I’ve been thinking about exposing the boys to math through 3d printing lately. Today I decided to explore making Paula Beardell Krieg’s cube shapes with them. Here’s the exploration the boys did back in March when we first got them:
Even though we’ve played a bit with these shapes before I still thought that thinking through these yellow and pink shapes would be a fun challenge. The project turned out to be a tiny bit harder than I thought it would be, but it still was a nice conversation.
We started by first looking at the three pyramids that can come together to make a cube and continued by looking at what happens when you slice those shapes in half.
In the last video the boys were thinking about trying to describe these shapes by describing the lines that formed the edges. At the beginning of this video I told them that this particular approach was going to be tough since they didn’t know how to write equations of lines in 3 dimensions.
So, I had them continue to search for properties of the shapes that they could describe.
The boys were still struggling to find some ideas about the shape that went beyond the lines on the boundary, but we kept looking.
My older son hit on the idea that the shape was made from “stacking squares on top of each other.” We spent the rest of the video exploring that idea.
Now that we had the idea about stacking squares we went to Mathematica to try to create the shape. It took a few steps to move from the ideas about the squares to generating the code for the shape. We didn’t get all the way there during this video, but we did figure out how to make a cube.
Unfortunately I had to end the video since the camera was about to run out of memory.
While I was getting the videos off the camera the boys worked on how to change the cube shape to the pyramid shape. It was a good challenge for them and they got it. We talked about that shape for a bit and then moved on to the challenge of creating the “pink” and “yellow” shapes that Paula Beardell Krieg created from paper.
We had a little bit of extra time today and it was fun to walk through this challenging problem. I think creating shapes to 3d print is a really fun way to motivate math with kids. Can’t wait to use the printed shapes in a project tomorrow!
Yesterday I was able to watch the Global Math Project presentations (well, most of them) via the Facebook Live feed. Hopefully those videos will be preserved here:
One tank that caught my eye was given by Henry Segerman. I’d guess that his work and Laura Taalman’s work account for at least 80% of what I know about exploring math through 3d printing.
As I write this post there are 96 prior posts with the “3D Printing” tag on my blog. 3D Printing is still pretty new, and I think many people around math are only starting to see its use in education. Segerman’s talk made me want to throw together a list of fun projects that we’ve done just in case anyone is looking for a starting point after seeing his talk.
Some of my original thoughts on exploring math through 3d printing can be found in this blog post from March 2014 which features two really neat videos from Brooklyn Tech and Laura Taalman:
(1) James Tanton’s Geometry Problem and 3d printing
Since this blog post was inspired by a talk a James Tanton’s Global Math Project, it seems appropriate to kick it off with a project inspired by Tanton:
What is surface area of figure formed by all points within a distance 1 from a regular tetrahedron with faces of area 1?
(2) Hard to highlight just one project that Segerman Inspired, so here’s the first of 2
One of the Segerman’s examples in yesterday’s talk was about bubbles. He showed a few complicated bubble examples but there are simple ones that are amazing, too. Here’s an example showing that the “bubble” formed by dipping a tetrahedron in soap is the same shape as a 4-dimensional shape:
(6) Exploring connections between algebra and geometry
3d printing can come in handy for looking at math ideas that previously you could only study on paper or on the computer screen. For example, a common algebra mistake is to think that:
Here’s what these two surfaces look like:
Here’s two projects exploring these algebra ideas with the boys:
(8) 3d printing can be a fun way to review ideas from elementary geometry
In his talk yesterday Segerman mentioned a few prints that his undergraduate students created. As he showed this projects he talked about how the creation process really helps students understand and explore the underlying math.
In the project below, creating the shape of the tile helped me review and explore equations of lines with the boys:
After getting some intuition from this problem we extended the problem to 4 dimensions using Taleb’s approach. The prints were really fun to play with and it is amazing to hear kids talk about these shapes that come from 4 dimensions:
But 3d printing can help you see even more advanced ideas. Here’s a cube inside of a dodecahedron, for example:
and, of course, many (most!) of examples that Henry Segerman showed in his talk yesterday are perfect for showing how 3d printing can help everyone experience some advanced ideas in mathematics.
I’ll end with the project we did yesterday, which is a delightful example of how 3d printing can help you explore a math idea:
[Note: 10:30 am on Oct 7th, 2017 – had a hard stop time to get this out the door, so it is published without editing. Will (or might!) edit a bit later]
About two years I found an amazing design by Steve Portz on Thingiverse:
Today we revisited the idea. We began by talking generally about the volume of a cylinder:
The next part of the project was heading down the path to finding the volume of a cone. I thought the right idea would be to talk first about the volume of a pyramid, so I introduced pyramid volume idea through snap cubes.
Also, I knew something was going a little sideways with this one when we were talking this morning, but seeing the video now I see where it was off. The main idea here is the factor of 3 in the division. Ignore the height h that I’m talking about.
Next we looked at some pyramid shapes that we’ve played with in the past. The idea here was to show how three (or 6) pyramids can make a cube. This part was went much better than the prior one 🙂
The ideas here led us to guess at the volume formula for a cone.
