A terrific example for calculus students from Nassim Taleb and Alexander Bogomolny

I saw a wonderful exchange on twitter yesterday on a problem posted by Alexander Bogomolny:

At first this problem didn’t really jump off the page as a good first year calculus example, but then I as the solution that Nassim Taleb posted:

I’m a tiny bit time constrained this morning and can’t get the Taleb tweet to embed right, so here’s the solution a second time just in case the embedding remains broken:

Taleb

So, Taleb reduces the difficult-looking limit and sums to two integrals. The ideas underlying this reduction are both beautiful on their own and fundamental in calculus.

A few questions that I think would be worth discussing with calculus students are:

(1) [this one was discussed in the twitter thread] Why did the integrals start at 0, and does that matter?

(2) Why is ratio of the integrals equal to the ratio of the sums? This answer to this question is related to the answer to (1). It is also an excellent way to reinforce some of the main ideas behind Riemann sums.

(3) Probably less mathematically interesting, but a good challenge exercise for students is evaluating Taleb’s integral formulation of the problem using l’Hospital’s rule. I say “less mathematically interesting” because you have to evaluate the same integrals in both approaches, but the approach via l’Hospital’s rule allows you to discuss the Fundamental Theorem of Calculus and also review the chain rule. The arithmetic here requires you to be extra careful, but I think the other ideas outweigh the annoying arithmetic.

Too bad that the school year is over – but this is a great example to keep in your back pocket for next year’s calculus classes!

Kids looking at “4d cubes”

In our project last weekend we looked at a fun probably problem posted by Alexander Bogomoly. Our approach to the problem was to look at 3d printed versions of the shape:

Shapes

Here are the two projects:

Working through an Alexander Bogomolny probability problem with kids

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

During these project the boys thought one of the shapes looked a lot like a version of a 4 dimensional cube – specifically Bathsheba Grossman’s “Hypercube B” (seen in the picture below in red):

img_1711

For today’s project I thought it would be fun for the kids to talk about the connection with the 4d cube in more detail.

Here’s how I explained the idea to my younger son:

After that introduction I gave him the camera – here’s what he had to say:

Finally, I gave my older son the same instructions off camera. Here’s what he had to say about the shapes with the camera:

Fun little project – it is always interest to hear what kids have to say about slightly unusual shapes.

Using Poker to motivate some basic counting ideas

Both of my son’s have recently gotten interested in poker. My older son has also been working through Art of Problem Solving’s Introduction to Counting & Probability book, so I thought I’d use poker to motivate a few counting problems tonight.   The goal wasn’t to get too complicated, rather just to explore a few basic ideas.

Here’s how I introduced the project.  At the end they decided we’d count the ways to get (i) a 4 of a kind, (ii) a flush, and (iii) two pairs.

First we tackled counting the ways to get 4 of a kind. My younger son’s approach was pretty interesting even though he overlooked one piece of the counting.

Next we counted the number of ways to get a flush. The boys wondered if we should count the straight flushes separately, so we tackled the both problems:

Finally we tackled the challenge of counting the number of ways to get exactly two pairs. Here my younger son’s approach was pretty interesting again and led to a nice conversation about over counting.

So, a fun little project. It nice to find a little counting motivation from games they are already playing.

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

In yesterday’s project we were studying a fun probability question posed by Alexander Bogomolny:

That project is here:

Working through an Alexander Bogomolny probability problem with kids

While writing up the project, I noticed that I had misunderstood one of the
geometry ideas that my older son had mentioned. That was a shame because his idea was actually much better than the one I heard, and it connected to several projects that we’ve done in the past:

Paula

 

Learning 3d geometry with Paula Beardell Krieg’s Pyrmaids

Revisiting an old James Tanton / James Key Pyramid project

Overnight I printed the pieces we needed to explore my son’s approach to solving the problem and we revisited the problem again this morning. You’ll need to go to yesterday’s project to see what leads up to the point where we start, but the short story is that we are trying to find the volume of one piece of a shape that looks like a cube with a hole in it (I briefly show the two relevant shapes at the end of the video below):

Next we used my son’s division of the shape to find the volume. The calculation is easier (and more natural geometrically, I think) than what we did yesterday.

It is always really fun to have prior projects connect with a current one. It is also pretty amazing to find yet another project where these little pyramids show up!

Working through an Alexander Bogomolny probability problem with kids

Earlier in the week I saw Alexander Bogomolny post a neat probability problem:

There are many ways to solve this problem, but when I saw the 3d shapes associated with it I thought it would make for a fun geometry problem with the boys. So, I printed the shapes overnight and we used them to work through the problem this morning.

Here’s the introduction to the problem. This step was important to make sure that the kids understood what the problem was asking. Although the problem is accessible to kids (I think) once they see the shapes, the language of the problem is harder for them to understand. But, with a bit of guidance that difficulty can be overcome:

With the introduction out of the way we dove into thinking about the shape. Before showing the two 3d prints, I asked them what they thought the shape would look like. It was challenging for them to describe (not surprisingly).

They had some interesting comments when they saw the shape, including that the shape reminded them of a version of a 4d cube!

Next we took a little time off camera to build the two shapes out of our Zometool set. Building the shapes was an interesting challenge for the kids since it wasn’t obvious to them what the diagonal line segments should be. With a little trial and error they found that the diagonal line segments were yellow struts.

Here’s their description of the build and what they learned. After building the shapes they decided that calculating the volume of the compliment would likely be easier.

Sorry that this video is a little fuzzy.

