includes
accuracy), it does make the job much
mathematicians) believe that climate
harder.
change is occurring, but this is certainly
Finally, a very real problem in climate
not a universally held opinion. One
science is distinguishing between cause
reason for this is that predicting the
and effect (for example does a rise in
climate is genuinely hard. As the great
scientist Niels Bohr famously remarked, Carbon Dioxide cause a rise in
temperature or is it the other way
“It is difficult to predict anything,
especially about the future.” There are a round), and distinguishing between
human made change and natural
number of (mathematical) reasons why
variations.
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
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
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.”
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
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.
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 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.
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.
Figure 3: Predictions of future temperatures from various climate centres around the world.
Hugh’s Views
Modelling and the Climate
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
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.
By NASA (The Project Apollo Image
Gallery) Public domain], via Wikimedia
Commons
By NASA (The Project Apollo Image
Gallery) Public domain], via Wikimedia
Commons
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.
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.
Hugh’s Views
Modelling and the Climate
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.
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!
Click image to view video
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
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.
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 ‘multistage modelling’ into
the university
curriculum for those
STEM
undergraduates who
need them, in a way
that is most suitable
to their needs.”
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.
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.)
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.
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
After graduating in
provide more opportunities for students
mathematics at
to undertake this throughout their
Cambridge University,
mathematics education.
Paul qualified as a
teacher. He has had
Nrich has a
experience in a
collection of
variety of school
both short and
based roles including
longer modelling
coordinating
problems.
professional
development and
curriculum leadership. One particular favourite of mine is the
He was Deputy Head ‘Where to Land’ Problem. Chris is
of an international
swimming in a lake, 50m from the
school for 8 years and shore. Her family are 100m along the
has considerable
shore. She'd like to get back to her
experience of
family as quickly as possible.
different mathematics
curricula worldwide.
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
1861
1862
1868
1873
1874
1875
1880
1882
1884
Time
286
273
268.8
268.6
266
264.25
263.2
259.4
258.4
1893
1895
1911
1913
1915
1923
1931
1933
1934
1937
257.8
255.6
255.4
254.4
252.6
250.4
249.2
247.6
246.8
246.4
*IAAF data from 1913, amateur data pre-1913; only final record of any one year cited.
Challenge 2: Graph
290
Men’s world record in seconds
285
280
Time (s)
275
270
265
260
255
250
245
240
1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980
Year
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
300
Men's world record in seconds
280
Time (s)
260
240
220
200
180
1860
1880
1900
1920
1940
Year
1960
1980
2000
2020
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 a challenge
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.
Longer challenges
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.
Theme Park Queue
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: Facts and Figures
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
Starting a Marathon
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: Facts and Figures
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.
Earth Day: April 22nd
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?
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
Choose a challenge
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
Theme Park Queue
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.
London Marathon
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.
Acknowledgements
https://en.wikipedia.org/wiki/Four-minute_mile
http://www.flemings-mayfair.co.uk/blog/2015/04/17/spectators-guide-tothe-best-spots-at-london-marathon-2015/