m e i . o r g . u k
Curriculum Update
A level Mathematics
and Further
Mathematics
content
Final content for A
level Mathematics
and Further
Mathematics for
teaching from 2017
has been published.
Visit the links below to
download:
GCE AS and A
level mathematics
GCE AS and A
level further
mathematics
A level and GCSE
changes
For timelines of
changes to GCSE
and A levels,
including which
changes affect which
year groups, visit the
links below:
Timeline of GCSE
reforms
Timeline of AS
and A level reforms
Changes to
GCSEs, AS and A
levels that will affect
each current school
year group
I s s u e
4 4
F e b r u a r y
2 0 1 5
Valentine’s Maths
Conic sections by paper folding and
GeoGebra. Sue Pope delivered a
As our regular readers know, we love a session at the 2012 MEI Conference
topical theme and this month lends
on Folding Mathematics, from which
itself to some connections between
session notes are available to
maths and Valentine’s Day. In this issue download. The International
we will be looking at origami and paper Conference on Origami in Science,
folding, including a look at how to fold
Mathematics and Education (OSME),
and weave paper hearts for Valentine’s inspires researchers and provides a
Day relevance! We will also take a look platform for them to discuss and learn
at the wider uses of origami in
of the newest developments. Sessions
manufacturing and engineering.
at the 2014 6OSME included
classroom uses of paper folding.
Paper folding has a real place in the
mathematics classroom. An NCETM
You’ll find links to paper
Teacher Enquiry Funded Project,
folding classroom resources
Original research into geometry, a
later in this issue, including
mathematical investigation, was
Carol Knights’ latest
proposed when its participants met at
teaching resource Paper Folding and
the initial London NCETM Influential
Proof. Richard Lissaman also uses the
Maths Teachers’ Conference. “Our
Valentine’s theme in this month’s Crash
belief was that paper folding could be
Course, looking at the idea of an array
used to enhance students’
in Python and then thinking about using
understanding of geometry and possibly an array to find a pair of loved-up
wider areas of mathematics, such as
‘amicable numbers’!
algebra and proof, by taking a physical
approach.” A final report has been
In this issue
published on the group’s findings. The

Curriculum Update
use of Origami in the Teaching of
Geometry by Sue Pope describes how

February focus: Origami and
Origami was used as a source of
maths
mathematical problem solving in a
series of lessons with Year 6 and Year

Crash Course: Arrays in Python
7 children.

Site-seeing with... Debbie Barker
Paper folding has featured in several

Teaching Resource: Paper
mathematics conferences, including the
Folding and Proof
2014 MEI Conference, where MEI’s
Tom Button delivered a session about
Click here for the MEI
Maths Item of the Month
M4 is edited by Sue Owen, MEI’s Marketing Officer.
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.
Paper folding in the
maths classroom
Origami hearts
Easy Heart
Getting started with paper folding
in the maths classroom
Resources from the
British Origami Society
Origami Club has a
printable resource
that takes step by
step you through the
process of folding a
paper heart, as well
as an animation that
you can view at
various speeds.
Check Heart
By using paper that is
coloured on one side,
you can fold a heart
with a checked
pattern on it. This
helpful animation
can be viewed at slow
speed.
There are many more
Valentine’s origami
patterns and
animations on the
same site, along with
folded shapes,
numbers and maths
symbols.
Folding angles of 45
and 90 degrees
Nick Robinson shows
one of the many
methods of creating
angles of 45 and 90,
with non-raw sides.
This design can be used to teach many
things; accuracy, geometry, folding
techniques, combining polygons and
will serve practically as either a
bookmark or a setsquare.
Folding angles of 30
and 60 degrees
Ian Harrison’s method of
constructing angles of
60ºand 30º by folding is
based on the symmetry of an
equilateral triangle - one that has three
edges of equal length.
Folding optimal
polygons
David Dureisseix looks at
various ways of folding
the "optimal" pentagon &
hexagon (the largest
regular hexagon within a square of
paper).
Other resources include:
Fujimoto and
Fuse twist boxes
If you would like some
hands on origami
practice, the British
Origami Society is
holding its convention
in March in Birmingham.
