Extending Vsauce’s 4 dimensional shadow tweet a bit

Saw a fun tweet from Vsauce before we left for vacation:

He later shared the shape, too:

Using shadows is a an incredibly fun way to explore complex shapes. Henry Segerman gave an amazing talk about the idea last fall at MIT:


I think playing with these sorts of shadows is a great way to share complex shapes with kids, too (we also used a Zometool set!):

Playing with shadows inspired by Henry Segerman

4-dimensional Shadows

Playing with our Zometool model of Bathsheba Grossman’s “Hypercube B” was especially cool – you can see some of the same effects as in the Vsauce video, though I think the two shapes are a little different.

Though this project shows that there are a few different Zome versions of the shape, so maybe my blue-strut version isn’t all that different from Vsauce’s 4-d cube after all:

Bathsheba Grossman’s “Hypercube-B” part 2

Anyway, as Vsauce’s tweet shows, exploring shapes via shadows is fascinating. It is also a really fun way to introduce kids to shapes that they’ve probably never seen before!

When we were egged on by Katie Steckles

Saw this fun video from Katie Steckles yesterday:

We’ve used several ideas from Steckles for our projects. Her fold and cut video, in particular, led to several really fun projects that are collected in this blog post:

Our Math year in review part 2: Fold and Cut

The demonstration in her egg video seemed like a eggcellent  pre-Easter / pre-vacation project, so I had the kids watch and then we grabbed a pair of compasses and went to work.

My older son has a bit more experience with geometric constrictions, so I had him go first:

It turns out that this project is the first ever project related to geometry constructions for my younger son. It was definitely fun to see him doing all the work and also probably pretty lucky that he got to see how to draw the egg twice before making his attempt:

This was a really fun project.  I think this is an especially fun project for kids using a rule and compass pair of compasses for the first time!

Sharing Kelsey Houston-Edwards’s video about Pi and e with kids

Yesterday I a new video from Kelsey Houston-Edwards that just blew me away. At this point I don’t have the words to describe how much I admire her work. What she is doing to make challenging, high level math both accessible and fun for everyone is amazing.

The new video was about this question on Math Overflow from Erin Carmody:

If I exchange Infinitely many digits of Pi and E are the two resulting numbers transendental?

Before showing the boys Houston-Edwards’s video, I wanted to see what they thought about the question. So, we just dove in:

Next, I took a great warm up idea from Houston-Edwards’s video and asked the boys if they could find *any* two irrational numbers that you could use to swap digits and produce a rational number.

Now, with that little bit of prep work, we watched the new video:

After the video we talked about what we learned. I think just tiny bit of prep work we did really helped the boys get a lot more out of the video.

One of the fun little challenge questions from the video was to show that (assuming \pi and e differ in infinitely many digits, then you will produce uncountably many different numbers by swapping different digits. I didn’t expect that the boys would be able to construct this proof, so I gave them a sketch of how I thought about it (and hopefully my idea was right . . . . )

I think that kids will find the ideas in Houston-Edward’s new video to be fascinating. It is so fun (and sadly so rare) to be able to share ideas that are genuinely interesting to professional mathematicians with kids. As always, I can’t wait for next week’s PBS Infinite series video!

Sharing problem #3 from the European Girls’ Math Olympiad with kids

Yesterday I saw the great news that team USA won the European Girls’ Math Olympiad:

Flipping through the problems last night, problem #3 really caught my eye as one that math students might really enjoy because the solution is really cool. Here’s the problem:


This afternoon I thought it would be fun to talk through the problem with the boys. I have no expectation that they would be able to solve this problem – obviously! – but I really did think that a sketch of the solution would be really interesting to them.

I started by talking through the problem to make sure that they would understand it:

Once the boys understood the problem we dove into trying to solve it – where do you even begin – both boys said in the last video that the problem seemed impossible! Starting with some simple configurations with 2, 3, and 4 lines helped us see that the answer to the problem might be “no”.

To wrap up I showed the boys how you solve this problem via a coloring argument. The critical idea is that you can color the regions that are formed by the lines, with no two regions sharing a side having the same color – with just two colors. Once you have the coloring, there’s a fun little “aha” moment when you watch the path the snail takes . . . .

