# 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.

# Talking through two problems from the 2005 AMC 10

I really enjoy using old AMC problems to talk about math with the boys.

These two problems gave my older son some trouble today:

Tonight I had a chance to talk through these problems with them.

Here’s the probability problem:

Here’s the GCD problem:

# Talking about “sums of divisors” with kids and also a pi surprise

I didn’t do a very good job managing the time on this project today. The trouble is that there are lots of different directions to go with the ideas and we walked down a lot of different paths.

But – I think this is a great topic to show off the beauty of math and we end with an amazing connection between sums of divisors of integers and $\pi$.

The topic of sums of divisors of an integer came up in my younger son’s weekend enrichment math program yesterday. I thought it would make for a good topic for a project, so I gave it a go this morning.

The first part of the project was mostly about divisors and the kinds of questions that we could ask about them. A lot of the discussion here is about a question you can ask about the product of a number’s divisors:

Next we began to look at the sum of the divisors of a few different numbers. The boys noticed a few patterns – including a pattern in the powers of 2.

At the end we were looking to see if we could find patterns in the powers of 3.

It was proving to be a little difficult to find the pattern in the powers of 3, but we kept trying. After few ideas that didn’t quite help us write down the pattern, they boys had an idea that got us there.

At the end of this video I showed them that the sum of the divisors of powers of 6 was connected with the sum of the divisors of powers of 2 and powers of 3.

To wrap up I wanted to show some larger patterns in divisor sums, so we moved to Mathematica to play around a bit.

While I was doing the same playing around last night I accidentally stumbled on an amazing fact: As n gets large, the average of the sum of the divisors of the numbers from 1 to n is approximately $(\pi^2 / 12)*n$.

Number theory sure has some fun surprises 🙂

This is definitely a fun topic and also one that could be used in a variety of ways (arithmetic review, intro to number theory, computer math, . . . ). I wish that I’d presented it better. Probably it needs more than one project to really fit in all of the ideas, though.

# 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).