School and University Partnership for Educational Renewal in Mathematics An NSF-funded Graduate STEM Fellows in K–12 Education Project

University of Hawai‘ i Department of Mathematics

UHM Department of Mathematics superm@math.hawaii.edu

All these traditional foldings require a procedure. When this procedure is known, the reproduction of a folding does not require any investment of mathematical knowledge, but only of the memory and the skill. On the other hand, the discovery of a procedure requires the interpretation of more or less explicit information, and in particular taking into account elements related to the forms and geometrical transformations. The lesson of traditional folding must meet at least two requirements:

∗ Students must discover new folds within their reach but by ways that stretch their ability. 

∗ The teacher must be present to comment on the mathematical observations and thinking when the opportunity arises. 

o Stopping the action to observe, to describe what happens (eg: fold a side in itself produces a perpendicular fold, folding one side on an adjacent side produces a bisector, etc.) 

o Proposing to undo the fold to observe what is produced by the folds (for example, the making of quack produces only isosceles triangles and rectangles) 

A procedure can be shown via: ∗ Film: there are demonstrations on YouTube (search for “folding paper” or “folding 

plane”). The video goes by quickly, but you can see it several times. ∗ Photographs: simple books of models are intended for the children. ∗ Animated images: there are many examples on www.origami‐kids.com especially 

aircraft and boats. ∗ Drawings: there are conventional symbols to represent movements and folds.  

Only the simple origami figures are accessible to the elementary students. There are many different techniques available online or in different books, and websites. The paper airplanes are an inexhaustible source of experiments (a very good example on http://avionenpapier.pagesperso- orange.fr/avionde.htm). Here are four more simple foldings: • the quack • the boat • the frog • the lotus For each one of them, various supports make it possible to vary the difficulty level. Pre-K – 1st grade: The quack and the boat already provide enough challenges. Work with the teacher at a table near the computer. Going back and forth between the observation of the slideshow and the film, let the students observe, interpret, try, and try again even if it means folding the paper several times.

UHM Department of Mathematics superm@math.hawaii.edu

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2nd – 4th grade: The quack is generally already known, therefore use the lotus, the boat, or the frog, and the simplest models of airplanes. The boat and the frog can be folded from the sketch without difficulty. If necessary, the students can consult the slideshow first and then maybe photos with comments, and then leading up to the film. 4th grade – 6th grade: Folding from the sketches of the boat, frog, airplane, and other models. When the students feel comfortable with folding the boat, they should:

∗ Research what changes happen when you alter the proportion of the rectangle that you start with. (longer rectangles, or closer to square) 

∗ Boat or frog: students learn to fold with just the video.