Tag zometool

Working through an Alexander Bogomolny probability problem with kids

Earlier in the week I saw Alexander Bogomolny post a neat probability problem:

There are many ways to solve this problem, but when I saw the 3d shapes associated with it I thought it would make for a fun geometry problem with the boys. So, I printed the shapes overnight and we used them to work through the problem this morning.

Here’s the introduction to the problem. This step was important to make sure that the kids understood what the problem was asking. Although the problem is accessible to kids (I think) once they see the shapes, the language of the problem is harder for them to understand. But, with a bit of guidance that difficulty can be overcome:

With the introduction out of the way we dove into thinking about the shape. Before showing the two 3d prints, I asked them what they thought the shape would look like. It was challenging for them to describe (not surprisingly).

They had some interesting comments when they saw the shape, including that the shape reminded them of a version of a 4d cube!

Next we took a little time off camera to build the two shapes out of our Zometool set. Building the shapes was an interesting challenge for the kids since it wasn’t obvious to them what the diagonal line segments should be. With a little trial and error they found that the diagonal line segments were yellow struts.

Here’s their description of the build and what they learned. After building the shapes they decided that calculating the volume of the compliment would likely be easier.

Sorry that this video is a little fuzzy.

Having decided to look at the compliment to find the volume, we took a look at one of the pieces of the compliment on Mathematica to be sure that we understood the shape. They were able to see pretty quickly that the shape had some interesting structure. We used that structure in the next video to finish off the problem:

Finally, we worked through the calculation to find that the volume of the compliment was 7/27 units. Thus, the volume of the original shape is 20 / 27.

As I watched the videos again this morning I realized that my older son mentioned a second way to find the volume of the compliment and I misunderstood what he was saying. We’ll revisit this project tomorrow to find the volume the way he suggested.

I really enjoyed this project. It is fun to take challenging problems and find ways to make them accessible to kids. Also, geometric probability is an incredibly fun topic all by itself!

A project inspired by Steve Phelps’s Dissection tweet

I saw a neat tweet from Steve Phelps today:

We’ve done a couple of projects on the Rhoombic Dodecahedron before – here are three of them:

Using Matt Parker’s Platonic Solid video with kids

A 3D Geometry proof with few words courtesy of Fawn Nguyen

Penrose Tiles and some simple 3D Variations

After seeing Phelps’s tweet I thought it would be fun to see if the boys remembered how to find the volume of the shape. So, I built one out of our Zometool set and asked them what they knew about the shape.

Here’s what my older son had to say:

Here’s what my younger son had to say:

I’m glad I saw Phelps’s tweet – it was fun to revisit some of these old projects occasionally. Also, it was a nice reminder of how easy it is to share 3d shapes with kids using a Zometool set.

A surprise 30-60-90 triangle

Over the last couple of days we’ve done two projects that started from a couple of easy to state questions:

(i) Given some squares with area 1, how do you make a square with area 2?

(ii) Given some squares with area 1, how do you make a square with are 3?

Those project are here:

A neat and easy to state geometry problem

Some simple proofs of the Pythagorean Theorem

Tonight my older son is at a school event. That gave me time to do a fun little extension of these two projects with my younger son.

First I reviewed the original problems:

My son solved the 2nd problem above by making triangles with sides 1, \sqrt{2}, and \sqrt{3}. For this part of the project I wanted to show him a different triangle that has a side length of \sqrt{3} – a 30-60-90 triangle:

Now – for a little extra fun – we made a Zometool cube. That cube shows that the face diagonal (of a 1x1x1) cube has length \sqrt{2}. It also shows that the internal diagonal has length \sqrt{3}.

Here’s the surprise – if we extend basically the same geometry to 4 dimensions, we find that the “long” internal diagonal of a 1x1x1x1 cube has length 2, and that there’s a secret little 30-60-90 triangle hiding in the cube!

We did a similar project a few years ago:

Did you know that there is a 30-60-90 triangle in a Hypercube

It was nice to revisit this idea today 🙂

Extending Vsauce’s 4 dimensional shadow tweet a bit

Saw a fun tweet from Vsauce before we left for vacation:

He later shared the shape, too:

Using shadows is a an incredibly fun way to explore complex shapes. Henry Segerman gave an amazing talk about the idea last fall at MIT:

segerman

I think playing with these sorts of shadows is a great way to share complex shapes with kids, too (we also used a Zometool set!):

Playing with shadows inspired by Henry Segerman

4-dimensional Shadows

Playing with our Zometool model of Bathsheba Grossman’s “Hypercube B” was especially cool – you can see some of the same effects as in the Vsauce video, though I think the two shapes are a little different.

