# Showing that 1/40 is in the Cantor set is a great arithmetic exercise for kids

Yesterday we did a fun project on the Cantor set inspired by an amazing tweet from Zachary Schutzman:

That project is here:

Today we extended some of the ideas from that project by showing that the number 1/40 is in the Cantor set. Here’s how my son approached the problem – the idea he uses builds on the idea we talked about with the number 1/10 in yesterday’s project. I was happy to see that those ideas had stuck with him!

Now that we knew 1/40 was in the Cantor set, we talked about what other numbers of the same form must be in it. Although we don’t prove it (that’s what the paper in Schutzman’s tweet does), he’s now found all of the numbers with finite decimal expansions that are in the Cantor set

Finally, I wanted to go down a path relating these base 3 expansions to infinite series, but my son’s ideas took this last part in a slightly different direction. Which was fine and also fun. It really shows that kids can have fun exploring – and also have the capacity to have some great ideas about – infinite series.

These two projects have been really fun. I think the ideas about the Cantor set are great for kids to play around with!

# A fun fact about the Cantor set and a great arithmetic exercise for kids!

Yesterday I saw an amazing tweet about the Cantor set:

The amazing paper posted by Zachary Schutzman was in response to this question posed by Jordan Ellenberg:

I thought explaining some of the ideas about the Cantor set to my younger son and then having him play around with some fractions in base 3 would make a pretty fun project. So we tried it out tonight.

First we talked a bit about the Cantor set and he shared some initial thoughts:

Next I asked him to try to compute 1/4 in base 3. I always like projects like these with kids as they sneak in a little extra practice with fractions. Here’s his work:

Finally, I asked him to compute 1/10 in base 3 using an idea I mentioned at the end of the last video. After he did that, I asked him to find a few other fractions of the form k/10 that must also be in the Cantor set.

This was definitely a fun project. The math ideas here are slightly tricky, but hopefully the work here shows that the are accessible (and interesting!) to kids.

# 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:

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 around with the PCMI books

After seeing a plug for them on twitter I bought the PCMI books. They arrived yesterday:

The first book I picked up was Moving Things Around since the shape on the cover of the book is (incredibly) the same shape we studied in a recent project.

One more look at the Hypercube

I found a neat problem in the beginning of the book that by another amazing coincidence was similar to a (totally different!) problem we looked at recently:

Revisiting Writing 1/3 in binary

We started by talking about the books and the fun shape on the cover:

Now we moved on to the problem. It goes something like this:

Consider the number 0.002002002…. in base 3. What is this number? How about in base 4,5,7, and n?

We started in base 3 and the boys had two pretty different ways to solve the problem!

Next we moved on to base 4:

Now we moved to the remaining questions of base 5, 7 and N. Unfortunately I got a phone call I had to take in the middle of this video, so I had to walk away while the solution to the “N” part was happening.

We finished up with the challenge problem -> What is 0.002002002…. in base 2?

This is a pretty neat challenge problem ðŸ™‚

Definitely a fun start to playing around with the PCMI books. Can’t wait to try out a few more problems with the boys!

# Revisiting James Tanton’s base 3/2 exercise

Several years ago we played around with James Tanton’s base 3/2 idea:

Fun with James Tanton’s base 1.5

A tweet from Tanton reminded me about his project earlier this week. I was excited to revisit it and got a double surprise when my older son told me that he actually did it in his 7th grade math class last week! It is nice – actually amazing – to see Tanton’s work showing up in my son’s math class!

An unfortunate common theme with some of our recent projects is that they aren’t going as well as I hoped they would. Still, though, this was fun and I’ll have to spend a bit more time thinking about the last bit – how to write 1/3 using base 3/2.

We started by reviewing base 2 and, in particular, how you can play around with binary using blocks.

Next we looked at base 3/2. I’m sorry that this video runs 10 min – I definitely should have broken it into 2 pieces.

Finally we accidentally walked into a black hole. I assumed that writing 1/3 in base 3/2 wouldn’t be that difficult and that an easy pattern would emerge quickly. Whoops.

Turns out that no pattern emerges quickly, and even playing around on Mathematica for a bit after we turned off the camera we couldn’t find the pattern. The discussion facilitated by the work on Mathematica was great – at least my kids learned that (i) there are multiple ways to write a number in base 3/2, and (ii) there are easy sounding project that I can’t figure out!

I hope to revisit this part after I understand it better myself. Any help in the comments would be appreciated.

I really like this project and am sad that a little bit of stumbling around by us might have obscured the beauty of Tanton’s idea. Hope we’ll be able to revisit it soon.

# Pi in base 3

There’s a new – and amazing – video out about the seemingly crazy math fact that 1 + 2 + 3 + . . . . = -1/12:

At 16:53 in the video something amazing happens – a t-shirt with $\pi$ in base 3!! I ordered it immediately ðŸ™‚

I’m happy that I did because the boys were curious about how to do the calculation. Tonight we talked about it starting with a few easier examples first:

Next we moved to Wolfram Alpha to finish up the calculations:

It was nice that we were able to get 6 digits after the decimal point so easily. Fun little project coming from an awesome shirt ðŸ™‚

# Another great piece of math to share with kids from Richard Green

Saw this really cool post from Richard Green over the weekend:

I love unsolved problems that kids can understand! In this case what really jumped off the page was that there were so many different directions to go when sharing this problem with kids. I picked the first three ideas that came to mind and used them for a fun little project with the boys this afternoon.

