## More math with bubbles

Bubbles were just in the air this week!

and last night flipping through Henry Segerman’s math and 3d printing book I found these bubble project ideas:

So I printed two of Segerman’s shapes overnight and tried out a new bubble project this morning.

I started with some simple shapes from our old bubble projects – what happens when you dip a cube frame in bubbles?

The next shape we tried was a tetrahedron frame:

Now we moved on to two of Segerman’s shapes. These shapes are new to the boys and they have not previously seen what bubbles will form when the shapes are dipped in bubble solution.

If you enjoy listening to kids talk about math ideas, their guesses and descriptions of the shape are really fun:

The second shape from Segerman we tried was the two connected circles. We actually got (I think) a different shape than I’d seen in Segerman’s video above which was fun, and the boys were pretty surprised by how many different bubble shapes this wire frame produced:

Definitely a fun project. I tried a bubble project for “Family Math night” with 2nd graders at my younger son’s elementary school last year. Kids definitely love seeing the shapes (and popping the bubbles).

## Dave Richeson’s Knotted bubbles project

Saw this tweet from Dave Richeson last week which basically “had me at hello”:

We’ve done a few bubble projects in the past, so the boys were already familiar with the basic concept:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

More Zome Bubbles

Anyway, I ran out to home depot and got some wire and we made some knots. I had each of the boys make a trefoil knot and then make a random knot of their own choosing. In retrospect I wish I’d spent maybe just 5 minutes explaining some of the ideas in Richeson’s blog post – oh well, the excitement got the better of me 🙂

Here’s my older son playing with his trefoil knot and making a Mobius strip bubble. I love the “hey, I actually think I got it” moment:

Here’s him playing with the knot me made – in retrospect I’d argue for a knot that was slightly less complicated:

Next up was my younger son. First up was the trefoil knot and we got another great moment “I think this might be a Mobius strip” !!

Finally we made his own knot and explored. Again, I’d probably ask for a less complicated knot if I was doing this again:

So, that so much to Dave Richeson for posting his old project – this is an incredible project, and an especially great one for kids. The appearance of the Mobius strip is really quite an amazing little math miracle!

## More zome bubbles

I asked my older son to choose a project for today and he wanted to dip more zome shapes in bubbles. We’ve done a few of these project previously:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

My older son went first – his two shapes were a dodecahedron and an icosahedron:

My younger son made two neat shapes including a non-planar loop, or, as he said “a squigly decagon”:

## Trying out 4 dimensional bubbles

At the end of the project with my younger son this morning he remembered that we’d see some of the 4 dimensional shapes we were looking at in our Zome Bubble project. He went on to wonder if we dipped our 4 dimensional shapes in the bubble solution would we get a 5 dimensional shape. Well – We had to try that!

First, though, we looked at what happened when you dipped a cube and tetrahedron in bubble solution:

Next we tried the 4 dimensional shapes – what happens when you dip the zome versions of the 5-cell and the Hypercube into the bubble solution?

“I think we are going to see a 5 dimensional shape”

Here’s what happened:

I’m really loving just playing around with the 4 dimensional shapes with the boys. Soon we’ll move on to looking at the 4d version of Patrick Honner’s Pi Day project – can’t wait for that!

Finally, here’s the project from this morning that led to my younger son wondering about bubbles:

Sharing 4d shapes with kids

## 15 (+1 bonus) Math ideas for a 6th grade math camp

Saw an interesting tweet last week and I’ve been thinking about pretty much constantly for the last few days:

I had a few thoughts initially – which I’ll repeat in this post – but I’ve had a bunch of others since. Below I’ll share 10 ideas that require very few materials – say scissors, paper, and maybe snap cubes – and then 5 more that require a but more – things like a computer or a Zometool set.

