Paper Folding 7
Fatima Teem
As you can see above, you can generate the sequence of numbers 1, 2, 4, 8, 16, 32 and so on, just by folding the paper in half again each time. This means that there is an exponential relationship between the number of folds you have made and the number of areas created on the paper.
Here is an example of folding the paper around the centre to produce rotational symmetry. I worked with a student to produce snowflakes with 9 points, 12 points, and other points, after watching this interesting video by Vi Hart.
If you fold a paper in half a bunch of times, you can create a tesselation by cutting portions of the paper out. The number of folds and the size of the repeated portion of the tessellation have an interesting relationship.
For further resources on paper folding and mathematics, see this TED talk by Robert Lang, this book on the mathematics of paper-folding, and this useful PDF describing some geometry theorems that can be demonstrated through paper folding. See also this very interesting article on fraction flags (via @DwyerTeacher).
For each of the problems below one person should make the shape and then be convincing. Your partner is the skeptic. When you move to the next question switch roles. Start with a square sheet of paper and make folds to construct a new shape. Explain how you know the shape you constructed has the specified area.
I have great affection for this book. After looking for a book about folding paper for many years I at last stumbled across this book in Harrods book department during a trip to London with my then fiance to buy furniture for our future home. She bought the book for me and inscribed it with the date, 4th April 1956 and the words "Furniture Day at Harrods." In the circumstances I felt a little embarrassed by the dedication of the book which was To Fumble-fisted Fathers"!
My copy of the book is the hard-backed British edition published by Thames and Hudson in 1955. I later learnt that the book was originally published by Harcourt Brace and Company of New York in 1948, also in hard-back. The British and American editions are virtually identical except that the American edition has some information about Mme. Soong on the back of the dust cover which is omitted from the otherwise identical dust cover of the British Edition.
Paper-backed editions of the book were issued and I have one published in 1967 by World's Work Ltd., of Kingsworth, Surrey, in England. The Foreword has a final paragraph which reads:" I hope that my American friends, especially the Junior Americans, will find great interest and enjoyment in the art of paper folding." This was omitted in the original British edition, so I can only suppose that the paper-backed edition was taken from the American edition and not that of Thames and Hudson.
Several books, including the notable "Paper Toy Making" by Margaret Campbell in England and Joseph Leeming's "Fun with Paper" in the United States survived in print from before the Second World War, but as far as I know, Maying Soong's was the first book on paperfolding to be published after the war. As we all know, it was the first of very many others.
As the blurb on the 1948 American edition says, Maying Soong came from a distinguished pre-communist Chinese banking family. They were as much at home in the West as in China and Mme Soong received a British education in Shanghai. One of her main interests was music. She studied further in England, France and Switzerland. After the Communist take-over, Maying Soong settled in the United States. She married a brother of Mme Chiang Kai-Chek , the wife of the Chinese leader, who was forced out by the Communists and who continued to govern in Taiwan.
My original copy didn't have any biographical information and I was inclined at first to take the "Chinese" in the title with a pinch of salt. All of the models seemed to be either part of the repertoire of paper-folded models we knew in the West or apparently newly created.
Since then I have changed my views. Mme Soong certainly learnt her paperfolding in China and so "The Art of Chinese Paper Folding" is appropriate. I have for many years tried to gather information about paperfolding in China and I have come to the conclusion that the traditional kind of paperfolding has long been practised in China, just as it has in Japan and the West. From time-to-time models which are certainly Chinese inventions still percolate to the West. Certainly the Chinese do not appear to have developed the elaborate ceremonial folding of the Japanese, not the advanced adult folding of the kind seen in the Kayaragusa (which is often known incorrectly as the Kan no mado). But because the simpler traditional models appear to have been so international, we should hesitate before attributing their origin to Japan without further evidence.
It is of interest that "The Art of Chinese Paper Folding" contains instructions for what is known as the "Chinese" Pagoda. While not the first modular fold, it is certainly an early example of this style and shows that modular folding is not at all a recent invention.
