Learning from 3D Printing

About a month ago Patrick Honner linked to this video about a 3D printer at his school:

What I found particularly intriguing about this video was the potential educational uses of 3D printing.  Though I’d seen articles here and there about 3D printing, the focus always seemed to be what you could make rather than what you could learn.   The educational possibilities in the Brooklyn Tech video convinced me to get one.   We’ve had it for a little over week and are really having fun learning how to use it.   One of the best resources we’ve found so far is this amazing blog by James Madison University math professor Laura Taalman, aka @mathgrrl:


By coincidence, the Mathematical Association of America just this week released this video where Taalman talks about some of her experiences with students and 3D printing.  Her example about printing a set of Borromean rings was particularly fascinating to me.

Mostly as a result of playing around on the MakerHome blog, we’ve printed several different knots, a Sierpinski tetrahedron, a bunch of different polyhedra and some really neat hinged shapes (and this list is just what’s in front of me on the kitchen table right now!):


We’ve also made a few things on our own after learning from some of the instructions on the MakerHome blog as well as from this helpful video from Wolfram:


For example, from those two sources, and lots of trial and error, we were able to print out a hollowed out cube that illustrates the “Prince Rupert Problem” – a cube is actually able to pass through a second cube of the same size:

Rupert Side By Side

Rupert In

Much like the Brooklyn Tech and Taalman videos suggested, printing this example is filled with great ways for kids to see some interesting math.  I’m really excited to find more fun projects to do with the boys.  I think this is going to be a great tool to help them understand some pieces of math that may have previously been a little out of their reach right now.

Johnson Polyhedra

It feels as though the MakerHome blog by Laura Taalman (aka @mathgrrl )  is going to be a source of great fun for a long time.  The entry on March 6th provides several links to sites for printing 3D polyhedra and shows several examples of her prints:


The link to Johnson polyhedra grabbed our attention and we decided to print the mysterious object known as  J79 – the “Bigyrate dimished rhombicosidodecahedron” (it has two colors because our white spool ran out during the print):


I had not previously heard of the Johnson polyhedra.  For me finding out about objects like these is part of the great fun that the 3D printer brings – you can now hold these objects in your hand rather than just reading about them, so there’s more incentive talk about them with kids (and to just play around with them!).

And play we did.

After printing we decided to see if we could make the same shape with our Zometool set.  Here we actually ran into a little bit of difficulty when it appeared that we could only make about half of the shape.    Just before were going to call it a night, though, we noticed that the combination of a short and medium red piece was the missing link we were looking for.  Here’s the finished product:


These two videos give a better view of the objects:

Finally, if you would like more information on the other Johnson Polyhedra, both Wolfram Mathworld and Wikipedia sites have quite a bit of information:



Sue Van Hattum’s Optimization Problem

Last night John Golden included  one of our Family Math projects in a list of some neat math-related posts he’d seen recently.  Thanks, John, for including us in the Math Carnival:


Reading through some of the other projects in the list, I found a really neat activity by Sue Van Hattum and decided to try it out with the boys this morning.  The idea in this activity is to find the box with maximum volume that you can make using a sheet of paper:


Although this is really a calculus activity, it still seemed like there was plenty of interesting math for kids to talk through and I figured that I’d just show them the final optimization step on a computer.

The first step was an introduction to the problem.  We’ve talked a little bit about geometry, so the basic ideas behind how to find the volume of a box were at least something that they’ve heard before:

The next step was to go to the kitchen table and look at some examples.  Before we turned the camera on I had each kid prepare one of the sheets of paper we’d be using in this section.  It was really interesting to me to hear them talk about the patterns that they thought would be in these numbers, as well as where they thought the maximum would be.

The last step was going to the computer to see what the graph of our volume function would look like.   I’ve recently been talking about graphs of functions with my older son as part of his normal school work.  Some of the lessons I’ve learned from those discussions have made me want to talk a bit more about graphs with both of the kids, that was part of what attracted me to this lesson.  I though it was neat to see them connect the specific calculations we’d done in the prior step to the points on the graph and then see that the maximum volume actually occurred at a slightly different value of x than they were expecting.

Thanks again to John for including us in his post, and thanks to Sue for this great exercise.


Earlier today I attended the memorial service for Michael Goodgame, one of three students at Carleton College who were killed in a car accident last week.  Michael was traveling with his teammates on the Carleton men’s ultimate team to a tournament in Stanford, CA when the accident happened.   I did not know him, but having attended the funeral last year of a player I used to coach – Brute Squad’s Stephanie Barker – I felt that I needed to be there to today to help support the Carleton team.


Dealing with Stephanie’s death was one of the most difficult things that I’ve been through.  Though I was not coaching Brute Squad last year, the love you have for the players you work with does not fade, and it was terrible to feel powerless to help them through this horrible time.