So, a seemingly impossible problem has a really pretty and really instructive solution. I think the coloring idea is something that middle school and high school kids who are interested in math will really enjoy seeing.

Exploring the “Snapchat problem” with my kids

A few weeks ago a friend asked me this question:

“Say [our group]] has 26 people in it, but Snapchat has a limit of 16 in a group. How many groups would need to be created so that everyone is in a group with all of the [people in the original group]?”

I wrote about my solution to the problem here:

A fun math question about Snapchat from a friend

Last night I finally got around to going over the problem with the boys. When I first saw the problem I thought that it would be a really interesting problem for kids to explore. My kids are probably on the young side to fully understand the problem, though, so I started the project by making sure that I explained the problem really carefully. One thing that turned out to be a bit challenging for my younger son was distinguishing between the group of 26 people and the groups of 16 people:

At the end of the last video my older son wanted to start drawing some pictures. I let him draw in the 26 people off camera and then we started exploring how to make the groups. There were several interesting mathematical ideas that came up almost immediately. For instance: if we are going to minimize the number of groups, how many people should we try to put in those groups?

One thing that was really interesting to me here is that my younger son had a really hard time translating the words describing how he wanted to make the groups into what to draw in the picture.

We just paused the last video after we got to about 5 min and continued the discussion. I was a little bit stumped about what to do – my younger son had exactly the right idea and was saying the exact right words, but he just couldn’t figure out how to draw in some lines that represented those words. My plan was to simply let him figure it out.

Most of this video is my younger son struggling to find a way to draw the picture he wants. He eventually got it, though, which made me happy.

So, we’d just found a way to solve the problem with 4 groups, which was really cool – they found that solution faster than I did 🙂 The question now was could we do it with three?

Both kids had some pretty good intuition about why 3 would be impossible. I wanted to show them a slightly more mathematical way to think about it. Although it took a bit longer than I expected, it was fun to explain this proof to them.

I really like this problem and think that lots of kids learning math will really like it. The process you go through in solving this problem is a great example of problem solving in math.

A fun present from Paula Beardell Krieg

I met our friend Paula Beardell Krieg today while she was visiting Boston. She gave me an amazing shape that she’d made with paper. The shape builds on a few other shapes that we’ve studied in some recent projects:


So, tonight I showed the boys the shape and just let them play with it. Honestly, it is incredible to me that this shape is made out of folded paper – you can just sit and play with it forever!

My younger son played with the new shape first:

and here’s what my older son had to say:

For the last part of the project I wanted to show the boys a shape that Paula showed me earlier today. The trouble was that I’d not been able to make it again! I was actually hoping that one of the boys would make it accidentally . . . . but no.

Luckily I was able to make it fairly quickly after my older son finished playing. So, here’s the surprise shape:

Another wonderful project inspired by Paula – we are very lucky to have met so many great math folks on twitter!

Matt Enlow and Suzanne von Oy’s geometry problem

Saw a tweet from Matt Enlow today that led to a fun discussion and also a fun project tonight with the kids:

The last tweet in the conversation was a new Desmos activity from Suzanne von Oy showing how the problem worked:

I couldn’t wait to try out this problem with the boys tonight. We aren’t (obviously!) going to go into a lot of depth – this isn’t really a problem for 5th graders! But, I thought the boys would have some fun talking about it.

Since the problem is a pretty challenging one for kids to even understand, I started the project by trying to explain the problem carefully.

Next we tried to pick some points at random and then draw some triangles. If there are infinitely many equilateral triangles passing through these 3 points, it ought to be easy to draw one of them, right?

My older son went first. The cool thing for me in both this and the next video was seeing kids experience the problem and struggle with both trying to understand it and trying to solve it. There really is a lot of great geometry for kids here:

Next my younger son gave it a try. His approach was absolutely terrific to watch – I never would have approached the problem the way he did.

Next we went to play with Suzanne von Oy’s Desmos program. We got interrupted by the new puppy in the house across the street between leaving the living room and heading upstairs to play with the program, so I took the first minute of this video to review the problem again.

My older son went first again. He quickly found a picture that didn’t satisfy the conditions of the problem and that threw him for a little loop. Once we got past that, though, he seemed to have a much better understanding of the problem.