The first 4 are the ones I shared in response to the original tweet:

## (1) Fawn Nguyen’s take on the picture frame problem

This is one of the most absolutely brilliant math projects for kids that I’ve ever seen:

When I got them to beg

Here’s how I went through it with my younger son a few years ago:

## (2) James Tanton’s Mobius strip cutting exerciese

This is a really fun take on this famous scissors and paper cutting exercise:

You will honestly not believe what you are seeing when you go through Tanton’s version:

Here’s the link to our project:

James Tanton’s incredible mobius strop cutting project

## (3) Martin Gardner’s hexapawn “machine learning” exercise

For this exercise the students will play a simple game called “hexapawn” and a machine consisting of beads in boxes will “learn” to beat them. It is a super fun game and somewhat amazing that an introductory machine learning exercise could have been designed so long ago!

Intro “machine learning” for kids via Martin Gardner’s article on hexapawn

## (4) Katie Steckles’ “Fold and Cut” video

This video is a must see and it was a big hit with elementary school kids when I used it for “Family Math” night:

Here are our projects – all you need is scissors and paper.

Our One Cut Project

Fold and cut project #2

Fold and cut part 3

## (5) Along the same lines – Joel David Hamkins’s version of “Fold and Punch”

I found this activity in one of the old “Family Math” night boxes:

Joel David Hamkins saw my tweet and created an incredible activity for kids.  Here’s a link to that project on his blog:

Joel David Hamkins’s fold, punch and cut for symmetry!

## (6) Kelsey Houston-Edwards’s “5 Unusual Proofs” video

Just one of many amazing math outreach videos that Kelsey Houston-Edwards put together during her time at PBS Infinite Series:

Here is how I used the project with my kids:

Kelsey Houston-Edwards’s “Proof” video is incredible

## (7) Sharing the surreal numbers with kids via Jim Propp’s checker stacks game

Jim Propp published a terrific essay on the surreal numbers in 2015:

Jim Propp’s “Life of Games”

In the essay he uses the game “checker stacks” to help explain / illustrate the surreal numbers. That essay got me thinking about how to share the surreal numbers with kids. We explored the surreal numbers in 4 different projects and I used the game for an hour long activity with 4th and 5th graders at Family Math night at my son’s elementary school.

This project takes a little bit of prep work just to make sure you understand the game, but it is all worth it when you see the kids arguing about checker stacks with value “infinity” and “infinity plus 1” 🙂

Here is a summary blog post linking to all of our surreal number projects:

Sharing the Surreal Numbers with kids

## (8) Larry Guth’s “No Rectangle” problem

I learned about this problem when I attended a public lecture Larry Guth gave at MIT.  Here’s my initial introduction of the problem to my kids:

I’ve used this project with a large group of kids a few times (once with 2nd and 3rd graders and it caused us to run 10 min long because they wouldn’t stop arguing about the problem!). It is really fun to watch them learn about the problem on a 3×3 grid and then see if they can prove the result. Then you move to a 4×4 grid, and then a 5×5 and, well, that’s probably enough for 80 min 🙂

Larry Guth’s “No Rectangles” problem

## (9) The “Monty Hall Problem”

This is a famous problem, that equally famously generates incredibly strong opinions from anyone thinking about it. These days I only discuss the problem in larger group settings to try to avoid arguments.

Here’s the problem:

There are prizes behind each of 3 doors. 1 door hides a good prize and 2 of the doors hide consolation prizes. You select a door at random. After that selection one of the doors that you didn’t select will be opened to reveal a consolation prize. At that point you will be given the opportunity to switch your initial selection to the door that was not opened. The question is  -> does switching increase, decrease, or leave your chance of winning unchanged?

One fun idea I tried with the boys was exploring the problem using clear glasses to “hide” the prizes, so that they could see the difference between the switching strategy and the non-switching strategy:

Here’s our full project:

Exploring the Monty Hall problem with kids

## (10) Using the educational material from Moon Duchin’s math and gerrymandering conference with kids

Moon Duchin has spent the last few years working to educate large groups of people – mathematicians, politicians, lawyers, and more – about math and gerrymandering.  . Some of the ideas in the educational materials the math and gerrymandering group has created are accessible to 6th graders.

