Whenever I say this, the unfinished sentence in my head is ‘Fold-and-cut, baby! Fold-and-cut.’ I am totally weird.
The fold-and-cut theorem states states that any shape with straight sides can be cut from a single sheet of paper by folding it flat, possibly with many folds, and making a single straight complete cut.
I have been impatiently waiting to try this activity with others. Last night was the fourth in a series of teacher professional development workshops loosely based around developing mathematical thinking through puzzles and games. As Adelaide edges into winter and the days get shorter, it’s getting harder to rush out of class and head to an early evening PD session instead of a glass of wine on the couch at home. The semester is drawing to a close, and energy levels are low. I’m sure others feel the way I do, so I decided to open with a ‘slow math’ activity — gentle, therapeutic folding and cutting.
This post is doubling up as the list of resources for participants. We didn’t use everything listed below, but it’s all worth checking out.
We started with the first 2:30 min of Katie Steckles’ Numberphile video on the fold-and-cut theorem.
Then I gave out copies of pages 6-8 of Joel David Hamkins‘ handout. (You can get it at this page.) Printing each page on different colour paper helped distinguish between problems — and it looks pretty! Have lots of copies for participants in case they want to try again. Last night it turned out that two per person was enough, but with students I’d suggest a few more.
At the end we watched the rest of Katie’s video, including her demonstration of cutting all 26 letters of the alphabet, each with one cut. We also had a quick look at the more intricate fold patterns on Erik Demaine‘s site. Erik also lists mathematical papers about the fold-and-cut theorem. We spent just over 30 minutes on all these activities.
I had forgotten in the moment about this suggested progression (easy to hard) from Bowman Dickson: Equilateral Triangle → Square → Isosceles Triangle → Rectangle → Regular Pentagon → Regular 5 pointed Star → Scalene Triangle → Arbitrary Quadrilateral. (Read how his fourth graders went with it here.) I did remember that a scalene triangle is supposed to be reasonably tricky, so I threw that out as a challenge. I suggest having lots of blank paper and rulers available so that people can try their own shapes.
Patrick Honner has a thorough list of folding resources, including fractal one-cuts, which I’m keen to explore further. Kate Owens has a detailed agenda for a four-hour teacher workshop on paper folding with links to standards. I noted her advice about needing strong hole punches for ‘one-punch’ activities, which is why I bypassed those particular challenges last night. Mike Lawler explored this as a Family Math project with his sons; videos and details here. Finally, the Art of Mathematics suggests it as a good ‘first day’ activity. See links at the bottom of this page of other suggestions of what to do on the first day of semester.
Have you done these activities? Do you have more resources to share? I’d love to hear about your experience.
Update (27 July 2016)
Having done this with pre-service teachers this week, I have two more resources to add:
- An five-page handout (updated on 10 June 2021). The first three pages with symmetric shapes (from JD Hamkins). The final two pages (from Patrick Honner) include irregular shapes and a Koch curve.
- Chapter 3 of the free book ‘Art & Sculpture’ from Discovering the Art of Mathematics. This has a fabulous sequence of questions.