My younger son went next and eventually found an arrangement of the points that didn’t work at all. That was actually a really cool surprise ( we’ll deal with that surprise in the next video).

So, we got a wonderful surprise in the last video when we stumbled on an arrangement of the three points that didn’t seem to have any equilateral triangles passing through them.

Talking about what went wrong was a fantastic little surprise and it really made this project for me.

This was a super fun project. Thanks so much to Matt Enlow and Suzanne von Oy for sharing both the problem and the Desmos activity. Math twitter is amazing!

10 complex, rich tasks to recommend after seeing Geoff Krall’s Shadowcon talk

Earlier today I saw a link to Geoff Krall’s Shadowcon talk. In part of the talk he made the case for sharing complex, rich tasks with kids. I might not have the exact words right because I’ve completely failed to find the link to the Shadowcon talks I saw this morning so I can’t find the video to the talk again 😦

But, hopefully, though I do have the general idea right and I wanted to share some tasks that I think fit the bill.

Some of the tasks I’ve discussed before in this blog post:

10 Pretty Easy to Implement Math Activities for Kids

but I think they are worth sharing again!

(1) A fun question from a friend about math and Snapchat

Here’s the question my friend asked:

“Say [our group]] has 26 people in it, but Snapchat has a limit of 16 in a group. How many groups would need to be created so that everyone is in a group with all of the [people in the original group]?”

I decided to write up the thoughts I had while working through the problme:

A fun math question about Snapchat from a friend


I think working through this problem would make for a fantastic activity for kids.

(2) A great geometry question from David Butler with an incredible Desmos activity from Suzanne von Oy

I saw this question posted on twitter by David Butler:

It made for a terrific project with the boys:

A Neat Geometry Problem I saw from David Butler

Following the project, Suzanne von Oy posted an activity on Desmos that looked at the problem:

Our project using Suzanne von Oy’s amazing Desmos Activityfor David Butler’s geometry problem

I love this problem and think it makes for a fantastic project for kids learning geometry.

(3) Kelsey Houston-Edwards’s Proof video

The new math video series from Kelsey Houston-Edwards is amazing, and the video about proofs is off the charts.

There are several problems in the video that are accessible to kids and, actually, definitely worth sharing with them before they see the video. Our project using the problems from the video is here:

Kelsey Houston-Edwards’s Proof video is incredible

(4) James Tanton’s Mobius strip cutting project

For about 1 million reasons you need to get your hands on this book!

Flipping through the book last fall I found one of the most amazing projects I’ve ever seen. Kids from probably 5th grade through graduate school will go bananas for this one!


Here’s our project using Tanton’s ideas.

An absolutely mind blowing project from James Tanton

(5) Speaking of cutting projects . . .

Although these project are linked in our project on James Tanton’s Mobius strip cutting project, I think the ideas deserve a second mention.

First our the project using Katie Steckls’s “Fold and Cut” video:

There are dozens of ways to use the ideas in this video with students – here are a couple of things that we did.

Our one cut project

Fold and cut project #2

Fold and cut part 3

The second cutting project was written by Joel David Hamkins after seeing this “Fold and Punch” project:

Hamkins’s take on Fold and Punch will make you very happy!

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

(6) Sharing the Hyperuniform Distrobution with kids

I want every kid to have the opportunity to experience the world of math and science through Natalie Wolchover’s writing in Quanta magazine. Her article from last July introduced me to the “hyperuniform distributon”:

A Bird’s-Eye View of Nature’s Hidden Order

Screen Shot 2016-08-20 at 8.42.03 AM

We’ve done several projects using Wolchover’s articles. I love taking current ideas from math and science and sharing them with kids and Wolchover’s writing makes that task really easy. Our project on the hyperuniform distribution is here:

Using a Natalie Wolchover article to talk about the hyperuniform distribution with kids

(7) Playing with Neural Networks

Seems like every day now there are news stories about data science, machine learning, and neural networks. The tensorflow playground is a great way to share basic ideas from those fields with kids.

A Neural Network Playground

Screen Shot 2016-07-23 at 2.42.59 PM

Our project using the tensorflow program is here:

Sharing a fun neural network program with kids

(8) Sharing Ann-Marie Ison’s math art with kids

Ann-Marie Ison’s art project on modular arithmetic is amazing:

We did two projects based on her work and I’ve also used the ideas in few talks with high school kids, too.

