Sharing Kelsey Houston-Edwards’s Pi video with kids

We really enjoyed watching Kelsey Houston-Edwards’s latest PBS Infinite Series video last night:

We’ve played around with a few ideas about altering Pi a few times previously. One was with Vi Hart’s “Pi = 4” video in this project:

A fun fractal project – exploring the Gosper curve

A second time was a super fun series of projects on 4-dimensional shapes inspired by a tweet from Patrick Honner. That complete series is here:

https://mikesmathpage.wordpress.com/2016/03/24/patrick-honners-pi-day-exercise-in-4d-part-5-the-120-and-600-cells/

Patrick Honner’s Pi day exercise in 4d part 5: The 120 and 600 cells

Today we explored the 2-dimensional idea in Kelsey Houston-Edwards’s video in 3-dimensions using 3d printing.

First, though, I asked the boys what they thought about the latest PBS Infinite series video:

Next we took a look at 6 different “spheres” in 3-dimensions which were defined using the different way of measuring distance Kelsey Houston-Edwards introduced in her video. It is really fun to hear kids talking about these shapes, and even more fun to be able to actually hold these shapes in your hand!

The last thing we did was look at the 6 spheres all together. My younger son noticed how much bigger the shapes got as you moved from the L-0.75 norm to the L-4 norm. My older son noticed that some of the shadows of the shapes looked pretty similar even though the shapes didn’t look the same at all:

This was a really fun project to prepare. It is really fun to show kids ideas from advanced math that the wouldn’t likely see in school. It is also really fun to hold these strange shapes in your hand! Lately we’ve been using the F3 program to help us make objects to print – it was a really lucky coincidence to see the new PBS Infinite Series video *after* I learned to use the new program. It took less than 5 minutes to make the .stl files for the 6 shapes 🙂

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Using Vi Hart’s hyperbolic space tweet with kids

Yesterday Vi Hart tweeted about an amazing program you can use to explore hyperbolic space:

https://twitter.com/vihartvihart/status/776891645149138944

The program, which (I think) came from joint work by Andrea Hawksley, Vi Hart, Henry Segerman, and Mike Stay, is here:

A program to explore hyperbolic space

The youtube video in Hart’s tweet shows her playing with the program and explaining a little bit about the different shapes you see. After watching that video I thought it would be fun for the boys to explore the program without any explanation of what was going on. I was really interested to see how a kid would react to seeing this 3 dimensional hyperbolic geometry for the first time.

Here’s what my older son had to say:

 

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

 

This is such a fun program to let kids play with. The boys noticed many strange properties of hyperbolic space without knowing they were looking at a strange new space. It really is amazing to have resources like this right at your fingertips!

Carl Sagan on the 4th dimension

In a comment on a prior post my friend Amy told me about this Carl Sagan talk about the 4th dimension:

 

I decided to have the boys watch the video and then talk to start our Family Math project for today. After they watched it we discussed the parts they found interesting.

My younger son thought these two things were interesting:

(1) How the apple appeared as it fell through “Flatland”, and also how the creatures in Flatland interacted with the apple.

(2) The other 2D world that was curved.

 

My older son found these parts interesting:

(1) The discussion of projecting images from one dimension to a lower dimension, and

(2) The idea that lower dimensional things would have a hard time, but not an impossible time, noticing higher dimensions.

 

Right at the end we talked about the similarity between Carl Sagan walking around the sphere and (i) Vi Hart’s “Wind and Mr. Ug” video and (ii) a bicycle trip around a Klein Bottle. These are two of my all time favorite videos – sadly, though, the Klein Bottle video is no longer on youtube 😦 Luckily Wind and Mr. Ug remains:

 

Definitely a fun little project today – always fun to hear the boys’ ideas about complicated math topics 🙂

Playing “The Witness” with kids is an amazing experience

I learned about a new game called “The Witness” from this Vi Hart tweet last week:

https://twitter.com/vihartvihart/status/692128282876186626

After seeing a few reviews and the game trailer, I thought I’d give it a try:

 

The game is really fun if you like solving puzzles, but the pleasant surprise about this game is how fun in is to play with kids. So far none of the puzzles are so hard that kids can’t solve them, though figuring out how to solve to solve each particular one is far from obvious!

I think another Vi Hart tweet has the best description of the game:

https://twitter.com/vihartvihart/status/693927287343837184

Here’s a little pic from our first time playing

   

and a video of my son solving one of the puzzles – which probably doesn’t give you the exact right impression of the game, but it is better than nothing:

and when my son was sick this morning we sat on the couch together trying to draw the right solution to some puzzles we were seeing (essentially) in a mirror.


The kids love the open-endedness of the game – so if we can’t figure out how to solve all of the puzzles in one section we can just go somewhere else. My younger son called this a “sandbox game” which is a term I’ve not heard before – but he says the ability to go to other places is just like playing in a sandbox. The also love the challenge of solving the puzzles.

