MEI Conference 2016
Paperfolding and Proof
Jane West
[email protected]
Further Mathematics Support Programme
Paper Folding
Isosceles
Triangle
A4
Paper
Fold edge
to edge
Fold edge
to fold
Fold edge
to edge
Fold edge
to edge
Kite (1)
A4
Paper
Equilateral
Triangle
•
Make a
fold across
the middle.
Make a fold so that
the bottom left
corner touches the
centre line and the
fold goes through
the top left corner
Fold edge
to edge
Fold corner
to corner
Fold so that the fold
Finally, fold over
lines up with the edge of the top as shown.
the paper already folded.
Kite (2)
Fold corner
to corner
Kite (2) - 1. Using angles and congruence
In the diagram ABCDEF is a sheet of A4 paper.
(i.e. the sides are in the ratio AB:AE = 1: )
A
B
ABCF is a square.
Three folds are made along the green dotted lines
1. Along BF so that A meets C
2. Along AO so that E meets C
3. Along EC so that D meets Y
Y
X
Prove that BCOX is a kite.
Assuming AB = 1
AC =
(AC is diagonal of unit square)
AE = AC =
(AE is long side of A4 sheet)
Triangle ACE is isosceles and angle EAC = 45°
C
F
O
D
E
AO is bisector of angle EAC (it's the fold line of E onto C) so
angle EAO = angle CAO = 22.5°
The same argument (by symmetry) gives
angle XBO = angle CBO = 22.5 °
①
②
angle AEC = angle ACE = 67.5° (triangle ACE is isosceles with 45° at the apex A)
angle ACB = 45° (AC is the diagonal of square ABCF)
angle OCB = 67.5° + 45° = 112.5°
In triangle FAX
angle AFX = 45° (FB is the diagonal of square ABCF)
angle FAX = 22.5° (see ①)
angle FXA = 180 - 45 - 22.5 = 112.5°
angle OXB = angle FXA = 112.5° (opposite angles)
angle OCB = angle OXB = 112.5°
Now consider triangles OCB and OXB
angle OCB = angle OXB (see ③)
angle XBO = angle CBO (see ②)
angle XOB = angle COB (each triangle adds up to 180°)
side OB is common to both triangles
triangle OCB is congruent to triangle OXB (angle, side, angle)
In particular, OX = OC and BX = BC, so BCOX is a kite.
③
Further Mathematics Support Programme
Pentagon
Make the following Pentagon
D
A
Start with a sheet of A4 paper.
Fold corner C up to corner A.
B
C
A
D
F
Lay BE along DF to create a mirror line.
Mark the crease and open back out.
B
E
A
B
D
E
F
Fold BE to lie on the crease.
Fold DF to lie on the crease.
Although this looks like a regular pentagon it is not actually regular.
Prove it is not regular.
Further Mathematics Support Programme
Pentagon Proof
A
D
ϑ
F
B
The original sheet of paper is A4. Give it
dimensions of
.
So
and
The paper was folded to match C with A.
So when opened up we know
E
1
D
A
The angle at A of the pentagon is
ϑ
Call the length
F
In triangle ADF
B
C
This gives the top angle of our pentagon as
However the angles of a regular pentagon should all be
So this is not a regular pentagon.
.
.
Further Mathematics Support Programme
Interlude - Haga's Theorem
Starting with a square sheet of paper, it is easy to find the point half-way
along one edge by folding.
What about finding a third?
A
D
Starting with a square sheet of paper, find the
point half-way along one edge by folding and
marking the point with a crease.
Fold the bottom left-hand corner up to meet
this mark.
B
A
Now consider the
C
B
and
.
D
Let
so
and therefore
E
So
F
and
are similar.
C
Assuming the
the paper
paper has
has side
side 1.
1.
let
Assuming
let
So
So
(using Pythagoras’ theorem)
But
But
is similar
similar to
to
is
So
So
i.e.
i.e.
