# Numberphile’s “Freaky Dot Patterns” video

Saw this neat tweet from Patrick Honner earlier in the week:

The video itself will blow you away:

I shared it with Dan Anderson yesterday who made a couple of computer versions of patterns from the video:

Sorry the video quality isn’t so great, but it was fun talking through these patterns with the kids. We started with the square pattern:

Then we moved on to the triangle pattern. It was surprisingly difficult for the kids to understand how to describe the rotations, but eventually they figured it out. I think I’ll revisit a bit more about rotations for a Family Math Project this weekend.

The patterns in the Numberphile video make a great project to talk through with kids. I can’t wait to try a few more ideas from their video.

# What I hope was a productive struggle

My son encountered a really great, but pretty tough, problem from an old AMC 10 yesterday:

I mentioned that I’d really like to see James Tanton make a video about this problem because he has a fantastic way of presenting problems, and also his approach tends to be different than mine.

But, rather than a professional mathematician working through the problem, here are the thoughts my son had (with a little direction from me in the last two videos).

Part 1 – a struggle to understand what’s going on. This is about a 7 minute talk where there are some good ideas, but we don’t get on to a path that leads to the solution:

After the first video my son went to school. We picked up talking about the problem when he got home and essentially started over. I encouraged him to try to think of the problem in a new way. Because he mentioned some symmetry arguments in the number of ways to place “n” black squares on the board, I asked him to look for symmetry in the geometry in this problem.

After studying the geometry for a bit, for the last part of this project we tried to find a systematic way to look at the possible arrangement that solve the problem.

So, a great problem and hopefully a productive struggle. There are so many great ideas hiding in this problem – finding those ideas is one of the things that makes this problem so interesting and so challenging.

# Looking at John Baez’s AMS Blog with kids

I’ve only got 10 minutes to write this before taking my son to his viola lesson, so . . . zoom zoom.

Saw a really neat blog post from John Baez today:

Given the recent fun we’ve had with Patterns of the Universe

I thought it would be fun to hear what my kids thought of the Hoffman-Singleton graph.

I’m excited to look at graphs a little more with the boys. We did one project a while ago:

Going through Joel David Hamkins’s Gralph Theory for Kids

time to do more 🙂 (sorry for all of the likely typos!)

# A review of “On the Dot” by Gamewright

Doing some prep work for Family Math night at my younger son’s school, I ran across a really neat game that had been used in prior years – On the Dot by Gamewright:

Sadly the game has been “retired” so I don’t think you can by it on Gamewright’s website, but I did see some for sale on Amazon.

Game play seems simple enough – you try to recreate a specific pattern of dots using 4 cards that you hold in your hand. Here’s my 4th grade son trying out a few examples:

After those examples, I asked him what he thought of the game and what math ideas he thought were part of the game:

I really like this game – it is easy to learn how to play and gets kids thinking about all sorts of math-related ideas from problem solving, to symmetry, to counting. It was also kind of fun to follow my son’s idea that the cards are not commutative.

So, a great game for kids. Sorry to hear that it has been retired, but get it on Amazon while you still can!

# A year’s worth of Family Math

We had a fun Zome project this afternoon:

With that project, we’ve hit Family Math 365 – a year’s worth of Family Math! We’ve come a long way since Family Math 1:

One of the most powerful influences for me along the way was this Numberphile interview with Ed Frenkel from about 2 years ago:

Actually quite a lot of what Frenkel said in that piece struck me, but especially his call for mathematicians to go out an show the world how beautiful math is. I probably don’t quite fit into Frenkel’s “professional mathematician” group, but I took up the call anyway.

I was thinking back to Frenkel’s video after we finished building out little Zome shapes today. The shapes aren’t just pretty, they can be used to show some amazing math. Really advanced stuff, actually – the idea that there’s no general solution for a 5th degree equation, for example, comes from looking at a group isomorphic to the symmetry group of the Dodecahedron. That’s probably not an idea for kids, though.

After re-watching Frenkel’s video today I decided to make our 365th Family Math project an interview with the boys rather than one of our normal math discussions. I was curious what they thought of the shapes and what math ideas they learned, or thought could be learned from these shapes.

They had some nice ideas. Listening to them makes me wish there was a way to share these fun shapes and some of the math ideas that went into building them with more kids.

Anyway, here’s what the boys had to say:

Can’t wait for the next 365 projects!

# Using Kate Nowak’s rotated parabola with kids

Saw a Kate Nowak exercise via a Dan Anderson tweet earlier in the week:

and wrote about an estimation problem coming from that picture that I thought was fun:

A fun estimate question inspired by Kate Nowak

I thought it would be fun to see what the boys thought about the rotated parabola, so this morning I showed them a few rotated parabolas and asked them what they thought:

My younger son was interested some of the pieces of area that were cut out by the rotated parabola. Funny enough, my older son was interested by a similar question a few years ago:

The area inside of a parabola

It was fun to explore their ideas about the different areas of the graph. We had a neat detour when my older son wondered what the graph of $y = x^2$ would look like of we were really zoomed out.

Finally, my older son was interested in what the parabola would look like under a variety of different rotations. The discussion here ended up being a neat surprise as what grabbed the boys’ attention was how to change the x- and y- coordinates following a rotation so that the graph would display correctly. I wouldn’t have thought to talk about that, but they were pretty interested in understanding how the coordinates changed. The funny thing is that walking down this path gets you really close to talking about trig functions.