Sorry that this one goes a little longer than usual, but you’ll see the kids remain totally engaged (and fascinated) all the way through. So much fun!

I started by simply sharing Green’s google+ post with them:

The first project based on this unsolved problem that I thought would be interesting to kids was looking to see if they could find numbers that were written with just 0’s and 1’s in base 2 and in base 3. To a mathematician this probably doesn’t seem to be that interesting of a problem, but the kids found it to be pretty neat. They were really excited when they discovered the pattern!

The second project I thought would be interesting took about 10 minutes. The idea in this part of the project is to see if we can find a pattern in the way to convert numbers from base 2 to base 4. It took a while for the kids to see the pattern, but they were really happy when they found it. Again, the connection here probably isn’t really that surprising to mathematicians, but it is amazing to watch kids see it for the first time:

The last project that I thought the kids would find interesting was finding the probability that a number written in base n would have just 1’s and 0’s. To simplify the project we just looked at 3 digit numbers. The kids had some really great ideas here and we got to explore a couple of different ideas and patterns.

At the end of the second video in this part we returned to talking about the ideas in the original problem.

As I said at the beginning, I love sharing unsolved math problems that kids can understand. The really nice thing about the problem Richard Green shared is that there are lots of neat properties of base number arithmetic that are closely connected to this problem. Talking through some of these properties is a fun way for kids to explore math, and maybe even get a tiny little glimpse of mathematical research. Definitely a fun afternoon ðŸ™‚

# All about that base – a fun exercise from Art of Problem Solving

Last night I stumbled on a great little exercise from Art of Problem Solving’s Introduction to Number Theory book, though I’d didn’t dawn on me how neat the problem was until later in the evening.

The problem itself is pretty easy to state:Â  Convert 100 from base 10 to base 9Â  (or from base 9 to base 8, or base 8 to base 7, and etc.)

What I realized late last night is that working through this problem allows us to find fun ways to connect arithmetic, algebra, and geometry.Â Â  So we revisited the exercise this morning,Â  starting off with a quick review of the original problem:

After the introduction, I wanted to reinforce a basic problem solving idea by starting with an easier problem.Â  This problem I had in mind was showing how the number 10 in one based could be represented in other bases.Â  My son suggested looking at the number 1 first, so we did both:

After talking about the easier problems for a little bit we switched from the whiteboard to the floor to see if snap cubes could help us see some geometry.Â  We reviewed converting the number 10 in one base to one base below to make sure that we understood how the snap cubes represented the different place values.Â  At the end of the video we took a quick peek at now to represent three digit numbers with the snap cubes.

Now comes the fun!Â  We know that 100 in base 5 converts to the number 121 in base 4.Â  Can we see that relationship in a geometric way with our snap cubes?

After seeing the neat geometric connection we returned to the board to talk about the algebraic connection.Â  This felt likeÂ  a really natural way to talk about some basic algebra, and my son seemed to be comfortable with the basic algebra we discussed here:

The next thing we talked through is the question that my son wondered about in the beginning – is there a relationship between these base conversions andÂ  Pascal’s triangle?Â Â Â  We have actually done a few similar projects before.Â  See here, for example:

Pascal’s Triangle and Powers of 11

I didn’t want to go into too much depth since we were already 30 minutes in to this talk, but we did spend some time looking to see if the next line of Pascal’s triangle – 1 3 3 1 – had a relationship with converting cubes of numbers to different bases:

Having found that Pascal’s triangle did indeed seem to help us understand how to convert cubes into different bases, we went back to the snap cubes to see if there was a geometric connection here, too:

So, a fairly innocent looking question from our Art of Problem Solving book leads to a really fun project connecting arithmetic, geometry, and algebra.Â  Super fun way to spend a morning.Â  The more time we spend in this book, the more I’ve come to appreciate how a little introductory number theory can be a neat way to build up number sense.

# Starting some fun new chapters in our books today

As luck would have it I started new chapters with both boys this morning.Â  In our number theory book we’ll be looking at different bases and in the geometry book we’ll be looking at right triangles.Â Â  Both are such fun topics and I’m super excited about spending the next few weeks exploring them.

I’ve touched on both of these topics previously with each kid.Â  Not really formally, but I’ve found doing calculations in different bases to be a great way to build number sense, for example.Â  We even did an old project about fractions and decimals in binary:

Fractions and Decimals in Binary

We also did a really fun project where we made a “binary adding machine” out of Duplo blocks.Â  This is a great activity for building up the idea of place value and is really one of my favorite math activities that we’ve ever done (the subtraction machine was especially fun!):

As for the geometry, we’ve touched on the Pythagorean theorem several times, even just by accident a few days ago in a snap cube project with my younger son:

It is exciting to be covering right triangles and the Pythagorean theorem more formally now, and so fun to be sitting with my son as he walks through one of the proofs of the theorem this morning.Â  It thought Art of Problem Solving made a nice choice in the proof they selected to use as an example.Â  Tlast chapter in the book was about similar triangles, and this proof helps you move forward with new right triangle material while reviewing the old similar triangle material at the same time:

Following our talk this morning we watched Numberphile’s excellent video with Barry Mazur explaining the “blob” Pythagorean theorem, because why not learn about the Pythagorean theorem from a Harvard math professor?:

Such fun topics!Â  Going to be a great couple of weeks leading up to Thanksgiving ðŸ™‚