## Alexander Bogmolny shared one of my all time favorite problems this morning

This tweet brought a big smile to my face this morning:

This is an absolute treasure of a 3d geometry problem, so if you’ve not seen it before definitely take some time to ponder it.

I asked my younger son to play around with the problem using our Zometool set. Here’s what he found:

I love that the Zometool set helps make this problem accessible to kids.

## Exploring some polynomial basics with kids

I asked the boys what they waned to talk about for a project today and got a bit of a surprise when my 6th grader suggested polynomials. It seems that the topic has just come up in his math class at school and he’s interested in learning a bit more.

To start the project I asked him what he knew about polynomials:

Next I asked my younger son to explain adding and subtracting polynomials, and then to try to see how to multiply them since he said that he didn’t know how to multiply in the last video:

Now I wanted to show an unusual property of polynomials that was relatively easy to understand. My hope was to show my older son something that he’d not seen before but also something that was still accessible to my younger son. I chose to show them a short exploration of a difference table for a quadratic

Finally, I showed how you could use the difference table to reconstruct a quadratic function if you knew the values of the function at three consecutive integers:

So, despite the surprise topic request, this was a fun little project. It was nice to be able to find a topic that you could explore if polynomials were “new to you” and still get something out of if polynomials were a familiar topic.

## Recursive functions and the Fibonacci numbers

My son asked me about recursive functions yesterday morning and I showed him Dan Anderson’s online tutorial:

Even though Dan’s resource covers just about everything ( ha ha ) I thought maybe there was still something we could discuss this morning. So, I talked about the Fibonacci numbers.

First we did a quick introduction:

Next I had both boys pick their own recursively defined functions – and I got pretty lucky with the choices!

Now I showed them one approach you can use to solve these recursive equations. For the purposes of showing this idea to kids I’m not worried about the background details, but rather using the idea for some basic exponent review. (and, sorry, I’m a little careless around 1:30, but luckily catch my error fairly quickly before the whole video is derailed):

Now that we found the neat relationship between Fibonacci numbers and the golden ratio, we finished the calculation and found an explicit formula for the Fibonacci numbers:

We finished up with by checking our new formula on Mathematica. I also showed them a lucky coincidence from twitter yesterday that relates to this project. That coincidence involved this problem posted by Alexander Bogonmlny:

And this portion of the solution posted by Nassim Taleb:

(unfortunately as I tried to zoom in on Taleb’s solution while filming the camera got way out of focus, so close your eyes for the last few minutes of this video 😦 ).

Even if the ideas for finding the explicit solution to these recursive equations is a bit advanced, I still think this is a neat topic for kids to see. It certainly is a fun way to get some nice algebra review.

## A nice probability coincidence today

This morning my older son and I talked through the following problem from the 2003 AMC 10b:

It turned out that an arithmetic problem is what led to his confusion on this problem, but discussing this problem was a nice opportunity to talk about discrete probability. At the end of our conversation I told him to always remember the underlying idea in discrete probability is simple -> count the cases that work and then count the total cases. It may not always be so easy to do, but it really is all that you have to do to solve the problem.

After he went off to school I this problem posted on Twitter:

This is a terrific problem, and it is really tempting to try to break the problem into pieces and essentially to try to solve it with recursion.

But remember the simple little idea I told my son -> count the cases that work and count the total cases. You’ll find a delightful solution to the problem!

## Sharing Annie Perkins’s Cairo pentagons with kids part 2

[sorry at the beginning that this post feels a little rushed. I wrote it during an archery class my son takes, but I forgot the power cord to my laptop and only had 20% battery at the start . . . . ]

Over the last week I saw two really neat videos from Annie Perkins on the Cairo pentagon tiling:

Yesterday I did a project with my older son on this shape of the pentagon. That project’s focus was on coordinate geometry:

Exploring Annie Perkins’s Cairo Pentagons with kids

Today I did a project with my younger son with 3d printed versions of the pentagons that I made today (after a few glorious fails . . . .). Sorry that the tiles don’t show up super well on camera when they are pushed together – I’d hoped that the white background with show through the gaps, but not so much 😦

Before starting I showed my son the two videos from Perkins and began the project by asking him to try to recreate the shapes he saw. He liked the tiling but ran into a little trouble trying to recreate it. It turns out that tiles also fit together in a way that doesn’t extend to a tiling of the plane. My son had a nice geometric explanation about why the shape he found wouldn’t extend to the full plane.

