Lesson Plan (coming soon)
For years there was a myth that paper could not be folded more than 7 times – in fact I remember trying at primary school with a huge roll of wrapping paper. We were so confident up until the 5th fold.
This myth has since been debunked – often in fact – but no more spectacularly than a high school student from the US by the name of Britney Gallivan. She too was given a challenge of folding a paper more than 7 times by her teacher. In fact the teacher set the challenge of 12 times, and yet she was still not put off by the feat. She incredibly managed to find a very luxurios and thin roll of toilet paper and a huge long (and we are talking huge) corridor so that she could roll it out and fold it a remarkable 12 times and secure a world record!
She didn’t stop there though – In fact she didn’t just start off by buying lots of luxury toilet paper – she used maths! She noticed that the biggest obstacle to folding paper is the curvature from the point of fold. This curvature is barely noticeable on the first couple of folds, but after a a while, it becomes very difficult to fold the paper due the curve created at the recent fold. The radius of this curve soon overtakes the length of the paper and so it becomes impossible. She derived a formula based on the thickness of the sheets that tells you how long the sheet needs to be to fold n times, or how many times you fold a paper of n length. You can see her explain more in this video:
This is not an unsolved problem like many others in this section – but it is a great story of a student being a record holder thanks to applying her maths to a problem.
Also – it is a fantastic introduction to exponential growth. I like to do this in a quick challenge with students as young as 6 or 7. Counting the amount of layers show why it is so diffcult to fold over ten times, and remarkably if you could fold paper 42 times – it would reach the moon! That is a good investigation to explore and one I have included in the lesson slides.
My Experience
Start with the challenge of folding a paper 7 times using an A4 piece of paper. Then declare that you were told that it is impossible to fold it more than 7 times. In my experience the next step is most often – bigger paper. So have some A3, or A2/A1 if possible, to hand.
Start asking questions – why is it so difficult to fold more than 7 times?
Then to bring in some maths – start discussing about thickness of the layers – and more importantly – the amount of layers that occur on each fold. Have the students document how many layers occur in each fold. Then you can describe how this is called exponential growth. You can even link this to COVID and it’s rapid spread across the world. Or recall the famous chess and rice story.
After completing this quick maths activity, I then talk about the great Ms Gallivan and her discovery, and finally expand on this activity with a look at theoretical folds. Something that mathematicians always do, as there is almost always more to explore in every area of maths, is to look at what could happen if we were to be able to keep folding.
Taking a piece of paper at about 0.8mm – you can work out how many folds it takes to have a thickness of say 1km, or to the distance of the Moon or Sun. Or, in a classic ratio question approach – try to work out a rough estimate of a piece of paper by folding it four times and measuring the height, and then dividing by 16.
Maths Beyond the Curriculum 