# An infinite descent problem with pentagons

The boys had a nice hiking trip to Mt. Osceola today.

When they got back we did a little follow up to yesterday’s project on infinite descent:

Infinite Descent with kids inspired by Jim Propp

That project came from reading the lastest Jim Propp blog post:

Fermatโs Last Theorem: The Curious Incident of the Boasting Frenchman

So, the kids were a little tired tonight, but we still managed to have nice (and short) project tonight discussing this problem from Propp’s blog:

“If you liked the proof that there is no integer-sided golden rectangle, you might try to prove, by similar means, that there is no regular pentagon whose five equal sides and five equal diagonals all have integer length.”

The Zometool set makes this project especially fun!

However, because I am also tired, tonight’s write up will be short (and not edited – I’m going to bed).

In part 1 we built some pentagons and tried to see if we could see a pattern in the lengths of the sides and the diagonals. I’d left the length of the sides of the larges pentagon as a mystery to help motivate finding the pattern:

The boys didn’t see the pattern so we moved to the whiteboard – when you write the lengths down, the patter is much easier to see:

So, next we moved back to the couch to test if the pattern we found worked – it did! My older son wanted say that the longs were an integer and that idea threw me off in our discussion here. To try to get things back on a more correct path (meaning, one that was more clear to me given that I was tired ๐ ), we went back to the witeboard after this part of the discussion:

ok – so back to the white board to see if the boys could tackle the infinite descent proof. They had the idea for how to start the proof in the last video – assume that we’ve found a regular pentagon with integer sides and diagonals. How do we find the smaller pentagons now?

It took a few minutes, but we got there ๐

It is fun seeing kids struggle through a puzzle like this one, and the Zome set is almost a miracle helper for this particular problem. Fun little post hiking project ๐

# Infinite descent with kids inspired by Jim Propp

I saw another neat post from Jim Propp a few days ago:

Fermatโs Last Theorem: The Curious Incident of the Boasting Frenchman

The part that caught my attention as something that kids might find fun was the discussion of “infinite descent.” It reminded me a little of a Zometool project that we did last year:

Fibonacci Spirals and Pentagons with our Zometool set

We sure had a lot of Zome fun in the new house before we got furniture ๐

To get going with the “infinite descent” discussion today I had the boys build another Fibonacci spiral shape from the Zometool set. We talked about the shape and my younger son remembered the connection to the Golden Rectangle, which was a nice surprise.

This shape is actually really fun to build because you don’t really get a sense for how quickly the shape grows until you try to fit it in your living room ๐

My son remembering a connection with the Fibonacci Spiral and the Golden Rectangle made for a nice transition to the next part of the project. The first thing we needed to do was talk about what it means to be “similar” in geometry.

For the last part of the project we stepped through the infinite descent argument showing that the Golden Ratio cannot be the ratio of two integers.

The argument requires a little bit of algebra so I went through it slowly to make sure that my younger son could follow it. Hopefully he was able to follow it – and I think he did because at the end he asked about non-integer values for the ratio.

Pretty fun to start the project with a spiral that grows larger and larger and end with a spiral that shrinks and shrinks!

Another fun project inspired by Jim Propp’s blog. I originally wanted to look at the pentagon question mentioned in Propp’s post, but I think we’ll explore that question next weekend.

# Nesting Platonic Solids

My friend Matt Gorbet showed me this amazing platonic solid video yesterday:

My initial reaction was to wonder if we could build it out of our Zometool set. Luckily we bought some extra green and half green struts last year and were able to figure out how to build it!

My second reaction was to wonder if the icosahedron at the center was the same icosahedron that we’d seen in our “dodecahedron folding into a cube” project:

A neat post from Simon Gregg showing that a dodecahedron can fold into a cube

Can you believe that a dodecahderon folds into a cube?