Having decided to look at the compliment to find the volume, we took a look at one of the pieces of the compliment on Mathematica to be sure that we understood the shape. They were able to see pretty quickly that the shape had some interesting structure. We used that structure in the next video to finish off the problem:

Finally, we worked through the calculation to find that the volume of the compliment was 7/27 units. Thus, the volume of the original shape is 20 / 27.

As I watched the videos again this morning I realized that my older son mentioned a second way to find the volume of the compliment and I misunderstood what he was saying. We’ll revisit this project tomorrow to find the volume the way he suggested.

I really enjoyed this project. It is fun to take challenging problems and find ways to make them accessible to kids. Also, geometric probability is an incredibly fun topic all by itself!

Jim Propp’s “Swine in a line” game part 2

Last week I saw a really fun new question from Jim Propp:

Here’s the first project that we did on the game:

Jim Propp’s Swine in a Line game

Today we returned to the game to see if we could make any more progress understanding how it worked.

First we reviewed the rules and decided on an initial approach to studying the game for today:

Their first idea was to try to keep the two ends open since they knew the result when you reached the position with only 1 and 9 open.

Now we tried to study a bit more. The kids were having trouble seeing a path forward, so I just let them play.

At the end of the last video we were studying a position with all of the pens filled except for 7 and 9. In this video we searched for a winning move.

Finally, we took one more crack at solving the game. They boys got very close to the main idea, about an inch away(!), but didn’t quite get over the line.

The boys were really interested in the game and we kept talking for about 30 min after the end of the project. During that talk they did uncover the main idea. After that we played several games where they followed the strategy and they were able win against me every time following that strategy. It was a fun way to end the morning.

Jim Propp’s “swine in a line game”

Saw this great new video from Jim Propp yesterday:

This morning I had the boys watch the video and then we spent maybe 15 min talking through the game and seeing what we could learn.

First I asked them what they thought after seeing the video:

Now we played the game and the boys made a couple of initial discoveries. You can see quickly why this is a fun game for kids to play around with:

Next we played the game one more time. We aren’t trying to solve the game in this project, just to try to learn a few things about it.

Finally, we wrapped up by talking about a few of the things they learned playing the game. This part didn’t quite go how I wanted, but it was still interesting to hear what they had to say.

I’m excited to play around with this game a bit more later in the week. It’ll be interesting to see if the boys can continue to make project towards the solution.

Playing with Dan Anderson’s complex map program

Dan Anderson made a really neat little complex mapping program today:

This program allowed us to do a fun continuation of the project that we did over the weekend:

Sharing Kelsey Houston-Edwards’s Complex Number video with kids

Looking at the complex map z -> z^2 with kids

So, with each kid tonight I had them just play with the program and then I played a game of tic tac toe with them for fun 🙂

Here’s my older son playing:

and here’s our tic tac toe game:

Here’s my younger son playing:

Finally, here’s our tic tac toe game

Definitely a fun project. Can’t wait to play with the program more – this is a really fun subject to share with kids.

Looking at the complex map z -> z^2 with kids

Yesterday we did a fun project using Kelsey Houston-Edwards’s compex number video:

Sharing Kelsey Houston-Edwards’s Complex Number video with kids

The boys wanted to do a bit more work with complex numbers today, so I thought it would be fun to explore the map Z \rightarrow Z^2. The computations for this mapping aren’t too difficult, so the kids can begin to see what’s going on with complex maps.

We started by looking at some of the simple properties. The kids had some good questions right from the start.

By the end of this video we’ve understood a bit about what happens to the real line.

After looking at the real line in the last video, we moved on to the imaginary axis in this video. The arithmetic was a little tricky for my younger son, so we worked slowly. By the end of this video we had a pretty good understand of what happens to the imaginary axis under the map Z \rightarrow Z^2.

At the end of this video my younger son noted that we hadn’t found anything that goes to the imaginary axis. My older son had a neat idea after that!

Next we looked at (1 + i)^2. We found that it did go to the imaginary axis and then we found two nice generalizations that should a bunch of numbers that map to the imaginary axis.

Finally, we went to Mathematica to look at what happens to other lines. I fear that my attempts to make this part look better on camera may have actually made it look worse! But, at least the graphs show up reasonably well.

It was fun to hear what the boys thought they’d see here versus their surprise at what the actually saw 🙂

I think this is a pretty fun project for kids. There are lots of different directions we could go. They also get some good algebra / arithmetic practice working through the ideas.

Sharing Kelsey Houston-Edward’s complex number video with kids

I didn’t have anything planned for our math project today, but both kids asked if there was a new video from Kelsey Houston-Edwards! Why didn’t I think of that 🙂

The latest video is about the pantograph and complex numbers:

Here’s what the boys thought about the video:

They boys were interested in the pantograph and also complex numbers. We started off by talking about how the pantograph works. With a bit more time to prepare (and probably a bit more engineering skill than I have), building a simple pantograph would make a really fun introductory geometry project.

Next we talked about complex numbers. We’ve talked about complex number several times before, so the idea wasn’t a new one for the boys. I started from the beginning, though, and tried to echo some of the introductory ideas that Kelsey Houston-Edwards brought up in her video.

To finish up today’s project we looked at some basic geometry of complex numbers. The specific property that we looked at today was multiplying by i. At the end of this short talk I think that the boys had a pretty good understanding of the idea that multiplying by i was the same as rotating by 90 degrees.

Complex numbers are a topic that I think kids will find absolutely fascinating. I don’t know where (if at all) they come into a traditional middle school / high school curriculum, but once you understand the distributive property you can certainly begin to look at complex numbers. It is such a fun topic with many interesting applications and important ideas – many of which are accessible to kids. Just playing around with complex numbers seems like a great way to expose kids to some amazing math.