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6TH GRADE 6TH GRADE THE Powered By Docstoc
					                                   6TH GRADE
                              THE ART OF MATH
                              Origami: paper folding

     Students will discuss the Japanese culture and how the art of origami reflects this
     culture. We will also discuss the history of origami.
     Students will participate in creating a polyhedron cube, each student will be
     responsible for creating 1 or more sections that will contribute to the cube.
       Students will begin to understand how geometry and origami are closely related,
       showing symmetry, angles and mirror images, area, congruency, volume and
       many other concepts.

Polydedron- A polyhedron (plural polyhedra or polyhedrons) is often defined as a
geometric solid with flat faces and straight edges
 Cube- A solid with six congruent square faces also called facets or sides.. A regular
hexahedron. Each face has four equal sides and all four interior angles are right angles.
 Geometry- Geometry is one of the oldest sciences. Initially a body of practical
knowledge concerning lengths, areas, and volumes.
 Symmetry- generally conveys two primary meanings. The first is an imprecise sense of
harmonious or aesthetically pleasing proportionality and balance;[1][2] such that it reflects
beauty or perfection. The second meaning is a precise and well-defined concept of
balance or "patterned self-similarity" that can be demonstrated or proved according to the
rules of a formal system: by geometry, through physics or otherwise.
 Angles- A shape, formed by two lines or rays diverging from a common point (the
 Area- The amount of two-dimensional space. Area is also used to measure the outermost
surface of an object.
 Congruency- if two shapes are congruent, they are the same (shape and
Volume- is how much three-dimensional space it occupies,
INFORMATION: Like many things in Japanese culture, origami (from "oru" meaning
to fold, and "kami" meaning paper) has its origins in China. It is believed that paper was
first made, and folded, in China in the first or second century. The earliest records of
origami in Japan date to the Heian Period (794-1185). It was during this period that
Japan's nobility had its golden age and it was a time of great artistic and cultural
advances. Paper was still a rare enough comodity that origami was a pastime for the elite.
Paper was folded into set shapes for ceremonial occasions such as weddings. Serrated
strips of white paper were used to mark sacred objects, a custom which can still be seen
in every shrine to this day.

It was in the Edo Period (1600-1868) that much of today's popular traditional culture
developed as forms of entertainment for the merchant classes and the common people.
Kabuki and ukiyo-e are just two examples and origami also gained poularity. By the mid-
19th century, 70 or more different designs had been created. But aside from its
ceremonial use, its popularity has been in decline since the Meiji Period (1886-1912) and
the modernization of Japan.

In the mid-1950s, 11-year old Sasaki Sadako developed leukemia as a result of her
exposure to radiation as a baby during the atomic bombing of Hiroshima in 1945.
Tradition held that if you made a senbazuru (a thousand paper cranes) and made a wish
after completing each one, your wish would come true. Sadako set about making the
tsuru, wishing for her own recovery. But as she continued, she began to wish instead for
world peace. She died when she had made only 644 and her school friends completed the
full number and dedicated them to her at her funeral. The story helped inspire the
Children's Peace Memorial in Hiroshima and a statue of Sadako in Seattle. Each year
on Peace Day (August 6th), thousands of origami tsuru are sent to Hiroshima by chidren
all over the world. There are too many folding steps in making a tsuru for me to describe
simply here and lots of sites already provide this and many other ideas (see the links

In more recent times, the Internet has helped spread the word about Japanese culture,
both the long-hidden aspects and the things that western people had heard of but knew
little about. Origami is one such facet that lends itself to the visual medium. Designs can
be explained in line diagrams or photos and, with practice, can be mastered by anyone.
The next step, as with any art form, is to find a topic or field that appeals and develop
your own style. In the words of Yoshizawa Akira, the 'acknowledged grandmaster of
origami, the father of modern creative origami':

ELEMENTS AND PRINCIPLES: line, shape, balance, form,

MATERIALS: origami paper cut to 3 inch squares

GLE’s: I3c, II1a, II2a, III1a, IV2a, V1a
Origami Polyhedra - nuwen.net
In all chaos there is a cosmos, in all disorder a secret order


First I will detail how to make a single piece (step-by-step) and then I will outline how to
assemble them into a model of a polyhedron.

Paper type and paper size: 3 INCH SQUARES

Accuracy issues and making the squares:

Basically, All folds should be absolutely accurate and creased sharp; all measurements
must be absolutely accurate and all cuts must be absolutely accurate. If you make a small
error, the pieces will not be too adversely affected, but remember that you are making
a large number of (supposedly) absolutely identical pieces. If you are inaccurate in
your folding, cutting, or measuring, the pieces will be irregular and will not fit together

Folding a single piece:

Solid lines mark valley folds, and no mountain folds are involved anywhere. It will be
obvious; don't worry. Also, do not label your square in any manner while folding it. The
only time a pencil is needed is when dividing up a sheet of paper into squares to be cut.
All folds can be made without any markings; my markings are to make my instructions
clear.) Also, when I refer to spinning the paper 180 degrees around, I mean rotating it
around on the table. You actually will need to turn the paper over near the end of the
process, but I'll make it clear then.

