Math for eight-year-olds: graph theory for kids!

This morning I had the pleasure to be a mathematical guest in my daughter’s third-grade class, full of inquisitive eight- and nine-year-old girls, and we had a wonderful interaction. Following up on my visit last year (math for seven-year-olds), I wanted to explore with them some elementary ideas in graph theory, which I view as mathematically rich, yet accessible to children. Cover

My specific aim was for them to discover on their own the delightful surprise of the Euler characteristic for connected planar graphs.

Page 1


We began with a simple example, counting together the number of vertices, edges and regions. For counting the regions, I emphasized that we count the “outside” region as one of the regions.


Then, I injected a little mystery by mentioning that Euler had discovered something peculiar about calculating $V-E+R$.  Could they find out what it was that he had noticed?

Page 2+3

Each student had her own booklet and calculated the Euler characteristic for various small graphs, as I moved about the room helping out.Page 5Page 4








Eventually, the girls noticed the peculiar thing — they kept getting the number two as the outcome!  I heard them exclaim, why do we keep getting two?  They had found Euler’s delightful surprise!

Page 6+7

The teachers also were very curious about it, and one of them said to me in amazement, “I really want to know why we always get two!”

Page 8


With the students, I suggested that we try a few somewhat more unusual cases, to see how robust the always-two situation really was. But in these cases, we still got two as the result.


The girls made their own graphs and tested the hypothesis.Page 9Page 10+11


Eventually, of course, I managed to suggest some examples that do in fact test the always-two phenomenon, first by looking at disconnected graphs, and then by considering graphs drawn with crossing edges.Page 13Page 12








In this way, we were led to refine the $V-E+R=2$ hypothesis to the case of connected planar graphs.

Now, it was time for proof.  I was initially unsure whether I should actually give a proof, since these were just third-graders, after all, and I thought it might be too difficult for the students. But when the teachers had expressed their desire to know why, they had specifically encouraged me to give the argument, saying that even if some students didn’t follow it, there was still value merely in seeing that one can mount an argument like that.  Excellent teachers!

The idea of the proof is that $V-E+R=2$ is true at the start, in the case of a graph consisting of one vertex and no edges. Furthermore, it remains true when one adds one new vertex connected by one new edge, since the new vertex and new edge cancel out.  Also, it remains true when one carves out a new region from part of an old region with the addition of a single new edge, since in this case there is one new edge and one new region, and these also balance each other. Since any connected planar graph can be built up in this way by gradually adding new vertices and edges, this argument shows $V-E+R=2$ for any connected planar graph. This is a proof by mathematical induction on the size of the graph.

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Page 16


Next, we moved on to consider some three-dimensional solids and their surfaces.  With various polyhedra, and the girls were able to verify further instances of $V-E+R=2$.

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Page 18+19


The girls then drew their own solids and calculated the Euler characteristic.  I taught them how to draw a cube and several other solids pictured here; when the shape is more than just a simple cube, this can be a challenge for a child, but some of the children made some lovely solids:Page 21Page 20








In the end, each child had a nice little booklet to take home. The images above are taken from one of the students in the class.

Graph Booklet
What a great day!

You can find a kit of the booklet here: Math for eight-year-olds: graph theory for kids!

See also the account of my previous visit:  Math for seven-year-olds: graph coloring and Eulerian paths.


My new book:

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

41 thoughts on “Math for eight-year-olds: graph theory for kids!

  1. Maybe it’s time you collect these into “Math for children, that adults can enjoy too!” (because surely next year you’ll have a post about math for nine year olds, and so on).

    The topics could cover basic graph theory, perhaps very rudimentary explanation about Hilbert’s grand hotel, basic Ramsey theory or other combinatorics.

    If you use a font which looks like crayons, and nice colors, it could really stick.

  2. Ok, now do diff eq for kids (because i’m an adult and really struggling with it and graph theory was even harder to get my brain around)

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    • If you think about the case of a cube, say, then you can imagine puncturing one of the faces and stretching the cube out so that it lays flat in the plane. The face that you punctured becomes the “outside” region upon doing that. The same is true for all the other polyhedra that we considered — you open up one of the faces and lay the whole thing completely flat, so that the region you opened up becomes the outside region in the plane. One can also imagine the cube drawn on the surface of a sphere, in which case any one of the faces can be viewed as the “outside” region. It is interesting to consider the Euler characteristic for polyhedra that form a torus (like a donut) or more complicated shapes. You no longer always get 2 in these more general cases.

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      • It’s a great title and I wouldn’t change it one bit : ) I guess my parent comment was tangentially hinting at how deficient my schooling was and how much I appreciate you filling in the gaps for me : )

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  6. This is absolutely wonderful. I have been convinced for some time now that graph theory is accessible to kids starting at a young age – what a shame that most kids don’t get to encounter it! Here is a post about me introducing a (very basic) graph theory concept to my daughter when she was 4 . She is now 6, and I haven’t done too much follow-up graph theory. This post might have just inspired me to come back to it!

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  10. Thank you so much, this is just what I was looking for. My son (age 9) has come across the Utilities problem (probably because I gave him How Long is a Piece of String to read by Eastaway and Wyndham) and wanted to know WHY K_3,3 isn’t planar (although he didn’t phrase it that way). My Graph Theory is a little rusty, so I was searching for some accessible material I could go through with him and found this, which is spot on, as once we have established Eulers formula he will easily be able to apply it for himself to see why. (As an aside, I’ve never come across someone suggesting giving one house water access by having it’s own well in the garden before, I found that quite creative).

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