Tag Jim Propp

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

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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

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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.

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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:

dodecahedron fold

Here’s the full project:

Can you believe that a dodecahedron folds into a cube?

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A review of “The Sherlock”

We received a nice gift from Jim Propp earlier this month. With the kids off of school for a snow day today, it seemed like a good time to open it up.

My younger son played with it for a while and then I wanted to have him show how the game worked. Here are his initial thoughts about the game:

Here’s the first example of a puzzle solve. You’ll see that even having solved it once before it is still not necessarily so simple:

Here’s a second solve example – this one goes pretty quickly

Finally, we wrapped up by having him show some of the other pieces and me asking him to talk about what he thinks the main ideas are for this puzzle. Interestingly he doesn’t think that it is a math puzzle, but rather a logic puzzle 🙂

So, thanks to Jim Propp for giving us this really nice puzzle game.

A quick hexaflexagon project thanks to Jim Propp

My younger son had been interested in hexaflexagons lately. He saw them Martin Gardner’s “The Collossal Book of Mathematics” first and then I bought him a few books thanks to a suggestion from Annie Perkins:

Then, by lucky coincidence, Jim Propp made this fun video:

After watching that video with my son this morning, he took Propp’s suggestion and cut the corners off of one of his hexaflexagons. Here’s what he had to say:

Even tonight I’m not sure what he was getting at when he was talking about the hexagons moving when the shape was flexed. I might revisit that with him another time, but it was fun to hear him talking about what was going on with this new shape.

I’m happy that he’s been having fun with these shapes. Just when I thought there wasn’t any more he could do with them, Propp’s video opened up a

Sharing Jim Propp’s base 3/2 essay with kids part 2

I’m going through Jim Propp’s piece on base 3/2 with my kids this week.

His essay is here:

Jim Propp’s How do you write one hundred in base 3/2?

And the our first project using that essay is here:

Sharing Jim Propp’s base 3/2 essay with kids – Part 1

Originally I wanted to have the kids read the essay and give some of their thoughts for part 2, but I changed my mind on the approach this morning. Instead I asked each of them to answer the question in the title of Propp’s essay -> How do you write 100 in base 3/2?

Propp points out in his essay that his approach to base 3/2 via chip firing / Engel machines / exploding dots is not what mathematicians would normally consider to be base 3/2. The boys are not aware of that statement, though, since they have not read the essay yet.

Here’s how my younger son approached writing 100 in base 3/2. The first video is an introduction to the problem and, from knowing how to write numbers like 100 (in base 10) in other integer bases.

I think the first 3 minutes of this video are interesting because you get to hear his ideas about why this approach seems like a good idea. The remainder of this video plus the next two videos are a long march down the road to discovering why this approach doesn’t work in the version of base 3/2 we are studying:

So, after finding that the path we were walking down led to a dead end, we started over. This time my son decided to try to write 100 as 10×10. This approach does work!

Next I introduced the problem to my older son. He also started by trying to solve the problem the same way that you would for integer bases, though his technique was slightly different. He realized fairly quickly (by the end of the video, I mean) that this approach didn’t work:

My older son needed to find a new approach, and he ended up finding an idea different from my younger son’s idea to find 100 in base 3/2. His idea was to use chip firing:

I thought that today’s project would be a quick reminder of how base 3/2 works (at least the version we are studying). That thought was way off base and was completely influenced by me knowing the answer! Instead we found – by accident – a great example of how to explore a challenging problem in math. Sometimes the first few things you try don’t work, and you have to keep trying new things.

Definitely a fun morning!

Sharing Jim Propp’s base 3/2 essay with kids

Jim Propp’s essay on base 3/2 is fantastic:

Here’s a direct link to his blog post in case the twitter link doesn’t work:

Jim Propp’s How do you write one hundred in base 3/2?

and here are links to our two prior base 3/2 projects:

Fun with James Tanton’s base 1.5

Revisiting James Tanton’s base 3/2 exercise

I’m hoping to have time to spend at least 3 days playing around with Propp’s latest blog post. Today we had 20 min free unexpectedly in the morning and I used that time to introduce two of the ideas. They haven’t read the post, yet, but instead I started by having them watch Propp’s short video about the binary Engel machine:

After watching that video I had the boys recreate the idea with snap cubes on our white board. Here’s that work plus a few of their thoughts on the connection with binary:

Next I challenged the boys to draw the base 3/2 version of the machine. After they did that we counted to 10 in base 3/2 and talked about what we saw:

I was happy that the boys were able to understand the idea behind the base 3/2 Engel machine. With the work from today giving them a nice introduction to some of the ideas in Propp’s essay, I think they are ready to try reading the essay tomorrow. It’ll be interesting to see what ideas catch their eye. Hopefully we can do another short project on whatever those ideas are tomorrow morning.

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.

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.

Revisiting a counting project for kids that I learned from Jim Propp

For our math project today we returned a tiling idea that is a really fun idea for kids to explore. Here are a few of our prior projects on the subject:

A fun counting exercise for kids suggested by Jim Propp

Counting 2xN domino tilings

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Today the plan was to look at 2xN tilings first and then move on to tilings of 3xN rectangles with 3×1 dominoes.

We stared by exploring some simple 2xN cases and looked for patterns:

In the first video we counted the number ways that we could tile 2×1, 2×2, 2×3, and 2×4 rectangles with dominoes. Now the boys noticed the connection with the Fibonacci numbers and we tried to find and explanation for why the Fibonacci numbers seemed to be showing up here. The nice thing is that the boys pretty much got the complete explanation all on their own.

Now we moved on to counting the tilings of a 3xN rectangle using 3×1 “dominoes” – what would be different? What would be the same?

One really interesting thing here is that my older son and younger son came up with different ideas for how to count the general arrangement.

So, in the last video my older son had a counting hypothesis that I couldn’t quite understand. In the beginning of this video I have him explain his process more carefully. The surprise was that for the 3×6 case we were looking at next both of their counting procedures predicted the same number of domino tilings.

In this part of the project we tried to follow both procedures to see how they worked.

Having sorted out the counting procedure in the last video, we now looked carefully at the 2xN and 3xN tiling procedures and saw that we could compute the number of tilings for the 2×100 and 3×100 cases if we wanted to.

I’d love to come up with more counting projects for kids. These projects are accessible to young kids and I think shows of some really fun ideas from advanced math that kids probably don’t usually see in school.

Fun with rotopo

Saw this neat tweet from Jim Propp yesterday:

After playing the game for a just a few minutes I knew that my kids would love it.

Here’s each of their reaction to seeing and playing the game.

My younger son first:

My older son next:

So, definitely a fun little game for kids. They need to be fairly fluent with the arrow keys on the keyboard, but that’s really all that’s required. Definitely some fun puzzles to solve!