August 2017
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## 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.

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!

## A problem discussed by Martin Gardner, Grant Sanderson, Bjorn Poonen, and the AMC 10 :)

My son was working on some old AMC 10 problems today and this problem gave him a bit of trouble. The problem is #18 on the 2007 AMC 10 b:

We’ve studied a similar problem before in these three projects:

A Strange Problem I overheard Bjorn Poonen discussing

Bjorn Poonen’s n-dimensional sphere problem with kids

A fun surprise in Bjorn Poonen’s n-dimensional sphere problem

Related problems seem to be following me around this week, though. I saw one version when I bought Martin Gardner’s book at the Omaha public library book sale:

And another version when I watched Grant Sanderson’s latest video:

For tonight’s project we worked through the problem and talked about some of the fun ideas that come into the problem in higher dimensions.

Here’s the introduction to the problem:

Next my older son showed his solution to the problem. There were a few stumbles, but he got through it:

Next I had my younger son try to solve a similar problem – he also struggled a bit but eventually found the ideas necessary to solve it:

The next problem we looked at was the version of the problem in 3 dimensions. Once we solved that problem the boys had a good guess of how the problem would look in 4 dimensions!

Finally we played around with the strange ideas that come into play in 10 and 1206 (though I think we said 1063 in the video):

So, definitely a fun problem – from a nice 2d geometry problem to totally non-intuitive problem about higher dimensional spheres. Funny to run into it three times in the last week!

## Playing with Jim Propp’s essay on Arthur Engel

Jim Propp’s August 2017 blog post is absolutely terrific:

Prof. Engel’s Marvelously Improbably Machines

Even though we are visiting my parents in Omaha, I couldn’t resist having the boys watch the video in the essay and then play with the challenge problem.

Here’s what they thought after watching the video – the nice thing is that they were able to understand the problem Propp was discussing [also, I shot these videos with my phone, so they probably don’t have quote the quality or stability of our usual math videos]:

Next I had them play the game that Propp explained in his video. The idea here was to make sure that they understood how the [amazing!] solution to the problem shown in the video:

Next we tried the challenge problem from the essay. I almost didn’t do this part of the project, but I’m glad I did. It turned out that there were a few ideas in the Propp’s video that the boys thought they understood but there was a bit more explanation required. Once they got past those small stumbling blocks, they were able to solve the problem.

I’m really excited to dive a little deeper into the method of solving probability problems that Propp explains in his essay. What makes me the most excited is that the method came from someone thinking about how to explain probability to kids.

The last video shows that understanding Engel’s method does take a little time. Once kids get the general idea, though, I think they’ll find that applying to a wide variety of problems is pretty easy. It is amazing how such a simple method can made fairly complex probability problems accessible to kids.

## Talking through the birthday problem with kids

Today is my birthday – I figured there was no better way to celebrate than to talk through the birthday problem with the boys.

We’ve talked about it before – here’s what they remembered:

Now . . . how to we tackle this problem? Getting started proved to be a little difficult:

Now that we figured out how to approach the problem, we dove into the calculations. This step also proved to be challenging, but eventually they were able to see how the calculation worked.

Finally we wrote a little [ bug filled ] program on Mathematica and played around with the results.

It is always fun to share a famous problem with the boys.

## Talking a bit more about my son’s probability problem

Yesterday we did a fun project on a probability problem / game my son was working on. The game involves rolling three 10-sided dice and adding up the numbers. Repeat the process until you’ve seen one of the sums 40 times. Yesterday 15 was the winner:

Here’s yesterday’s project:

A probability and stats problem with dice my younger son was working on today

I wrote a short program on Mathematica to play my son’s game 1,000,000 times. I was interested to see how each of the boys would interpret the results.

Here’s what my older son had to say:

Here’s what my younger son had to say:

It is interesting to hear what kids have to say about the various probabilities and distributions. The results of the 1,000,000 simulations are probably pretty surprising. This problem that my son made up is actually a pretty fun problem to explore with kids.

## A probability and stats problem with dice my younger son was working on today

When I got up this morning my younger son was playing some sort of dice game in the kitchen. An hour later he was still rolling dice so I finally asked him what he was doing:

It turns out what he was actually trying to was find the first sum that would appear 40 times, but I only understood that later.

This seemed like an easy activity to turn into a project, so we got started by having him explain what he was doing:

Next we turned to Mathematica to play around a little bit with the problem. I had to explain some terms first (and sorry I had part of the screen out of view for a bit). After explaining those terms we looked at the distribution of the sums:

Finally we wrapped up by taking a very deep dive into the distribution of the sum of three 10 sided dice. The kids were able to understand the probability of getting a 3 or a 30, and then we talked about a few of the other probabilities that Mathematica was showing us.

Later in the morning my son finished his game. 15 was the first roll to appear 40 times.

It was really fun to base a project on a math problem that my son came up with on his own.