Sharing Kelsey Houston-Edwards’s bridge video with kids

Kelsey Houston-Edwards published another fantastic video this week:

By coincidence my kids had been making domino runs this week and I was already planning on doing this bridge activity with the boys – perfect timing! Even better, Houston-Edwards’s video shows that this activity is a great way to introduce kids to basic mathematical modelling. That topic has been on my mind quite a bit this week too because of the new Grant Sanderson calculus series.

So, we watched the video on Friday night and talked about some of the ideas this morning.

Here’s a short introduction to the problem and a bit about what the boys remembered from the video:

Next we moved to the problem of trying to use some basic math to describe what’s going on with these bridges. Although they’d seen the math modelling in Houston-Edwards’s video previously, the modelling ideas were not the first ideas that came to their mind. Instead they were able to solve the first step in the bridge problem. Instead they were able to just see that solution.

With the algebraic solution not being quite the first thing that came to their mind, I decided to dive into that solution for the bridge with 3 bricks. The nice thing about the 3 brick bridge is that the numbers are still not that complicated.

Once we had some equations written down we talked about various different approaches to solving them. My younger son found a pretty clever way to solve these equations without too much algebraic effort.

Finally, I had the boys make some bridges on the table rather than the slightly cheating way that we did in the first video. We had a bit of debate off camera about whether or not their top bricks were fully off the table, but they were certainly very close 🙂

Definitely a fun project. I’m going to try to do more of these modelling tasks in the next month or so. Right now I’m not completely sure where to find good introductory modelling tasks for kids, but hopefully solving that challenge will be a fun project for me!

Sharing Kelsey Houston-Edwards’s Cryptography video with kids

I’m falling way behind on Kelsey Houston-Edwards’s video series, sadly. Her “How to Break Crytography” video is so freaking amazing that it needed to be first in line in my effort to catch up!

So, this morning I watched the video with the boys. We stopped the video a few times to either work through some of the math, or simply to just have me explain it a bit. Overall, though, I think this video is not just accessible to kids, but is something that they will find absolutely fascinating.

Here’s what my kids took away from it:

Next we went upstairs to write some Mathematica code to step through the process that Houston-Edwards described in her video. In this video we (slightly clumsily) step through the code and check a few small examples:

When I turned the camera off after the last video my younger son asked a really interesting question -> Why don’t we just use Mathematica’s “FactorInteger[]” function?

We talked about that for a bit in this video and then tried to use Shor’s algorithm to find the factors of a number that was the product of two 4 digit primes.

So, we had the camera off for a little over a minute after the last video, but the good news is that Mathematica did, indeed, finish the calculation. It was a nice (and somewhat accidental) example of how quickly this algorithm runs into trouble.

The cool thing, though, is that it did work 🙂

Definitely a fun project, though it does require a bit more computer power than most of our other projects. I’m happy to be catching up a little on Kelsey Houston-Edwards’s video series – it really is one of the best math-related things on the internet!

Sharing Kelsey Houston-Edwards’s video about Pi and e with kids

Yesterday I a new video from Kelsey Houston-Edwards that just blew me away. At this point I don’t have the words to describe how much I admire her work. What she is doing to make challenging, high level math both accessible and fun for everyone is amazing.

The new video was about this question on Math Overflow from Erin Carmody:

If I exchange Infinitely many digits of Pi and E are the two resulting numbers transendental?

Before showing the boys Houston-Edwards’s video, I wanted to see what they thought about the question. So, we just dove in:

Next, I took a great warm up idea from Houston-Edwards’s video and asked the boys if they could find *any* two irrational numbers that you could use to swap digits and produce a rational number.

Now, with that little bit of prep work, we watched the new video:

After the video we talked about what we learned. I think just tiny bit of prep work we did really helped the boys get a lot more out of the video.

One of the fun little challenge questions from the video was to show that (assuming \pi and e differ in infinitely many digits, then you will produce uncountably many different numbers by swapping different digits. I didn’t expect that the boys would be able to construct this proof, so I gave them a sketch of how I thought about it (and hopefully my idea was right . . . . )

I think that kids will find the ideas in Houston-Edward’s new video to be fascinating. It is so fun (and sadly so rare) to be able to share ideas that are genuinely interesting to professional mathematicians with kids. As always, I can’t wait for next week’s PBS Infinite series video!

