KinderGraph Theory

Jenny is sick and Ellie had homework, so Kenzie and I were left alone to have our own fun. We got out a small whiteboard (one of many made from a giant 4′ x 8′ sheet purchased at Lowes and cannibalized into whiteboards of various sizes).

Starting with an idea from the wonderful Socks are Like Pants, Cats are Like Dogsby Malke Rosenfeld and Gordon Hamilton, we decided to build “pictures” that showed how many letters different names had in common, beginning with just two: KENZIE and ELLIE.

KENZIE and ELLIE diagram.

We connect them with three lines because they share two E’s and one I. Once she had the hang of it we added the name of her friend: EMILY. Kenz then completed the picture.

KENZIE, ELLIE, and EMILY diagram.

We played around with various other three-name combinations that included ones where one name wasn’t connected to the other two, no names were connected, and one where I challenged her to find three different names (of kids she knew) that had the most connections possible. We also looked at four-name games. I thought that Kenz would struggle to see that there were six comparisons with four names but she didn’t.

After a while, we changed the game slightly. Kenzie learned that these were actually maps three-island countries, such as the one with the Island of Kenzie, the Island of Ellie, and the Isle of Emily above. The connecting lines were bridges so that people could travel between the islands and across the ocean surrounding the islands.

The initial question we explored was whether or not a resident of the country living on a given island, could leave that island and walk on every bridge exactly once (without having to swim or take a boat). We explored this by erasing the dry-erase lines. Kenzie got tired of erasing and decided to number them somewhat like the picture below (I added arrows for clarity in the diagram here although we didn’t do this on the whiteboard).

Walking all bridges exactly once.

In the case above, starting on the Island of Kenzie and traversing in the sequence indicated above leads to crossing each bridge exactly once and ending on the Island of Emily. We had some cases where it wasn’t possible. One case (and I leave the picture to you) showed that it was not possible to do this starting from one of the islands but it was possible from the other two. I counted the number of bridges connecting each island and Kenzie noted right away that the Island for which it was not possible had an odd number of bridges and the other two had an even number of bridges.

We tested the odd-even theory with an example that showed that this didn’t explain everything (can you think of a counter example?).

We finished our investigation with another challenge, at this point abandoning the name-letter connections and using simple dots to represent the islands. I pointed out that people wanted to end up where they started so that they could have dinner and go to bed in their homes after a long day of bridge walking. It’s easy to do in cases such as this:

Starting and ending at the same island.

But it doesn’t work so well in this case:

Starting at the top doesn’t end there.

And we counted the number of bridges at each island again:

Counting bridges.

At this point we were over an hour into the investigation and my Kenzergartner declared that she was bored. I was fine with that, having developed some fluency with the idea of a graph (per Graph Theory) and digraphs (or DIrected Graphs) as well as modest investigations into paths and circuits and their properties. On her own, she posited the importance of odds and evens to determining if an Eulerian Path (visit each bridge exactly once) or, better, an Eulerian Circuit (…and end where you started) was possible.

Graphs are easy to draw and can be made to represent many many things so are a natural representational model for really young kids. The stories created around vertices (islands or names) and edges (connecting lines) provide a supportive context that is easily abandoned after working with the graphs becomes comfortable (as when we shifted from named islands to simple dots).


One Reply to “KinderGraph Theory”

  1. I love how natural that transition was from pretend-play with the story to more abstract dot representations. It’s okay to stay fully immersed in a story (a lot of parents doubt if it’s okay – I think it is). Yet abstractions, if they emerge, can help us create many new stories. Sweet investigation – thank you for sharing!

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