You can read about the experiences of
a previous novice attendee in this New
Scientist article.
Ian Short’s nrich article Paper Folding
- Models of the Platonic Solids
describes how to build models of the
Platonic solids using sheets of paper. A
resource with nets and animations is
available from Maths Is Fun.
Mathigon is a
visually stunning
interactive site
for teachers and
students, with a
section about
Mathematical
Origami:
“Explore the beautiful world of Origami
and Mathematics – from Platonic and
Archimedean solids to stars and
compounds. Containing stunning
photographs, folding instructions and
mathematical explanations.”
Although the article is aimed at younger
students, the section Ten Teaching
Techniques in The Japan Society’s
Origami in the Classroom: Where
Every Child Counts! is useful for all
age groups.
Further reading about paper folding
in the classroom:
Origami: mathematics in creasing
Mathematics of paper folding
The Miura-Ori map
Origami
Origami and Math
Mathematical Art Galleries
Paper folding
on a larger scale
Woven paper
hearts
Although these are
really intended as
Christmas
decorations in
Sweden, these paper
heart baskets would
work well for
Valentine’s Day.
Instructions and
templates can be
downloaded from
Mirkwood Designs.
There is also a
YouTube video
demonstration that
may help to guide you
through the weaving
process.
Triaxial Heart
Once you’ve got the
hang of the weaving
technique, here’s a
more challenging
Triaxial Heart from
papermatrix to try.
Fractal sponges
Crowdsourcing Origami Sculptures, a
2014 6OSME abstract, looks at projects
where volunteers worked collaboratively
to build very large origami sculptures by
linking thousands of traditional business
card cubes, such
as the fractal
sponges built in last
year’s
MegaMenger:
“The MegaMenger may be the largest
fractal ever built, made out of over a
million business cards and
encompassing the entire Earth.”
Although MegaMenger is now
complete, the MegaMenger website has
some instructions and printable
PDFs you can use to print some fractal
cards. See also Wired.com’s article
about the Snowflake Sponge designed
by MIT-trained engineer Dr. Jeannine
Mosely.
Engineering in Origami
Robert Lang is one of the best
origamists of the modern times. He
uses mathematics and engineering
principles to fold intricate designs. In his
highly recommended Ted Talk The
Math and Magic of Origami, not only
does Lang give a fascinating insight into
the history of origami, and how it has
developed with the application of
mathematics. He describes the
principles of crease patterns used in
origami. Lang’s illustrations of models
created from hundreds of folds using a
single piece of paper are quite amazing!
Lang explains how to break down the
process of folding the paper into a
simple step by step process. He
developed a free computer program
called TreeMaker that calculates the
crease pattern from a simple stick
figure.
He also explains how the principles of
origami can be employed in other uses
beyond the aesthetic: “Surprisingly,
origami and the structures that we've
developed in origami turn out to have
applications in medicine, in science, in
space, in the body, consumer
electronics and more.” Lang goes on to
talk about some of these applications.
Origami in Engineering
As the Engineering Council says, “To
most people origami purely means the
art of folding paper. However, today's
engineers are using this ancient oriental
technique to help solve some of the
world's great challenges.”
Here are links to just two examples:
A solar array that can be tightly
compacted for launch and then
deployed in space to generate power
for space stations or satellites
A complex folding pattern of 28
creases that form the design for a
rigid bag with a closed bottom
Crash Course:
Arrays in Python
A maths and
computing puzzle
column written by
Richard Lissaman
This column
provides an
introduction to the
programming
language Python
using maths puzzles
as motivation to
learn code!
In this month’s
Valentine’s day
special we’ll look at
the idea of an array in
Python and then think
about using an array
to find a pair of lovedup ‘amicable
numbers’!
The task
An array is just a list
of numbers in a
particular order.
Python commands
creating arrays and
operating on arrays
are shown on the left
of the screen grab
opposite.
Try entering the code
(see right) in the
Codecademy
Python workspace
and running it for
yourself.
Crash Course icon: Cropped
from original Python wallpaper
by Cazembe used under CC BY
1) In the first line we are declaring our list or array. We have called it myarray.
At the moment it is empty, it is a list of precisely no things!