Here’s our project using these math and gerrymandering educational materials:

Sharing some ideas about math and gerrymandering with kids

## (11) This is a computer activity -> Intro machine learning with Google’s Tensorflow playground.

This might be a nice companion project to go along with the Martin Gardner project above. This is how I introduced the boys to the Tensorflow Playground site (other important ideas came ahead of this video, so it doesn’t stand alone):

Our complete project is here:

Sharing basic machine learning ideas with kids

## (12) Computer math and the Chaos game

The 90 seconds starting at 2:00 is one of my all time favorite moments sharing math with my kids:

The whole project is here, but the essence of it is in the above video:

Computer math and the chaos game

## (13) Another computer project -> Finding e by throwing darts at a chess board

This is a neat introductory probability project for kids. I learned about it from this tweet:

You don’t need a computer to do this project, but you do need a way to pick 64 random numbers. Having a little computer help will make it easier to repeat the project a few times (or have more than one group work with different numbers).

Here’s how I introduced the project to my kids:

Here’s the full project:

Finding e by throwing darts

## (14) Looking at shapes you can make with bubbles

For this project you need bubble solution and some way to make wire frames. We’ve had a lot of success making the frames from our Zometool set, but if you click through the bubble projects we’ve done, you’ll see some wire frames with actual wires.

Here’s an example of how one of these bubble projects goes:

And here’s a listing of a bunch of bubble projects we’ve done:

Our bubble projects

## (15) Our project inspired by Ann-Marie Ison’s math art:

This tweet from Ann-Marie Ison caught my eye:

Then Martin Holtham created a fantastic Desmos activity to help explore the ideas:

It is fun to just play with, but if you want to see how I approached the ideas with my kids, here are our projects:

Using Ann-Marie Ison’s incredible math art with kids

Extending our project with Ann-Marie Ison’s art

## (16) Bonus project!!A dodecahedron folding into a cube

This is a an incredible idea from 3d geometry.

We studied it using our Zometool set – that’s not the only way to go, but it might be the easiest:

Here’s the full project:

Can you believe that a dodecahedron folds into a cube?

## Ten 3D Printing math projects to help students explore math

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:

The Global Math Project’s Facebook page

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:

Learning from 3D Printing

Here are some sample projects:

(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:

Here are some of the shapes we printed as we explored what the shape itself looked like:

and here are the two projects that we’ve done exploring this problem

James Tanton’s geometry problem and 3d printing

Revisiting James Tanton’s Tetrahedron Problem

(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:

(3) A second idea from Segerman – exploring shadows

One of Segerman’s most beautiful creations is on the cover of his book:

It is incredibly fun to have kids explore this shape:

Here’s the project we did after seeing Segerman give a talk last fall:

Playing with Sahdows inspired by Henry Segerman

Here’s a link to all of our project inspired by him:

All of our Henry Segerman-inspired projects

(4) There is also no way to limit Laura Taalman’s work to one example.

Here’s a project where we explored some of here 3d printed knots – one of which was featured in Segerman’s talk yesterday:

Playing with some 3d printed knots

(5) Here’s another project inspred by Taalman – tiling pentagons

Taalman’s 3d printed tiling pentagon designs are one of the most amazing pieces at the intersection of math and education:

We’ve used them for several projects including making cookies!

Here’s that project

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

and here’s a link to all of our projects inspired by Laura Taalman:

All of our projects inspired by Laura Taalman

(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:

$(x + y)^2 = x^2 + y^2$

Here’s what these two surfaces look like:

Here’s two projects exploring these algebra ideas with the boys:

Comparing x^2 + y^2 and (x + y)^2 with 3d printing/a>

Comparing Sqrt(x^2 + y^2) and Sqrt(x^2) + Sqrt(y^2) with 3d Printing

(7) 3D printing can also be surprisingly useful for studying 2d geometry

We’ve done a few neat projects in this area.

(i) Which triangle has larger area, a 5-5-6 triangle or a 5-5-8 one?