Random walks with kids

A week or so ago my older son did a short project on random walks out based on a page in Patters of the Universe:

Returning to Patterns of the Universe

By coincidence that week Kelsey Houston-Edwards’s new video was about random walks. So, we watched her video after that project:

Today my younger son is sick and wasn’t up to participating in a project. So, I thought it would be fun to revisit the random walk project and dive in a little deeper since my older son was a little more familiar with that topic.

I started by asking him what he remembered about random walks from the prior project and from the PBS Infinite Series video. One thing that he remembered is that 2d random walks do tend to return to where they started, but 3-d ones tend not to.

We started looking at specific random walks by studying a 1-dimensional random walk. We created a random walk by rolling dice and didn’t get quite what we were expecting, but that result led to a fun conversation:

In the last video we got more even numbers than we were expecting, so we decided to continue on to see if the walk would return to 0. Obviously we kept rolling even numbers . . . .

Next we moved on to studying a 3d random walk (and, of course, now rolled lots of odd numbers 🙂 )

We created the 3d random walk with snap cubes and it was pretty neat to see the shape that emerged from the dice rolls.

Despite the unexpected outcome with the even and odd rolls this was a fun project. I’d like to think a little more about how to make some random walk 3d prints. My guess is that those prints would be really fun to share with kids.

Kelsey Houston-Edwards’s ‘Proof’ video is incredible

Kelsey Houston-Edwards’s latest video is amazing:

It absolutely blew me away. The kids have a late karate class today so I had to wait a few extra hours to use the new video for a project – WORST WAIT EVER!!!

To start the project I had the boys look at three of the problems in the video to see what approach each kid would take to prove the mathematical statement. My younger son is in 5th grade and my older son is in 7th grade, so I’m not expecting perfect proofs (by any stretch of the imagination) but rather just looking for their ideas.

Here’s the first problem – can you cover an 8×8 checker board with 2×1 dominoes if you remove two opposite corners:

The next problem was to show that the sum:

1 + 3 + 5 + \ldots + (2n - 1) = n^2

Here’s what they had to say – both ideas here were really interesting and used arithmetic. I was excited to see their reaction to the geometric proof in the video:

The next problem was to show that ” n choose 2″ was equal to 1 + 2 + 3 + \ldots + (n-1).

My younger son had a nice idea to start small and work his way up. He got stuck so I helped him a little. As in the last video, my older son did the proof by calculating.

After working through these three problems we watched the new video together. The problem about the L-shaped tile covering the 2^n x 2^n grid caught my youngers son’s eye. That led to a short discussion of induction.

The problem about breaking the stick into 3 parts and forming a triangle caught my older son’s eye. He reconstructed the cool proof from the video. I’d like to show him some alternate proofs from geometric probability, too, since they are all so fun!

I’m really enjoying the math videos that Houston-Edwards is making. This one is especially amazing. How great would it be for every math class in the country to watch her video tomorrow! I think it would change the way that kids see math.

Exploring different L^p versions of the torus

A few weeks ago we did a fun project on L^p spheres after watching Kelsey Houston-Edwards’s video on different ways of measuring distance:


Sharing Kelsey Houston-Edwards’s Pi video with kids

Playing around a little with our 3d printing software last night made me want to try a similar project with a torus in various L^p metrics. I made 5 different ones and set the printer to print them overnight . . . and the print failed. Boo 😦

So, I’m re-printing them to use for a project this afternoon, but for now the project with my younger son just used the shapes on the computer.

Here’s what he thought about the usual torus and the torus in L^1

Next we moved on to looking at the torus in the L^3 and L^5 metrics:

Finally, we looked at some of the shapes when p was not an integer. We looked at p = 0.75, 1.5, and 1.05.

Using the computer program was a nice way to save the project after the print failed. I’m really hoping that the 2nd time is a charm with the print and we can explore the 3d printed shapes this afternoon!

Sharing Kelsey Houston-Edwards’s singularity video with kids

We’ve been enjoying going through Kelsey Houstin-Edwards’s new video series. This week’s was a bit more advanced than some of the prior ones, but I gave it a shot with the kids anyway. I tried to focus on connecting the ideas about singularities in the video with some of the 3D printed shapes we’ve been studying from Henry Segerman’s new book.

