Working through a tough geometry problem with my younger son

My younger son is working through Art of Problem Solving’s Introduction to Geometry book this school year. He works for about 30 min ever day and seems to really enjoy the challenge problems. Today the last challenge problem from the chapter on similar triangles gave him some difficulty.

I could tell he was having trouble understanding the problem and I asked him what was wrong. His answer was interesting -> they didn’t give any side lengths.

The problem is a pretty good challenge problem for kids learning geometry, but I thought talking through it tonight would make a good project. So, here’s the problem and a short introduction to what was giving him trouble:

In the next part of the project we began to solve the problem. There are (I think) two critical ideas -> (i) finding all of the similar triangles, and (ii) finding the parts in the diagram which have the same length.

It takes a few minutes for my son to find all of the relationships, but he does get there. Despite being a little confused, his thought process is really nice to hear.

Now we moved to the last side to see if we could find another relationship that would simplify the equation that we are hoping will be equal to 1.

After we had that last relationship he was able to see how the expressions in the equations corresponded to various side lengths in the picture. From there he was able to see why the sum was indeed 1.

I like this problem a lot and am happy that my son wanted to struggle with it.

Studying Tetrahedrons and Pyrmaids

[had to write this in more of a hurry than usual as 30 min of my morning was spent fishing for a dropped retainer that fell through a gap in our bathroom floor . . . . so sorry for the quite write up, but this project is a really fun way to get to hear a younger kid think about 3d geometry]

There were two really nice math ideas shard on twitter this week and I had no idea that they were related.

The first was a famous problem shared by Alexander Bogomolny:

I did a fun Zometool project with my younger son using the problem:


That project is here:

Alexander Bogomolny shared one of my all time favorite problems this morning

Then came Patrick Honner’s appearance on the My Favorite Theorem podcast:

I shared some of the ideas from the podcast and subsequent twitter follow up with my older son:


Sharing Patrick Honner’s My Favorite Theorem appearance with kids

Today – with just my younger son – I looked at a surprising connection between these two projects. We started by reviewing the Pyramid / Tetrahedron problem and then trying to guess the relationship between the volume of the two shapes.

Sorry that the lighting is so awful in these videos – unfortunately I only noticed after we were done.

Next I showed him the larger Tetrahedron with the inscribed octahedron. Although the main point of today’s project wasn’t Varignon’s theorem, I explained the theorem and asked my son to find some of the inscribed squares.

This connection was pointed out by Graeme McRae in this tweet:

At the end of the last video my son was starting to think about how volume scales. Since that’s an important point for this project I wanted to have all of those thoughts in one video.

It is interesting to hear how he tries to reconcile his mathematical thoughts about the volume of the two shapes with what he sees right in front of him.

Finally, we wrapped up by trying to find the relationship between the volume of the small tetrahedron and the volume of the pyramid.

I’m happy that my son is not convinced that the mathematical scaling arguments are correct. I can also say that holding these two objects in your hand it really does not look like the pyramid has twice the volume. Can’t wait to follow up on this.

Making one of Annie Perkins’s drawings from Zometool

I saw a neat tweet from Annie Perkins last night:

It seemed like a great Zometool project, so this morning we cleared all the furniture out of the living room and gave it a go. My older son had something else going on today, so there were only two of us working on this project.

Here’s what my younger son thought of Perkins’s drawing:

Here are his thoughts after he completed the shape that he guessed would be the main building block for the project:

Here are his thoughts after we were just over half way done:

Finally, here’s are his thoughts on the completed project:

It is always fun to hear what kids have to say about shapes – and this project was a nice way to hear how my son’s thoughts about a fairly complicated shape evolved as we built the shape.

We’ve done a few other project like this one – this two old projects inspired by Kate Nowak and Anna Weltman comes to mind:

Talking about Kate Nowak’s shape

Anna Weltman’s loop-de-loops part 2

Using our Zometool set to replicate mathematical drawings has been a great – and totally unexpected – way to explore math ideas.

Sharing Patrick Honner’s My Favorite Theorem appearance with kids

Patrick Honner was a terrific guest on the My Favorite Theorem blog today:

We’d done a blog post – inspired by Patrick Honner, obviously – about Varignon’s Theorem previously – Varignon’s Theorem – inspired by this tweet:

After listening to today’s My Favorite Theorem episode I wanted to do a follow up project – probably this weekend – but then I saw a really neat tweet just as I finished listening:

Well . . . I had to build that from our Zometool set and ended up finding a fun surprise, too. I shared the surprise shape with my older son tonight and here’s what he had to say:

What a fun day! If you are interested in a terrific (and light!) podcast about math – definitely subscribe to My Favorite Theorem.

Alexander Bogomolny 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:

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(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:

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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 🙂

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Tiling 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.

Here’s a pic from the Wikipedia page on the Cairo tiles:

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!