The Plastic Number

Last night I got an idea from Ian Stewart’s amazing book:

The idea was about a sequence of integers related to the “plastic” number.

I started the project by building a very simple shape out of our Zometool set and asking the kids what they thought about it. One lucky surprise was that my older son guessed the recurrence relation for the sequence we’d be studying!

Just listening to my introduction now I totally butchered the definition of the plastic number – sorry about that . . . .

Next we built several more triangles and studied the sequence more carefully. My older son confirmed the recurrence relation he saw previously and my younger son found a different one!

Also – sorry about the lighting.

Finally – we checked out the Wikipedia page for the Plastic Number and then explored the equations relating to the two recurrence relations that we found previously.

Sorry about the bumbling around in Mathematica – I needed the screw up trifecta 🙂

This was a fun project. I think it would be a neat one for kids learning about factoring, too, so they could see how the two equations we studied in the last video relate to each other.

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Using 3d printing to share 4-dimensional spheres with kids

A few weeks back we did a project on 4-dimensional spheres intersecting a different sorts of 3d worlds:

What if Flatland wasn’t a plane!

Last night I got around to 3d printing some of the shapes from that project:

Today we talked through the idea of how objects from higher dimensions “look” as they pass through lower dimensional shapes. We started by talking about the idea from Flatland – a 3d sphere passing through a 2d plane. After that we moved on to talking about what the intersections would look like if the sphere was passing through a plane that was creased in to a “V” shape:

Next we moved on to talking about a 4d sphere intersecting the same sorts of objects – a flat 3d space and a “V” shaped one. To create the “V” shape, I just assumed that the 4th dimension – call it w – had a value equal to the absolute value of the x-coordinate.

Next we looked at the 3d printed shapes I made last night. These shapes show a few different stages of a 4-d sphere passing through the “V” shaped 3 dimensional space:

Finally, rather than looking at 4d sphere passing through a “V” shaped 3d space, we went and looked at the shapes made when a 4d sphere passes through a 3d space that is bent like a parabola. So, using my language from above, the 4th coordinate in the space, w, is set equal to x^2.

The shapes here are really cool and also pretty surprising.

Our second facets project

We did this about a week ago and I never got around to publishing it. Both kids are sick today so it seemed like a good day to revisit the old movies.

Our first projects with our Facet set is here:

Our Facets have arrived!

The kids have enjoyed making little creations with the Facets ever since.

Here’s how their 2nd projects looked – my younger son went first. It is so fun to hear a week later how many different topics we covered in the talk about the ring he made:

Here’s my older son’s work. He made an interesting shape that we tried to extend with the camera off. Unfortunately that shape was too heavy for the magnets and collapsed. We got a little lucky, though, and some interesting shapes survived the collapse and we turned them into new, fun shapes:

I’m really happy with the Facets and can’t wait to do more projects with them!

Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d folded fractals

Several weeks ago Quanta magazine published an article about incredible new work being done by Laura DeMarco and Kathryn Lindsey:

3-D Fractals Offer Clues to Complex Systems

Along with the article was a fun little paper folding contest that was won by Yoshiaki Araki. Here’s the starting fractal made using Mathematica:

julia

I was so excited to see Araki’s tweet and printed the shape that night!

The first thing I did was have the kids play with it and describe what they saw:

As a follow up to that initial exploration, I had each of the boys try the folding project from Quanta Magazine’s contest. It is a pretty challenging project for kids, but here’s how it came out. My older son first:

Next my younger son:

I’m really interested in learning more about these folded fractals. I would especially love to understand the basics of how the shapes fold up. Can’t wait to play with the ideas her more.

Paula Beardell Krieg’s intersecting squares

Saw a fun sequence of tweets from Paula Berdell Krieg last week:

Then . . . we received an envelope!

Next we compared the shape to the 3d prints it was based on:

After this I showed the boys the tweet above that shows how to unfold the shape into a cube and they were able to recreate the procedure:

Definitely a fun shape to explore – thanks to Paula for sending it to us! It really is amazing how much geometry you can explore just by folding paper 🙂

K-1 Family Night 2017

Here’s my plan for the K-1 Family Math nights for this year:

For the first project I’m going to borrow Lior Patcher’s idea about the 4-color from this incredible blog post:

Unsolved Problems with the Common Core

I have 30 min for the 3 projects with the kindergartners and an hour with the 1st graders, so I think for the younger kids we’ll just do one coloring sheet. The sheets I’m going to use come from an old post from Richard Green discussing a really neat result about tiling octagons. The result is a pretty deep result from geometry, but with the side benefit of hopefully producing images that young kids will enjoy seeing and coloring with 4 colors. I learned about Green’s post from Patrick Honner about 2 years ago:

Our project using Green’s post is here:

Using a Richard Green Google+ post to talk about geometry with my son

Here are the two images octagon tilings I’ll use in the project.

image2-copy

image1

Next we’ll move on to making bubble shapes with our Zometool set. As I write this, at least, I think I’ll dip the shapes in the bubbles myself. I’m worried that letting a room full of younger kids loose on a container full of bubble solution will end up with bubble solution everywhere. Also the zome parts are small and I won’t be able to supervise all of the kids on my own. Anyway, we’ve done a few zome bubble projects and my kids and the neighborhood kids have really enjoyed them. The shapes are really incredible to see and trying to guess what the bubble shapes will look like is a fun challenge for kids. Here are a couple of our old zome bubble projects:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

More Zome Bubbles

Zometool

Finally, with the kindergartners I’m going to do the paper folding project that we did for our original Family Math. I did the same project with the kindergartners last year and it went reasonably well (assuming that you set your expectations on the “me dealing with 30 6 year olds” setting!). I’ve got the first graders in a week, so I’ve got a bit of time to think through a replacement project to avoid duplicating last year’s work. Here’s the project as we did it in 2011:

Sharing Kelsey Houston-Edwards’s singularity video with kids

We’ve been enjoying going through Kelsey Houstin-Edwards’s new video series. This week’s was a bit more advanced than some of the prior ones, but I gave it a shot with the kids anyway. I tried to focus on connecting the ideas about singularities in the video with some of the 3D printed shapes we’ve been studying from Henry Segerman’s new book.

Also, I’m just getting over a few days with the norovirus, so sorry if this one (including the write up) has a bit less energy than usual.

Here’s the latest PBS Infinite Series video:

Here’s what the boys took away from the video:

Next we looked at a couple of the shapes that Henry Segerman has made to study with shadows. We were able to see (eventually) that the shadow of the north pole would be a point at infinity – or a singularity.

At the end of this video we started looking at a torus, and the conversation took a very interesting topological turn.

So, we landed on a question of what different shapes might be a torus. It took a bit of time to straighten out this idea, but after a few minutes we came to an agreement on what a torus was.

After that we saw that we didn’t have the same singularity problem trying to create a map that we had on the sphere.

After talking about the torus we spent the rest of this video talking about the pseudosphere which has more than one singularity.

So, another great video from Kelsey Houston-Edwards. It was fun connecting her ideas with some of the 3d prints we’ve been studying lately.