Lillian Oppenheimer, who sought out paperfolding wherever it might be found, told me that she tried to communicate with Mme Soong, but without any real success. She put it down to the elevated society to which Mme Soong belonged. It appears that Lillian and Maying Soong never met.
Some years ago I was bold enough to write to Mme. Soong and received a charming letter in reply. She confirmed that some of the models in her book were her own design and she also sent me diagrams for two of her models which were not included in the book.
I am teaching exponential functions in my Algebra II classes this week. And I just came back from this teaching conference, where one of the sessions included a few handouts of the types of problems that this one charter school uses. And lucky for me, one was on exponential growth and decay.
I know this is supah late, but I wanted to say I used this in my College Algebra and it worked well! I blogged about it here: -epsilon-delta.blogspot.com/2012/03/exponential-functions-folding-and.html
Any number of household items can be used for scoring the paper. Nails, exact-o blades, compass points, safety pins, the list goes on... The important thing is that implement chosen does not tear the surface of the paper or cut through it. For this instructable I used the end of a T-pin in combination with the body of a mechanical pencil.
Before starting on the project, I did a few tests to compare the effectiveness of a nail and the T-pin. I found that the T-pin tore the rough side of the paper I was using but worked well on the smooth side of the paper, providing a crisp, sharp edge. The nail worked fine but made a less defined edge.
When making curved folds I find its best to use a medium weight paper as the paper tends to stay in its desired shape better. For my piece I used Canson Mi-teintes drawing paper, which is available at most art stores.
Begin by drawing a series of curves on the paper until you are happy with the design. Using a scoring implement, follow the lines to create a guide for the curved folds. The process of folding is quite easy but it may take some time to develop familiarity with it. Start at an end of one of the curves and begin folding little sections of the curve at a time until you have folded along the entire length. I find that things can get difficult once multiple curves have been folded. In these cases it may be best to flatten the whole piece of paper and work on the lines that were troubling. When the piece is complete try placing it in different lighting conditions that highlight the folds.
As lovely as these stars are, what really caught my attention was the way Tobias showed how to use paper folding to make a pentagon from a square. This square-to-pentagon transformation was in a separate video, and since it will take me about two days to forget everything I saw in the video I drew out the directions.
Came back to it today. Part of my problem was misunderstanding the points fold, which finally I really looked at. The other thing was an idea that helped. The first point I folded I then unfolded so it could be the last point already half done.
Years and years ago, when I was still in college studying to become a math teacher, I had a professor who said we could attend the Detroil Area Council of Teachers of Mathematics conference instead of doing one of the projects for class. I do not remember much about that class or what we learned but I remember the conference.
More specifically, I remember one of the sessions that I attended. The session was held in the library at Lake View (I think) High School in St. Clair Shores, Michigan. I remember that I sat in the back of the class. There was a morning session and an afternoon session--I attended both. I remember leaving with tons of ideas and have used many of them over the years. (The second session was called 'scrap paper math'.)
The steps that follow this are from this session of the DACTM conference. I do not remember the instructor's name or I would give him the credit that he so richly deserves.
I have made dozens of these over the years. They make very nice Christmas ornaments. Even made from old math papers they look pretty. Since it is totally hollow inside and can be disassembled without tearing the paper, I have used the 20 pointed star to wrap small gifts.
In the class, the instructor modeled the procedure that he used with his students. I can speak from experience--it works. He was folding paper as he gave instruction but he was also talking and asking questions. He loved asking questions about fractions and the size of different angles. He kept using math vocabulary and encouraging us to use the vocabulary in our answers. The only way to really get comfortable with technical terms is to use them repeatedly.
Start with a square sheet of paper--he used scrap paper. If there is anything that a teacher has a lot of, it is scrap paper. He had brought stacks of old worksheets, news letters, even student work that he had lying around. You do not have to go buy origami paper. All you need is a nice paper cutter--try the art room or the main office. You can use scissors but it takes longer and the cuts are not as straight. I usually mass produce a whole box of squares using whatever paper is available.