I mentioned in the previous post that I was considering doing some modular Sonobe origami in an upcoming workshop for middle school students. Wondering if this is a good idea, and thinking that I had better have some backup plans, I decided to make a list of origami models that I have used in school workshops in the past.
These simple models are nice for the very young and for beginners. Start with a triangle made from cutting standard origami paper along a diagonal.
A bit of playing around with the triangles should be enough for you to figure out how to fold examples like the ones above, and maybe even to come up with a few of your own. The instructions for the sailboat are on the OrigamiUSA diagrams page.
This is a very nice toy model from OrigamiUSA. See this post for some notes on it. Whenever I have used this with younger kids, I have always brought some pre-folded ones with me for them to play with and draw on.
A traditional model using a square origami paper, the instructions are also available from OrigamiUSA. It makes a nice pocket to put your frog into. See this post for more on this model.
The origami t-shirt is a model that I use to start off workshops for older students - for younger children I usually bring some paper that has had the initial folds completed, so they can get to the finished product in a few folds - this motivates them to go back and do the whole model themselves. You can find the many origami shirts online, the one I use is one by Gay Merrill Gross that appeared in the first issue of Creased magazine.
Maybe its because we don't send letters much that this model does not grab everyone as overly exciting. I like it, and it is good to have some origami that you can do with standard letter paper (lots of these available in schools). See this post about the crease pattern. Also from the OrigamiUSA page.
This is the first and only origami model that I learned as child and I enjoy sharing it with children. It's another model (like the frog and the cup) that you can play with, and like the envelope it can be made from standard letter paper. The diagram is available on OrigamiUSA. The boat becomes even more playful when it is used as "storigami" in the tale of The Captain's Shirt. Happily destroying your boat while reciting the tragic story teaches an essential lesson about the impermanence of the origami art form.
I had success with this model in a workshop for grade 7 and 8 students. There is a very nice overview of it here, and the model was also featured in Creased magazine under the name "origami blowtop," but I have not found the issue.
Its modules are the first few folds of the waterbomb, another popular model for beginners. The waterbomb base has also made its way into origami tessellations (see here). If I were to do a tessellation in a workshop, it would be the waterbomb.
All of the lesson plans for Creased magazine's "Teachers' Corner" are available online here. The picture frame is presented here in lesson 2. The blinz base used to create the picture frame is know to many children as the preliminary folds for the fortune teller or cootie catcher - another origami toy that is always fun to make.
I've not yet used this in a workshop, but have enjoyed showing people how to fold it. You can make these from those pesky magazine inserts, or maybe from bus transfers. The model comes from Robinson's book "The Origami Bible." I have a post about it here.
When considering how to present these models, and how to conduct workshops for young people, I've looked into the "origametria" approach described in this paper by Miri Golan and Paul Jackson. Perhaps because I have only done origami with students in workshop settings where the emphasis is not on the mathematical value (which I hope is intrinsic anyway), rather than as part of a regular classroom program, I have not followed the origametria principles very closely. However, their advice on using positive language and respecting the students's work and efforts are essential in any setting - any feeling students have about what they experience will outlive anything mathematical they happen to notice about the creases.
The resources on OrigamiUSA and Creased are great, but keep in mind that very few beginners are good at reading origami diagrams, and most children are put off by them. I like to have the instructions available for the students, but most of them learn the pieces by following along with you, and having you (and others) repeat the folds in front of them. A document camera is a plus, but you have to be willing to walk around and demonstrate the folds, and recruit helpers who are ahead of the others to assist you.
Update: A handout used for the workshop, which did include a little modular origami, is here.
I'm prepping for an origami workshop for middle school students, and am thinking about focusing on modular origami with sonobe units. In modular origami, many folded pieces of paper are assembled to make a model. Usually, all the pieces of paper (called units) are folded the same way. Like other forms of origami, modular origami is generally done without glue or scissors, so the pieces need to fit together so that they lock in place - putting the pieces together often feels like weaving or braiding. In past workshops for children at this age I have used a simple waterbomb octahedron (like this), but the sonobe presents a bit of a challenge - I might fall back on something with less frustration potential. On the other hand, that frustration is part of the lesson that sonobe teaches - it always seems like it just is not going to come together, and then it does.
There are some good instructions for modular origami projects using the sonobe unit out there - for folding the unit this one is good, and for assembly I like this one.
The version of the unit that I like to use is a simpler one than is presented in the instructions that I have found online, and is taken from Origami for the Connoisseur, by Kasahara and Takahama (not really being a connoisseur, it is one of the very few things I can put together from this book). I've attempted some diagrams for this below.
The unit has a few less folds than the standard version, and this makes a difference when you are making 30 or more of them.
The Simple Sonobe Unit
Sonobe Unit Assembly
Update: a handout based on these diagrams is here.
"Let none be ashamed to learn, for a good work requireth good counsel."
- Albrecht Dürer, 1520
Mathematics and Art, sometimes thought to be opposite poles of human experience and activity, can sometimes sit down together in conversation.
Sometimes we might think of the conversation as more of a lecture - with Mathematics providing some technical advice. Perspective drawing is the prime example - where mathematics allows artists to create realistic images, or else to subvert the expectations of realism (as in anamorphic art). Art does reply - for these mathematical tools, it repays the favor by providing Mathematics with inspiring teaching and learning opportunities (in the case perspective drawing, see this essay by mathematician Annalisa Crannell).
Art may ask questions of Mathematics - after all, math is not merely a tool; it's often the inspiration for Art's work. Geometry, often in the form of polygons, polyhedra, and tessellations, makes many appearances in art from the old (see for example the work of Albrecht Druer, or this essay on Raphael's famous painting The School of Athens), to the new (check out the latest Bridges mathematical art galleries). Art that springs from Math can, in turn, say something back to Math - providing motivation in math education (see most posts from mathmunch), and inspiration for working mathematicians. A big part of what gets me excited about the recreational math that I do are the pretty pictures (I've been putting pictures from this blog that I like over on this tumblr for a while).
The book Beautiful Geometry gives us a rare opportunity to listen in on an extended and fascinating dialog between Math and Art: here they are talking like old friends, sharing jokes, and discussing other subjects like history and literature through the pairing of brief and interesting essays by mathematician Eli Maor and beautiful illustrations by artist Eugen Jost.
You will likely want to join in the conversation - the essays and artwork in Beautiful Geometry are inspiring and motivating - prompting me, at least, to try to play around with their ideas. For example, the images below are from a Geometer's Sketchpad iteration based on Jost's artwork entitled 3/3 = 4/4, which accompanies Maor's essay on geometric series (Chapter 30).
These images represent some early partial sums of the series below - can you see how?
Jost's piece Pentagons and Pentagrams (Chapter 22), similarly had me reaching for GSP:
Although some of Jost's pieces allow themselves to be mimicked by us amateurs (armed with appropriate software), many are true art works, conveying an aesthetic that keeps them from being mere diagrams (while still saying something substantive about mathematics).
Several of the topics that Maor writes about are ones I've looked at before (Lissajous figures, means, hypocycloids, figurate numbers), but even in these somewhat familiar areas the essays and illustrations are nudging me to look back and explore some more. (I even learned something new about quadrilaterals: connecting the midpoints of a quadrilateral always yields a parallelogram - Chapter 3).
Jost's art and Maor's articles are going to inspire many of us to continue experiencing mathematics through beauty, and to look at art with a greater understanding of the math that helps to make it beautiful.