The second project is here and has a link to the first one:

Extending our porject iwth Ann-Marie Ison’s art

The second project also includes a link to a fantastic Desmos program to explore the modular arithmetic designs created by Martin Holtham:

(9) Larry Guth’s “No Rectangles” problem

Screen Shot 2016-01-26 at 7.41.58 AM

This is another problem I’ve used over and over again in talks with kids. It is a super fun (and pretty rare) example of a problem that is interesting to research mathematicians and also accessible to kids.

The problem itself is pretty easy to understand – if you have an NxN grid, what largest number of squares in that grid that you can color in without 4 of those squares forming the corners of a rectangle?

The 3×3 and 4×4 cases are accessible to young kids (I did this project with 2nd and 3rd graders last year). The older the kids the larger the grid you can explore 🙂

Here’s our project on the problem:

Larry Guth’s “No Rectangles” Problem

(10) Fawn Nguyen’s picture frame problem

I don’t know that I have much to add from the write up in “10 Pretty Easy to Implement Math Activities for Kids” post above. This is fantastic project for kids and an incredibly creative take on a problem that everyone in math has seen 100 times.

Fawn Nguyen’s When I got them to beg

For this project you just need some scissors and paper. Here’s a video of my younger son working through the project 4 years ago:

Honestly, Fawn’s project is one of the most amazing and creative math projects for kids that I’ve ever come across.

Comparing Sqrt(x^2 + y^2) and ( Sqrt(x^2) + Sqrt(y^2) )

Last week we used 3d printing to compare (x + y)^2 and x^2 + y^2:

3d prints

That project is here:

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

My younger son is still sick today and not able to participate in a math project, so I chose a slightly more algebraically complicated comparison to look at with just my older son -> \sqrt{x^2 + y^2} and \sqrt{x^2} + \sqrt{y^2}

Here’s what the shapes look like:

I started the project by reviewing the original project in this series just to remind my son about how we thought about the 3d surfaces in the prior post. He remembered most of the ideas, fortunately, so the introduction was fairly quick.

After the introduction we talked about some basics of the algebra we were going to encounter in this project, namely that \sqrt{x^2} = |x|. This part all by itself is a difficult concept to understand and the bulk of the video below was spent talking about it.


With the difficult part of the algebra behind us we moved on to talking about the surface z = x^2 + y^2. What does this surface look like?

I really enjoyed the discussion here – the question is actually a pretty challenging one for a kid to think through.


Next we tried to figure out what the surface z = \sqrt{x^2} + \sqrt{y^2} would look like.

I think it takes a while to get used to working with graphs of the square root function. My son struggled a bit here to figure out the shape here. Hopefully that struggle helped him


Now I revealed the shapes and let my son discuss the properties of the shapes now that he could hold them in his hand. There were a few surprises, which was nice 🙂


I’m really happy about this series of projects. It is fun to explore the variety of ways that 3d printing can help kids explore math.

Random walks with kids

A week or so ago my older son did a short project on random walks out based on a page in Patters of the Universe:

Returning to Patterns of the Universe

By coincidence that week Kelsey Houston-Edwards’s new video was about random walks. So, we watched her video after that project:

Today my younger son is sick and wasn’t up to participating in a project. So, I thought it would be fun to revisit the random walk project and dive in a little deeper since my older son was a little more familiar with that topic.

I started by asking him what he remembered about random walks from the prior project and from the PBS Infinite Series video. One thing that he remembered is that 2d random walks do tend to return to where they started, but 3-d ones tend not to.

We started looking at specific random walks by studying a 1-dimensional random walk. We created a random walk by rolling dice and didn’t get quite what we were expecting, but that result led to a fun conversation:

In the last video we got more even numbers than we were expecting, so we decided to continue on to see if the walk would return to 0. Obviously we kept rolling even numbers . . . .

Next we moved on to studying a 3d random walk (and, of course, now rolled lots of odd numbers 🙂 )

We created the 3d random walk with snap cubes and it was pretty neat to see the shape that emerged from the dice rolls.

Despite the unexpected outcome with the even and odd rolls this was a fun project. I’d like to think a little more about how to make some random walk 3d prints. My guess is that those prints would be really fun to share with kids.