2) In these lines we add the numbers 10 and 20 to our list. So after
myarray.append(10) our list is just the number 10 and then after
myarray.append(20) our list becomes 10, 20. The order is important here, every
time we append a number it gets put on the end of our current list.
3) Here we can see how to refer to things in myarray. The number in the very
first position of myarray is referred to as myarray[0]. So the first number is in
the zeroth position! This is a common convention in computing and even in
mathematics where you might see the first term of a sequence given the name
x0. The second number in our list is called myarray[1].
4) Now we join on the numbers 30, 40, 50, 60, 70, 80, 90, 100 to our array using
a for loop (see the last column for more about loops). Finally we print out all the
things in our myarray list.
There are lots of other commands for interacting with arrays. You can read about
them here.
Now we’re going to get Python to help us find a pair of amicable numbers.
Amicable numbers are two different positive integers so that the sum of the proper
divisors of each is equal to the other. We could ask whether 12 and 20 are a pair
of amicable numbers?
The sum of the proper divisors of 12 is 1 + 2 + 3 + 4 + 6 = 16.
This is not equal to 20…
The sum of the proper divisors of 20 is 1 + 2 + 4 + 5 + 10 = 22. T
This is not equal to 12…
So 12 and 20 are not a pair of amicable numbers…
(Cont on next page)
Crash Course:
Python challenges
Amicable Numbers However there is a pair of amicable numbers n, m in which n and m are both less
You might like to look
at these resources
after trying this
month’s problem:
than 300. Let’s use Python to find them!
The code below is one way to set about this:
Valentine’s Day:
Amicable Numbers
Amicable Numbers
video
Friendly Number
Friendly Numbers,
Solitary Numbers,
Perfect Numbers
Friendly Number
Perfect, Friendly,
and
Amicable numbers
If you run this code there will be no output but try to work out what it does:
First a function called sumoffactors is defined. To understand this function you
need to know that int(n/2) is the largest integer less than or equal to n/2.
So int(2.8) = 2. This function has a return command. This means that once the
function has done its work it will send back a value, in this case the value of the
variable total. So sumoffactors(10) is the value of total if the function is run with n
= 10.
Can you see that sumoffactors(10) will be the sum of the proper divisors of 10?
Then an array is created called sumfactorsarray. The first two numbers in the
array are set to be 0 and 1. After that the for loop then adds the sums of all the
proper factors of the numbers from 2 to 300 inclusive. This means that
sumfactorsarray(20) is the sum of all the proper factors of 20 and
sumfactorsarray(233) is the sum of all the proper factors of 233.
Problem of the month
Can you, in Python, add to the code above to check sumfactorsarray to find the
pair of amicable numbers n, m in which n and m are both less than 300 and print
them to the screen?
The solution is available to download from the Monthly Maths web page.
Site seeing with…
Debbie Barker
Each month a
different member of
MEI staff will share a
couple of their
favourite resources it might be some
software, a website, a
printable download, a
book, etc.
As part of the online element of the
Teaching GCSE Mathematics (TGM)
course we have been looking at
Transformations this month. I start by
sharing one of my favourite starters,
Mobius Transformations Revealed.
The column is
intended to be a quick
read - but we’d love
to hear from you if
you’d like to give
feedback or tell us
how you used the
I like its gentle style and I like how
resources.
students react to the simplicity, the
music and the surprises. The pace
This month’s
resources are shared means that students have time to jot
down their responses on mini
by Debbie Barker,
whiteboards to whatever prompts you
the MEI Coordinator
for CPD in teaching provide:
at Key Stage 3 and

Which transformations do you
Key Stage 4.
recognise?

What mathematical terms are you
Debbie designs and
manages MEI’s
reminded of?
Teaching GCSE

What questions do you have?
Mathematics (TGM)
course.
I think it sets the scene brilliantly for an
approach to transformations of “Here’s
the change, how can we mathematise
it?” I seek to provide a situation where
students experience the need for a
description, consider possible solutions
and to wonder what the conventions
might be.
