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## Some beautiful geometry in a challenge problem from Alexander Bogonolny

I did the project below with the boys on Sunday before they went off to camp for a week. The idea wasn’t to get into heavy math, but rather just a relaxed walk through some fun shapes. We got one detail wrong in the 4th video which I was sort of kicking myself for, but then I saw a tweet from Nassim Taleb showing some of the geometry in a different problem that Alexander Bogonolny had posted and it made me realize the connection between the algebra and geometry in our problem was still fun to show:

So, despite the error I thought I would publish the project anyway.

Here’s the original problem:

Below are the videos showing our walk through the geometry. First, though, here’s the quick introduction to the problem:

After that intro we looked at the region described by the constraint in the problem. We have to thicken up the region a little bit using the absolute value function in order to see it, so the Mathematica code looks a bit more complicated than in the problem, but that extra complexity is just to make the picture easier to see.

One cool thing about our discussion here is that my younger son thought there should be 3 fold symmetry in the shape because there was 3 fold symmetry in the equation ðŸ™‚

Now we looked at the situation in which the surface achieves the maximum value subject to the constraint in the problem. My younger son made the nice observation that the two surfaces appeared to be “blending together” at certain points. That “blending” is an important idea in Lagrange Multipliers – though, don’t worry, we aren’t going down that path today.

Next we looked at the minimum value of the surface subject to the constraint in the problem. The error I made here was accidentally reversing the two surfaces. The fixed surface – the one describing the constraint – is now on the outside rather than the inside.

Finally, I asked the kids to pick a value smaller than 45/4 for the curve so that we could see what happened. Unfortunately they picked 7 which is too small – there’s no surface! – so they chose 10 and that allowed us to see that the shrinking surface inside of the original shape. Also we can see fairly clearly (after some rotation) that the two shapes do not intersect.

Definitely a fun project showing the boys a beautiful side of a really challenging problem.

## Working with the PCMI books part 2: coloring an octahedron

Last week we got the PCMI books:

Our first project involved a neat problem about understanding the number 0.002002… in different bases:

Playing around with the PCMI books

Today I was looking for another fun problem and found another problem that I thought would make a fun project:

Barbara has an octahedron, and she wants to color its vertices with two different colors. How many different colorings are possible? By “different” we mean that you can’t make one look like the other throu a re-orientation.

I started by introducing the problem and asking the kids what their initial ideas were:

They had a couple of pretty good ideas including some basic ideas about symmetry. Using those ideas we began counting the different colorings:

We counted the cases in which 3 vertices were black and 3 vertices were red. This case proved to be tricky, but going through it slowly got us to the correct answer.

Finally, as a fun little extension, I asked them to find the number of ways to color the faces of a cube with two colors. Having solved the octahedron problem already, this one went pretty quickly, and they even noticed the connection between the two problems ðŸ™‚

I like this problem. I’m glad that the boys were able to see some of the basic ideas. When you add more colors the counting gets much more difficult and some pretty advanced math comes into play. The number of colorings with “n” colors is:

$(n^6 + 3n^4 + 12n^3 + 8n^2) / 24$

The different terms correspond to different symmetries of the cube / octahedron. We’ll have to wait a few more years to cover the complete details ðŸ™‚

## Working through an AMC 8 geometry problem

My younger son was working through the 1989 American Junior High Mathematics Exam this morning and got stuck on this problem:

Here’s a link to the entire exam on Art of Problem Solving’s website:

The 1989 AJHME on Art of Problem Solving’s website

I thought this problem would make for a nice project since there are a couple of good mathematical ideas in it, so we sat down to talk about it. My younger son talked through his approach first:

My older son went next and had a different approach:

To wrap up we talked about how the answer would change if the problem was set up with a slightly different arrangement of the cubes and the boys found their way to an important idea in geometry:

I’m really happy that the old AMC problems are available – they are a wonderful resource to use to find challenging but accessible problems for kids.

## Playing around with the PCMI books

After seeing a plug for them on twitter I bought the PCMI books. They arrived yesterday:

The first book I picked up was Moving Things Around since the shape on the cover of the book is (incredibly) the same shape we studied in a recent project.

One more look at the Hypercube

I found a neat problem in the beginning of the book that by another amazing coincidence was similar to a (totally different!) problem we looked at recently:

Revisiting Writing 1/3 in binary

We started by talking about the books and the fun shape on the cover:

Now we moved on to the problem. It goes something like this:

Consider the number 0.002002002…. in base 3. What is this number? How about in base 4,5,7, and n?

We started in base 3 and the boys had two pretty different ways to solve the problem!

Next we moved on to base 4:

Now we moved to the remaining questions of base 5, 7 and N. Unfortunately I got a phone call I had to take in the middle of this video, so I had to walk away while the solution to the “N” part was happening.

We finished up with the challenge problem -> What is 0.002002002…. in base 2?

This is a pretty neat challenge problem ðŸ™‚

Definitely a fun start to playing around with the PCMI books. Can’t wait to try out a few more problems with the boys!

## Working through “Euler’s Gem” with kids

A few weeks ago I stumbled on Diave Richeson’s book Euler’s Gem:

Although the book is not intended for kids, it is written for a general audience and I thought the boys might enjoy working through the book slowly. I’ve been having them read one chapter per day and they are really having fun with the ideas.