Now that we’d talked about the volume formulas for a cone and a cylinder, we could use the 3d print to guess at the volume formala for the sphere.
With all of that prep work behind us, we took a shot at pouring water through the print. It worked nearly perfectly 🙂
I am really happy that Steve Portz designed this amazing 3d print. It makes exploring some elementary ideas in 3d geometry really fun!
Steven Strogatz had great series of tweets about math education earlier in the week. These two have stayed in my head since he posted them:
I know that last year Strogatz taught a college level course similar to the one he is describing in the tweets. We even used a couple of his tweets about the course material for some fun Family Math activities. For example:
Unusual intuitive argument for why A= pi r^2 for a circle, found by one of the tables in our #math exploration class. I love these surprises pic.twitter.com/dch9PfmynZ
So, thinking back to projects like those got me thinking about all sorts of other ideas you could explore in an appreciation course. At first my ideas were confined to subjects that are traditionally part of pre-college math programs and were essentially just different ways to show some of the usual topics. Then I switched tracks and thought about how to share mathematical ideas that might not normally be part of a k-12 curriculum. Eventually I tried to see if I could come up with a (maybe) 3 week long exploration on a specific topic. I chose folding and thought about what sort of ideas could be shared with students.
Below are 9 ideas that came to mind along with 30 second videos showing the idea.
(1) A surprise book making idea shown to me by the mother of a friend of my older son:
(2) Exploring plane geometry through folding
We’ve done many explorations like this one in the last couple of years – folding is an incredibly fun (and surprisingly easy) way for kids to explore ideas in plane geometry without having to calculate:
Here’s one introductory example showing how to find the incenter of a triangle:
(3) The Fold and Cut theorem
Eric Demaine’s “fold and cut” theorem is an fantastic bit of advanced (and fairly recent) math to share with kids. Our projects exploring “fold and cut” ideas are here:
[space filled in with random words to get the formatting in the blog post right 🙂 ]
(5) An amazing cube dissection made by Paula Beardell Krieg
We’ve also done some fun projects with shapes that I wouldn’t have thought to have explored with folded paper. Paula Beardell Krieg’s work with these shapes has been super fun to play with:
Seemingly simple ideas about folding and bending can lead to pretty fantastic mathematical objects! These objects are another great reminder of how 3d printing can be used to make mathematical ideas accessible.
(8) Getting to some more advanced work from Erik Demaine and Joseph O’Rourke
As hinted at early with the Fold and Cut theorem, some of the mathematical ideas in folding can be extremely deep:
(9) Current research by Laura DeMarco and Kathryn Lindsey
Finally, the Quanta Magazine article linked below references current research involving folding ideas. The article also provides several ways to share the ideas with students.
The two blog posts below show my attempt to understand some of the ideas in the article and share them with kids. The video shows some of the shapes we made while studying the article.
So, these are just sort of ideas that popped into my head thinking about one part of a math explorations class. Feels like you could spend three weeks on folding and expose kids to lots of fun ideas that they’d (likely) never seen before.
For today’s project I wanted to have the boys focus on the approach that Nassim Taleb used to study the problem posed by Alexander Bogonolny. That approach was to chop the shape into slices to get some insight into the overall shape. Here’s Taleb’s tweet:
So, for today’s project we followed Taleb’s approach to study a 4d space similar to the space in the Bogomolny tweet above. The space is the region in 4d space bounded by:
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To start the project we reviewed the shapes from the project inspired by Kelsey Houston-Edwards’s hypercube video. After that we talked about the equations we’d looked at in the project inspired by Alexander Bogomolny’s tweet and the shape we encountered there:
Next we talked a bit about the equations that we’d be studying today and I asked the boys to take a guess at some of the shapes we’d be seeing. We also talked a little bit about absolute value which briefly caused a tiny bit of confusion.
The next part of the project used the computer. First we reviewed Nassim Taleb’s approach to studying the problem posed by Alexander Bogomolny. I think it is really useful for kids to see examples of how people use mathematical ideas to solve problems.
The 2d slicing was a fascinating way to approach the original 3d problem. We’ll use the same idea (though in 3d) to gain some insight on the 4d shape.
One fun thing about this part of the project is that we encountered a few shapes that we’ve never seen before!
Finally, I revealed 3d printed copies of the shapes for the boys to explore. They immediately noticed some similarities with the hypercube project. It was also really interesting to hear them talk about the differences.
At the end, the boys think that the 4d shape we encountered in this project will be the 4d version of the rhombic dodecahedron. We’ve studied that shape before in this project inspired by a Matt Parker video:
At the end of that project we were looking carefully at how you would find the volume of a rhombic dodecahedron in general. Today I wanted to move from the general case to the specific and see if we could calculate the volume of our shapes. This tasked proved to be much more difficult for the boys than I imagined it would be. Definitely a learning experience for me.
Here’s how we got going. Even at the end of the 5 min here the boys are struggling to see how to get started.
So, after the struggle in the first video, we tried to back up and ask a more general question -> how do we find the volume of a cube?