So by
by folding
folding
So
square.
square.
and so
so
and
in half
half we
we can
can find
find the
the point
point which
which is
is of
of the
the length
length of
of the
the
in
Further Mathematics Support Programme
Haga's Theorem - General Case
A
B
D
This can be generalised for any fraction.
Suppose you fold the corner B of a unit square
to meet the top edge at a point which is
from A.
E
F
In
and
are similar.
using Pythagoras’ theorem
and
So
As before
are similar.
i.e.
The mid-point of
fraction).
will be
of the length of the square (i.e. the next
As a result any fraction of the form can be constructed by folding alone.
Indeed a 'spiral' of successive fractions can be folded by rotating the paper
through
each time.
For more information on this see https://plus.maths.org/content/folding-numbers
or search for 'Haga's Theorems Origami'
Paper folding
and Proof
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?
Isosceles Triangle
Start with a piece of A4 paper.
Fold as shown edge to edge.
Isosceles Triangle
Fold as shown, again edge to edge.
Isosceles Triangle
Turn the shape over. It looks like an isosceles
triangle. Prove that it is isosceles.
Kite (1)
Start with a piece of A4 paper.
Fold as shown edge to edge.
Kite (1)
Fold as shown.
Kite (1)
Turn it over. Prove that this is a kite.
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?
Kite (2)
Start with A4 sized paper. Fold as shown below.
Kite (2)
Fold corner to corner, as shown.
Kite (2)
Fold corner to corner, as shown.
Kite (2)
This is a kite – or is it?
‘Regular’ Pentagon
Start with a sheet of A4 paper.
Fold corner C up to corner A.
Lay BE along DF to create a
mirror line.
Mark the crease and open
back out.
Fold BE to lie on the crease.
Fold DF to lie on the crease.
This looks regular but isn’t.
Prove it.
Hint 1
The top angle is made up of two smaller angles.
Find out what these are and add them together.
Haga’s Theorem
You can easily find half way along one side of a
square by folding.
Can you find a third of the way along a side by
folding?
Find the length of the side
AE.
Now find the length of DF.
Does this help?
Why not use the shapes in KS3 and then
introduce the proofs in KS4?
Instructions for the shapes
can all be found at this link
Paper folding instructions
Equilateral Triangle Hints
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.
Teacher Notes
Teacher notes: Paper Folding
Here we have 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.
Teacher notes: Paper Folding
The next activities require students to create specific common shapes
and then prove whether or not they are the shapes they appear to be.
The first and second require a knowledge of Pythagoras’ theorem and
surds, the third requires basic understanding of trigonometry including
exact values.
Hints are given for the equilateral triangle, 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 explanation 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: Isosceles Triangle
To show that BEC is isosceles we
need to show that BE = BC.
A
B
Using the fact that the ratio of the
sides of ‘A’ paper is 1:√2 we can say
that AB is of length 1 and the length
of BC is √2.
Because of how we did the first fold
the length of AE is also 1 so using
Pythagoras’ theorem in ABE we find
that BE is also √2.
So BE=BC and BEC is isosceles.
E
D
C
Teacher notes: Kite (1)
To show that BDEG is a kite we can
show that BG=BD and DE=EG.
Again we can say that AB is of length 1
and the length of BD is √2.
A
B
Using Pythagoras’ theorem in ABG we
can deduce that BG is √2. So BG=BD.
FG=√2-1, as does FE.
So DE = 1-(√2-1) =2-√2.
Using Pythagoras’ theorem
EG² =(√2-1)² x2=6-4√2.
So EG=√(6-4√2) = 2-√2.
So BG=BD and DE=EG and therefore
BDEG is a kite.
G
F
C
D
E
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°
?
Kite (2)
• There are three proofs on separate handouts.
Acknowledgements
Thanks to Carol Knights for the basics of this PPT which appeared in
MEI’s Monthly Maths in February 2015
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
About MEI
• Registered charity committed to improving
mathematics education
• Independent UK curriculum development body
• We offer continuing professional development
courses, provide specialist tuition for students
and work with industry to enhance mathematical
skills in the workplace
• We also pioneer the development of innovative
teaching and learning resources