One of the things that helped me, though, were the words of Jason Adams, Stephanie’s friend and college coach at Northeastern.  He talked about how different people have different ways of dealing with tragedy and that no one way was wrong.   Outside of the memorial service, a short conversation with Brute Squad’s Sara Jacobi made me want to write about some of the players I’ve coached, why they’ve inspired me, and why I loved them so much.  I wrote about Molly McKeon and Gwen Ambler of Riot, and Blake Spitz and Peri Kurshan of Brute Squad.  Writing about these players helped me deal with Stephanie’s death.  I think that’s why I’m writing now.

I arrived at today’s memorial service about an hour early so I could have time to sit and reflect.  For a while I was the only one sitting in the pews, and the pastor came over to talk to me.  We chatted for a bit about how to help people deal with these tragedies.  I told her about Jason’s words and how they helped me last year.   She liked what he had to say.

I don’t know how Michael’s father had the strength to speak at this memorial, but he spoke beautifully and his words have struck me much like Jason’s words did.

He spoke about the words that Achilles’s mother said to him in the Iliad.  After a little searching when I got home, I think he was paraphrasing this passage (section 9.410):

For my mother Thetis  the goddess of the silver feet tells me

I carry two sorts of destiny toward the day of my death.  Either,

if I stay here and fight beside the city of the Trojans,

my return home is gone, but my glory shall be everlasting;

but if I return home to the beloved land of my fathers,

the excellence of my glory is gone, but there will be a long life

left for me, and my end in death will not come to me quickly.

He didn’t quote this passage directly  though, but rather focused on the choice Achilles had of a short life with glory or a long but uneventful life that would be “forgotten in two generations.”  Those last words really struck me for some reason.  I did not understand where he was going, but what he said next was powerful.

For him the implication of Thetis’s message to her son was that the small acts of kindness done over many years do not have lasting meaning or glory.   She was wrong, he said, and his son Michael was proof.  He thanked all of Michael’s teachers, mentors, and friends for all of their kindness and love towards his son over the years, and said that he had become the incredible kid that everyone was there to celebrate because of them.    It was a beautiful message of love.

After the service I spent some time talking with Carleton’s coach who was doing his best to help his players cope.  They are all getting on a plane at 6:00 am tomorrow to attend two more memorial services in Minnesota.    I mentioned Jason’s words from last year to him, and how those words had inspired me to write about some of the players I’ve coached.  We talked about how we love these kids.

As I was driving home my thoughts drifted away from the Iliad to Wordsworth’s “Tintern Abbey.”  His words reminded me of the love I have for the players I’ve worked with and how just thinking about them is uplifting to me:

                                                           if this

Be but a vain belief, yet, oh! how oft –

In darkness and amid the many shapes

Of joyless daylight; when the fretful stir

Unprofitable, and the fever of the world,

Have hung upon the beatings of my heart –

How oft, in spirit, have I turned to thee.

How often indeed, and today especially.  And the end of the poem captures how, just because of them, the last 6 years have been so special to me:

. . . . these steep woods and lofty cliffs,

And this green pastoral landscape, were to me

More dear, both for themselves and for thy sake.

Hug your teammates.


Using Manipulatives for a fun twist on Dan Meyer’s geometry problem

About a week ago Dan Meyer posted an interesting geometry problem on his blog and asked for some open-ended feedback.  Over the next few days he provided some additional thoughts / content and the resulting discussion was fascinating.


I’ve written two prior blog posts about the problem and an extension that was particularly interesting to me.  Thanks to an Evelyn Lamb post on Twitter today I saw something new that reminded me of Dan’s problem and made me think of a way to introduce a similar problem to younger students using manipulatives.

Each day since the end of August  @mathgrrl has been posting a 3D printing project on the MakerHome blog.  The March 2nd post was on Haberdasher’s Problem:


These 3D prints from the Haberdasher problem give younger students a way to gain access to Dan’s problem without the need to know any geometry formulas at all.  We just need to remove the circle from the original problem and replace it with an equilateral triangle.    The new problem would go something like this, but, of course, I’m not suggesting to use this specific abstract mathematical language with younger kids:

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the perimeter of an equilateral triangle. Find P such that the area of the square and triangle are equal.

Here’s what I think I’d do (and I would love to have been able to run through this with my own kids, but a stomach bug had other ideas, unfortunately):

(1) Given the kids these manipulatives and let them play around:



It is pretty amazing that you can form both a square and an equilateral triangle from these hinged shapes.  I think it would be fascinating to hear kids who haven’t had much exposure to geometry discuss the various relationships between the shapes (including concepts of area and perimeter) using whatever mathematical language is appropriate for them.  The goal isn’t precision or geometry formulas just discussion.

(2) The next would be to introduce the new version of Dan’s problem without the mathematical formality.  Several of the commentors on Dan’s original post suggested the idea of cutting a string to form two shapes of equal areas.  That formulation of this new problem would work really well here, I think.  There’s a new and pretty interesting math challenge at this step relating to scaling.  If the string happens to be exactly as long and the combined perimeters of our square and triangle, the kids already have the solution in front of them.  However, if the string is a different length, the problem hopefully will lead to  a neat discussion about ratios and scaling.