After running into a little difficulty in the last video, he started over with a new strategy. That new strategy involved putting the tiles together in groups of two and fitting those groups together. This method did lead to a tiling that he thought would extend to the full plane.

Definitely a fun project. You can see some links to other tiling projects we’ve done in yesterday’s project with my older son. Tiling is definitely a topic you can have a lot of fun with on a few different levels – from younger kids talking about the shapes they see, to older kids learning how to describe the equations of the boundary lines and coordinates of the points. Making the tiles is a fun 3d printing project, too.

## Exploring Annie Perkins’s Cairo Pentagons with kids

I saw a great tweet from Annie Perkins a few days ago:

I thought it would be a fun idea to add to the list of our growing list of pentagon projects. At this point I’ve lost track of all of them, but they got started with this amazing tweet from Laura Taalman:

Using Laura Taalman’s 3d Printed Pentagons to talk math with kids

and the most recent project (I think!) is this one:

Evelyn Lamb’s pentagons are everything!

Oh, and obviously don’t forget pentagon cookies 🙂

After seeing Perkins’s tweet I started down the path of making the Cairo tiling pentagons but super unluckily had a typo in my printing code. At least my cat made good use of the not-quite-Cairo pentagons:

So, while I wait for the correct pentagons to print, I thought I’d talk about the special shape of the Cairo tiles with my older son. One of the neat things about all of these pentagon projects is getting to talk about geometry with kids in sort of non-standard, non-textbook way. Tonight’s conversation was about coordinate geometry using the properties of the Cairo pentagon.

Wikipedia’s page on the Cairo tiling pentagon

To start the project I drew the shape on our board and asked my son to find the coordinates of the points. This is a bit of an open ended question because you have to know the lengths of the side so know the coordinates – I was happy that he noticed that problem (and, just to be 100% clear, I don’t know for sure if there are restrictions on the sides for the Cairo tiling – I’ll learn that when the new pentagons finish printing – ha ha).

Here’s how he started in on the problem:

For the second part of the project he had to make one more choice for a side length, and then he was able to find the coordinates of all of the corners of the pentagon.

One of the great (and happy) surprises with math and 3d printing is that you get neat opportunities to explore 2d geometry. Some of our old projects exploring 2d geometry with 3d printing are here:

Using 3d printing to help kids learn algebra and 2d geometry

I’m excited to play with the Cairo tiles when they finish printing tonight. Hopefully the 2nd time is a charm!

## Revisiting Joel David Hamkins’s “Graph Theory for Kids”

A few years ago we did a fun project with Joel David Hamkins’s “Graph Theory for Kids”:

Going through Joel David Hamkins’s “Graph Theory for Kids”

Here’s the link to Hamkins’s notes for the project:

Graph Theory for Kids

This project was also inspired by the project we did yesterday on the graph isomorphism theorem:

Sharing Lazlo Babi’s graph isomorphis talk with kids

For today’s project I printed two copies of Hamkins’s booklet and had the boys work through it on their own. After they were finished, we talked through the project after they were finished. Here’s the conversation broken into 4 parts – as you’ll see, Hamkins has made an absolutely fantastic project for kids:

Part 1: An introduction to graphs and one surprising property

Part 2: Looking at some more complicated or “extreme” examples and also illustrating how some of the more complicated graphs make for nice counting exercises for kids

Part 3: Now a few examples that the kids made on their own – this part led to a nice discussion about crossings