Some 3D Geometry for Pamela Rawson

The last video in the 1st project shows that there is an icosahedron hiding inside of the shape you get when you fold the dodecahdedron:

I wondered if that icosahedron was the same as the one we were seeing in the nested platonic solids. The kids weren’t sure (and neither was I!), so we built the folded shape with the same size dodecahedron at took a closer look:

The last thing we did was connect the orange balls to form the new icosahedron, and – incredibly! – it was exactly the same size as the first one!! How amazing is that – the icosahedron inside of the nested Platonic solids is exactly the same one as you get when you fold up the large dodecahedron.

As fun of a project as we’ve ever done ๐

# An interesting introduction to completing the square

My older son started a new section in Art of Problem Solving’s Introduction to Algebra book today. The chapter begins to look at quadratics in a bit more depth. Given the question my son picked for today’s movie I assume that one of the topics will be completing the square. It is always interesting to see how a kid approaches an advanced topic before really knowing much about that topic.

The question asks you to find the minimum value of $x^2 + 10x - 7$. Here’s his work:

After he finished his work I gave a really basic introduction to completing the square using this problem as an example:

# A great complex number question from Art of Problem Solving

My son picked a great problem from Art of Problem Solving’s Introduction to Algebra to talk through today:

A lot of people on Twitter thought the problem was fun and many people commented on the difference between a geometric and an algebraic approach.

A geometric approach is beyond my son’s understanding right now, but the algebraic approach has so many great lessons going for it – many more than you think if you aren’t generally around kids working through math problems:

After we were done I talked through some of the geometry for just a little bit – I don’t think introducing the geometry solely through this problem is such a great idea, but I wanted to try anyway. Here’s how it went:

Finally, as an extra shout out to AoPS – here’s one of my favorite math videos of all time – Richard Rusczyk illustrating how powerful a geometric approach to complex numbers can be with a stunning solution to an old math contest problem:

# Simplifying complex numbers

My older son moved on to a new chapter in his algebra book today – complex numbers. We’ve done work with complex numbers before, and even by coincidence this weekend in our Mandelbrot set projects:

An introduction tot he Mandelbrot set for kids

Playing with Dan Anderson’s Mandelbrot program

My son chose an interesting problem about simplifying complex numbers for our video today. The problem ask you to simplify the expression (5 + 3i) / (3 – 4i).

I was pleasantly surprised by his approach to the problem:

We had not discussed any of the problems or examples in the section, so it probably isn’t much of a surprise that my guess about the approach he would take to the problem was wrong. I thought it would be useful to show him what an approach similar to “rationalizing the denominator” would look like:

I love how different these two approaches look at first glance. In fact, it is pretty surprising that you arrive at the same answer. Feels like this would be a fun example to introduce matrix inversion.

# Playing with Dan Anderson’s Mandelbrot program

After seeing our introductory Mandelbrot project from yesterday:

An introduction tot he Mandelbrot set for kids

Dan Anderson sent us a link to a wonderful set of Mandelbrot Set-related resources on his programming page:

Today I had the boys play around with the “Mandelbrot Zoom” program:

Dan Anderson’s program “Mandelbrot Zoom”

It is really fun to hear what the kids have to say playing around with the Mandelbrot set.

My younger son was fascinated by the “mini Mandelbrots” that he saw as we zoomed in:

My older son wondered if the Mandelbrot set was connected. This problem actually took many years to resolve (it is) but it was fun to hear what he meant by “connected.”

So, a fun little project just talking about and exploring the Mandelbrot set.

# An introduction to the Mandelbrot set for kids

Last night I was writing about “beautiful math” for kids:

When I asked my younger son what he thought was the most beautiful math he’d seen, he replied “fractals” and specifically mentioned the Mandelbrot set.ย  We haven’t done a project about the Mandelbrot set, so it seemed like a good idea to talk about it today.

As I say in the introduction below – it isn’t just a pretty picture, there is some really cool math.

So, I started the project by talking (or probably more accurately stumbling) through an explanation of the map that defines the Mandelbrot set.ย  After that, we worked through a few examples:

Having looked at 0, -1, and 1, we now moved on to looking at some complex numbers. The next numbers we tried were i and -i. It turns out that both of these numbers are part of the Mandelbrot set, but the calculations are slightly (really, just slightly) more complicated.

