m e i . o r g . u k
Curriculum Update
GCSE
Ofqual has published
its research into
SAMs for the new
GCSE examinations
and has announced
that the exam boards
will be publishing
adapted SAMs,
approved by Ofqual,
by the end of June.
Ofqual is consulting
on some technical
requirements for the
new GCSEs.
A level
Ofqual’s working
group on A level
assessment has met
twice and will be
reporting to Ofqual on
its findings in July.
There will be sessions
at the MEI conference
with the latest
information on A level
changes and plans
for MEI’s new A
levels.
Core Maths
Core Maths launch
videos are available
online for teachers
who want to know
more about our Core
Maths qualifications.
I s s u e
Using the news as a starting point
for exploring maths and
promoting discussion
4 7
J u n e
2 0 1 5
Although no longer
included in the NCETM
Secondary newsletter,
it’s worth looking at archived ‘It’s in the
News’ resources; these explore a range
of mathematical themes in a topical
context. The resource is not intended to
be a set of instructions but a framework
which you can personalise to fit your
classroom and your learners.
The use of real
world news items
is a great way to
get relevant
resources and to use as starting points
to include maths in discussions. This is
particularly relevant to Core Maths,
however the use of topical material can
‘Engaging
be very motivating for mathematics
Mathematics for All
students at all key stages. The
Learners’, produced in
2009 by the Department
Education World article ‘Why Teach
for Education, draws
Current Events?’ says: “For children to
together the experience
become competent lifelong learners,
of teachers and their
they must learn how to use nonfiction
learners in a series of
materials to expand their knowledge
case studies. It gives guidance and
base, solve problems, and make
support on how to make mathematics
decisions.”
more exciting, meaningful and relevant
In his 2012 research paper ‘Talking
for all learners.
about maths’, Tony Pye (University of
East London) “looks at the premise that
In this issue
children are encouraged to talk and the
reasons why this might be effective in

Curriculum Update
helping them understand concepts
better by involving the input of their

June focus: Using the news as a
starting point for exploring
peers.” Through using George Pólya’s
maths and promoting discussion
ideas (see also Monthly Maths,
October 2014) as a basis for problem
Crash Course: Using Python to
solving, Pye reports: “We realised that
simulate polling
this was in many ways a more powerful

Site-seeing with... Stella Dudzic
way of teaching and pupils were more
engaged with maths than previously,

KS4 Teaching Resource: Plotting
when they had been simply completing
Data
textbook exercises or worksheets.”
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.
Using news items
as starting points
New KS4
classroom
resource
At the end of the
newsletter is a Key
Stage 4 teaching and
learning resource:
KS4: Plotting data
developed by Carol
Knights. This edition
looks at a range of
visual representations
of data.
The focus for these
activities is on making
sense of graphics,
and although this
doesn’t necessarily
relate to any
particular curriculum
content objective, the
process of
understanding
information presented
in novel ways and
interpreting the data
is a useful real-life
skill as well as
involving many
aspects of problem
solving.
The resource can be
downloaded from the
Monthly Maths web
page.
Where do I look?
Teachers and students have a wide
variety of information options, including
television, radio, the Internet, and print
sources such as newspapers and
magazines. It’s good to keep a log of
potential news items; just copying and
pasting the links onto a Word document
can save you time searching for that
‘thing’ you heard about’!
Newspapers and magazine (both
printed and digital editions) might
provide starting points for discussions
and ways to teach maths topics using
real world situations.
Some newspapers offer
free daily digital
downloads, which offer a
convenient way to scan
through while on public
transport, etc.
It’s also useful to
skim through
newspapers,
including the free
editions, in order
to identify useful articles that could be
scanned and used in the classroom as
a ‘real’ source of information on which
to base a discussion.
Alex Bellos writes a maths blog for
the Guardian.
The IMA produces the e-16+
Newsletter, aimed at students studying
maths at Highers, AS or A levels. The
newsletter shows “the usefulness,
interest, beauty and applications of
mathematics in so many areas.”
Audio and video news
podcasts/broadcasts can
provide motivating starting points
Numberphile produces many
YouTube videos about maths, often in
reaction to something topical.
