A great counting problem for kids from the AMC 8

This problem from the 2015 AMC 8 gave my son some trouble today. Actually quite a bit of trouble:



I’m not 100% sure what caused the difficulty. It might be that once you start thinking about this problem one way that it is hard to switch. Whatever the cause, though, we had a really good conversation about the problem.

Here’s his original approach that is incorrect:

So, after finding out that the answer in the last video was incorrect, we went to try to find the error. He found it pretty quickly.

After that I tried to explain an alternate approach to the problem. Unfortunately my explanation ended up causing quite a bit of confusion:

In the last part of our discussion I tried to dig my way out of the hole I created in the last video.

Even watching this video after the fact, I’m not sure what was the original source of his confusion. There was definitely some difficulty going from 4 parallel edges to 6 pairs of parallel edges.

By the end of our conversation he was able to walk through the argument, but I think that I’ll revisit some similar problems with him just to be sure the main ideas have sunk in.

I think this short project is a nice example of how old contest problem can help kids learn math. For me anyway, it is really challenging to come up with good problems and the fact that all of the old AMC problems are available for kids to work through is an incredibly helpful resource. Hopefully I can find some similar counting problems on other old AMC contests.


Going through three AMC 8 problems

My younger son has been doing a little practice for the AMC 8. Yesterday three problems from the 2013 exam gave him a little trouble. We went over them together.

The first was a tricky geometry problem. Both the words and ideas needed to solve this problem were new to him.

Next up was a challenging counting problem – we broke this into two pieces. This is a great counting problem for kids. In the first part we found out how to calculate the answer, but didn’t finish the calculation:

In the second part we talked about strategies to finish the calculation:

Finally – a fantastic geometry problem. It has a few little traps in it, but my son found a nice solution.

I love using the old AMC contest problems to help the boys see math that is both fun and challenging. These problems were really fun to talk through.

Intro “machine learning” for kids via Martin Gardner’s article on hexapawn

Last month I had the nice surprise of finding Martin Gardner’s book The Colossal Book of Mathematics at the Omaha Public Library’s book sale:

I’ve been flipping through the book and thinking about how to share some of the ideas with the boys.  Chapter 35 – “A Matchbox Game-Learning Machine” really caught my attention.  In particular, Gardner’s discussion of the game “hexapawn” inspired me to try this introductory “machine learning” idea with kids.

I had the boys read the (approximately) 2 pages on the game and the approach and then we talked through the game to make sure they understood it:

Next we started playing. We were very lucky to have a coffee table that allowed us to easily show the 24 cases and their snap cubes. This video shows the first two times through the game. I hope that it shows that playing through the game is something that is accessible to kids:

The next part shows 3 more turns of the game. My main reason for showing these three turns is so you can see some of the parts that kids find challenging. I think these parts are a big part of what makes sharing Gardner’s idea with kids so fun. The pattern matching and the general walk through the game keeps their attention while they learning about machine learning.

Next we played for a while with the camera off. After a while the kids (and the computer) learned something:

Next we played a bit more with the camera off and the before long the “computer” learned to win the game every time. Amazing!

In the last 3 min of this video the boys talk about some of the things that they learned in this project.

This is one of the most fascinating projects that we’ve ever done. It does require a bit of set up and probably a bit more careful supervision than usual to make sure that the kids don’t go down the wrong path, but wow is there a lot to learn here. I think that opening the door for kids to see how computers / machines might “learn” is an amazingly valuable lesson.

Playing with some mathy art ideas this morning

Yesterday my older son spent the afternoon at a friend’s house. When I went to pick him up I learned that his friend’s mom is an amazing artist:

Among the dozens (maybe 100’s) of pieces of work around the studio were some patterns on pieces of cloth. I asked how they were made and she told me that it was a simple process involving sharpies and alcohol. I knew instantly what we’d be doing for our math project this morning 🙂

Here’s a quick introduction to the project:

Now, with the camera off, I had the boys make their own patterns. Here’s my younger son’s description of what he made and what he thought he’d see when we added the alcohol:

Next up was my older son. He’d also experimented with a few different patterns and a few different markers.

This was a really fun project. I think this project would be fun for a wide variety of of kids – from elementary school kids (with supervision) up through high school. It is neat to hear the kids describe the shapes and, of course, the shapes and patterns themselves are really cool!

