Today I wanted to boys to explore a bit more. The plan was to explore one basic property together and then for them to play a bit on the computer individually.

Here’s the first part -> Looking at what happens when you compute the continued fraction for a rational number:

Next I had the boys go the computer and just play around.

Here’s what my younger son found. One thing that made me very happy was that he stumbled on to the Fibonacci numbers!

Here’s what my older son found. The neat thing for me was that he decided to explore what continued fractions looked like when you looked at multiples of a specific number.

So, a fun project overall. Continued fractions, I think, are a terrific advanced math topic to share with kids.

— Alexander Bogomolny (@CutTheKnotMath) June 27, 2017

and the mathematical point that caught my eye was the question -> which positive integers are close to being integer multiples of ?

One possible approach to this question uses the idea of “continued fractions.” I learned about continued fractions from my high school math teacher, Mr. Waterman, who taught them using C. D. Olds’s book.

So, today I stared off by talking about irrational numbers and reviewing a simple proof that the square root of 2 is irrational:

Next we talked about why integer multiples of irrational numbers can never be integers. This I think is an obviously step for adults, but it took the kids a second to see the idea:

Now we moved on to talk about continued fractions. I’m not trying to go into any depth here, but rather just introduce the idea. I use my high school teacher’s procedure: split, flip, and rat ðŸ™‚

We work through a simple example with and also see that the first couple of fractions we see are good approximations to .

With that background work we went up to use Mathematica to explore different aspects of continued fractions quickly. One thing we did, in particular, was use the fractions we found to find multiples of that were nearly integers.

Finally, we wrapped up by using continued fractions to find good approximations to , and a few other numbers.

Definitely a fun project, and one that makes me especially happy because of the connection to Mr. Waterman. Hopefully the boys will want to play around with this idea a bit more tomorrow.

After the project I printed 12 of the pentagons and had the kids play with them today. See Kaplan’s post for some historical notes about the pentagon. The historical importance is probably too advanced for kids to appreciate, but what they can appreciate is that this pentagon can be surrounded in multiple ways. I had the boys play around to see what they could find.

Here’s what my younger son found:

Here’s what my older son found:

This project was really neat. I think making shapes like the one in Kaplan’s post is a great way for kids to review (or even get introduced to!) both equations of lines and some elementary geometry. Also, as always, it is extremely fun for kids to explore ideas that are interesting to professional mathematicians ðŸ™‚

The picture in the middle part of the post looked like something that kids could understand:

For our project today I thought it would be fun to talk about how to make the polygon tile in the above picture. After we understand how to describe that polygon, we can 3d print a bunch of the tiles and talk more about the idea of “surrounding a polygon” with these tiles tomorrow.

This project is a fun introduction to 2d geometry (and especially coordinate geometry) for kids. We also use the slope / intercept form of a line when we make the shape.

We got started by looking at Kaplan’s post:

Next we began to talk about how to make the shape – the main idea here involves basic properties of 30-60-90 triangles. My older son was familiar with those ideas but they were new to my younger son.

We also talk a little bit about coordinate geometry. The boys spend a lot of time discussing which point they should select to be the origin.

In the last video we found the coordinates of 3 of the points. Now we began the search for the coordinates of the other two. We mainly use the ideas of 30-60-90 triangles to find the coordinates of the first point.

The 2nd point was a bit challenging, though:

The next part of the project was spent searching for the coordinates of the last point. The main idea here was from coordinate geometry -> finding the coordinates of the middle of the square. The coordinate geometry concepts here were difficult for my younger son but we eventually were able to write down the coordinates of the final point:

We were running a little long in the last video, so I broke the video into two pieces. The last step of the calculation is here:

After finding all of the coordinates we went upstairs to make the shape on Mathematica. We used the function “RegionPlot3D” that allows us to define a region bordered by a bunch of lines. Below is a recap of the process we went through to make the shape and a quick look at the shapes in the 3d printing software:

This isn’t our first 3d printing / tiling project. Some prior ones are linked in a project we did last month after seeing an incredible article by Evelyn Lamb:

Yesterday my younger son was playing a dice game (explained in the first video) that reminded me a bit of the coupon collection problem. I thought it would be fun to try out that problem with the boys this morning. We were a little low energy, but I think it was still a good project. I’ll have to figure out how to revisit it to make sure the points stuck.

Here’s the introduction, including the game my son was playing:

Next we worked through one case of the problem – rolling dice trying to collect 6 “coupons”. My older son thought it would take 15 rolls and my younger son thought it was take 20.

Now I tried to help the kids dive into the math. We ended up going down a path that was much more complicated than I intended. I’m not sure why I made the choice that I did here, but . . . it happens sometimes ðŸ™‚

So, at the end of the last video we were caught in a seemingly complicated infinite series. I tried to explain why the expression we had on the board had to be equal to one. Then I tried to explain why the expected number of rolls had to be greater than one. The explanation here is a disaster, though.

Now that things had gone totally off the rails, I tried to pull it back. Luckily things did go better, and it was easier for the boys to see the expected number of rolls when there were fewer open slots.

Finally I wanted to show the kids how the ideas we talked about here would apply to a more difficult problem – say 100 coupons. We got off on the wrong foot here, but we eventually saw how the ideas we’d talked about previously applied.

Despite the low energy and going doing a path that was a bit too complicated, I think this is a fun problem for kids to study. It looks very difficult initially, but through a bit of calculation (and maybe a bit of hand waving) we can break it down into some smaller problems that we are able to solve. Putting the solutions of those smaller problems together, we can show that the solution to the original coupon collection problem isn’t too hard to understand.

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:

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!

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:

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):

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.

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. Â The goal wasn’t to get too complicated, rather just to explore a few basic ideas.

Here’s how I introduced the project.Â 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.

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:

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!

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

Numbers x,y,z are distributed uniformly on [0,1]. Find the probability of max{x,y,z} âˆ’ min{x,y,z} â‰¤ 2/3 (ARMLC 2016, #7) .

— Alexander Bogomolny (@CutTheKnotMath) June 14, 2017

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!