At the middle of this video we produced a crude map of the Mandelbrot set with snap cubes, and then at the end we discussed a little bit about how a computer program to plot the Mandelbrot set would work.

Now we moved to the computer to study the Mandelbrot set in Mathematica. Luckily Mathematica has a function – MandelbrotSetPlot[] – that makes this part of the project pretty easy for us. In this part we talked a little bit about what happens when you vary the number of iterations, and also what happens when you zoom in.

Determining the coordinates for zooming in was also a nice little mathematical discussion with the boys.

The coordinates to zoom in on turned out to be a more interesting topic than I was expecting. With the camera off we had a long discussion about the coordinates of the location that they wanted to see more carefully. I’m sorry I turned the camera off, actually, but oh well.

I love the discussion and general thoughts from the kids about the shapes we were seeing here. My younger son is right – this really is beautiful math for kids to see!

# MoMath’s “Beautiful Math” collection, and some “beautiful math” for kids

The museum of math has put together a nice (and growing!) collection of videos of mathematicians talking about beautiful math:

Watching the videos it struck me that they were aimed at adults and older kids, but I thought it would be easy to show some beautiful math that younger kids could appreciate, too. I thought of two of my favorite projects with the boys and then asked each of them to tell me what they thought was the most beautiful math that they’d seen.

My two favorites:

(1) The Chaos game:

Our project is here:

Computer Math and the Chaos Game

and my favorite part starts about 2:30 into this video and goes for about 45 seconds:

(2) John Conway’s “Amusical” version of the Collatz Conjecture

Our project is here:

The Collatz Conjecture and John Conway’s “Amusical” Variation

and my favorite part is at the end where we convert the “amusical” process to music. The music starts around 2:00 – prior to that is just explaining Conway’s Collatz process in the Mathematica code. I love it when my younger son says “I didn’t know you could hear 20”:

(3) My older son’s choice for beautiful math – the 4th dimension

We’ve done a couple of projects related to the 4th dimension – here are a few:

Carl Sagan on the 4th Dimension

Sharing 4d-shapes with kids

which had a fun connection to our Zometool / bubble project. Around 1:00 in the video is a great moment – “who knew that bubbles could find the center of a tetrahedron?”

That comment from my son led to this wonderful drawing posted on twitter:

Another really fun higher dimensional problem for kids who know the Pythagorean Theorem is a neat problem I learned from Bjorn Poonen:

Talking through Bjorn Poonen’s N-dimensional Sphere Problem with kids

(4) My younger son told me that he thought fractals were the most beautiful math that he’d seen.

We’ve done several fractal projects for kids, too. Here are a few:

A fun fractal project – exploring the Gosper curve

After this projects, were really luck to receive some laser cut Gosper curves from Dan Anderson to play with:

Dan Anderson’s Gosper Curves

Using the Koch Snowflake to introduce fractals

Using Matt Parker’s Menger Sponge video to talk fractions with kids

One of my favorite moments from these projects happens around 3:00 in the video below as the boys stretch out a 3d printed Peano curve into (nearly) a straight line:

So, I’m really happy to see the Museum of math putting together a collection of mathematicians talking about beautiful math. I can’t wait to see more videos in the collection! Hopefully some of the videos and projects above can help younger kids see some beautiful ideas in math, too.

# Two neat factoring problems for kids

My son is in the review section of chaper 11 of Art of Problem Solving’s Introduction to Algebra book.ย ย ย  I asked him to chose a problem for our movie today and he chose a “greater than / less than” problem that connects to factoring.ย  It is a nice way for kids to see a connection between arithmetic and algebra:

Since that problem went fairly quickly, I grabbed one of the example problems from earlier in the chapter that had caught my attention. This one is a really neat example of how algebra can help solve an arithmetic problem:

A fun morning – I really love the problems in the Art of Problem Solving books!