Part 4: Some 3d shapes and a really fun observation from my older son about the sphere

## Sharing Lázló Babai’s Graph Isomorphism talk with kids

I’ve been going through some of the videos of the invited talks from the Joint Math meetings and trying to figure out how to share them with kids. The tallks and our first two projects are here:

Sharing an idea from Alissa Cran’s JMM talk with kids

Sharing Federico Ardilla’s JMM talk about the permutahedron with kids

Today we looked at Lázló Babai’s talk on graph isomorphisms:

I started by explaining what “graph” meant in this context and then exploring a few examples:

I decided to break the final conversation at the end of the last video into two pieces so that the boys wouldn’t feel rushed talking about the last two graphs:

Before moving on to a pretty challenging example, I decided to show them a connection between shadows and graphs – the conversation here was actually much more fun than I expected!

Next we moved on to a pretty challenging example of two isomorphic graphs. The boys did a nice job showing that these two graphs are indeed isomorphic.

To wrap up we looked at a more complicated example from Babai’s talk. From that example I think the boys were able to see that the graph isomorphism problem can be pretty hard!

This was a fun project. I think that kids really enjoy introductory graph theory. Another fun project we’d done in the past on the subject is here:

Going through Joel David Hamkins’s “graph theory for kids”

Hopefully we’ll do more soon 🙂

## Sharing Federico Ardila’s JMM talk with kids

This is the second in a little project I’m doing with the JMM talks. Some of the invited talks were published earlier this week:

I’m definitely enjoying the talks, but also wondering if there are ideas – even small ones – that you can take from the talks and share with kids. My hope is that kids will enjoy seeing ideas and concepts that are interesting to mathematicians.

The first project came from Alissa Cran’s talk:

Sharing an idea from Alissa Cran’s JMM talk with kids

Today I tried out an idea from Federico Ardila’s talk with my younger son (who is in 6th grade). The idea related to an interesting shape called the “permutahedron.”

We began with a quick explanation of the idea and looked at some simple cases:

Next we moved to building the permutahedron that comes from the set {1,2,3}. At the end of the last video, my son speculated this shape would have some interesting symmetry. We used our Zometool set to build it.

One thing I’m very happy about with this part of the project is that building this permutahedron is a nice introductory exercise with 3d coordinates for kids.

Finally, we talked about the permutahedron that comes from the set {1,2,3,4}. My son had some interesting thoughts about what this shape might look like. Then I handed him a 3d printed version of the shape and he had some fun things to say 🙂

The 3d print I used is from Thingiverse:

Permutahedron by pff000 on Thingiverse

Definitely a fun project for kids, I think. Making the hexagon was fun and also a nice little geometric surprise. Exploring the 3d printed shape was also really exciting – it is always great to hear what kids have to say about shapes that they’ve never encountered before.

## Sharing an idea from Alissa Crans’s JMM talk with kids

Yesterday videos of several of the invited JMM talks were published:

I was skimming through the talks to see if there might be anything fun to share with kids and came to this slide in Alissa Crans’s talk “Quintessential quandle queries”

I had a hunch that kids might find it strange that an expression like $A B A^{-1}$ would be something of interest to mathematicians, so I decided to see what the boys thought:

When we played with numbers, the expression $A B A^{-1}$ was just $B$. So now I tried a few non-number ideas with A and B representing certain moves on a grid. The first set of moves seemed to behave just like the numbers did, but the second set of moves produced a little surprise:

Now we looked at a completely new situation -> in the video below A and B will represent moves of a Rubik’s cube.

Here I got a really fun surprise when the boys saw that doing repeated applications of the “move” $A B A^{-1}$ was actually really easy to describe mathematically 🙂

Definitely a fun little project. There’s no real need to show the boys the complete talk – they don’t need to learn the complete content of the talk. It was fun to show them an idea that is interesting to mathematicians, though, and especially fun to give them a peek at some simple operations that don’t commute.