# An introductory mathematical induction problem I learned thanks to Kelsey Houston-Edwards

Yesterday Annie Perkins asked about resources to help a student understand mathematical induction:

Looking for some resources I was reminded of this incredible video that Kelsey Houston-Edwards made in 2017 – the proof by induction example starts around 5:25:

Today I thought it would be fun to have the boys go back and work through that problem. I started by introducing the problem after we’d made some L shapes out of snap cubes. The boys were able to understand the problem and solve the 1×1 and 2×2 cases:

Now we moved to the 4×4 case. My older son started by looking at different cases – which was a good way to solve the specific 4×4 case. At first it was hard to see how to generalize this approach, though my younger son had some nice ideas about how to extend it:

The next step was a deep dive into the 4×4 case and looked carefully at the exact problem we were trying to solve. Here the boys did a great job of seeing how to extend the solution of the 2×2 case to the 4×4 case.

Finally, we looked at how to extend the 4×4 case to the 8×8 case, and then how to extend to all cases of the problem. We ended the project by discussing how / why mathematical induction works on problems like this one:

# My talk at the 2018 Williams College math camp

[had to write this in a hurry before the family headed off for a vacation – sorry that this post is likely a little sloppy]

Yesterday I gave a talk at a math camp for high school students at Williams College. The camp is run by Williams College math professor Allison Pacelli and has about 20 student.

The topic for my talk was the hypercube. In the 90 min talk, I hoped to share some amazing ideas I learned from Kelsey Houston-Edwards and Federico Ardila and then just see where things went.

A short list of background material for the talk (in roughly the order in the talk is):

(1) A discussion of how to count vertices, edges, faces, and etc in cubes of various dimensions

This is a project I did with my kids a few years ago, and I think helps break the ice a little bit for students who are (rightfully!) confused about what the 4th dimension might even mean:

Counting geometric properties in 4 and 6 dimensionsf

(2) With that introduction I had the students build regular cubes out of the Zometool set I brought. Then I gave them some yellow struts and asked them to construct what they thought a hypercube might look like. From the prior discussion they knew how many points and lines to expect.

To my super happy surprise, the students built two different representations. I had my boys talk about the two different representations this morning. Funny enough, they had difference preferences for which was the “best” representation:

Here’s what my older son had to say:

Here’s what my younger son had to say:

At the end of this section of my talk I showed the students “Hypercube B” from Bathsheba Grossman (as well as my Zometool version):

(3) Now we moved on to looking at cubes in a different way -> standing on a corner rather than laying flat

I learned about this amazing way to view a cube from this amazing video from Kelsey Houston-Edwards. One of the many bits of incredible math in this video is the connection between Pascal’s triangle and cubes.

Here are the two projects I did with my kids a after seeing Houston-Edwards’s video:

Kelsey Houston-Edwards’s hypercube video is incredible

One more look at the hypercube

After challenging the kids to think about what the “slices” of the 3- and 4-dimensional cubes standing on their corners would be, I showed them the 3D printed versions I prepared for the talk:

Here are the 2d slices of the 3d cube:

Here are the 3d slices of the 4d cube:

(4) Finally, we looked at the connection between cubes and combinatorics

Here is the project I did with my older son after seeing Ardila’s video:

Federico Ardila’s Combinatorics and Higher Dimensions video is incredible!

I walked the students through how the vertices of a square correspond to the subsets of a 2-element set and then asked them to show how the vertices of a cube correspond to the subsets of a 3-element set.

There were a lot of oohs and ahhs as the students saw the elements of Pascal’s triangle emerge again.

Then I asked the students to find the correspondence between the 4-d cubes they’d made and subsets of a 4-elements set. I was incredibly happy to hear three different explanations from the students about how this correspondence worked – I actually wish these explanations were on video because I think Ardila would have absolutely loved to hear them.

(5) One last note

If you find all these properties of 4-D cubes as neat as I do, Jim Propp has a fantastic essay about 4 dimensional cubes:

Jim Propp’s essay Time and Tesseracts

By lucky coincidence, this essay was published as I was trying to think about how to structure my talk and was the final little push I needed to put all the ideas together.