Now we grabbed a ruler and measured the side length of the cube. This task also had a few tricky parts -> do we include the zome balls, for example. But now we were making progress!
Finally we turned to finding the volume of one our our 3d printed rhombic dodecahedrons. We did some measuring and found how many of these shapes it would take to fill our zome shape and how many it would take to fill a 1 meter cube.
So, a harder project than I expected, but still fun. We’ve done so much abstract work over the years and that makes the concrete work a little more difficult (or unusual), I suppose. I’m happy for this struggle, though, since it showed me that we need to do a few more projects like this one.
It seemed as though this one could be just as fun. I started by introducing the problem and then proposing that we explore a simplified (2d) version. I was excited to hear that the boys had some interesting ideas about the complicated problem:
Next we went down to the living room to explore the easier problem. The 2d version, , is an interesting way to talk about both absolute value and lines with kids:
Next we returned to the computer to view two of Nassim Taleb’s ideas about the problem. I don’t know why the tweets aren’t embedding properly, so here are the screen shots of the two tweets we looked at in this video. They can be accessed via Alexander Bogomolny’s tweet above (which is embedding just fine . . . .)
The first tweet reminded the boys of a different (and super fun) project about hypercubes inspired by a Kelsey Houston-Edwards video that we did over the summer:
The connection between these two projects is actually pretty interesting and maybe worth an entire project all by itself.
Next we returned to the living room and made a rhombic dodecahedron out of our zometool set. Having the zometool version helped the boys see the square in the middle of the shape that they were having trouble seeing on the screen. Seeing that square still proved to be tough for my younger son, but he did eventually see it.
After we identified the middle square I had to boys show that there is also a cube hiding inside of the shape and that this cube allows you to see surprisingly easily how to calculate the volume of a rhombic dodecahedron:
Finally, we wrapped up by using some 3d printed rhombic dodecahedrons to show that they tile 3d Euclidean space (sorry that this video is out of focus):
Definitely a fun project. I love showing the boys fun connections between algebra and geometry. It is also always tremendously satisfying to find really difficult problems that can be made accessible to kids. Thanks to Alexander Bogomolny and Nassim Taleb for the inspiration for this project.
The shape sort of stuck in my mind and last night I finally got around to making two shapes inspired by Phelp’s shape. My shapes are not the same as his – one of my ideas for this project was to see if the boys could see that the shapes were not the same.
So, we started today’s project by looking at the two shapes I printed overnight. As always, it is really fun to hear kids talk about shapes that they’ve never encountered before.
Next we looked at Phelp’s tweet. The idea here was to see if the boys could see the difference between this shape and the shapes that I’d printed:
Finally, we went up to the computer so that the boys could see how I made the shapes. Other than some simple trig that the boys have not seen before, the math used to make these shapes is something that kids can understand. We define a pentagon region by 5 lines and then we vary the size of that region.
I’m not expecting the boys to understand every piece of the discussion here. Rather, my hope is that they are able to see that creating the shapes we played with today is not all that complicated and also really fun!
This was a really fun project – thanks to Steve Phelps for the tweet that inspired our work.
During the project the kids had a little trouble counting the verticies, edges, and faces of one of the complex shapes. We solved the problem with our Zometool set, but I wanted to try a different approach and printed the shapes again:
So, with these shapes I went through the project again. First a quick review:
Next, now that we have shapes that fit together, can we count the faces, verticies, and edges?
My younger son was still having a little bit of trouble seeing the number of edges, so we slowed down a bit:
Finally we did a quick recap of how the cube helped us. I was trying to get the boys to think about the shape without touching it, but wasn’t super successful.
This was a fun 2nd look at the F – E + V = 2 formula. We’ll be doing more projects based on Richeson’s book throughout the summer.
It is such a delightful read that I thought the kids might enjoy it, too, so I had them read the introduction (~10 pages).
Here’s what they learned:
Next we tried to calculate Euler’s formula for two simple shapes – a tetrahedron and a cube:
After that introduction we moved on to some slightly more complicated shapes – an icosahedron and a rhombic dodecahedron. The rhombic dodecahedron gave the kids a tiny bit of trouble since it doesn’t have quite the same set of symmetries as any of the Platonic solids:
Now we tried two very difficult shapes:
We studied these shapes last week in a couple of projects inspired by an Alexander Bogomolny tweet:
I suspected that this part would be difficult, so I had them count the faces, edges, and verticies of the two shapes off camera. Here’s what they found:
So, since the boys couldn’t agree on the number of verticies, edges, and faces of one of the shapes, I had them build it using our Zometool set to see what was going on. The Zometool set helped, thankfully. Here’s what they found after building the shape (and we got a little help from one of our cats):
Definitely a fun project. It was especially cool to hear the kids realize that the shape they were having difficulty with was (somehow) a torus. Or, as my older son said: “In the torus class of shapes.” I’m excited to try to turn a few other ideas from Richeson’s book into projects for kids.
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, is an interesting way to talk about both absolute value and lines with kids:
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