Cutting a double Möbius strip

Saw this sequence of tweets last week and thought that the paper cutting examples from the video in video linked by Patrick Honner would make for a great activity to go through with kids. I even recruited a few kids besides my own to try out the exercise.

Here are the four shapes that we cut in half:


The main purpose of the exercise was to hear kids talking about the shapes, and especially what they thought the shapes would turn into when cut. So, not as much discussion as usual, just ideas.

First up – my kids going through the four exercises:





Next up – a 2nd grade son of one of our friends:





I goofed up the construction of the double Möbius strip (you have to twist the strips the opposite way to make this work), so the 4th video is really our second time through, though 1st time with the correct shape.

Finally – a 4th grade daughter of one of our friends:





I loved hearing what all the kids had to say, and all of them seemed to really love these activities. Would love to go through this exercise with a larger group of kids – it would be amazing to hear all of their ideas about what shapes they thought they would see, and well as their descriptions of the shapes after we cut them. Lots of fun and lots of great opportunities to talk and share math ideas with kids here!

Two interesting problems from the Julia Robinson math festival

A comment on yesterday’s blog pointed me to some fun problems from the Julia Robinson Math Festival:

Julia Robinson math festival problems

It was exciting to see these problems and I really enjoyed played around with them a little bit yesterday afternoon. With nothing specific planned for our Family Math today I thought picking two of them to work through with the boys would make for a fun project this morning. Since my younger son is just starting to learn some basic geometry, I picked two of the problems that didn’t require any advanced geometry, but I’ll definitely keep the complete list in mind when I’m looking for other geometry projects in the future.

The first problem we looked at was problem #5 in the link above:

A circle of radius 15” intersects another circle, radius 20”, at right angles (see below). What is the difference of the areas of the non-overlapping portions.

On thing that made this problem attractive is that had some similarities with the problem we looked at yesterday. The boys found a nice solution and also avoided the trap set by the problem writers!

The second problem I picked from the above link was #8 – a neat Pythagorean identity that I’d not seen before.

This problem gave them a bit more difficulty, though the did remember a few things from Numberphile’s Blob Pythagorean Theorem video, so that was nice to hear. After struggling to figure out how to get started, they eventually decided to check out what would happen in a 3-4-5 triangle – a great way to get started!

The interesting thing about this video, though, is just listening to the kids trying to get their arms around the problem. Similar to many previous examples, the path to the solution isn’t a straight line.

We paused the last video because we’d gone over 5 minutes, but we just turned the camera on and off. The next video just picks up on the calculation we were doing for the 6-8-10 triangle. At the end of the last video there was a bit of confusion between the radius and the diameter of the circles. With that confusion out of the way we get to the solution of the problem without too much trouble:

So, thanks to Ben for pointing out this great list of problems from the Julia Robinson Math Festival. Can’t wait to use a few more of these problems in future math projects!