# 15 (+1 bonus) Math ideas for a 6th grade math camp

Saw an interesting tweet last week and I’ve been thinking about pretty much constantly for the last few days:

I had a few thoughts initially – which I’ll repeat in this post – but I’ve had a bunch of others since. Below I’ll share 10 ideas that require very few materials – say scissors, paper, and maybe snap cubes – and then 5 more that require a but more – things like a computer or a Zometool set.

The first 4 are the ones I shared in response to the original tweet:

## (1) Fawn Nguyen’s take on the picture frame problem

This is one of the most absolutely brilliant math projects for kids that I’ve ever seen:

When I got them to beg

Here’s how I went through it with my younger son a few years ago:

## (2) James Tanton’s Mobius strip cutting exerciese

This is a really fun take on this famous scissors and paper cutting exercise:

You will honestly not believe what you are seeing when you go through Tanton’s version:

Here’s the link to our project:

James Tanton’s incredible mobius strop cutting project

## (3) Martin Gardner’s hexapawn “machine learning” exercise

For this exercise the students will play a simple game called “hexapawn” and a machine consisting of beads in boxes will “learn” to beat them. It is a super fun game and somewhat amazing that an introductory machine learning exercise could have been designed so long ago!

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

## (4) Katie Steckles’ “Fold and Cut” video

This video is a must see and it was a big hit with elementary school kids when I used it for “Family Math” night:

Here are our projects – all you need is scissors and paper.

Our One Cut Project

Fold and cut project #2

Fold and cut part 3

## (5) Along the same lines – Joel David Hamkins’s version of “Fold and Punch”

I found this activity in one of the old “Family Math” night boxes:

Joel David Hamkins saw my tweet and created an incredible activity for kids.  Here’s a link to that project on his blog:

Joel David Hamkins’s fold, punch and cut for symmetry!

## (6) Kelsey Houston-Edwards’s “5 Unusual Proofs” video

Just one of many amazing math outreach videos that Kelsey Houston-Edwards put together during her time at PBS Infinite Series:

Here is how I used the project with my kids:

Kelsey Houston-Edwards’s “Proof” video is incredible

## (7) Sharing the surreal numbers with kids via Jim Propp’s checker stacks game

Jim Propp published a terrific essay on the surreal numbers in 2015:

Jim Propp’s “Life of Games”

In the essay he uses the game “checker stacks” to help explain / illustrate the surreal numbers. That essay got me thinking about how to share the surreal numbers with kids. We explored the surreal numbers in 4 different projects and I used the game for an hour long activity with 4th and 5th graders at Family Math night at my son’s elementary school.

This project takes a little bit of prep work just to make sure you understand the game, but it is all worth it when you see the kids arguing about checker stacks with value “infinity” and “infinity plus 1” 🙂

Here is a summary blog post linking to all of our surreal number projects:

Sharing the Surreal Numbers with kids

## (8) Larry Guth’s “No Rectangle” problem

I learned about this problem when I attended a public lecture Larry Guth gave at MIT.  Here’s my initial introduction of the problem to my kids:

I’ve used this project with a large group of kids a few times (once with 2nd and 3rd graders and it caused us to run 10 min long because they wouldn’t stop arguing about the problem!). It is really fun to watch them learn about the problem on a 3×3 grid and then see if they can prove the result. Then you move to a 4×4 grid, and then a 5×5 and, well, that’s probably enough for 80 min 🙂

Larry Guth’s “No Rectangles” problem

## (9) The “Monty Hall Problem”

This is a famous problem, that equally famously generates incredibly strong opinions from anyone thinking about it. These days I only discuss the problem in larger group settings to try to avoid arguments.

Here’s the problem:

There are prizes behind each of 3 doors. 1 door hides a good prize and 2 of the doors hide consolation prizes. You select a door at random. After that selection one of the doors that you didn’t select will be opened to reveal a consolation prize. At that point you will be given the opportunity to switch your initial selection to the door that was not opened. The question is  -> does switching increase, decrease, or leave your chance of winning unchanged?

One fun idea I tried with the boys was exploring the problem using clear glasses to “hide” the prizes, so that they could see the difference between the switching strategy and the non-switching strategy:

Here’s our full project:

Exploring the Monty Hall problem with kids

## (10) Using the educational material from Moon Duchin’s math and gerrymandering conference with kids

Moon Duchin has spent the last few years working to educate large groups of people – mathematicians, politicians, lawyers, and more – about math and gerrymandering.  . Some of the ideas in the educational materials the math and gerrymandering group has created are accessible to 6th graders.

