Perfect thirds www.rigb.org
The Royal Institution of Great Britain Mathematics Masterclasses
9 th October 2009
Warning: this document contains comments for teachers in italics. All the solutions are explained in more detail in the
pdf file of the Project Origami book included in this pack.
Warm up
Q1: Can you fold a square of paper out of an A4 sheet?
Notes for teachers: depending on your time you may or may not use this question. In our trials we noticed that many pupils needed some warming-up with paper folding as they may not remember some basic things.
Q2: Discuss with your pair how you did it and why it works.
Notes for teachers: this can create an opportunity to revise concepts and vocabulary like 45˚ angle, right angle, angle bisector, symmetries of the square, perpendicular, parallel, diagonal, etc and how these concepts relate to the paper folding done.
Folding fractions
Q3: Can you fold an A4 sheet of paper in halves? Can you fold a sheet of paper in quarters? What other fractions can you fold perfectly?
Notes for teachers:
 1/2,1/4, 3/4, 1/8, 3/8, ... any multiple of a power of 1/2
 challenge the pupils to fold 1/3, 1/5, 1/7 until they find it hard; don’t reveal just yet that the rest of the activity provides a method to fold 1/3 perfectly
 this can be used in a way of comparing methods that give approximations of other fractions (say 1/3 or 1/5) and methods that provide an exact value
Ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
Further questions? Contact ssantos@ri.ac.uk
.
Page 1 of 8

Perfect thirds www.rigb.org
The Royal Institution of Great Britain Mathematics Masterclasses
9 th October 2009
Notes for teachers:
 perform the fold without telling children what you are doing, just let them watch you
 get one volunteer to come up and check that your fold is in fact 1/3 – some pupils want to measure with a ruler, you want them to think of better ways like folding the rest of the paper in 3 parts using the first third as reference
 you then tell them that the experiment gives them reasons to conjecture this is one third, but how can they know that it is not 0.6 or another number close to 1/3? You can tell them about numerical confirmation versus ‘knowing for sure’, the latter being achieved with a mathematical proof, mathematics being the only science where you can achieve certainty, knowing for sure that something is true
 next you lead them to try the fold themselves and study it as in the questions below.
Fold a square of paper following the procedure below.
Note for teachers: you can do it again slowly so that they follow and you can also use the GeoGebra applet included projected on the board.
1) 2) 3)
Q1: What is the length you have folded after step 3?
Q2: Can you explain how it works?
Notes for teachers:
 some pupils will need to check that the fold is in fact 1/3 – some pupils want to measure with a ruler, you want them to think of better ways like folding the rest of the paper in 3 parts using the first third as reference
 folding this gives them reasons to conjecture this is one third, this is an opportunity of addressing the difference between checking and knowing for sure: to know for sure you need a mathematical proof
 the next steps break down the investigation
Ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
Further questions? Contact ssantos@ri.ac.uk
.
Page 2 of 8

Perfect thirds www.rigb.org
The Royal Institution of Great Britain Mathematics Masterclasses
9 th October 2009
Observe the red and blue triangles below: justify that they are similar triangles.
Note for teachers: the two opposite sides of the square are parallel, pupils can highlight which angles are corresponding, etc and derive similarity of triangles.
Compare the base of the red and blue triangles.
Note for teachers: the base of the blue triangle is half the base of the red triangle, the vertical line shown is a crease for half of the square.
What can you say about their heights? If the side of the square is one unit length, what is the height of the blue triangle?
Note for teachers: the height of the red triangle is twice the height of the blue, dividing the side of the square in three parts and so the height of the blue triangle is 1/3 units if the side of the square is 1unit long.
Note for teachers: some pupils prefer to look at other triangles as below looking at the ratio between the vertical and
horizontal sides of the smaller blue triangle on (a), the smaller red triangle on (b), and the ratio between the vertical
sides of the blue and the red triangles on (c). a a a b c
How does this help with working out the folded length?
Note for teachers: in the diagram below the blue triangle is isosceles (since the diagonal of the square is at 45˚ from the base) and so the base and the vertical side have the same length. Since the height is 1/3, the base is 1/3.
a a
Ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
Further questions? Contact ssantos@ri.ac.uk
.
Page 3 of 8

Perfect thirds www.rigb.org
The Royal Institution of Great Britain Mathematics Masterclasses
9 th October 2009
Q2 (continued): Can you explain how it works?
There are other ways of working it out.
Step 1: set the origin at the bottom-left-hand corner and use coordinate axis (bottom and left edges). Write down the equations of the straight lines l
2
and l
3
through the point P.
P
P l
2 l
3 l
3 l
1
Note for teachers: l
2
has equation y=1-x and l
3
has equation y=2x.
Step 2: use simultaneous equations to work out the horizontal coordinate of the point P.
Note for teachers: l
2 has equation y=1-x and l
3 has equation y=2x. l
2
: y=1-x 2x=1-x 3x=1 x=1/3 y=2/3 . l
3
: y=2x , y=2x , y=2x ,
P(1/3 , 2/3) has horizontal coordinate 1/3.
Ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
Further questions? Contact ssantos@ri.ac.uk
.
Page 4 of 8

Perfect thirds www.rigb.org
The Royal Institution of Great Britain Mathematics Masterclasses
9 th October 2009
Ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
Further questions? Contact ssantos@ri.ac.uk
.
Page 5 of 8

Perfect thirds www.rigb.org
The Royal Institution of Great Britain Mathematics Masterclasses
9 th October 2009
Q3: On the same square you folded earlier, use the folds made to fold the line of equation x=1/3.
Note for teachers:
Q4: On a fresh square of paper, fold the lines with the equations below.
Hint: decide on the origin, x-axis, y-axis before starting the fold, setting the origin, x and y axes as before.
 y = x
 y = 2x
 y = 4x
 y = 8x
 y = 1-x
 y = 3x
Note for teachers: for y=3x we intend children to use the construction above and fold y=3x as the diagonal of the rectangle with sides 1 and 1/3 units.
Ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
Further questions? Contact ssantos@ri.ac.uk
.
Page 6 of 8

Perfect thirds www.rigb.org
The Royal Institution of Great Britain Mathematics Masterclasses
9 th October 2009
Q5: We now want to divide a length in five equal parts. Generalise the method studied earlier to fold fifths. Explain how it works.
Note for teachers: fold the rectangle with dimensions 1 and ¼, fold its diagonal: this is the line y=5x. The intersection between y=1-x and y=5x gives us the desired fraction. Ask children to interpret the equations y=4x and x=1/5 as folds. y=1-x 4x=1-x 5x=1 x=1/5 y=4x , y=4x , y=4x , y=4/5 .
Q6: Generalise the method to divide a length in any number of parts. Explain why it works.
Note for teachers: below is an example of a sequence of folds to obtain 1/6, there are other ways like bisecting the 1/3
fold. In general, to obtain 1/n we can use the fold for 1/(n-1) as shown in the simultaneous equations below. The argument using similarity of triangles also holds for the general case.
Ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
Further questions? Contact ssantos@ri.ac.uk
.
Page 7 of 8

Perfect thirds www.rigb.org
The Royal Institution of Great Britain Mathematics Masterclasses
9 th October 2009 y=1-x (n-1)x=1-x nx=1 x=1/n
y=(n-1)x , y=(n-1)x , y=(n-1)x , y=(n-1)/n .
... 1/6, 1/5, 1/4, 1/3, 1/2
Ideas for the activity and the worksheet were taken from the book Project Origami by Thomas Hull.
Further questions? Contact ssantos@ri.ac.uk
.
Page 8 of 8