TED Talks “shares the best ideas
from the TED Conference with the
world, for free: trusted voices and
convention-breaking mavericks, icons
and geniuses, all giving the talk of their
lives in 18 minutes.”
TED-Ed creates lessons as
an extension of TED’s
mission of spreading great
ideas, e.g. The math behind
Michael Jordan’s legendary hang
time, and Football physics: The
"impossible" free kick
If you’re not already a Twitter
user, register and look for
people posting about topical
maths and ‘follow’ them. Their tweets
will appear in your news feed; you don’t
have to tweet if you don’t want to!
Search using hashtags such as
#maths, and you will find accounts
such as Maths Jam, tes Maths,
NRICH maths, MMP - maths.org,
MathsCareers Website, and many,
many more. You’ll be surprised how
many ideas you will find in this way!
Look for news accounts to follow, so
that you scan through each day to see if
there are any news stories that might
lend themselves to being adapted for
use in your maths classroom.
Resources to help
get you started
Key Stage 3
Post 16
Bowland Maths Case
Studies use a rich
investigational style to
explore topical events
and real life
situations, to develop
“thinking, reasoning
and problem solving
skills and put
substance into the
Key Concepts and
Processes or problem
solving and
mathematical
reasoning.” The
material is aimed at
Key Stage 3, but can
be adapted for a wide
range of ages and
abilities. Topical
resources include:
Critical Maths Free
Resources, developed by
MEI with DfE funding, are
available through free
subscription. These resources are
designed for post-16 students at level 3,
and will be especially useful for Core
Maths classes.
Torbury Festival:
Interactive lessons
that ask pupils to
apply their
mathematical
knowledge to
overcome challenges
from floods to
escaped cattle to over
-excited crowds
storming the stage.
Water Availability:
Pupils examine ways
to compare the
availability of water
fairly between three
countries, to
determine which is
most in need.
Working in groups,
pupils assemble the
relevant data for one
country and use it to
argue for resources
for that country.
Level 3 Core Maths Resources,
developed by MEI with funding from
OCR, can be accessed free of charge
by completing an online subscription
form. Teachers can also access the
OCR (MEI) Level 3 Tutor Area for
Quantitative Problem Solving and
Quantitative Reasoning Forum
(subscribe and log in to view), where
ideas for resources and problem solving
are shared.
Integrating Mathematical Problem
Solving (IMPS) resources, developed
by MEI and funded by The
Clothworkers' Foundation, can also be
accessed free of charge. They are
designed to help teachers of
mathematics and teachers of other
subjects at A level to teach relevant
aspects of mathematics and statistics,
showing how they are used in solving
real problems. There is also an IMPs
News Forum (log in to view).
Secondary/FE
The Nuffield
Foundation provides
resources for teaching the use of
mathematics and statistics. The
resources are self-contained and can
be used for any secondary/FE lessons
where the context or skills are relevant.
Resources are divided into three levels
and can be used to support a wide
variety of qualifications. Topical
resources include:
Music Festivals: Students use
weather data to consider which month
would be the best to hold an outdoor
music festival. They practise using
either a calculator in Statistics mode or
a spreadsheet to calculate mean and
standard deviation values.
Currency Conversion: An
introduction to conversion graphs and
direct proportionality in the context of
currency conversion, reading and
plotting graphs and calculating
gradients.
Sightseeing Tour: This resource can
be used as a classroom activity or an
assignment. It involves students setting
up their own network as the basis for a
sightseeing tour.
TES Mathematics
Teaching Resources
offer a mixture of types of
resource shared mostly by teachers,
including resources created/developed
to be topical, for example:
Factorising Quadratics Hannah's
Sweets MegaPack
Hannah's Sweets: Conditional
Probability
Running a Holiday Company
Challenge
Tour de France Maths: Problem
Solving Activities
You can also look in the month-bymonth Topical Resources Index
relating key celebrations, awareness
days and action weeks in 2015.
Resources to help
get you started
Information is
Beautiful
This site is
“dedicated to distilling
the world’s data,
information and
knowledge into
beautiful, interesting
and, above all, useful
visualizations,
infographics and
diagrams.”
The data is available
both as a
spreadsheet on the
website and as a
Google doc.
An article from April
that might interest
students is How
much do music
artists earn online?
