Fold and Cut Theorem and IGP

The Fold and Cut Theorem is about having a shape on the paper and folding it in a way that eventually you use a single cut to remove that shape or figure from the paper. For example, if the shape is a square you can use four straight cuts, but by using the Fold and Cut Theorem you are only using one cut if you fold the paper as shown  below:

[insert pic of square here]

The Fold and Cut Theorem has been used and discussed widely and there is a YouTube video by Numberphile describing it here: Fold and Cut Theorem.

Also, I have seen TFCT used at math festivals like the Julia Robinson Math Festival (JRMF). Another variation of TFCT  for elementary schools was created by Joel David, who is a professor at the University of Oxford. He used that activity at a school with nine year old kids. You can read it about it on his website.  He even created a booklet for it and you can download it here.

In 2017 I found there is a very nice way to create an activity about IGP and TFCT. The activity goes like this:

  1. Find an IGP such that it only has two or three simple shapes (motifs).
  2. Print one simple shape on a single sheet of letter paper and make 20 copies (depending on the number of students you have).
  3. Ask students to use only one cut to remove the shape from the paper.
  4. When they get the shape they need, tile it on a wall or sticky-note paper.

It’s true I designed  this activity for IGP, but it should not be limited to IGP. Actually, if you start with some simple pattern like a Greek or Chinese pattern it will be even better. If you use  white paper then the kids will also have the opportunity to color their tile before they stick it on the wall or the large paper. This way the activity will be more personalized.

Maybe someone will ask: where is the math here? Well, there is a lot of math going on here. For example when you fold the paper you are using symmetry. In the case of the square you folded across three lines of symmetries and sometimes it will be nicer if you are less helpful and let the kids figure out where they need to fold the paper, or at least when you do the folding across the symmetry line just ask them why it works. I did it this way before and some of them figured it out after some trial and error. Especially since the shapes of IGP are mostly symmetric, it’s hard not see it that way.

Bellow will give some examples of Fold and Cut Theorem and IGP

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