                                     A completed piece.

This is your final goal for this section: to make a completed piece. This process is
actually very easy and quick; once mastered, it ought to take you one minute per piece.
(For the purposes of photographing this process, I used a 3-inch square; you should use a
1.5-inch square. This is why my photos will have more blue lines on them than your
squares will have.) So, start with a 1.5-inch square of paper:
                               A freshly cut square of paper.

Make a precise and creased fold lengthwise. Here is what I mean:

                                Dividing the square in half.

                     Here is the process halfway through completion.

The actual purpose of this fold is just to give you a reference to make the next two folds.
Unfold the paper and lay it flat. Take the bottom edge of the paper and fold it to the
center crease; then spin the paper 180 degrees and do the same. Here is what I mean:
                              The folds that you'll be making.

 Here is the process halfway through completion (both folds are shown simultaneously;
                     you should make them one at a time, of course).

Okay. Now, unfold the paper and lay it flat. (You will be folding the paper here again;
you just need to do some things in the meantime. I'll refer to these folds, rather
uncreatively, as "the second and third folds" later on.) Take the bottom-right corner of the
paper and fold it into a triangle so that what was the left side of the paper now lies on top
of the second fold you made. Leave that folded, spin the paper 180 degrees and make the
same fold. Here is what I mean:

                                   Folding two triangles.
This is the traditional fold you make when producing a (lousy) needlenose paper airplane.
Now, take the bottom-right corner of the paper and make another needlenose-type fold.
That means bringing the fold that you just made to lie exactly on top of the second fold
you made. Then rotate the paper 180 degrees and make the same fold. The following
image's bottom-right corner shows the end result of this process; the upper-left corner
shows it halfway through completion.

                              Another "needlenose" type fold.

That fold was hard to describe but easy to perform; it's used in the production of lousy
needlenose paper airplanes everywhere. Now is the time to remake the second and third
folds you made:

Okay. Now, take the bottom-left corner of the paper and fold it so that what was the left
edge of the paper now lies on top of the top edge of the paper, producing a triangle, like

Rotate the paper 180 degrees and repeat. A parallelogram! Now, you must tuck in that
large triangle fold into the paper. I have no way to easily describe this in words. Here is
what I mean:
                   The left fold is tucked in, while the right fold is not.

Then rotate the paper 180 degrees and tuck in the other fold, resulting in:

Good. Now flip the paper over and rotate it so that it looks like this:

                                 The backside of the paper.

Fold the bottom point of the paper straight up to meet another vertex of the
parallelogram, like this:

Then rotate the paper 180 degrees and repeat, producing this:
Okay. Now you need to give the paper a bend in the middle. (This is actually a mountain
fold, but I could have you flip the paper over again and make a valley fold; so what?)
You will end up with this:

Now, as you can see, you have a finished piece:


Making models:

1. The cube. A boring cube. The easiest to construct, it takes 6 pieces.
2. The octahedron. (A stellated octahedron, actually.) Takes 12 pieces. Not difficult.
3. The icosahedron. (A stellated icosahedron.) Takes 30 pieces. Also not difficult.
4. The stellated truncated icosahedron. Takes 270 pieces... I think. Difficult (though not
overly so), but incredibly time-consuming.

I suggest that you start off with the cube and work onwards. Now that you have enough
pieces constructed to make the model of your choice, you need to learn the basics of
model construction. A piece has two sharp corners and two pockets, which allow them to
interlock. Here are two pieces placed to illustrate this:
And here they are locked together, corner in pocket:

Here is a third piece, placed over the first two:

And here the third piece is locked in:

There is a free corner and free pocket that can be locked together. Doing so necessitates
forming the three pieces into a three-dimension configuration that I call a peak:
It is vitally important to understand what I mean when I say "peak", because peaks are the
founding blocks of your models. (Although you should assemble your models piece-by-
piece and not make a bunch of 3-piece peaks and then assemble the peaks. The former
works; the latter won't. Example: the cube contains three peaks, but only requires six
pieces. Solution: pieces can form more than one peak. Trust me: go piece-by-piece.) Now
you should be able to assemble a cube. Here is a cube, pictured with a peak at the center
of the image:

                                         A cube.


Updated a long time ago.

  1. procedure was followed
  2. time was used wisely
  3. participation in group projects
  4. created at least 1 successful polyhedron for individual grade

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