1. Teaching the Geometry of the Plane Using Origami
2010 Singapore Math Challenge
Teaching the Geometry of the Plane Using Origami
Hello, everyone!
My name is Masashi Sanae. I teach mathematics at Ritsumeikan High School.
Since I am not so good at English, I will have Mr. Elliff, who is also from Ritsumeikan High School, assist me today.
In this teacher’s session, I’d like to speak about ‘Teaching the Geometry of the Plane Using Origami’.
2010 Singapore Math Challenge
Masashi Sanae & Stephen Elliff
Ritsumeikan High School

2. 2010 Singapore Math Challenge
Origami
Toady I will introduce the teaching plan of geometry using Origami.
Do you know Origami?
Origami is the folding of paper to make objects, and it is a classic children’s activity in Japan.
The left photograph is a "Crane" made by using Origami, and the figure on the right is the expanded figure .
In Japan, people often give hospitalized people 1000 of these cranes as encouragement.
These cranes are called ‘Senbazuru’ in Japan.
We can make various figures by using origami besides the "Crane".
However, origami is not only used as child's play.
It is possible to actually use it as a teaching tool in mathematics education,
especially in the field of geometry.
Using origami in geometry. is called "origamics“.
I ‘d like to introduce the basics of this concept today.
ORIGAMICS

3. The Basic Drawing Method
2010 Singapore Math Challenge
The Basic Drawing Method
①perpendicular bisector
②perpendicular to
line segment through a point
③angle bisector
First of all, I’ll talk about the basic drawing method.
Plain geometry is studied from junior high school to the first year of senior high school in Japan.
A ruler and compass are used to produce the basic drawing method in plain geometry,
the perpendicular bisector, the perpendicular line to a line segment through a point and the angle bisector,.
In the case of the perpendicular bisector, we draw two circles from both ends of the segment with the same radius,
and then draw a straight line that passes through the intersections.
As for the other two drawing methods, we can also draw them by using a ruler and compass .

4. 2010 Singapore Math Challenge
Basic Folding
①perpendicular bisector
②perpendicular to
line segment through a point
③angle bisector
If we do these things with origami, we can perform them as so.
Take the two ends of a line segment, and fold the paper so that the two ends are on top of each other.
The line created by the fold is the perpendicular bisector.
You can also create the perpendicular to a line segment through a point, or the bisector of an angle through similar folds.
As you can see, the process is simpler than when using a compass and ruler.
It is also easier for students to grasp the concept.

5. 2010 Singapore Math Challenge
Incenter
One of the first activities in the class involves triangle centers.
Choose one vertex and take the two sides.
Fold the paper so that the two sides are on top of each other.
Do the same with the other vertices.
The point where the folds meet is the incenter.
If you fold the paper with the incenter as the vertex, the three sides lay on top of each other.
And most importantly, you can easily see that these three lines have the same length.
Thus you can create the circle which is inscribed within the triangle.

6. 2010 Singapore Math Challenge
Circumcenter
As well as the incenter, we can create the circumcenter.
Choose one side and take the two ends.
Fold the paper so that the two ends are on top of each other.
Do the same with the other sides.
The point where the folds meet is the circumcenter.
If you fold the paper with the circumcenter as the vertex, the three folds lay on top of each other.
You can easily see that the three lines have the same length.

7. 2010 Singapore Math Challenge
Centroid
The next activities in the class involves a centroid.
If you create a centroid,
you can easily see that the length between the midpoint of one side and the vertex can be divided into three sections.

8. 2010 Singapore Math Challenge
Excenter
Now we consider the excenter as in the figure.
Choose vertex A and take the two sides AB and AC.
Fold the paper so that the two sides AB and AC are on top of each other.
And then take the line BD and fold it so that BD is on top of BC.
Do the same with CE so that it is on top of CB.
If you fold the paper with the excenter as the vertex, the lines BD, CE and the edge BC lay on top of each other.
You can easily see that the line segments from the excenter to the lines BD,BC and CE have the same length.

9. Application - All Sides Overlap
2010 Singapore Math Challenge
Application - All Sides Overlap
We can fold a polygon
so that all sides overlap
Using the techniques we have just learned I give you the following proposition.
We can fold a polygon so that all sides overlap.