A nice little triangle puzzle

A few follow ups to the triangle puzzle

https://mikesmathpage.wordpress.com/2017/03/01/a-nice-little-triangle-puzzle/

(ii) A neat geometry idea from Patrick Honner

Here’s how we used 3d printing to explore this triange:

Inequalities and Mr. Honner’s Triangles

(iii) A neat geometry problem shared by Tina Cardone

Here’s how I explored this problem with 3d printing

A Cool Geometry Problem Shared by Tina Cardone

which led to a fun and unexpected follow up:

A Follow Up to Our Tina Cardone Geometry Project

(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:

Sharing a Craig Kaplan post with kids

(9) 3d printing can also make abstract math / advanced problems accessible

A few months back I saw this problem shared by Alexander Bogomolny:

Nassim Taleb’s look at the problem on Mathematica made me think that the problem could be shared with kids:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

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:

Here’s that project:

Extending our Bogomolny / Taleb project from 3 to 4 dimensions

(10) Using 3d printing for calculus and beyond

I’m written a few posts and done a few projects about how to use 3d printing to explore some basic ideas from Calculus.

That collection of posts is here:

Posts about 3d printing and calculus

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:

Revisiting the Volume of a Sphere with 3d printing

## Playing with 3d printed knots from Mathematica

Yesterday I learned that Mathematica has a wide variety of knots that you can 3d print. We’ve done a few knot projects in the past. Here are 3 of them:

Playing with some 3d printed knots

Dave Richeson’s knotted bubbles project

I thought that actually being able to hold the printed versions of so many different knots in your hand was going to be a game changer for knot projects, though. So, I printed a few as test cases and had the boys look at them.

My older son went first:

My younger son went next – he had a couple of things to say, but wanted to point out some of the knots in Colin Adams’s book, so we cut this video a little short so that we could go find the book:

After finding the book we were trying to match one of the printed knots with the knot in the book that he had wanted to print. The knot he wanted to print had 8 crossings and the one that we thought matched it turned out to have 7. Whoops – we had the wrong knot 🙂 A good accidental lesson that comparing two knots isn’t super easy!

I’m really looking forward to trying more projects with these prints. There are a little over 30 different knots with 8 or fewer crossings. It’ll probably take a week to print them all, but that’ll be a fun collection to have for future knot projects!

## 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.

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

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:

## Our math year in review

I’m sick and not working today but instead have been sort of day dreaming about all of the math the boys and I worked on this year. The sheer amount of absolutely great ideas that people are sharing via blogs, twitter, or otherwise makes it incredibly easy to find fun projects. Here are some of my memories from 2016.

Again, though, I’m sorry that this likely reads so poorly. Despite being so sick it was fun to write this and think back through the year.

(1) Sharing math that I saw from professional mathematicians

Two of my favorites in this category came from Bjorn Poonen, Eugenia Cheng and Laura Taalman:

The project we did after seeing Poonen’s problem about n-dimensional spheres was the most viewed math project for kids on my blog this year:

Bjorn Poonen’s n-dimensional sphere problem with kids

A neat video from Eugenia Cheng inspired me to revisit an old post from Laura Taalman and do a project on “tiling pentagon cookies”

At the risk of failing to mention lots of people in my current Dayquil-induced state, I’m also incredibly grateful to these professional mathematians who have inspired tons of our projects with the math they’ve shared:

Jim Propp

Steven Strogatz

Evelyn Lamb

Dave Richeson

and one of the neatest things that happened to me all year was when Joel David Hamkins created an amazing “fold and punch” activity based on an activity that I found in some old material from “Family Math Night” at my younger son’s school:

Math for nine year olds: fold, punch and cut for symmetry

(2) James Tanton

I have to give him his own category because the work he is doing to share math with kids (and everyone, really) is astonishing.

This project on cutting Mobius strips that I saw in his book “Solve This” is one of the most incredible math projects I’ve ever seen:

An absolutely mind-blowing project from James Tanton

Here are all of our project inspired by Tanton – you can basically pick one at random and have an amazing math conversation with kids (though if you don’t want to pick at random the candy dividing one is really cool!):

Project inspired by James Tanton

(3) Laura Taalman and Henry Segerman’s work in 3d printing

The work that Taalman and Segerman are doing with math and 3d printing is stunning. I mentioned one of projects inspired by Taalman above – there are dozens’s more:

Projects inspired by Laura Taalman

Segerman published a new book about math and 3d printing this year.  I was incredibly lucky to be able to bring the boys to a talk he gave about his work:

Projects inspired by Henry Segerman

(4) Sharing new and / or popular math ideas with kids.