Also, I’m just getting over a few days with the norovirus, so sorry if this one (including the write up) has a bit less energy than usual.

Here’s the latest PBS Infinite Series video:

Here’s what the boys took away from the video:

Next we looked at a couple of the shapes that Henry Segerman has made to study with shadows. We were able to see (eventually) that the shadow of the north pole would be a point at infinity – or a singularity.

At the end of this video we started looking at a torus, and the conversation took a very interesting topological turn.

So, we landed on a question of what different shapes might be a torus. It took a bit of time to straighten out this idea, but after a few minutes we came to an agreement on what a torus was.

After that we saw that we didn’t have the same singularity problem trying to create a map that we had on the sphere.

After talking about the torus we spent the rest of this video talking about the pseudosphere which has more than one singularity.

So, another great video from Kelsey Houston-Edwards. It was fun connecting her ideas with some of the 3d prints we’ve been studying lately.

Sharing Kelsey Houston-Edwards’s Markov Chain video with kids

The latest video from Kelsey Houston-Edwards is incredible:

I love the problem she’s presenting – it seems completely impossible to someone hearing it for the first time, but after her video kids can solve similar problems immediately!

We started the project today right after we watched the video. The first thing I did was ask the kids about what they just saw:

Next we tried to solve the rook problem that Houston-Edwards posed at the end of the video. The kids were able to solve it pretty quickly 🙂

To wrap up, we took a crack at solving the same problem with a bishop. This took a little longer, but was a great lesson. Hopefully this slightly more challenging problem allowed some of the ideas from Houston-Edwards’s video sink in:

I’ve been loving this new series of math video since it started, but this one was extra impressive for me. It is really cool to see someone explain such a challenging problem in essentially a public lecture setting. Can’t wait to see where this series goes in 2017!

Sharing Kelsey Houston-Edwards’s Pi video with kids

We really enjoyed watching Kelsey Houston-Edwards’s latest PBS Infinite Series video last night:

We’ve played around with a few ideas about altering Pi a few times previously. One was with Vi Hart’s “Pi = 4” video in this project:

A fun fractal project – exploring the Gosper curve

A second time was a super fun series of projects on 4-dimensional shapes inspired by a tweet from Patrick Honner. That complete series is here:

Patrick Honner’s Pi day exercise in 4d part 5: The 120 and 600 cells

Today we explored the 2-dimensional idea in Kelsey Houston-Edwards’s video in 3-dimensions using 3d printing.

First, though, I asked the boys what they thought about the latest PBS Infinite series video:

Next we took a look at 6 different “spheres” in 3-dimensions which were defined using the different way of measuring distance Kelsey Houston-Edwards introduced in her video. It is really fun to hear kids talking about these shapes, and even more fun to be able to actually hold these shapes in your hand!

The last thing we did was look at the 6 spheres all together. My younger son noticed how much bigger the shapes got as you moved from the L-0.75 norm to the L-4 norm. My older son noticed that some of the shadows of the shapes looked pretty similar even though the shapes didn’t look the same at all:

This was a really fun project to prepare. It is really fun to show kids ideas from advanced math that the wouldn’t likely see in school. It is also really fun to hold these strange shapes in your hand! Lately we’ve been using the F3 program to help us make objects to print – it was a really lucky coincidence to see the new PBS Infinite Series video *after* I learned to use the new program. It took less than 5 minutes to make the .stl files for the 6 shapes 🙂

Sharing Kelsey Houston-Edwards’s binary video with kids

Kelsey Houston-Edwards released a new math video last week:

So far we’ve been able to use all of her videos for great weekend projects. This video had a fun little surprise because we’d seen the problem she talks about in a (seemingly) totally different context – an old magic set! Once we dug out that out magic set from under my younger son’s bed we started the project 🙂

Here’s what the boys had to say about the video:

Before jumping to the challenge problems, we looked at the old magic trick:

Finally, we tried to answer the two challenge problems from Houston-Edwards’s video. I’m sorry this got a little rushed at the end – I’d not noticed that we were out of batteries! We finished with about 10 seconds to spare!

The two challenges are great problems for kids to think through – the boys found a few interesting patterns even though the relationship with powers of 2 was a little hard for them to see.