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46 At Right Angles | Vol. 2, No. 3, November 2013
ORIGAMICS: Activities based on exploration, conjecture and proof
in the classroom
by Kazuo Haga
An ‘Origamics’
Activity: X-lines
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Keywords: Kazuo Haga, origamics, paper folding, exploration, conjecture, proof, dynamic
geometry, Geogebra
Dr. Kazuo Haga is a retired professor of biology at the University of
Tsukuba, Japan. During his career as a biology professor, while waiting
for his experiments to progress, he used to while away the time doing
paper-folding and noting his mathematical findings through these
paper-folding sessions.
He devised a set of activities and classified them under the name
‘Origamics’ (coined by him) as the end product was different from
Origami. Unlike Origami, his exercises don’t produce paper models
but rather they lead to the study of the effects of the folding and seek
patterns.
Haga’s Origamic activities require students to explore simple,
geometric properties found when we fold paper in prescribed ways.
The aim of these activities is to give students easy-to-explore paper- folding puzzles so that they can experience a micro-version of the three
stages of mathematical research: exploration, conjecture and proof.
Here we look at one such activity from the chapter “X-Lines with lots of
Surprises”.
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At Right Angles | Vol. 2, No. 3, November 2013 47
Observe the following procedure:
Step 01:
Take an arbitrary point on the upper edge of a
square sheet of paper.
Step 02:
Place the lower left vertex onto the arbitrary point
and unfold. We obtain one creased line.
Step 03:
Place the lower right vertex onto the same point
and unfold. We obtain two X-shaped creases. We
shall call the pair of creases obtained as X-creases.
Repeat this procedure on different pieces of paper
with different starting points.
Different X-creases will be obtained by different
starting points, and therefore the position of the
point of intersection may vary.
Now take one piece of paper. Make a vertical book
fold to obtain the vertical midline of the square.
Do likewise with your other X-creases.
What do you observe from your various X- creases?
It seems that regardless of the starting point, the
intersection falls on the midline!
We now state our first observation:
The points of intersection of the X-creases fall
on the vertical midline.
Pile up the pieces of paper which you used to
make X-creases, and hold the pile up to the light.
What’s your observation?
You see that the points of intersection seem to
vary up and down along the midline, although
within a small range.
We state our second observation:
The points of intersection of the X-creases lie
along the midline and lie below the centre of the
square within a certain range.
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48 At Right Angles | Vol. 2, No. 3, November 2013
Next select a starting point and the corresponding
X-creases. Draw a line from the intersection point
to the starting point. Also draw lines from the
intersection point to the lower vertices of the
square. Then fold along an X-crease and hold the
paper to the light. It comes out that two of the
spokes coincide. Repeat with the other X-crease.
It appears that the third spoke also has the same
length!
We state our third observation:
The distances from the point of intersection
to the starting point and to each of the lower
vertices are equal.
We’ll leave it as an exercise for the reader to prove
the first and the third observation. Proofs will be
provided in the next issue of At Right Angles.
Please note you can use any Dynamic Geometry
Software such as Geogebra to simulate the above
mentioned paper folding exercise.
A B.Ed. and MBA degree holder, SHIV GAUR worked in the corporate sector for 5 years and then took up
teaching at the Sahyadri School (KFI). He has been teaching Math for 13 years, and is currently teaching the
IGCSE and IB Math curriculum at The Gandhi Memorial International School, Jakarta. He is deeply interested in
the use of technology (Dynamic Geometry Software, Computer Algebra System) for teaching Math. His article
“Origami and Mathematics” was published in the book “Ideas for the Classroom” in 2007 by East West Books
(Madras) Pvt. Ltd. He was an invited guest speaker at IIT Bombay for TIME 2009 and TIME Primary 2012. Shiv is
an amateur magician and a modular origami enthusiast. He may be contacted at shivgaur@gmail.com.
Reference
[1] ORIGAMICS: Mathematical Explorations through Paper Folding, Kazuo Haga (World Scientific Publishing Co. Pvt. Ltd)