Mazes, Maps, Moebius Strips 1 – Insides and Outsides


Here are quick notes on the first meeting of my new math circle, code name “Mazes, Maps,  and Moebius Strips” inspired by the book “Insides, Outsides, Loops and Lines” by Herbert Kohl, “Without Words: Mathematical Puzzles to Confound and Delight” by James Tanton, and drawing on the techniques from the “Avoid Hard Work!” by James Tanton, Maria Droujkova and myself.

There’s some kind of virus making rounds. 2 of the 5 kids didn’t attend. We had the Rocket Boy, L, and E.


First question: What do these 5 shapes – a circle, a hand, a Xmas tree, an arrow, and a tree with grass around it – have in common?

E: They are made out of lines

L: These are all closed shapes or figures

RB: They have insides and outsides

And then they were stuck thinking of anything else. So I showed another drawing – a hand wearing a ring. Is it like the other five or different?

L said that it was different because it had “an inside inside an inside”. RB generalized that all the other shapes only have 1 inside and 1 outside. But the hand with a ring had 3 insides.

Next question: So what makes a shape have more than 1 inside?

The boys (yes, again all boys in the circle, sigh) were stumped. E started drawing. I suggested they should try drawing shapes that have the same properties as the 5 shapes on the large sheet of paper.

RB and L very carefully drew shapes with just one “inside”. E, the youngest in the group, drew this (the golden shape on the top left):


Perfect! Just what we needed. How many insides does this shape have? Why? Aha!

Next, it was time to jump into a secret garden to collect some objects. We stood on the outside of the fence. How many times do we have to climb over the fence to get to Moby Snoodles, the ever-so-joyful whale?


E and L both wanted to know if they should follow a straight path. RB assumed anything goes attitude and drew curved paths which meant he always had to climb over the fence exactly once. I suggested to keep that in mind, but also check what would happen if he followed a straight path.

RB and L completed the tables and saw the pattern. E wasn’t familiar with odd and even yet, so I had to do a quick demonstration using markers.


Next was a different garden. Again, we started on the outside, but now the things we needed to collect were also on the outside. RB suggested to just walk around. E wanted to go straight to each resource and bring it back. L decided to go along straight lines from resource to resource. How many times did they have to cross the fence now?

And then the time was up and we had to stop. A completely optional homework was to draw more puzzles like this (if we have enough, we can have a puzzle exchange) and to try to figure out the mysterious symbol in the bottom right corner of each page.


One thought on “Mazes, Maps, Moebius Strips 1 – Insides and Outsides

  1. […] To begin with we did a bit of a review – looked at the collection of shapes, discussed what they shared in common. Then I gave them a more difficult island puzzle. Interestingly, everyone approached it by either tracing the lines with their fingers, coloring in the ocean or “walking” the map with their fingers sounding off “land, water, land, water, land, water”. Neither RB, L or E mentioned evenness or oddness that they discovered at the last meeting. […]

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