Math for nine-year-olds: fold, punch and cut for symmetry!



Today I had the pleasure to visit my daughter’s fourth-grade classroom for some fun mathematical activities. The topic was Symmetry!   I planned some paper-folding activities, involving hole-punching and cutting, aiming to display the dynamism that is present in the concept of symmetry. Symmetry occurs when a figure can leap up, transforming itself through space, and land again exactly upon itself in different ways.  I sought to have the students experience this dynamic action not only in their minds, as they imagined various symmetries for the figures we considered, but also physically, as they folded a paper along a line of symmetry, checking that this fold brought the figure exactly to itself.

The exercises were good plain fun, and some of the kids wanted to do them again and again, the very same ones.  Here is the pdf file for the entire project.

To get started, we began with a very simple case, a square with two dots on it, which the girls folded and then punched through with a hole punch, so that one punch would punch through both holes at once.

Fold punch and cut-page-001


Fold punch and cut-page-002IMG_20160222_071724Next, we handled a few patterns that required two folds. I told the kids to look for the lines of symmetry and fold on them.  Fold the paper in such a way that with ONE punch, you can make exactly the whole pattern.

Don’t worry if the holes you punch do not line up exactly perfectly with the dots — if the holes are all touching their intended target and there are the right number of holes, then I told the kids it is great!

IMG_20160222_071730Fold punch and cut-page-003The three-fold patterns are a bit more challenging, but almost all of the kids were able to do it.  They did need some help punching through, however, since it sometimes requires some strength to punch through many layers.


Fold punch and cut-page-004With these further patterns, some of the folds don’t completely cover the paper. So double-check and make sure that you won’t end up with unwanted extra holes!



The second half of the project involved several cutting challenges.  To begin, let’s suppose that you wanted to cut a square out of the middle of a piece of paper. How would you do it?  Perhaps you might want to poke your scissors through and then cut around the square in four cuts.  But can you do it in just ONE cut?  Yes!  If you fold the paper on the diagonals of the square, then you can make one quick snip to cut out exactly the square, leaving the frame completely intact.



Fold punch and cut-page-005IMG_20160222_071659

IMG_20160222_071844Fold punch and cut-page-006Similarly, one can cut out many other shapes in just one cut.  The rectangle and triangle are a little trickier than you might think at first, since at a middle stage of folding, you’ll find that you end up folding a shorter line onto a longer one, but the point is that this is completely fine, since one cut will go through both.  Give it a try!


Fold punch and cut-page-007



IMG_20160222_072122Here are a few harder shapes, requiring more folds.




Fold punch and cut-page-008It is an amazing mathematical fact, the fold-and-cut theorem, that ANY shape consisting of finitely many straight line segments can be made with just one cut after folding.  Here are a few challenges, which many of the fourth-graders were able to do during my visit.

IMG_20160222_072257What a lot of fun the visit was!  I shall look forward to returning to the school next time.

In case anyone is interested, I have made available a pdf file of this project.  I would like to thank Mike Lawler, whose tweet put me onto the topic.  And see also the awesome Numberphile video on the fold-and-cut theorem, featuring mathematician Katie Steckles.

See more of my Math for Kids!

A Mathematician's Year in Japan, by Joel David Hamkins, available on Amazon Kindle Books






21 thoughts on “Math for nine-year-olds: fold, punch and cut for symmetry!

      • I’ve likened symmetric models to buyer’s remorse: you begin with a model, and you keep regretting the generic you’ve chosen at the “generic filters store”. At the end you are left with a bunch of sets which make a model of ZF, but not necessarily ZFC.

        I don’t know if it’s possible to cut it in a single cut. But surely the template of a symmetric model is the part which does not “really” depend on the choice of generic, and thus impertinent to this choice, and only those names which are interpreted the same way by “most” generics.

        Maybe you should wait with this until they are 11 years old… 🙂

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  3. This is awesome! The exploration and the video. Can’t wait to show my students. They will dive right into this. Thanks for sharing!

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  5. The opposite ideas also are interesting to explore: mark fold lines on the paper and indicate either a location for a hole to be punched or a line to be cut. Can they work out what will result?

    • Yes, I had actually done that with a few of the students, and it works well. You don’t have to mark the fold lines, but just actually fold the paper, and then unfold it, and mark one hole. Where will the other holes be? The students draws all the holes, and then actually does it, to check how well they did.

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  9. GREAT !
    Would you have an objection if I translate this article in French & put it on my blog -with your refs and links as usual- ? (I’ll post a comment with the link under your post like previously)

    You might remember I have translated in french some times ago some of your posts
    * graph coloring, chromatic numbers, and Eulerian paths and circuits :
    * graph theory for kids! :
    My kids and people around me have unsurprisingly highly appreciated these 2 articles, I’d like to keep nurturing them…
    Congratulations for this work and kind regards !

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  13. I love this activity. I find myself wondering if it would be enhanced or destroyed by incorporating patty paper into the exploration. That is, putting these designs onto a translucent paper before folding will emphasize the overlay, and probably add clarity to some of the symmetries. It’s not clear to me if that would also take some of the fun out of the project; I suspect it wouldn’t.

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