“Do digital music
streaming services rip
off artists? Taylor
Swift thought so,
asking that her stuff
be removed from
Spotify.”
The blog keeps you
in touch with the
latest visualisations,
and you can follow
Information is
Beautiful on Twitter.
The NRICH Project aims
to enrich the
mathematical
experiences of all learners. The feature
Enriching the secondary curriculum
complements the Secondary
Curriculum page for teachers. This
feature contains some articles about
rich tasks and how they can be used to
inspire your students.
GCSE
The Shell Centre is
making many of its
publications available as free
downloads, including Extended Tasks
for GCSE Mathematics. Each of the
eight topic books contains a lead task,
discussed in detail with teacher's notes
and examples of student work, as well
as a number of secondary tasks,
providing ideas for other assignments.
Where there’s life, there’s maths
offers a range of materials designed to
support students as they pursue
applications tasks. These applications
tasks are intended to stimulate
students' interest in, and understanding
of, the world in which they live.
AQA GCSE Maths has
free topical resources:
Football World Cup, General Election,
Icelandic Volcano, Winter Olympics scroll down the resources page to view
and download.
On the following pages as
well as resources
recommended by another member of
staff, you will find resources written by
two of our staff members, using recent
news items as their basis.
Richard Lissaman has written the
Crash Course resource this month in
response to the interesting results of
the recent general election. You can
find more teaching resources about
voting systems in the October 2013
edition of Monthly Maths and its
associated resource How do we
decide?
Carol Knights has written the KS4
teaching resource Plotting Data based
on recent news reports about
earthquakes and diets. This looks at a
range of visual representations of data.
The focus for the teaching and learning
activities is on making sense of
graphics, and although this doesn’t
necessarily relate to any particular
curriculum content objective, the
process of understanding information
presented in novel ways and
interpreting the data is a useful real-life
skill as well as involving many aspects
of problem solving.
You can find see other interesting ways
of using and visualising data on the
Gapminder website, which promotes
“increased use and understanding of
statistics and other information about
social, economic and environmental
development at local, national and
global levels.”
The site includes a section for
educators who want to use Gapminder
in their education, as well as videos and
downloads. You can follow Gapminder
on Twitter.
Crash Course:
Using Python to
simulate polling
A maths and
computing puzzle
column written by
Richard Lissaman
In this edition we’ll use Python to simulate polling for elections. As we saw in the
recent UK general election, polls can be surprisingly inaccurate! There might be
many reasons for this including the way that the sample of voters is selected for the
poll. We’ll leave these to one side below and look at how a poll result might differ
from an actual result purely because it is based on a sample (i.e. because it
doesn’t include everyone who will vote).
In the code below the first line allows us to access a set of commands associated
with randomness - specifically the command random.shuffle which appears
further down the code.
This column provides
an introduction to the
programming
language Python
using maths puzzles
as motivation to learn
code!
You may wish to
review Crash Course
in the February
edition of Monthly
Maths, which covered
arrays, before looking
at the code opposite.
Crash course April/
May problem –
solution can be
downloaded from the
Monthly Maths web
page, or click the link
above.
The solution to this
month’s Crash
Course Problem of
the Month appears on
the Monthly Maths
web page.
An array called voter of 10 values is set up. Initially the first seven values are 1 and
the last three values are 0. But towards the end of the code the array gets shuffled
and you can see the new order of the values printed on the right below. Hopefully
it’s clear how random.shuffle works from this example.
In what follows on the next page our method for sampling will be to shuffle the
array which records all the voters’ intentions and then take our sample from the
start of the shuffled array. An example may help! Read on…
Imagine the following scenario. A town contains 1000 voters. Suppose 510 would
vote for party 1 and 490 would vote for party 0 in the upcoming election. The code
on the next page simulates taking 10 polls, each of 10 people and using their
voting intentions to predict the result of the election.
Crash Course:
Polling predictions
Problem of the
month
The Collatz
conjecture states
that, for any choice of
positive integer for the
first term x0, the series
defined as follows:
If xn is even then
xn + 1 = xn /2.
If xn is odd then
xn+1 = 3xn + 1
includes the value 1.