Today I asked each of them to talk about what they’ve learned through the first 5 chapters. Here’s what my younger son had to say:

Here’s what my older son had to say:

It is fun to hear what they are taking away from this book and also really nice to hear that both of them really do like the book. I’ve not tried an experiment like this one before, but the book is so well written that I really do think that with a little bit of help here and there kids can understand most of it.

## Exploring induction and the pentagonal numbers

Yesterday we did a fun project based on this tweet by James Tanton:

That project is here:

Exploring a neat problem from James Tanton

During the project yesterday we touched on mathematical induction and also on the pengatonal numbers. Today I wanted to revisit those ideas with slightly more depth.

We started with a quick review of yesterday’s project:

Now we looked at a mathematical induction proof. The example here is:

$1 + 3 + 5 + \ldots + (2n - 1) = n^2$

(the nearly camera ran out of batteries, that’s why this part is split into two videos)

Here’s the 2nd part of the induction proof after solving the battery problem:

To wrap up the project we went to the living room to build some shapes with our Zometool set. The Zome shapes really helped the boys make the connection between the numbers and geometry.

The boys really liked this project. In fact, my younger son spent the 30 min after we finished making the decagonal numbers ðŸ™‚

## Exploring a neat problem from James Tanton

I didn’t have an specific project planned for today and was lucky enough to see a really neat problem posted by James Tanton:

I didn’t show the tweet to the boys because I thought finding the patterns would be a good exercise for kids. We started with the k = 0 case. This case is also good for making sure that kids understand the basics of functions required to explore this problem:

Next we looked at the k = 1 case.

Next we looked at the k = 2 case and then my younger son made a really fun little conjecture ðŸ™‚

At the end of the last video my younger son thought that the k = 3 case might produce the pentagonal numbers. I had to look up those numbers ( ðŸ™‚ ) while the camera was off, but I found them and we checked:

We ended by looking at Tanton’s challenge problem -> what happens when k = -1? I had the boys take a guess and then we looked at the first few terms and the boys were, indeed, able to solve the problem!

The boys had a lot of fun playing around with this problem and I was really excited they found a different pattern than the one Tanton was asking for!

## One my time through F – E + V = 2

We did a fun project earlier in the week inspired by Dave Richeson’s book:

That project is here:

Looking at Dave Richeson’s “Euler’s Gem” book with kids

During the project the kids had a little trouble counting the verticies, edges, and faces of one of the complex shapes. We solved the problem with our Zometool set, but I wanted to try a different approach and printed the shapes again:

So, with these shapes I went through the project again. First a quick review:

Next, now that we have shapes that fit together, can we count the faces, verticies, and edges?

My younger son was still having a little bit of trouble seeing the number of edges, so we slowed down a bit:

Finally we did a quick recap of how the cube helped us. I was trying to get the boys to think about the shape without touching it, but wasn’t super successful.

This was a fun 2nd look at the F – E + V = 2 formula. We’ll be doing more projects based on Richeson’s book throughout the summer.

## Looking at Dave Richeson’s “Euler’s Gem” book with kids

I stumbled on this book at Barnes & Noble last week:

It is such a delightful read that I thought the kids might enjoy it, too, so I had them read the introduction (~10 pages).

Here’s what they learned:

Next we tried to calculate Euler’s formula for two simple shapes – a tetrahedron and a cube:

After that introduction we moved on to some slightly more complicated shapes – an icosahedron and a rhombic dodecahedron. The rhombic dodecahedron gave the kids a tiny bit of trouble since it doesn’t have quite the same set of symmetries as any of the Platonic solids:

Now we tried two very difficult shapes:

We studied these shapes last week in a couple of projects inspired by an Alexander Bogomolny tweet:

Working through an Alexander Bogomolny probability problem with kids

Connecting yesterday’s probability project with a few old 3d geometry projects

I suspected that this part would be difficult, so I had them count the faces, edges, and verticies of the two shapes off camera. Here’s what they found:

So, since the boys couldn’t agree on the number of verticies, edges, and faces of one of the shapes, I had them build it using our Zometool set to see what was going on. The Zometool set helped, thankfully. Here’s what they found after building the shape (and we got a little help from one of our cats):

Definitely a fun project. It was especially cool to hear the kids realize that the shape they were having difficulty with was (somehow) a torus. Or, as my older son said: “In the torus class of shapes.” I’m excited to try to turn a few other ideas from Richeson’s book into projects for kids.

## Revisiting writing 1/3 in binary

A few years ago we did a project about writing 1/3 in binary:

Writing 1/3 in Binary

Earlier this week my older son was working on a probability problem about flipping coins and that problem reminded me of that old project. So, today we revisited that old project. We also got a really fun surprise at the end.

Here’s the introduction – what to the boys remember about binary and, in particular, about writing 1/3 in binary?

Next I asked the boys to come up with a similar problem. They suggested trying to write 1/6 in base 5:

For what I thought would be the last part of the project I suggested that we take a look at the number 0.01010101… in base 5. They boys solved this problem pretty quickly based on what they learned in the last video and then my older son suggested a new problem -> what is 0.1001001001…. in base 2?

After I turned off the camera, just for fun I showed the boys what $\pi$ in base 2 was (using Wolfram Alpha). I noticed a fun connection with the number we just saw, so we added one final note to this project:

So, a fun project with a neat little surprise at the end. I think project like this are a great way for kids to expand their own ideas about numbers.