So, a fun morning project even if math need to compute or calculate the rotation of the graph is obviously way, way over their heads. But, since the picture is actually pretty simple, there’s still plenty of interesting things to talk about with kids, AND plenty of stuff that kids might be interested in!

# A counting example inspired by a Cristina Milos tweet

Saw this nice tweet from Cristina Milos earlier today:

It gets to something that I’ve been seeing with my kids that I don’t really understand. I’ll say right from the start that I may not have the situation exactly right, so take the next part with that in mind. But . . .

It seems to me that up until this year the boys would approach similar problems in a similar way. I was always struck that even with a 2 1/2 year age difference, their approaches were so much alike.

This year, though, my older son has started incorporating more sophisticated methods in his solutions. In the language in Milos’s tweet – he seems to have acquired (some of, of course) the schematic knowledge needed for efficient problem solving.

As I said above, I don’t know what caused him to start using these new approaches, but right now their two approaches to problem solving are noticeably different. I picked a problem that gave my older son a little trouble today to illustrate the effects I’m seeing.

The problem is from the 2005 AMC 8 and asks you to count the number of triangles that you can form from 6 dots:

Problem 21 from the 2005 AMC 8

My older son uses counting techniques and choosing numbers:

By comparison my younger son draws all 18 triangles:

There’s obviously nothing wrong with my younger son’s approach – and he even sorts the triangles into groups of congruent triangles, which is neat. His solution is less sophisticated, though. Since until very recently it seemed like the two kids had similar approaches to most problems, I’m interested to try to understand where the difference in approach suddenly came from.

# What learning math sometimes looks like: counting arrangements

A problem from an old MOEMS test gave my son a lot of trouble this morning. It was one of those times where you start looking at a problem one way and it is so hard to move away from that viewpoint.

It did make for a nice project this morning, though. Here’s the problem:

Sara places four books on a shelf. The blue book must be somewhere to the left of the green book. The red book must be somewhere to the left of the yellow book. In how many different orders can Sara place the books?

We started talking about the problem using snap cubes – one idea he has here is that the arrangements will come in pairs. You have to be careful with that idea, though, because you might already accidentally have a pair.

He also tries some counting ideas, but we also need to be a little careful to make sure the arrangements we are counting satisfy the conditions of the problem.

/

Next we try to find a systematic way of counting the arrangements. The first thing we try is to find all of the arrangements with the yellow book in the first slot.

After a long conversation about that case, he had an easier time understanding the cases where the green cube is in the first slot.

/

To wrap up the conversation this morning I tried having him look at other ways to divide the 6 cases into two groups of 3. He noticed that we could look at the blue and red cubes instead.

/

So, a tough problem for my son. Hopefully this conversation helped him see a few ways that looking at patterns help you count. His original focus was on finding the number of different possibilities for each slot, but that’s a tough way to approach this problem. An alternate approach that we didn’t cover today involves picking two slots out of the four for the green and blue books – I’ll leave that approach until the next time we talk about this problem.

# Frank Farris’s patterns

A couple of weeks ago Evelyn Lamb’s article Impossible Wallpaper and Mystery Curves introduced me to Frank Farris’s work. On Saturday I stumbled on his book at Barnes and Noble:

I was excited to try out some of his ideas with the boys even though they use complex numbers and exponentials which are over their heads. We did the whole exploration this morning using Mathematica.

To start, we just explored the exponential function.

Next we moved to looking at sums of two exponential functions. The boys were surprised by the graphs and we played around with a few more examples. They had some interesting ideas about what the pictures looked like, and I’m glad that the pictures also reminded them a little of Anna Weltman’s loop-de-loops.

Next we moved on to sums of three exponential functions motivated by the idea of trying to produce another kink in the loop. There was a little discussion at the beginning of this part of the talk about complex variables. I thought going down this path was going to be too difficult to explain, so I tried to bring the conversation back to the sums. I love the ideas they had about symmetry here.

Next we looked at Farris’s “mystery” shape and played around a bit more with the ideas. These shapes also led to fun conversations about symmetry:

Finally, I let the kids just play around. As I was writing up this project I got a “hey dad, come here and look at this cool shape” call:

So, despite the math underlying these shapes being a little over their heads, the kids seemed to really enjoy seeing these shapes. I loved hearing their ideas and I loved seeing them play around with the ideas for a long time after we turned off the camera.

Also, Farris’s book is absolutely amazing – you’ll love the ideas and the presentation, and probably most of all the incredible pictures he creates from the ideas!

# Extending Anna Weltman’s loop-de-loops with Frank Farris’s “Creating Symmetry”

We’ve had a ton of fun in the last couple of weeks with Ann Weltman’s loop-de-loop ideas:

Here are two of the projects that we did:

Anna Weltman’s loop-de-loops

Anna Weltman’s loop-de-loops part 2

Last night we stopped at a Barnes & Noble after dinner and I found a book that Evelyn Lamb had written about last month:

Here’s Evelyn Lamb’s piece:

Impossible Wallpaper and Mystery Curves by Evelyn Lamb

The book is absolutely wonderful and has so many cool examples. I’d hoped it would be easy to make some of the graphs in the book using Mathematica, and after a little documentation reading to kick off some rust, it wasn’t too hard:

At least visually the curves you can make from the idea in Farris’s book remind me of the loop-de-loops. I don’t really think it will be that productive to talk in detail with the boys about exponentials and graphing in the complex plane, but I do think they will like seeing the pictures and talking about them. I’m excited to show them some of the ideas from the book tomorrow morning.