As an example, here is a GeoGebra
activity for starting to think about
enlargements. Students are able to
move the point and the slider and are
asked to reproduce the following
pictures on their screen (both the point
and slider are invisible in these target
pictures):
Typical classroom dialogue would
follow along the lines of:




The point is important; it’s called
the centre of enlargement.
What do we need to know about
the centre of enlargement?
How might we describe that
information?
The slider is important too...
In my experience, many students start
to ask questions associated with the
slider taking values other than positive
integers…
A happy ending?
Parametric Heart
Spreadsheet
The Stable Marriage Problem
Do you ever wish
your spreadsheets
were more romantic?
This resource from
Think Maths will help
to make them so!
“Using parametric
equations, you can
construct a perfect
heart shape, and this
Excel file has
everything you need
to understand how
the equations give the
shape, as well as
allowing you to edit
the parameters and
see how that changes
the resulting plot.
“There's also a
second sheet with the
equations for a
cardioid. The
spreadsheet can be
given to students
directly, or kept as a
cheat-sheet while
they make their own.
“Great as part of a
lesson on parametric
equations, or just a
fun activity to play
around with.”
In this Numberphile video, Dr Emily
Riehl asks if you should accept a
marriage proposal or keep waiting for a
better offer? Men and women are asked
to compile a list of four prospective
suitors, in order of preference.
She explains how an algorithm will find
a solution to the problem of finding the
suitors most likely to result in stable
marriages for all – a process of
tentative engagements that can be
broken as prospective suitors find a
better choice.
Once everyone is engaged, some to
their first choice and some to their
fourth, this provides the solution to a
stable marriage.
The Happy Ending Problem
In another Numberphile video,
Prof Ron Graham looks at convex
polygons – how many points must your
put on a piece of paper (without any
three being in a straight line) to
guarantee a convex polygon of a set
number of sides (an n-gon)?
The conjecture is shown opposite. It
has been proved for n=4, n=5 and n=6.
New classroom resource
In the following pages is a teaching
and learning resource: Paper Folding
and Proof, developed by Carol
Knights, MEI Extension and
Enrichment Coordinator. The
PowerPoint can be downloaded from
the MEI Monthly Maths Magazine
web page.
This month’s edition has 4 activities
which link paper folding and proof.
These support the development of
reasoning, justification and proof: a
renewed priority within the new
National Curriculum.
They are presented in approximate
order of difficulty.
Square Folds
For each of these challenges,
start with a square of paper.
1. Fold the square so that
you have a square that
has a quarter of the
original area.
How do you know this is a
quarter?
Square Folds
2. Fold the square so that
you have a triangle that is
a quarter of the original
area.
How do you know this is a
quarter?
3. Can you find a completely
different triangle which has
a quarter of the area of the
original square?
Square Folds
4. Fold to obtain a square which has half the
original area. How do you know it is half the
area?
5. Can you find another way to do it?
Pythagoras Unfolded
Start with a square of paper.
Put a dot a short distance
from one vertex - about a third
of the side length works well,
not too close as you’re going
to have to fold from it.
Put dots exactly the same
distance from the other
vertices, working around the
square as shown.
Pythagoras Unfolded
Fold to join the dots as
shown.
What shape is this?
How do you know?
Pythagoras Unfolded
Mark three other dots as
shown, using the same
distance as you used for
the previous ones.
Pythagoras Unfolded
Create folds as shown.
Notice that two of the
folds don’t go all the way
across.
It’s easiest if you do the
vertical one first as this
gives the guide line for
where to finish the other
folds.
Pythagoras Unfolded
For the next slides, locate the relevant
lengths or shapes on your folded square.
Pythagoras Unfolded
a
Pythagoras Unfolded
b
Pythagoras Unfolded
c
Pythagoras Unfolded
What are the dimensions of this triangle?
Pythagoras Unfolded
What is this area?
c2
Pythagoras Unfolded
What is this area?
a2+b2
Pythagoras Unfolded
Explain what this sequence of diagrams shows:
Thinking about how you folded the shapes, can
you explain why the yellow and green triangles
will fit as shown?
Pythagoras Unfolded
Can you now link it all together and explain how
this is a proof of Pythagoras’ Theorem?