(i.e. it eventually
reaches the value 1,
after which it will be
periodic with values
1, 4, 2, 1, 4, 2, 1, 4, 2,
1,…)
For example
If x0 = 13 this
sequence is:
Notice that only six out of the ten polls predict the correct result (a win for party 1).
Problem of the month
A town in which 1000 people are eligible to vote is about to have an election.
Each voter has the choice between voting for Party 1 and voting for Party 0.
13, 40, 20, 50, 5,
16, 8, 4, 2, 1, 2, 4,
1, 2, 4,…
Assume that the true voting intentions of the entire town are such that x% of
voters will vote for Party 1 and (100-x)% will vote for Party 0. For example if x = 70
then party 1 would receive 700 votes and party 0 would receive 300 votes and
party 1 would win.
X9 = 1.
a)
Write a function called poll which takes x (as above) and m (the sample
size) as inputs and prints to the screen the predicted result of the election
based on a poll of m people.
So poll(70,100) will return the predicted result (in the form ‘predicted win for
party 1’ or ‘predicted win for party 0’ or ‘predicted tie’) based on a poll of 100
voters from a total of 1000 voters of whom 700 would definitely vote for
party 1 and 300 would definitely vote for party 0.
b)
By repeatedly using your function, make a judgement about which of these
polls in incorrect most often on average:

a poll of 10 people from a town with 1000 voters in which x = 55

a poll of 100 people from a town with 1000 voters in which x = 51
For which starting
integer x0 with
1 ≤ x0 ≤ 200 does the
sequence above
take longest to
reach the value 1?
For this value of x0
what is the smallest
n such that xn = 1?
Site seeing with…
Stella Dudzic
Stella Dudzic is
Programme Leader
(Curriculum and
Resources) for MEI
where she leads on all
aspects of curriculum
and qualification
development in
secondary
mathematics. She has
taught mathematics in
secondary schools for
22 years and was a
head of faculty before
taking up her current
post in 2006. Stella
has written two
revision guides for A
level Statistics, is
responsible for the
development of
resources for Core
Maths and she has
been the Royal
Statistical Society Guy
Lecturer.
Stella and Neil
Sheldon will be
presenting the MEI
Conference plenary
‘Using Technology
To Explore Large
Data Sets’. Stella will
also deliver sessions
on Florence
Nightingale and
MEI's new
Mathematics and
Further Mathematics
A levels.
One of my favourite websites is Figure
this! (Math challenges for families) –
an American site. They make good
discussion questions at KS3.
Here is an example:
Each challenge includes a hint and an
indication of contexts where this
mathematics is used. There is also a
suggested answer and you can search
the challenges by maths topic. The
thing I really like about them is that
because they are written to be suitable
for families, they start with a problem
and the mathematical thinking comes
out of that rather than trying to think of
something that uses the mathematics
that students are currently learning.
designed to help students understand
and interpret graphs and were written at
a time when developments were taking
place to explore how to assess
investigation and problem
solving in examinations – the
introduction to the book
sounds very topical even
though it was written in 1985.
“The aim of this series of
Modules is gradually to
introduce into the
examination questions that
will encourage a balanced
range of classroom activities.
It is particularly concerned
with those activities
highlighted by the Cockcroft Report:
problem solving, practical mathematics,
discussion and open investigation.
A few new mathematical techniques
may occasionally be introduced, but the
main concern is to broaden the range of
skills developed to include those
strategic skills which are essential if
pupils are to be able to deploy their
technical skills when tackling unfamiliar
problems both in and outside
mathematics.”
Because they are written for families,
they are generally accessible for
students who haven’t learnt a lot of
mathematics content yet and most of
the challenges involve problem solving.
One of my all-time favourite teaching
resources is the Language of Functions
and Graphs from the Shell Centre – I
remember using it to teach when it first
came out and now you can download
it for free online! The resources were
Above is a question from the classroom
materials. As well as a wealth of
teaching materials, the resource
includes teaching suggestions and
solutions. It was great value when we
had to pay for it and now even better
value!
Plotting data
Representing data visually often helps people
to see patterns or trends or to look for
differences.
It’s sometimes easier to look at a graph or
chart than to look at a tables of values.
On the following slides there are some
interesting ways of representing data.
Ternary plot
The first plot is called a Ternary plot.
This plot is used when each item to be plotted
has 3 pieces of connected information about it to
represent.