Erica Klarreich and Natalie Wolchover are doing amazing science (and especially math!) journalism work at Quanta Magazine. Oh to have had writing like theirs around when I was a kid!

I’m so happy to see pieces like Klarreich’s article on Maryna Viazovska’s sphere packing result

In fact, when I asked my older son what his favorite math memory was from 2016 he said it was learning about sphere packing. Yay!

Wolchover’s article about hyperuniform distributions blew me away and led to a really fun project with the boys:

Using a Natalie Wolchover article to talk about hyperuniform distributions with kids

I don’t think it is possible to overstate the importance of Klarreich’s and Wolchover’s writing. They are going to influence a generation of young mathematicians and physicists.

Another fun math-related item that got a lot of attention this year was Sugihara’s “ambiguous cylinder”:

We really had fun playing with this shape and I want to give a special thanks to Dave Richeson and Brenda Landis for sharing a 3d print of the shape.

Playing with Sugihara’s “ambiguous cylinder”

(5) The sphere packing problem reminded me of the new PBS Infinite Series work that Kelsey Houston-Edwards is doing.

Holy cow are these videos amazing! Here’s just one example:

Wwe are one behind because of the holidays, but each of Houston-Edwards’s videos has inspired a really fun project. Her videos are great tools to use to share math with kids.

Projects inspired by Kelsey Houston-Edwards

(6) Three projects from twitter that completely blew my mind:

There are actually a couple of projects that Simon Gregg’s tweet inspired. The main picture is this one (which always has weird embedding problems, so sorry it isn’t aligned correctly):

Prepping for this project to make sure that we could do it with our Zometool set was really fun, too:

A neat post from Simon Gregg

Much like the James Tanton “cutting a Mobius strip” project above, the idea is to try to guess what the shape is going to look like when you unzip it!

This was a great project with the boys and I also used it for a talk to a high school math camp at Williams. If you play with the Desmos program below your mind will be blow, too 🙂

(7) We played with more math-related art, too:

Paula Beardel-Kreig’s “Puff Boxes” were incredibly fun:

Playing with Paula Beardell-Krieg’s Puff Boxes

And Henry Segerman’s 3D Printing book introduced me to the work of Bathsheba Grossman, and we explored several of her creations:

Our projects with Bathsheba Grossman’s work

(8) Our Zometool work

I know there are lots of ways to spend money on math-related games, books, and toys in general. Building up a good Zometool set is my #1 recommendation. The opportunities to play and learn and study are endless!

Here are all of our Zometool projects:

Our projects with our Zometool set

I particularly recommend the bubble projects and Nesting Platonic Solids

(9) Dan Anderson’s Gosper Curves

When I asked my younger son what his favorite project from this year was he said that it was playing with the Gosper Curves. He really likes fractals!

We got a nice surprise in April when Dan Anderson sent us some laser-cut versions of the Gosper island shape:

Playing with Dan Anderson’s Gosper Curves

We did a few more Gosper-related project (including a Zome one) which are here:

Our Gosper projects

And, of course, we did a million projects inspired by ideas that Dan shares on twitter:

Our Projects Inspired by Dan Anderson

(10) Patrick Honner’s Pi Day exercise

On March 14th Patrick Honner shared a fun little “Pi Day” exercise:

This terrific project inspired me to try it out in 4 dimensions. That led to a fun multi-day project with my older son as we search for which 4-dimensional platonic solid was the most spherical (according to Honner’s definition).

This project combined ideas from geometry, Zometool, and 3D printing.

Here’s a collection of the projects:

Patrick Honner’s Pi Day Exercise in 4 dimensions

Honner’s “pi day” exercise is a perfect example of why I love all of the sharing of math ideas that people are doing these days. Not in a million years would I have come up with an idea like that – luckily he did, though, and it turned out to be a really fun way to explore more than just 3d objects!

It really was a great year in math for us. Can’t wait to see what 2017 brings.