This proof started with a square in which you
created triangles in a certain way; would it work
for any right-angled triangle?
Equilateral Triangle
Start with a piece of A4 paper. Fold as shown.
Equilateral Triangle
Pick up the bottom left hand vertex and fold so
that the vertex touches the centre line and the
fold being made goes through the top left vertex.
Equilateral Triangle
Pick up the bottom right hand vertex and fold so
that the fold being made lines up with the edge
of the paper already folded over.
Equilateral Triangle
Finally, fold over the top as shown. Folding away
from you helps keep the triangle together.
Equilateral Triangle
This looks like an equilateral
triangle, but how can we be
sure that it is?
Unfolding the shape gives us
lots of lines to work with, but
all we really need is the first
couple of folds.
It may help to start with a
fresh piece of paper and
simply make these two folds.
Equilateral Triangle
Drawing round these edges
will also be helpful.
This is what you should have.
What can you work out?
Equilateral Triangle
Hint #1
The triangle is created
as shown; we need to
show that 2 of its
angles are 60°
Equilateral Triangle
Hint #2
Can you find the
angles in this triangle?
Equilateral Triangle
Hint #3
Think of the length of the
short edge as 2x.
Equilateral Triangle
Hint #4
When you fold the vertex
in, what distance is
shown?
Equilateral Triangle
Hint #5
What dimensions of
this triangle do you
know?
Equilateral Triangle
Hint #6
The fold line is a line of
symmetry of the grey
shape.
Regular Pentagon
Start with a piece of A4 paper.
Put two diagonally opposite vertices together
and fold as shown.
Regular Pentagon
Turn the shape clockwise so that it looks like
this:
Regular Pentagon
Crease along the line of symmetry, then open it
back out.
Regular Pentagon
Fold so that the red edge meets and aligns with
the red fold.
Regular Pentagon
Fold the other side (marked in red) in a similar
way.
Regular Pentagon
This looks like a regular
pentagon, but how can
we be sure that it is?
Folding can be
inaccurate, but with
mathematics we can
prove whether it is
regular or not.
Regular Pentagon
Unfolding the shape gives us lots of lines to work
with, but all we really need is the first fold.
It may help to start with a fresh piece of paper
and simply make the first fold.
Regular Pentagon
You need to know that the dimensions of a
piece of A4 paper are 297mm by 210mm.
What is the size of the internal angle of a
regular pentagon?
Can you work out
the size of the
angle at the top of
the shape?
Regular Pentagon
Hint #1
The top angle is made up of two smaller angles.
Find out what these are and add them together.
Regular Pentagon
Hint #2
Draw this line on before opening it out
Regular Pentagon
Hint #3
Regular Pentagon
Hint #4
Why are these the same length?
Teacher notes: Paper Folding
This month’s edition has 4 activities which link paper folding and proof.
These support the development of reasoning, justification and proof: a
renewed priority within the new National Curriculum.
They are presented in approximate order of difficulty.
Discussion and collaboration are key in helping to develop students’
skills in communicating mathematics and engaging with others’
thinking, so paired or small group work is recommended for these
activities.
The first ‘Square Folds’ is suitable for many ages and abilities since a
range of responses with different levels of sophistication are possible.
The second activity looks at a paper-folding proof of Pythagoras.
Teacher notes: Paper Folding
The third and fourth activities require students to create specific
common shapes and then prove whether or not they are regular. The
third requires a knowledge of basic trigonometry, the fourth requires
basic understanding of trigonometry and Pythagoras and the ability to
expand brackets and solve linear equations (a quadratic term appears,
but is balanced by an equivalent one on the other side of the equation).
Hints are given, but it is helpful to allow students to grapple with the
problems by not showing these too soon. Once a hint is given, it’s
useful to wait a while before showing another. Discussion and
explanaiton are key.
One option is to print the hint slides out on card (2 or 4 to a sheet) and
just give them to students as and when they need them rather than
showing them to the whole class.
Teacher notes: Square Folds
There are a range of possible responses for each of these. One of the
key concepts that can be developed through this activity is an
appreciation of what is meant by proof.
The progression through:
• Convince yourself
• Convince a friend
• Convince your teacher
is a good structure to use to encourage students to think about ‘why’
something is as they think it is, rather than just responding with ‘you
can see that it is’.