A second project from the Wrong but Useful podcast

Yesterday afternoon I was listening to rest of the latest (as of August 31, 2017) Wrong but Useful podcast. That podcast is here:

The Wrong but Useful podcast on Itunes

A little project we did from a “fun fact” mentioned in the first part of the podcast is here:

Exploring a fun number fact I heard on Wrong but Useful

The second half of the podcast was a really interesting discussion of math education. One thing that caught my attention was comparing math education to music education and the idea of having students do “math recitals.”

Another part that caught my attention was a problem used mainly to see the work of the students rather than the specific answers. That problem is roughly as follows:

Find two numbers that multiply to be 1,000,000 but have the property that neither is a multiple of 10.

Here’s how my younger son approached the problem – it was absolutely fascinating to me to see how he thought about it.

Here’s what my older son did. Much more in line with what I was expecting.

Fun little project – definitely check out the Wrong but Useful podcast if you like hearing about math and math education.

Exploring a fun number fact I heard on Wrong but Useful

I was listening to the latest episode of Wrong but Useful today:

The Wrong but Useful podcast on Itunes

During the podcast the following “fun fact” came up -> \ln(2)^5 \approx 0.16.

I thought exploring this fact would be a fun activity for the boys and spent the next 30 min daydreaming about how to turn it into a short project. I also wanted the project to be pretty light since today was the first day of school for them. Eventually I decided to explore various expressions of the form \ln(M)^N via continued fractions and see what popped up.

We started by looking at the approximation given in the podcast. During the course of the discussion we got to talk about the relationship between fractions and decimals:

Now we looked at some powers of \ln(3) until the phone rang. We found a neat relationship with the 5th power. This relationship was also mentioned in the podcast.

While I was on the phone I asked the boys to explore a little bit. Here’s what they showed me when I got back.

Oh, wait – EEEk – I just noticed writing this up that we counted back incorrectly in this video. Whoops! Here’s the number we thought we were exploring -> \ln(12)^{15} is very nearly equal to 850,454 + 19,118 / 28207.   The next approximation that is better is 850,454 + 33,481,089 / 49,398,529.

You can see in the pic below that the 19,118/28,207 is accurate to 12 decimal places!

Sorry for this mixup.

Continued Fraction

Next they showed me one more good approximations that they found -> \ln(8)^{18} is nearly an integer. After that I tried to show them one I found but we ran into a small technical problem, so no need to watch the rest of the video after we finish with \ln(8)^{18}.

Finally, I got the technical glitch fixed and showed them that \ln(11)^2 is approximately 5 3/4. The next better approximation is 5 + 1,907 / 2,543

So, a fun little number fact to study. Sorry for the bits of the project that went wrong, but hope the idea is still useful!

Extending our Alexander Bogomolny / Nassim Taleb project from 3 to 4 dimensions

Last week I saw really neat tweet from Alexander Bogomolny:

The discussion about that problem on Twitter led to a really fun project with the boys:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

That project reminded the boys about a project we did at the beginning of the summer that was inspired by this Kelsey Houston-Edwards video:

Here’s that project:

One more look at the Hypercube

For today’s project I wanted to have the boys focus on the approach that Nassim Taleb used to study the problem posed by Alexander Bogonolny. That approach was to chop the shape into slices to get some insight into the overall shape. Here’s Taleb’s tweet:


So, for today’s project we followed Taleb’s approach to study a 4d space similar to the space in the Bogomolny tweet above. The space is the region in 4d space bounded by:

|x| + |y| + |z| \leq 1,

|x| + |y| + |w| \leq 1,

|x| + |w| + |z| \leq 1, and

|w| + |y| + |z| \leq 1,

To start the project we reviewed the shapes from the project inspired by Kelsey Houston-Edwards’s hypercube video. After that we talked about the equations we’d looked at in the project inspired by Alexander Bogomolny’s tweet and the shape we encountered there:

Next we talked a bit about the equations that we’d be studying today and I asked the boys to take a guess at some of the shapes we’d be seeing. We also talked a little bit about absolute value which briefly caused a tiny bit of confusion.

The next part of the project used the computer. First we reviewed Nassim Taleb’s approach to studying the problem posed by Alexander Bogomolny. I think it is really useful for kids to see examples of how people use mathematical ideas to solve problems.

The 2d slicing was a fascinating way to approach the original 3d problem. We’ll use the same idea (though in 3d) to gain some insight on the 4d shape.

One fun thing about this part of the project is that we encountered a few shapes that we’ve never seen before!