Here’s our project using these math and gerrymandering educational materials:

Sharing some ideas about math and gerrymandering with kids

## (11) This is a computer activity -> Intro machine learning with Google’s Tensorflow playground.

This might be a nice companion project to go along with the Martin Gardner project above. This is how I introduced the boys to the Tensorflow Playground site (other important ideas came ahead of this video, so it doesn’t stand alone):

Our complete project is here:

Sharing basic machine learning ideas with kids

## (12) Computer math and the Chaos game

The 90 seconds starting at 2:00 is one of my all time favorite moments sharing math with my kids:

The whole project is here, but the essence of it is in the above video:

Computer math and the chaos game

## (13) Another computer project -> Finding e by throwing darts at a chess board

This is a neat introductory probability project for kids. I learned about it from this tweet:

You don’t need a computer to do this project, but you do need a way to pick 64 random numbers. Having a little computer help will make it easier to repeat the project a few times (or have more than one group work with different numbers).

Here’s how I introduced the project to my kids:

Here’s the full project:

Finding e by throwing darts

## (14) Looking at shapes you can make with bubbles

For this project you need bubble solution and some way to make wire frames. We’ve had a lot of success making the frames from our Zometool set, but if you click through the bubble projects we’ve done, you’ll see some wire frames with actual wires.

Here’s an example of how one of these bubble projects goes:

And here’s a listing of a bunch of bubble projects we’ve done:

Our bubble projects

## (15) Our project inspired by Ann-Marie Ison’s math art:

This tweet from Ann-Marie Ison caught my eye:

Then Martin Holtham created a fantastic Desmos activity to help explore the ideas:

It is fun to just play with, but if you want to see how I approached the ideas with my kids, here are our projects:

Using Ann-Marie Ison’s incredible math art with kids

Extending our project with Ann-Marie Ison’s art

## (16) Bonus project!!A dodecahedron folding into a cube

This is a an incredible idea from 3d geometry.

We studied it using our Zometool set – that’s not the only way to go, but it might be the easiest:

Here’s the full project:

Can you believe that a dodecahedron folds into a cube?

# Sharing Kelsey Houston-Edwards’s Axiom of Choice video with kids

Kelsey Houston-Edwards has a new video out about the Axiom of Choice:

The video is amazing (as usual) and I wanted to be able to share it with the boys. This one is a bit hard than usual – the topic is pretty advanced to begin with and is also pretty far outside of my own knowledge – but we gave it a shot.

Here’s what the boys thought after seeing the video:

Next we reviewed how Houston-Edwards divided the numbers from 0 to 1 into buckets. The boys didn’t quite have the details right, but that actually made talking through the idea pretty easy – I learned from their explanation what points needed to be re-emphasized.

Now we talked through the really challenging part of the video -> creating the set with no size. Given the challenge of explaining this idea to kids, I’m pretty happy with how the conversation went here. Also, I only finally understood the argument myself while I was explaining it to them!

Now we backed away from the complexity of the Axiom of Choice and reviewed two other slightly easier ideas that came up in our discussion. Here we discuss why $\sqrt{2}$ is irrational:

Finally, we wrapped up by discussing why the rational numbers are countable:

Although kids will have a hard time understanding all of the ideas that Kelsey Houston-Edwards brings up in her Axiom of Choice video, I think it is fun to see which ideas grab their attention. The idea that you can have a set that doesn’t have a size is pretty amazing. I was pretty happy with how things went today – exploring the ultra complex idea first and then backing off to discuss slightly easier ideas involving infinity. Definitely a fun set of ideas to plant in the minds of younger kids 🙂

# 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!

# 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.

# One more look at the hypercube

We’ve done two projects on hypercubes after seeing Kelsey Houston-Edwards’s latest video. Those two projects are here:

Kelsey Houston-Edwards’s hypercube video is incredible!

Revisiting Kelsey Houston-Edwards’s hypercube video

The video that inspired those two projects is here:

After posting the second project our friend Roy Wiggins shared a video that he made several years ago:

After seeing Wiggins’s video I thought it would be fun to use in one more project with the boys. While they were at school today I printed the set of shapes corresponding to the intersection of the hypercube and 3d hyperplanes that you can see in both videos above. When the boys got home I talked about the shapes with them individually. In addition to the 3d prints, I had some zometool shapes set up to help understand the more traditional way to view and understand a hypercube.