Demonstrating by folding to show that different sections are equal may
be acceptable, or you may wish students to consider the dimensions of
the shapes.
Teacher notes: Square Folds
Some possible answers:
1. A square, a quarter of the area.
The lengths of the sides are ½ the original.
𝑥 𝑥 𝑥2
× =
2 2
4
2&3. A triangle, a quarter of the area.
1 𝑥
𝑥2
×𝑥 =
2 2
4
Teacher notes: Square Folds
4&5 A square, half the area.
Folding the vertices of the original square into the
centre is one way to demonstrate that it is half
the original area. Using Pythagoras theorem to
show that the new square has side length √2
is also possible.
This one is harder to justify.
The diagonal of the new square
is x.
Using Pythagoras’ theorem, the
side length of the square must
be
𝑥
2
so the area is
𝑥2
2
Teacher notes: Pythagoras Unfolded
Although it would be possible to skip straight to the diagrams, students
having something they’ve folded in front of them helps them to see and
understand exactly what’s going on. They can then annotate and/or
colour the square and stick in their books or folders.
Identify that the bottom triangle is a,b,c.
Show that a2+b2=c2
The proof will work for any right-angled
triangle. Create 4 copies of the required
triangle and arrange them so that a ‘long’ and
‘short’ side form the edge of a square as
shown (makes no difference which is short
and which is long for isosceles triangles).
Teacher notes: Equilateral Triangle
Making the equilateral triangle is relatively straight-forward and can be
used with all students during work with shape. Do they tessellate. Fold
the vertices in to the centre to make a regular hexagon etc.
The more challenging aspect of this activity is proving that it is indeed
an equilateral triangle.
Assuming the length of the blue side is 2x.
The red line aligns with the blue side
when the fold is made, so it is also 2x.
In the right angled triangle shown, Opp is
x and Hyp is 2x, therefore the angle
shown is 30°
Teacher notes: Equilateral Triangle
Making the equilateral triangle is relatively straight-forward and can be
used with all students during work with shape. Do they tessellate. Fold
the vertices in to the centre to make a regular hexagon etc.
The more challenging aspect of this activity is proving that it is indeed
an equilateral triangle.
Assuming the length of the blue side is 2x.
The red line aligns with the blue side
when the fold is made, so it is also 2x.
In the right angled triangle shown, Opp is
x and Hyp is 2x, therefore the angle
shown is 30° and the other angle in the
triangle must be 60°
Teacher notes: Equilateral Triangle
Looking at the grey kite, the fold line is a line of symmetry. This gives
the angles shown. Since the angle sum of a quadrilateral is 360° the
missing two angles must each be 60°.
30°
30°
30°
30°
60°
?
90°
?
Teacher notes: Regular Pentagon
The ‘top’ angle consists of a right angle and the angle shown.
Since the interior angle of a pentagon is 108°, if the marked angle is
18° then it might be a regular pentagon, if it’s not 18° then it definitely
isn’t regular.
Teacher notes: Regular Pentagon
The red line is created when the fold is made.
Teacher notes: Regular Pentagon
The dimensions of the triangle
are as shown.
Using Pythagoras’ theorem:
(297-x)2 = 2102+x2
2972-594x+x2=x2+2102
594x =2972-2102
594x = 44109
x= 74.26
Teacher notes: Regular Pentagon
The dimensions of the triangle
are as shown.
Using trigonometry:
74.26
tan 𝜃 =
210
𝜃 = 19.47°
This means that the top angle
is 109.47°, so the pentagon is
not regular… despite the title
of the activity!
Acknowledgements
Ideas for ‘Square Folds’ from http://youcubed.stanford.edu/task/paperfolding-fun/
Images from: http://kimscrane.com/images/EF21K3.jpg
http://www.muji.eu/images/products/l/4547315453153_l.jpg
http://fc00.deviantart.net/fs71/i/2010/257/0/b/origami_paper_pattern_by
_tseon-d2ypu5c.jpg
Pythagoras activity from T. Sundara Row (1905) Geometric Exercises
in Paper Folding.