An example on the next slide is about soil types.
Ternary plot
Can you work out
how to read
information from this
plot about soil types?
Ternary plot
The plot is built
around 3 ‘axes’.
What would a soil
which is 25% sand,
30% clay and 45%
silt be known as?
Ternary plot
25% sand
30% clay
45% silt
Ternary plot
4 soil types are
plotted, what is the
composition of
each?
Ternary plot
For each of the 4 soils, what’s the sum of the
percentages?
Will it always be this? Why?
This is an important point about Ternary plots:
they can only be used when the three categories
are the only three possible ones, and so they
always have to add up to 100%.
There aren’t many situations where this is the
case, which is perhaps one reason that we don’t
often see these plots.
Food and calories
Calories in food come from 3 main sources:
carbohydrates, proteins and fats, so each food
can be broken down according to the proportion
of calories coming from these 3 sources.
As an approximate guide:
• 1g of carbohydrate has 4 calories
• 1g of protein has 4 calories
• 1g of fat has 9 calories.
Food and calories
A popular big burger contains 25g of protein,
46g of carbohydrate and 29g of fat.
How many calories come from each source?
What percentage of calories come from each
source?
On the next slide, the percentages have been
calculated for some popular foods.
Plot the values on a ternary plot.
Sources of calories in food:
percentage values
Chocolate
Pizza
Chips
Apple
Cheese
Yoghurt
Pasta
Carrot
Crisps
Tuna
Fat
49
33
45
3
74
47
6
0
57
7
Carbs
45
49
51
95
1
30
79
100
38
0
Protein
6
18
4
2
25
23
15
0
5
93
Sources of calories in food
Who might be
interested in
this sort of
information?
Chocolate
Yoghurt
Pizza
Pasta
Chips
Carrot
Apple
Crisps
Cheese
Tuna
Fat %
Plotting very large data
One of the issues with plotting data arises when
we need to represent both relatively small and
relatively large data at the same time.
An example of this occurs if we want to look at
the relative sizes of earthquakes and tremors.
First we need to understand a little bit about how
earthquakes are measured.
The Richter Scale
The Richter scale is a term that many people will
be familiar with – however, this is not strictly
accurate as earthquakes are now measured
using the Moment Magnitude Scale (MMS).
The Richter scale was found to be less reliable
for earthquakes measuring more than 7…
… but what does a ‘7’ actually mean?
The MMS Scale
The numbers refer to
the amount of energy
released by he
earthquake as
shown.
How much more
powerful is a MMS 7
earthquake than an
MMS 5 one?
MMS Approximate Energy (joules)
1
2 000 000
2
63 000 000
3
2 000 000 000
4
63 000 000 000
5
2 000 000 000 000
6
63 000 000 000 000
7
2 000 000 000 000 000
8
63 000 000 000 000 000
9
2 000 000 000 000 000 000
10
63 000 000 000 000 000 000
The MMS Scale
It may be surprising that it is estimated there are
over 500,000 earthquakes a year, of which only
100,000 are felt by humans since many of them
are MMS 1 or 2.
Seismologists record all activity and to help look
for trends they sometimes plot the data too,
hoping they will find ways of predicting events so
that people can prepare.
Last month there were over 500 which
registered at MMS 4 or more.
The MMS Scale
A selection of
earthquake data from
around the world
during the last month
are given.
Plot them, putting the
date on the horizontal
axis and the joules
released on the
vertical axis.
Date
Approximate Energy
(joules)
12/05
6.32 x 1010
13/05
2.52 x 1011
17/05
7.96 x 1012
18/05
6.32 x 1010
19/05
7.01 x 1014
22/05
1.42 x 1015
25/05
8.93 x 1010
30/05
1.26 x 1014
02/06
1.26 x 1011
04/06
6.32 x 1013
The MMS Scale
You will probably have found this a difficult
task! If not an impossible one…
The bigger numbers are so much larger than
the smaller ones that it’s not easy to fit them
on the same axes.
If there are lots of ‘smaller’ values – as there
are with earthquakes - the scale needs to
allow people to see the differences, but
then the large numbers are off the scale.
The MMS Scale
One possibility is to use
semi-logarithmic graph
paper.
This has a linear scale on
the horizontal axis.