Though this project shows that there are a few different Zome versions of the shape, so maybe my blue-strut version isn’t all that different from Vsauce’s 4-d cube after all:

Bathsheba Grossman’s “Hypercube-B” part 2

Anyway, as Vsauce’s tweet shows, exploring shapes via shadows is fascinating. It is also a really fun way to introduce kids to shapes that they’ve probably never seen before!

A Zometool follow up to our cuboctohedron project

Earlier in the week we studied the cuboctahedron:

cubeoctahedron

That project is here:

Playing wiht the Cuboctahedron

Also earlier in the week I saw these shapes displayed in the MIT math department:

The chance encounters with these shapes this week gave me the idea to revisit them today and see if we could build them with our zometool set. The second shape, I think, is mislabled in the MIT display case – or maybe they are just using a less common name. The usual name is the icosidodecahedron, and it is also a shape we’ve seen before:

 Snowman

I started the project today by showing the shapes to the boys and asking what they knew about them:

Then we went to the living room to build the shapes. The only tricky part is that the cuboctahedron needs green struts. As always, the wonderful thing about the Zometool set is that you can go from seeing these shapes on a page to holding them in your hand almost immediately!

The last part of the project was building the dual shape of the cuboctahedron. I wasn’t sure if the zome set would let us do this since you can’t exactly find the center of the triangles with zome – but we did catch a lucky break! The dual is also a shape we’ve seen before 🙂

This project was really fun – exploring geometry with our Zometool set is one of my favorite activities!

Taking about Kate Nowak’s shape

Saw this neat drawing from Kate Nowak the other night:

I was interested to see if we could make the shape from our Zometool set, and . . . .

The boys really enjoyed making the shape last night and both also made several comments about how interesting it was. This morning we talked about it a bit. Both kids focused on symmetry. I spent a bit more time with my older son exploring the different kinds of symmetry, but it was great to hear what both kids had to say. It really is an amazing shape!

Younger son first:

Older son next:

This was a really fun project. The shape didn’t take that long to build, which was lucky. It is always fun to be able to pull out the Zome set to explore something that we saw on Twitter 🙂

Using 3d printing to talk symmetry with kids

We’ve done a lot of projects relating to platonic solids and dodecahedrons in particular. A really neat fact about dodecahedrons is that you can use the verticies to put 5 cubes inside!

It isn’t just a mathematical “fun fact” either – the symmetry groups involved play roles in important mathematical theorems.

For today’s project I wanted to explore one cube in a dodecahedron and look at the relationship between the rotations of the cube and the rotations of the dodecahedron.

We started by looking at the dodecahedron by itself:

Next we moved to looking at the cube in the dodecahedron and studied what rotating the dodecahedron did to the cube:

Finally we looked at some 3d printed models that we made to see if these models helped us explore the rotations a bit more:

I was a little disappointed that I made the 3d printed models a bit too small, but I still like how this project went. I’m going to try again with some slightly larger models with my older son.

A project inspired by an AMC 12 octagon problem

The problem pictured below from the 2003 AMC 12 gave my son some trouble:

Screen Shot 2016-12-26 at 9.58.08 AM.png

We talked through it together a few days ago, but I thought it would be fun to try to do an octagon-inspired math project today.

We started with the problem and then talked a bit about a 3d print we found on Thingiverse:

Next we took a look at a version of the 3d printed shape that we made from our Zometool set. You can’t make a regular octagon with a Zometool set, and the fact that our shape didn’t have a regular octagon led to a good discussion:

For the last part of the project we tried to find the volume of our truncated cube.

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:

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!

Revisiting our Zometool Snowman

When we first moved into our house we did a couple of fun and large Zometool projects because we didn’t have any furniture 🙂

This week I saw a fun tweet from Eli Lubroff that reminded me of one of those projects:

Here’s a part of that old project 🙂

Snowman

Today we revisited that old snowman and had the boys talk about each of the Archimedean solids in the shape. This is a fun project – not just because the shapes themselves are cool – but you get a nice opportunity to talk about counting and symmetry. You’ll see in the videos that my older son is a bit more comfortable with the idea, but my younger son seems to catch on by the 3rd video.

Here’s a link to all of the Archimedean solids on Wikipedia:

The Archimedean Solid page on Wikipedia

And here’s our project:

First the bottom of the snowman – the Truncated Icosidodecahedron

Next was the Rhombicosidodecahedron

Next was the Icosidodecahedron

Finally the Archimedean Solid Snowman 🙂 Two years later and he still fits!

Definitely one of my all time favorites and a really fun way to discuss counting and symmetry!