This Scrooge feels the need to point out that this only proves that the sum of the 3rd & 4th triangular numbers is a square #. #bahnumbughttps://t.co/KR4dALQIaL

Shortly after seeing the tweet my younger son and I were playing Othello. The combination gave me the idea for today’s project.

We started by talking about the triangular numbers and why consecutive triangular numbers might sum up to be a perfect square. My older son’s idea of how to think about triangular numbers was computational rather than geometric.

Now we moved to the Othello board and looked at the geometry. My younger son found two different geometric ideas which was fun.

Finally, I gave the kids a challenge to try to find another geometric version of the identity. This question was a bit more challenging that I intended it to be, but we eventually got there and even saw how our new picture related to the sum formula that my older son used in the first video:

The second is the Torus knot from Segerman’s new book Visualizing Mathematics with 3D Printing.

We started the project today by just talking about the knots. Comparing the two knots that are actually identical was useful in refining the language they used to talk about knots.

Next they wanted to try to compare the two identical knots by looking at their crossings. My older son had the idea of assigning a +1 to every “over” crossing and a -1 to every “under” crossing. My younger son noticed that this counting method should always produce a net 0 because we counted the over and under crossing for each crossing exactly once.

New we tried to compare Segerman’s torus knot to Taalman’s rolling knot. Here we used the “tangle” from Colin Adams’s book Why Knot?

One fun thing that came up by accident in this video is an amazing shadow cast by Taalman’s knot – that was a really fun surprise.

Unfortunately, it proved to be a bit difficult to get the tangle back together so we had to pause the video at the re-connect the tangle off camera. It is really neat, though, to watch kids try to make a copy of a knot.

Once we got the tangle connected we started the next video. Since the tangle can move around, it isn’t that hard to manipulate the tangle from the form Segerman’s knot to the form of Taalman’s knots. In fact, it happened more or less by accident!

As I mentioned above, it is actually a pretty difficult task for the kids to describe the features of the knots when they compare them – even with a knot as simple as the trefoil knot. I think one of the neat parts of this particular project is working on using more precise mathematical language.

So, a fun project. We have a new 3d printer and I’m really excited about using many more 3d printing ideas from Taalman and Segerman to explore math with the boys.

Although I’ve been traveling a bit for work this week the relationship between learning math and intuition has stayed in my head. Sometimes my thoughts have drifted to and old blog post about a problem from the European Girls’ Math Olympaid:

That post, in turn, was inspired by an old post by Tim Gowers where he “live blogged” his work while he solved a problem from the International Mathematics Olympiad.

It can be really hard for anyone to know what math intuition looks like because everyone sees polished solutions way more often than they see the actual process of doing math.

That’s part of the reason I make the “what a kid learning math can look like” posts – so everyone can see that the path kids (or anyone!) actually takes to the solution of a problem is hardly ever a straight line:

The other thing on my mind this week has been some old AMC 10 problems that have really given my older son some trouble. These are pretty challenging problems and require quite a bit of mathematical intuition to solve.

So, I’d like to make the same challenge with these problems that I made with the problem from the European Girls’ Math Olympiad – “live blog” yourself solving one of these problems. Post the though process rather than a perfect solution. Let people see *where* your mathematical intuition came into play.

After playing the game for a just a few minutes I knew that my kids would love it.

Here’s each of their reaction to seeing and playing the game.

My younger son first:

My older son next:

So, definitely a fun little game for kids. They need to be fairly fluent with the arrow keys on the keyboard, but that’s really all that’s required. Definitely some fun puzzles to solve!

Had a great night with the boys tonight. My older son was working on some old AMC 10 problems and we talked through one that stumped him for his movie:

It was #15 from the 2013 AMC 10a:

Next I spent some time with my younger son. He’s been studying the basics of lines using Art of Problem Solving’s Introduction to Algebra book plus a little bit of Khan Academy (when I’ve been traveling for work). I asked him what he’d learned so far and loved his response. It was a great reminder of the joy of learning new math ideas for the first time:

Everyone always talks about thinking through problem carefully – here’s a great problem and a great opportunity to give some examples of that kind of deep thinking.

Anyway, I ran out to home depot and got some wire and we made some knots. I had each of the boys make a trefoil knot and then make a random knot of their own choosing. In retrospect I wish I’d spent maybe just 5 minutes explaining some of the ideas in Richeson’s blog post – oh well, the excitement got the better of me 🙂

Here’s my older son playing with his trefoil knot and making a Mobius strip bubble. I love the “hey, I actually think I got it” moment:

Here’s him playing with the knot me made – in retrospect I’d argue for a knot that was slightly less complicated:

Next up was my younger son. First up was the trefoil knot and we got another great moment “I think this might be a Mobius strip” !!

Finally we made his own knot and explored. Again, I’d probably ask for a less complicated knot if I was doing this again:

So, that so much to Dave Richeson for posting his old project – this is an incredible project, and an especially great one for kids. The appearance of the Mobius strip is really quite an amazing little math miracle!

Here’s what my kids thought was neat about the video:

My younger son thought the “central sphere” problem was fun, and my older son thought the shape of the n-dimensional spheres was fun. I originally intended to talk about both this morning, but our talk about the shape of the spheres took enough time for one project.

So, below is our initial look at the shape of the spheres. There’s a lot of nice introductory geometry (and fractions!) in the discussion. Also, I made the choice to talk about 1/2 the length of the long diagonal – that choice sort of confused the kids, so I’d focus on the full length if I was doing this again.

Finally, we talked about how the diagonal changes as you go up in dimension. This is a fairly straightforward application of the Pythagorean theorem, so it isn’t that hard to talk about. The boys saw the pattern fairly quickly.

Then I introduced the volume formula for even-dimensional spheres and we calculated the ratio of the volume of a 30-dimensional sphere to the 30-dimensional box it is inscribed in.

I’m super excited for this new series of videos from Kelsey Houston-Edwards, and I can’t wait to share the next one with my kids!

After seeing the re-tweet from Mathjams I thought it would be fun to try this project, too. I ran to Joann Fabrics and found some 48 inch zippers and my wife helped me figure out how to make the shape. She did the sewing and commented that she was surprised that the pins we had holding the shape together started on once side and ended up on the other side! So, there’s a neat “what do you notice” math and sewing project here, too. See also:

I had each kid go talk about the shape separately so that the other kid’s ideas wouldn’t spoil theirs. Here’s what my younger son had to say as he played with the shape:

and here’s what my older son had to say:

I love this shape and exploring it is a great math project for kids (and probably for everyone!). Thanks to Mathsjam and to Andrew Taylor for sharing it.

I’m starting to think about what to do for the Family Math nights at my younger son’s school this year. During the day today I 3d printed 2 of Laura Taalman’s Peano Curves to see if that might somehow make a fun project for a group of 4th and 5th graders.

The plan tonight was to have each kid talk about the curve (they’ve seen it before) and see what they thought was interesting. My older son went first:

Then my younger son:

For the last part of the project we took the curve off the base and stretched it out (almost) into a line:

I think there’s a fun project here – these take a long time to make, but I think with a week or two of printing prep that there’s a good 45 minute project for 4th and 5th graders in here somewhere.