On the vertical axis each
‘cycle’ of numbers
represents 10 times the
value of the previous
cycle.
The MMS Scale
It could be:
1 2 3 4 5 6 7 8 9 10
The MMS Scale
It could be:
1 2 3 4 5 6 7 8 9 10
then 20 30 40 50 60 70 80
90 100
The MMS Scale
It could be:
1 2 3 4 5 6 7 8 9 10
then 20 30 40 50 60 70 80
90 100
then 200 300 400…
The MMS Scale
It could be:
1 2 3 4 5 6 7 8 9 10
then 20 30 40 50 60 70 80
90 100
then 200 300 400…
Note the ‘overlap’ at 10,
100, 1000 etc. and note that
the lines get closer
together.
The MMS Scale
For our purposes, the first
cycle will be:
x1010
and then x1011
and then x1012
Label the axes and plot the
values.
Plotting Large Data
This type of graph has uses in science, for those
looking at bacterial growth, the spread of infection,
charting planets and distances and many others.
Using a scale which gets ten times bigger
is common, however, sometimes a different scale
is used, such as on the following slide.
It’s just important to remember that it’s not a linear
scale…
Plotting Large Data
From the graph, can you determine:
• Which country has the biggest increase in
cases in any three day period?
• For Chile (the pink line) when is the biggest
increase in cases?
Teacher notes: Plotting Data
This edition looks at a range of visual representations of data.
The focus for these activities is on making sense of graphics, and
although this doesn’t necessarily relate to any particular curriculum
content objective, the process of understanding information presented
in novel ways and interpreting the data is a useful real-life skill as well
as involving many aspects of problem solving.
Teacher notes: Ternary Plots
Make it ‘girl friendly’:
This activity could be used with a wide
range of students as the only previous
knowledge required is expressing one
value as a percentage of another.
Ask students to discuss it
with a friend before giving
an answer
Slide 4
Give students plenty of time to look at this
to try to make sense of it.
Slide 7
Dot colour % Sand
% Clay
% Silt
Blue
12
64
24
Pink
41
35
24
Yellow
75
15
10
Black
15
0
85
Teacher notes: Ternary Plots
Slide 10
Fat: 29 x 9 = 261
Carbohydrates: 46 x 4 = 184
Protein: 25 x 4 = 100
Total calories: 545
Fat:
48% Carbs: 34% Protein: 18%
Slide 11
A blank ternary plot is available as a separate download.
Teacher notes: Plotting large data
For this activity students will ideally be familiar with standard index
notation. This activity involves quite challenging concepts.
Through trying to plot the raw data, students should come to the
realisation that ‘something different’ is needed.
Slide 18
Students should attempt to plot the data and should soon realise that it
is very difficult to do. Frustration and ‘failure’ can be key to genuinely
appreciating the issues and understanding why an alternative is helpful.
If students do manage to plot the data ask them how accurate it is for
the lower values – are they able to distinguish between 6.32 x 1010
and 2.52 x 1011 ?
Teacher notes: Plotting large data
Slide 21 to 24
It is worth spending some time looking at the paper, perhaps asking
students to describe what they notice about it before showing them the
slides.
Slide 25
It should be relatively straight-forward to plot the data. No pattern
emerges, but that is usual – particularly since these earthquakes are
not from a single region. A separate file of a semi-logarithmic blank
graph with 6 cycles is available to download.
Slide 28
The USA has an increase in the last few days of approximately 8000
cases.
Chile has the steepest line between June 8th and 11th, but this
represents approximately 1500 cases, whereas between 23rd and 26th
June there are about 2000 more cases.
Acknowledgements
Ternary soil plot from:
http://stackoverflow.com/questions/12520003/representing-ternary-plotdata-for-lookups
Calories in food data from:
http://www.calorieking.com/foods/
Earthquake data from:
http://ds.iris.edu/seismon/eventlist/index.phtml
Semi-logarithmic paper from:
http://incompetech.com/graphpaper/logarithmic/
Blank Ternary Plot
Source: http://ambiente.usach.cl/jromero/diagrama_ternario.htm
MEI is a registered charity, number 1058911
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Free Logarithmic Graph Paper from http://incompetech.com/graphpaper/logarithmic/