Talking about “The Cat in Numberland”

Last we did a couple of projects based on Kelsey Houston-Edwards’s video about infinity:

Sharing Kelsey Houston-Edward’s Infinity video with kids

Extending our project on Kelsey Houston-Edwards’s infinity video

I got a comment from Allen Knutson on the 2nd project recommending using “The Cat in Numberland” to talk about infinity with kids. I ordered the book immediately and had the boys read it a few times this week. We got around to talking about it this afternoon.

Here’s their initial reaction to the book:

In the last video we I asked the boys for 3 ideas from the book that they wanted to talk about. They chose:

(1) When “Hilbert’s Hotel” is full, how do you fit one more person in?

(2) How about fitting in 26 more people?

(3) When you take away half the people how can the hotel still be full?

Here’s the explanation for part 1 – the idea here shows one strange thing about infinity!

Here’s part 2:

My older son got a little confused by the numbering of the hotel rooms in this video. The numbering of the rooms is hardly the main point, but it is nice to be able to review / revisit some counting ideas in this unusual context:

For part 3 we had a nice conversation about how you can form a bijection between the counting numbers and the non-negative even integers. That conversation went pretty fast so I asked the boys to each find another bijection and got really lucky when they picked two pretty cool ideas – powers of 2 and prime numbers.

The last movie ended with a question about whether or not the primes were infinite. This was also hardly the main point of the project, but turned out to be a fun way to end the conversation today.

So, thanks to Allen Knutson for pointing me to the book and to Kelsey Houston-Edwards for the Infinity video which has now led to three fun projects with the boys!

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A fun coincidence with an Eduardo Viruena creation

I got some great feedback from Eduardo Viruena on the project we did with one of his math designs:

A short project inspired by a Holly Krieger tweet

One of his other designs he pointed me to was this one:

A small stellated dodecahedron approximated by dodecahedra

Here’s his picture:

Dodecahedra.jpg

I printed it over the course of the day (took about 6 hours) and showed it to my younger son when he got home from school. Here’s he described the shape, including noticing one very interesting pattern that he thought would form an Archimedean solid:

It turns out that the shape he saw would indeed be an Archimedean solid. In fact, it the exact solid we did a project on a few weeks ago!

Here’s that project:

Revisiting our Zometool Snowman

Which was inspired by this tweet from Eli Luberoff:

https://twitter.com/eluberoff/status/801541820979167232

The Snowman is still up in our living room (which I’ll attribute half to coincidence and half to laziness . . . . ) so we looked carefully at the two shapes:

Amazing what kids notice when they look at mathematical objects!

Playing with Sugihara’s “ambiguous cylinder”

The amazing video below came out over the summer:

My favorite memory of the video is my 5th grader seeing it for the first time and saying “dad, that mirror is terrible.” Ha!

Over the summer we printed a few of the shapes and played around with them, but ended up giving them away. Today I was reminded of the shape by this tweet from Dave Richeson:

I wasn’t sure if the print he showed in his tweet was available, so I found a different one on Thingiverse and started the print so the boys could play with the shape again tonight:

Ambiguous Cylinder Illusion by Make_Anything on Thingiverse

As luck would have it, though, Brenda Landis did put the print from Dave’s tweet on Thingiverse:

Here’s the direct link:

Brenda Landis’s “Sugihara’s Circle/Square” on Thingiverse

So, now we had two of them to play with 🙂

I had my younger son play with them first:

My older son went next:

So, a super fun project and another great example of how 3d printing can help kids see some amazing math-related ideas. Super duper thanks to Dave Richeson and Brenda Landis for sharing this print.

A short project inspired by a Holly Krieger tweet

Saw this tweet from Holly Krieger this morning:

and, oh bother, the embedded tweet isn’t coming through. Here it is:

I thought it would be fun to talk about some sort of similar shape and stumbled on this neat design by eduardoviruena on Thingiverse:

eduardoviruena’s “Sierpinski cubes” on Thingiverse

sierpinski-copy

I sent the print to the printer as I was running out the door. Unfortunately in my haste I made the print much smaller than I intended. Oh well . . . we still got to have a fun conversation.

Here’s what he had to say about the eduardoviruena’s “Sierpinski cubes” – I was really interested in his description of how to make this shape:

Because the print was so small I wanted to see if he had any other thoughts when he saw a larger version on the screen:

I’m really excited to do more 3d printing projects like this one – hopefully giving kids shapes like this to play with will help them see a fun and exciting side of math 🙂

Extending our project on Kelsey Houston-Edwards’s Infinity video

Yesterday we did a project inspired by Kelsey Houston-Edwards’s latest math video:

Here’s a link to our project:

Sharing Kelsey Houston-Edward’s Infinity video with kids

Last night my younger son and I were talking a little bit more about the project and he asked me why Cantor’s diagonal argument for why the set of real numbers is larger than the set of Natural numbers doesn’t work for rational numbers!! Yes!!

We explored that question today. First we did a quick review of the diagonal argument (which was the last part of yesterday’s project) and then we began talking about the rational numbers:

Next we looked at what would happen if you applied the diagonal argument to the rational numbers:

After getting our arms around the diagonal argument when applied to rational numbers, we backed up and looked at the argument why rational numbers are countable.

Unfortunately I made an easy concept hard in this part of the project. I was trying to explain the “easy for me” idea that if a set that is larger than the rationals was the same size as the natural numbers, that meant the rationals must also be the same size as the natural numbers. My explanation started off terribly and went down hill . . . .

Finally we looked at one of the strangest consequences of all of this infinity stuff. In math language – the rational numbers have measure zero.

The idea here always blows my mind and is a really fun idea about infinity to share with kids.

Sharing Kelsey Houston-Edwards’s Infinity video with kids

The latest PBS Infinite Series video came out this week:

This is the 4th video in an incredible series from Kelsey Houston-Edwards. Our projects on the first 3 are here:

Sharing Kelsey Houston-Edward’s [higher dimensional spheres] video with kids

Sharing Kelsey Houston-Edward’s Philosophy of Math video with kids

Sharing Kelsey Houston-Edward’s Pigeonhole Principle with kids

I had the boys watch the new video together and started today’s project by asking them what they thought was interesting.

After hearing what the kids found interesting, we dove into the idea of bijections. We talked a bit about how a bijection has to work both ways using the bus idea from the video.

After the bus example we moved on to the example of the bijection between the points in an interval and points on the real line.

We finished up by talking about the bijection between the national numbers and the positive even integers.

Since we’ve done several prior projects where infinity played some role, the next thing I asked the kids was for some thing that they already knew about infinity – both things that they thought made sense and things that they thought didn’t make sense. The discussion and examples here were amazing – “no one knows what infinity divided by infinity is” 🙂

Finally, we wrapped up the project talking about why the infinity associated with the real numbers is larger than the infinity associated with the natural numbers.

I thought this would be a fun way to end the project since it was one of the key ideas in Houston-Edwards’s video:

So, another really fun project from the new set of math videos from PBS Infinite Series. I love this new series – can’t wait for the next one!

Playing with some 3d printed knots

Today we looked at some 3d printed knots designed by Laura Taalman and Henry Segerman.

Two are versions of Taalman’s “rocking knot” which we found here:

Laura Taalman’s Makerhome blog: Day 110 – the Rocking Knot

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.