10. 2010 Singapore Math Challenge
A Triangle
In the case of a triangle, you can already do it.
Because this is the case of the incenter..
Incenter

11. A Quadrilateral with One Open Side
2010 Singapore Math Challenge
A Quadrilateral with One Open Side
In the case of a quadrilateral with one open side as in the figure,
if you fold the paper with the excenter as the vertex, the three sides overlap.
All the sides are straight, so we can cut a straight line with scissors.
Excenter

12. A Quadrilateral with One Open Side
2010 Singapore Math Challenge
A Quadrilateral with One Open Side
Well, how do we fold the paper in the case of an uneven quadrilateral with one open side as in the figure.
We extend the side DC.
We name the intersection of AB, and the extended line segment of DC, point E.
If you fold the paper with the excenter of triangle BCE as the vertex,
you can see that the three sides DC, CB, BE overlap.
Excenter

13. Circumscribed Polygon
2010 Singapore Math Challenge
Circumscribed Polygon
In the case of a polygon that circumscribes a circle, it is obvious.
If you fold the paper with the incenter as the vertex, all sides overlap.

14. 2010 Singapore Math Challenge
Polygon
Well, how do we fold the paper in the case of a general polygon?
We consider the quadrilateral as in the figure.
First we extend the two sides AD, and BC.
We name the intersection, E.
If you fold the paper with the incenter of triangle ABE as the vertex,
you can see that the three sides AB, AD, BC overlap.
And then let’s consider the triangle CDE.
If you fold the paper with the excenter of triangle CDE as the vertex,
you can see that the three sides CD, CE, DE overlap.
That’s to say, all sides overlap.
All the sides are straight, so we can cut a straight line with scissors.

15. 2010 Singapore Math Challenge
Polygon
Well, how do we fold the paper in the case of an uneven polygon?
Let’s consider the quadrilateral as in the figure.
We extend the side BC.
We name the intersection of AD, and the extended line segment of BC, point E.
If you fold the paper with the incenter of triangle ABE as the vertex,
you can see that the three sides AB, BC, AE overlap.
And then we consider the triangle CDE.
If you fold the paper with the excenter of triangle CDE as the vertex,
you can see that the three sides BC, CD, DE overlap.
That’s to say, all sides overlap.

16. 2010 Singapore Math Challenge
Similarly we can fold all polygons so that all sides overlap.
But if the polygon is complex, it is very difficult to imagin.
For example, consider the swan as in the figure,
If you fold the paper properly,
you can surprisingly fold the paper so that all sides overlap!

17. 2010 Singapore Math Challenge
All sides overlap, so that we can cut a straight line with scissors.
And if you unfold the paper, you can see the swan!

18. 2010 Singapore Math Challenge
Geometry with Salt
These photographs are mountains made with salt.
You can easily create these salt mountains if you just pour salt on the polygon.
The mountain ridge is the fold line as you saw a little while ago.
Because I have little time, I can’t speak about other topics of Origami,.
But as you can see, Origami is a good tool for mathematics!

19. 2010 Singapore Math Challenge
Reference
M.Sanae’S HomePage (Masashi Sanae)
Origami and Mathematics （Konichi Kato）
・折り紙は多くの教師が教えているのですか？
折り紙を授業の中で用いている先生は多くはありません。
A few teachers used Origami in mathematics education.
・それは何故ですか？
日本のカリキュラムの中では，作図は定規とコンパスを用いて行う事を標準としているからだと思います。
The reason why a few teachers use Origami in mathematics education is as so.
In Japan, using a ruler and a compass is the standard drawing method.
・折り紙を用いることの利点は何ですか？
定規・コンパスを用いるより手軽にできることだと思います。例えば長さが等しい事実などを体験できる点です。また，角の三等分などは，定規とコンパスではできないのですが，折り紙を用いると可能です。利点の方が多いと思います。
The advantage of using Origami in mathematics education is that it is simpler than using a ruler and a compass.
So for example, you can create triangle centers easily.
You can’t divide an angle into three by using a ruler and a compass, but you can divide it by using Origami.
There are more advantages .
・デメリットは何ですか？
特にすぐに思い浮かぶ点はありません。
I can’t think any of the disadvantage off the top of my head.

20. Thank you for listening.
2010 Singapore Math Challenge
Thank you for listening.
Thank you for listening.
Thank you.