Tag geometry

3D Printing Paula Beardell Krieg’s dissected cube shapes

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!


Talking through 3 AMC 8 problems with my son

My son was working on the 1993 AMC 8 yesterday and had trouble with a few problems. Today we sat down and talked through those problem.

The first was problem 17:


Here’s what my son had to say:

Next was problem 19:


Here’s what he had to say about this problem:

Next was problem 24:


Here’s what he had to say about this problem:

I like these old contest problems. They lead to really nice conversations!

A mistake that led to a great conversation

My older son had a homework problem that asked him to find the area of the region bounded by the two equations:

(i) | 2x + 3y | \leq 6, and

(ii) | x - 2y | \leq 4

Mathematica’s picture of that shape is here:


He told me that he used Pick’s theorem and found that the area of the shape was 13 square units.

There’s just one small problem – you can’t use Pick’s theorem to find the area of this shape since the corners of the shape are not lattice points of the grid.

What to do . . . .

I wrote a quick little program that picked 100 million random points in the 10×10 square centered at (0,0) and tested whether or not they were part of the shape. That program found that 13.71% of the points were part of the shape – that was enough to convince him that the area might be larger than 13 square units.

Next I had him re-read Pick’s theorem to see what went wrong. He saw pretty quickly that the shape didn’t meet the condition of having the corners lie on lattice points.

I really wanted to try to find a way to make Pick’s theorem work with this shape.  I had him determine the y-coordinate for the far right corner.  The value was y = -2/7.

After finding that value, we had a good talk about scaling.  To make the new grid larger we had to *divide* the x and y coordinates in the equations by 7.  Here’s Mathematica’s picture of the new shape and grid (note that the x and y values run from -25 to 25 in this picture):


With this shape we are able to use Pick’s theorem to calculate the area.   We counted 40 grid points on the boundary without too much difficulty.   Counting the ones in the middle was a little bit more of a pain, so we wrote a short program to perform that calculation for us.  Note that we have to change the “less than or equal to” from the original equations to “strictly less than” since we want to be inside the shape:

So, we have 653 lattice points in the inside and 40 on the boundary.  Pick’s theorem tells us that the area is equal to the number of lattice points on the interior plus half the number of lattice points in the boundary minus 1.  That’s 672 units.    In the picture above, 1 unit is equal to 1/49 of a unit in the original picture, so the original area is 96 / 7 or 13 5/7.   Close to what he found originally, but not equal!

Along the way we also talked about alternate ways to find the area – the easiest being dividing the shape into two triangles with a vertical line through the middle.

I’m really excited about the discussion that we had tonight.  Funny how many important ideas in math can come up from a problem about absolute value and inequalities.




Thinking about a math appreciation class

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:

Here’s a link to that set of projects:

Steven Strogatz’s circle-area exercise part 2 (with a link to part 1)

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:

Our Patty Paper geometry projects

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:

OUr Fold and Cut projects

Here’s one fun fold and cut example:

(4) Exploring platonic solids with Laura Taalman’s 3d printed polyhedra nets

You can find Taalman’s post about these hinged polyhedra here:

Laura Taalman’s hinged polyhedra blog post on her Makerhome blog

And if you like the hinged polyhedra, here’s a gif of a dodecahedron folding into a cube!

dodecahedron fold

Which comes from this amazing blog post:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

[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:

Our projects based on Paula Beardell Krieg’s work

(6) And Paula didn’t just stop with one cube 🙂

(7) Two more of Laura Taalman’s prints

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.

Here’s Taalman’s blog post about the Peano curve:

Laura Taalman’s peano curve 3d print

(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.

Quanta Magazine’s article on DeMarco and Lindsey’s work

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.

Trying to understand the DeMarco and Lindsey 3d folded fractals

Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d Folded Fractals with kids

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.

Counting in 4 dimensions

Yesterday we did a neat project based on problem #12 from the 2015 AMC 8:


That project is here:

A great counting problem for kids from the 2015 AMC 8

Then I got a nice comment on the project from Alison Hansel:

So, for today’s project we extended the problem from yesterday to 4 dimensions.