Here’s how the talk went with my younger son:

And here’s what older son and I talked about:

This was definitely a fun series of projects. It was interesting to me that both kids really struggled to see how to explain the shape of a hypercube from these 3d shapes. Not that I was expecting it to be easy (!) but this alternate view of the hypercube proved to be much more difficult to process than I expected.

# Revisiting Kelsey Houston-Edwards’s Hypercube video

Last week Kelsey Houston-Edwards published a fantastic video about hypercubes – it is one of the best math videos I’ve ever seen:

Here’s our project on her video:

Kelsey Houston-Edwards’s hypercube video is incredible!

Today while the kids were at school I wrote a couple of Mathematica functions to replicate her results. Writing the code to make these shapes is actually a pretty fun exercise in linear algebra and trig, but that’s a little more than I felt like sharing with the kids just now 🙂

Instead I had them look at the shapes on the screen and tell me what they thought. The first video in each with each kid shows the shape made by a plane intersecting a 3d cube standing on its corners. The second video shows the 3d intersection of a hyperplane perpendicular to the long diagonal of a 4d cube intersecting that cube.

Here’s what my older son had to say:

(a) The 3d cube being cut by a (slightly thick) plane:

(b) The 4d cube being cut by the hyperplane:

Here’s what my younger son had to say:

(a) The 3d cube being cut by a (slightly thick) plane:

(b) The 4d cube being cut by the hyperplane:

This project was really fun, and, as I mentioned above, would also be a great programming project for kids learning linear algebra and trig. I’m 3d printing some of the shapes how, so playing with those shapes will be our project tomorrow!

# Kelsey Houston-Edwards’s hypercube video is incredible

The latest video that Kelsey Houston-Edwards released is one of the best math videos that I’ve ever seen:

This morning I used the video for a project with the boys. We watched (roughly) the first 11 minutes. I stopped the video before the Houston-Edwards revealed shapes associated with the 4D cube.

Here’s what the boys thought of the video:

Next we talked about how the rule for Pascal’s triangle related to the shapes in the video. This procedure is really fun to talk through with kids.

At the end of this part of the project we talked about the possible shapes that we would encounter in the 4th dimensional version of the problem.

Now that we had some of the basics sorted out for the 4th dimensional case, we tried to figure out the shapes. I won’t give away the answers in the text, but we were able to get the first one, but the 2nd one was pretty hard to see – the closest we got was that “it is sort of a twisting shape” 🙂

To wrap up we watched the remainder of Houston-Edwards’s video. The boys were surprised to learn what that final shape was.

At the end I reminded them of a prior project that was sort of similar – studying the so-called Prince Rupert Cube. Unfortunately the cube broke while I was reminding them about that project. Oh well . . . .

At this point I’m expecting Kelsey Houston-Edwards’s videos to be incredible, but she exceeds my expectations every time! Her videos are so much fun to share with kids – I can’t wait for the next one!

# Sharing Kelsey Houston-Edwards’s topology video with kids

Kelsey Houston-Edwards’s latest video is terrific:

This one is particularly easy to share with kids because there are several puzzles where she asks you to stop and think about the solution. I began the picture frame puzzle as the starting point for our project today.

The puzzle goes roughly like this:

A common way to hang a picture is to use two nails in a wall and run the wire around those two nails. Assuming the nails / wall are strong enough, if you remove one of the nails the picture will still hang. Is there a way to hang a picture with two nails so that if you remove either of the nails the picture will fall?

We took a shot at this puzzle using yarn and snap cubes. It was a good challenge for the boys:

In the last video we got the picture to fall once, but the boys weren’t quite clear what happened – but now they at least knew it was possible! Here we explored the idea more carefully:

Next we finished watching the video and then discussed what we saw (as I publish this post the video preview isn’t embedding properly, but is really just audio anyway):

Finally we looked at two sets of shapes that appeared in the video that we’ve looked at before. The first is a 3d print of Henry Segerman’s “Topology Joke” and the 2nd is a set of “rollers” that we’d made after seeing a Steven Strogatz tweet. The tweet and the roller project are here:

3d printing and rollers

Another fun project from Kelsey Houston-Edwards’s amazing math series. Sorry to be brief on this project, but I had to get this one out quick because of a bunch of activities going on today.