Finally, I revealed 3d printed copies of the shapes for the boys to explore. They immediately noticed some similarities with the hypercube project. It was also really interesting to hear them talk about the differences.

At the end, the boys think that the 4d shape we encountered in this project will be the 4d version of the rhombic dodecahedron. We’ve studied that shape before in this project inspired by a Matt Parker video:

Using Matt Parker’s Platonic Solid video with kids

I don’t know if we are looking at a 4d rhombic dodecahedron or not, but I’m glad that the kids think we are 🙂

It amazes me how much much fun math is shared on line these days. I’m happy to have the opportunity to share all of these ideas with my kids!

Calculating the volume of our rhombic dodecahedron

Yesterday we did a fun project involving a rhombic dodecahedron:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

At the end of that project we were looking carefully at how you would find the volume of a rhombic dodecahedron in general. Today I wanted to move from the general case to the specific and see if we could calculate the volume of our shapes. This tasked proved to be much more difficult for the boys than I imagined it would be. Definitely a learning experience for me.

Here’s how we got going. Even at the end of the 5 min here the boys are struggling to see how to get started.

So, after the struggle in the first video, we tried to back up and ask a more general question -> how do we find the volume of a cube?

Now we grabbed a ruler and measured the side length of the cube. This task also had a few tricky parts -> do we include the zome balls, for example. But now we were making progress!

Finally we turned to finding the volume of one our our 3d printed rhombic dodecahedrons. We did some measuring and found how many of these shapes it would take to fill our zome shape and how many it would take to fill a 1 meter cube.

So, a harder project than I expected, but still fun. We’ve done so much abstract work over the years and that makes the concrete work a little more difficult (or unusual), I suppose. I’m happy for this struggle, though, since it showed me that we need to do a few more projects like this one.

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

I woke up yesterday morning to see this problem posted on twitter by Alexander Bogomolny:

About a two months ago we did a fun project inspired by a different problem Bogomolny posted:

Working through an Alexander Bogomolny probability problem with kids

It seemed as though this one could be just as fun. I started by introducing the problem and then proposing that we explore a simplified (2d) version. I was excited to hear that the boys had some interesting ideas about the complicated problem:

Next we went down to the living room to explore the easier problem. The 2d version, |x| + |y| \leq 1, is an interesting way to talk about both absolute value and lines with kids:

Next we returned to the computer to view two of Nassim Taleb’s ideas about the problem. I don’t know why the tweets aren’t embedding properly, so here are the screen shots of the two tweets we looked at in this video. They can be accessed via Alexander Bogomolny’s tweet above (which is embedding just fine . . . .)



The first tweet reminded the boys of a different (and super fun) project about hypercubes inspired by a Kelsey Houston-Edwards video that we did over the summer:

One more look at the Hypercube

The connection between these two projects is actually pretty interesting and maybe worth an entire project all by itself.

Next we returned to the living room and made a rhombic dodecahedron out of our zometool set. Having the zometool version helped the boys see the square in the middle of the shape that they were having trouble seeing on the screen. Seeing that square still proved to be tough for my younger son, but he did eventually see it.

After we identified the middle square I had to boys show that there is also a cube hiding inside of the shape and that this cube allows you to see surprisingly easily how to calculate the volume of a rhombic dodecahedron:

Finally, we wrapped up by using some 3d printed rhombic dodecahedrons to show that they tile 3d Euclidean space (sorry that this video is out of focus):

Definitely a fun project. I love showing the boys fun connections between algebra and geometry. It is also always tremendously satisfying to find really difficult problems that can be made accessible to kids. Thanks to Alexander Bogomolny and Nassim Taleb for the inspiration for this project.

Dropping a slinky from the ceiling

I’m running a 3d print for tomorrow’s project that’s going to take most of the day, so I wasn’t sure what we’d do for a project today. Then the boys and a few of their friends started playing with a slinky and I remembered a fun little experiment with a slinky.

What happens when you drop an uncoiled slinky from the ceiling?

here’s a closer look at the moment my son releases the slinky:

Here’s a second drop in which we zoom in on the bottom of the slinky:

Finally, here’s the 10x slo mo for that drop:

After the drops we talked about what was going on. I think this is a hard one for kids to understand, but they had some interesting ideas. To test some of those ideas we dropped an eraser from the ceiling at the same time that we dropped the slinky and saw which hit the ground first. Then we did the same thing dropping the eraser from the bottom of the slinky to see which one hit first.

Fun little morning of experiments!