Here’s the introduction and a quick reminder of yesterday’s problem. I had both boys review their solutions and then we began to discuss how to approach the same problem in 4 dimensions.

Next we dove a bit deeper into how to approach the 4 dimensional problem. They boys thought a bit about the symmetry that a 4d cube would have and at the end (after a long and quiet pause) my younger son thought that looking at how a square turns into a cube might help us.

In studying how a square transforms into a cube, the critical idea is how 4 edges turn into 12 edges. This video is a little on the long side, but I think the discussion is really interesting. By the end the boys have found the main idea for how to count edges as you move up in dimension.

Next I brought out Henry Segerman’s 4-d cube model and compared the model to the ideas we’d developed up to this point.

An important idea from earlier in this project was that my older son thought that each edge of the 4d-cube would be part of two 3d cube “faces”. Using the model we were able to see that, in fact, each edge is part of 3 cubes.

Finally – with the 4 pieces of prep work behind us! – we were able to answer the AMC 8 question about a 3d cube in 4 dimensions. So . . . how many pairs of parallel edges does a 4d cube have? The answer is 112 🙂

Thanks to Alison Hansel for the great suggestion for how to extend yesterday’s project. I think her idea makes a great way to introduce kids to some simple ideas in 4d geometry.

Going through three AMC 8 problems

My younger son has been doing a little practice for the AMC 8. Yesterday three problems from the 2013 exam gave him a little trouble. We went over them together.

The first was a tricky geometry problem. Both the words and ideas needed to solve this problem were new to him.

Next up was a challenging counting problem – we broke this into two pieces. This is a great counting problem for kids. In the first part we found out how to calculate the answer, but didn’t finish the calculation:

In the second part we talked about strategies to finish the calculation:

Finally – a fantastic geometry problem. It has a few little traps in it, but my son found a nice solution.

I love using the old AMC contest problems to help the boys see math that is both fun and challenging. These problems were really fun to talk through.

Playing with some mathy art ideas this morning

Yesterday my older son spent the afternoon at a friend’s house. When I went to pick him up I learned that his friend’s mom is an amazing artist:

Among the dozens (maybe 100’s) of pieces of work around the studio were some patterns on pieces of cloth. I asked how they were made and she told me that it was a simple process involving sharpies and alcohol. I knew instantly what we’d be doing for our math project this morning 🙂

Here’s a quick introduction to the project:

Now, with the camera off, I had the boys make their own patterns. Here’s my younger son’s description of what he made and what he thought he’d see when we added the alcohol:

Next up was my older son. He’d also experimented with a few different patterns and a few different markers.

This was a really fun project. I think this project would be fun for a wide variety of of kids – from elementary school kids (with supervision) up through high school. It is neat to hear the kids describe the shapes and, of course, the shapes and patterns themselves are really cool!

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

I woke up yesterday morning to see this problem posted on twitter by Alexander Bogomolny:

About a two months ago we did a fun project inspired by a different problem Bogomolny posted:

Working through an Alexander Bogomolny probability problem with kids

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, |x| + |y| \leq 1, 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:

One more look at the Hypercube

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.

Playing with Three Sticks

I saw this tweet from Justin Aion at the end of July and immediately ordered the game:

When I returned from a trip to Scotland with some college friends the game was on the dining room table – yes!! Today we played.

In this blog post I’ll show how the game ships and two rounds of play (and we might not be playing exactly right) to show how fun and accessible this game is for kids.

First, the unboxing. The game comes out of the box nearly ready to play.

Here’s our first round of game play. I think we misunderstood one of the rules here, but you’ll still see that the game is pretty easy to play:

Here’s the 2nd round of play. I think we understood the rules better this time, which is good. You’ll also see how this game gets kids talking about both numbers and geometry:

Finally, here’s what the boys thought about the game:

I’m really happy that I saw Justin Aion’s tweet and now have this game in our collection. It is a great game for kids!

Steve Phelp’s 3d pentagon

Sorry that this post is written in a bit of a rush . . . .

I saw a neat tweet from Steve Phelps earlier in the week:

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.