Anyway, with all of the hype around the release of the game last week, I just wanted to say that if you are looking for a mathy game to play with kids, this game is for you!

A fun fractal project: exploring the Gosper curve

Over the last few days I’ve been preparing a project with the boys based a great fractal geometry example I found in this wonderful book:

BookCover

We finally got going with that project this morning. The starting point was watching this Vi Hart video which gives a “proof” that \pi = 4:

After we watched the video we sat down to talk about the strange result and what they thought was going on. They seemed to gravitate to the idea that the jagged edges were causing problems, but the fact that the zigs and zags were getting smaller and smaller – and would eventually have a height of 0 – was still a bit confusing:

After talking about the Vi Hart video I introduced the kids to the Gosper curve by showing them the figures in the book that inspired this example. We also made use of an amazing program that Dave Radcliffe shared when I asked for a little help on Twitter:

Playing around with this program really helped the boys see the first couple of shapes in the sequence that eventually leads to the Gosper curve. I definitely owe Dave a big favor!

The next part of the project was to build the first couple of shapes that lead to the Gosper curve out of our Zometool set. The initial hexagon was easy, obviously, but the shape at step 2 gave them a little difficulty. In the video below they talk about building the shapes and then explore a connection between the hexagon from step 1 and the shape from step 2. The fun part here is that the boys saw some of the important connections that lead you to the Gosper curve.

Next we built a level 3 shape. It was lucky that we had the program from Dave Radcliffe since that allowed the boys to a little more confident that they had the right shape. It is interesting to see the 6 new level 2 shapes surrounding the original level 2 shape. Too bad our living room isn’t big enough to make a level 4 shape!

One interesting comment from my younger son is that he thinks that as we increase the size we’ll get closer and closer to a shape that looks like the original hexagon.

For the final part of the project we used our 3d printer to make 7 of the (approximate) Gosper curves. Here’s the shape we used from Thingiverse (our shape is the 2nd of the three shapes, but I can’t get that one to link properly):

The Gosper Curve on Thingiverse (I printed the middle one)

The punch line for the project is the same punch line that caught my attention in the book – when you increase the linear size of the Gosper curve by 3, the area inside the curve increases by a factor of 7 rather than a by a factor of 9. Everything that the boys have learned about scaling up to this point is that area scales as the square of the linear factors, but fractals have a different property. Pretty amazing!

Also, sorry for not explaining the analogy between the two boundaries right. Felt as if I was wrong as I was explaining it, but didn’t see what I got wrong until just now.

As a fun end to the project, I showed them Dan Anderson’s modification of the Gosper Island shape – sort of a combination of the Sierpenski Triangle / Menger Sponge shapes and the Gosper Island:

So, a fun project giving the kids an introduction to fractal dimension – a concept that I never would have guessed could be made accessible to kids. Really happy to have had the luck of running into this fun idea last week.

Binary Trees and Pascal’s Triangle

We tried a new Italian restaurant last night.  It used paper to cover the tables and they let kids draw on the table covers with crayons which was a nice way to pass the time.  I was surprised to see that my younger son was drawing binary trees.  He said that he remembered them from an old Vi Hart video, which was a little strange since it is a Thangsgiving video.  Oh well, no telling what kids will remember:

I had a different project on tap for today’s Family Math, but when your kid is drawing binary trees on the table it is probably a sign, so plans changed!  We started our talk this morning with a quick review of what binary trees are, and then talked about a few simple properties that they have:

Next we build on the topic that we touched at the end of the last video – representing coin flips in a binary tree.  If we want to keep track of only the number of heads and tails that we’ve seen, some sequences that we’ve seen before make a surprising appearance in our little tree:

Next we moved on to showing a picture of how the binary tree can merge into Pascal’s triangle.  It was neat to see that the kids had seen how the “diamonds” would appear.  We also talked a little informally about why the pattern here is indeed the same as in Pascal’s triangle.  One of the other fun things we look at in this video is how the row sums that were easy to see in the binary tree carry over to this setting:

Finally, I wanted to show how this idea could help us solve a problem that they’d not seen before (though a pretty standard Pascal’s triangle problem).  The problem ask about counting different paths in a lattice.  We can think of the go right / go up choice as similar to the heads / tails coince from the binary tree example:

So,  a little doodling on our restaurant table cloth last night turned into a fun little Family Math talk.  Always fun to see what kids remember from things that they’ve seen (and when they remember them, too, I suppose!).

The number 0.11111. . . .

Over the weekend we did a fun Family Math project about decimals in other bases:

https://mikesmathpage.wordpress.com/2014/04/05/fractions-and-decimals-in-binary/

As I was putting my younger son to bed last night he asked me about decimals in base 4.  I don’t know why – our previous conversation had been about how to play the golf game on the Wii – but something about the weekend talk had obviously stuck with him.

This morning we had some fun with a specific number – 0.111111….. (repeating forever).  What does this number equal in different bases?  It was a fun and unusual way to review decimals:

the Vi Hart video we reference at the end is here:

I love mornings like this!