By funny coincidence the correct cut of the string divides it into two pieces roughly equal to 47% and 53% of the original string’s length – almost exactly the same as the two pieces for the original square and circle problem (with a twist!).  Maybe that fact would provide an interesting extension, too.

As I said in my prior posts, Dan’s found a really fun problem!

Fun with Möbius Strips

The interesting math topic that I saw in a couple of different places over the last two days involves Möbius strips, and a question from my youngest son about the Möbius strip this morning led to a fun conversation.

The thing that got him thinking about Möbius strips was this 3D printing page about the Umbilic Torus:


We followed the directions on this page yesterday evening and printed a small triangular cross section Möbius strip.  It is a really neat shape and my younger son had a lot of fun playing with it yesterday before he went to bed.  I hope we can figure out how to print an actual Umbilic torus, but we don’t yet have that level of 3d printing skill just yet.

After he went to bed I saw this wonderful video posted by Steven Strogatz on twitter:

I was really happy to see Strogatz’s post for a couple of different reasons.  First, several months ago someone else had posted about the double Möbius strip in association with a Martin Gardner celebration and I’d not been able to find the link again.  Unfortunately I’d misunderstood the setup so it was nice to see where I’d gone wrong.  Second, having just spent half an hour talking about 3D Möbius strips the exercise in this video was a great natural extension (which I’ll probably do with the boys this weekend).

With that background, I was really happy to have my son ask this question today when he got up:

“What happens if you do two half twists instead of one when you make a Möbius strip?”

That led to this fun little discussion which shows a surprising result from geometry:

This is one of my favorite examples in math!

Of course, no discussion of Möbius strips is complete without showing Vi Hart’s amazing “Wind and Mr. Ug” video:

Such a fun topic for kids.

Fine, Ed Frenkel, you convinced me . . . .

(Morning note – I wrote this way to fast trying to get out the door for work.  Sorry for the likely typos and formatting erros, will edit later)

Yesterday Ed Frenkel of UC Berkeley published this op ed in the LA Times:


He also followed up with this twitter link to a Richard Feynman video:

In the video Feynman talks about some conversations with his cousin about solving linear equations using algebra.  By unlucky coincidence, I’m right in the middle of that topic with my younger son right now so the video really did hit home.

Though I haven’t read Frenkel’s book “Love and Math” yet, it does seem that he and I are pretty much on the same page when it comes to math education.  Even to the point where I’ve been using Rubik’s cubes to teach all sorts of fun little math to the kids – ha.


I took Frenkel’s weekend posts to heart today and instead of further work on solving linear equations, I decided to cover a fun math application with my younger son -> maps.

First we took a close look at our wall map of the Earth and talked a little bit about the distortions in the map near the poles:

Next, using a lacrosse ball and a strip of paper, we walked through how you might try to make a map of a sphere on your own and the problems that you’d encounter:

Finally, we tried to recreate an amazing video that Henry Segerman made about stenographic projection:

The link where we found Segerman’s sphere (including his video which is much better than ours) is here:


Definitely a fun morning of math, so thanks to Ed Frenkel for the inspiration.  I hope he succeeds beyond his wildest imagination in convincing the world that there’s lots of fun math that is accessible to young kids.

Pascal’s and Sierpinski’s triangle

We’ve really been enjoying “The Math Book” by Clifford Pickover (sorry, I don’t know Latex well enough to embed the \alpha and \beta into the book title).  We started reading it last weekend and did a project on the Prince Rupert problem:


During the week I’ve been having the kids read a section of their choice and write a little one page report.  They’ve written on the Menger sponge, the Klein bottle, the Hilbert hotel, slide rules, the 15 puzzle, and Pascal’s triangle.  The short (one page, mostly) sections in the book allow the kids to read and the write about interesting math, so these short projects have been a lot of fun.
Yesterday my youngest son wrote about Pascal’s triangle, and my older son had an interesting comment on the pictures in the book – why was there a picture of Sierpinski’s triangle in the section about Pascal’s triangle?    Good question, and one that we attempted to tackle this morning in our weekend Family Math series.

The first step was a short talk about the basics of Pascal’s triangle.  It is a nice little arithmetic review for younger kids, and there are so many fun identities hiding in the triangle that you could talk about Pascal’s triangle many times without worrying about running out of material.  In fact, just this week Alexander Bogomolny at Cut the Knot posted this neat set of identities that I don’t remember ever seeing previously:

For now, the infinite series math is a little over our heads, but there is still plenty of interesting math for kids in Pascal’s triangle:

After this short little discussion of Pascal’s triangle and how it worked, I showed them how you could simplify the triangle and get something that starts to look like Sierpinski’s triangle:

Always fun to play around with Pascal’s triangle, and if you are looking for a book that can help kids see some really fun math, get your hands on Pickover’s new math book as fast as you can!