Tag number theory

Sharing a new result about the Cantor set with kids

Earlier in the week I saw a tweet announcing a new (and really cool!) result about the Cantor set:

The new result is that any number in the intein the weekrval [0,1] can be written as the product x^2 * y where x and y are members of the Cantor set.

After reading the paper, I thought that it would be really fun to try to share some of the ideas with kids. The two ideas I wanted to highlight in the project today were (i) the geometric ideas in the construction of the Cantor set, and (ii) the interpretation of the Cantor set in base 3.

I started with a question about base 3 -> how do you write 1/2 in base 3?

Now we looked at constructing the Cantor set by removing intervals. The boys had lots of interesting ideas about what was going on

Next we looked at the incredible property that you can make any number in the interval [0,2] by adding two numbers in the Cantor set. This ideas here were a little harder for my younger son to understand than I was expecting, so I ended up breaking the discussion into two parts.

I think the ideas here are fun for kids to think through – how do I pick a number from one set and a second number (possibly from a different set) to add up to a specific number.

Here’s part 1:

and part 2:

Finally, we took a peek at the result from the paper -> how does multiplication work? This was also a fun discussion. The ideas necessary to see why you can find three numbers from the Cantor set that multiply to any number in [0,1] are obviously way out of reach for kids. However, seeing why the multiplication problem is difficult is within reach.

It is always a real treat to find math that is interesting to mathematicians to share with kids. I think talking through some of the ideas related to this new result about the Cantor set makes for an amazing math project for kids!
 

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Sharing Nassim Taleb’s dart probability problem with kids

Last week Nassim Taleb posted a fun probability problem on Twitter:

I “live blogged” my work on this problem at the link below and eventually found the solution (though after a long detour):

Sort of live blogging a solution to a problem posed by Nassim Taleb

One of the boys had to leave early this morning for a school event, so I was looking for a quick project. With some of the work I did in Mathematica on Taleb’s problem still up on my computer screen, I decided to run through the problem with the boys. The point here wasn’t for them to figure out the solution, but rather to see a neat example of counting techniques used to solve a challenging problem.

I started by explaining the problem and asking them to take a guess at the answer. The boys also had some interesting thoughts about the probability of the balls all ending up in different boxes.

Next we went to Mathematica to walk through my approach to solving the problem. In talking through my approach these ideas from number theory and combinatorics come up:

(1) Partitions of an integer,
(2) Binomial coefficients,
(3) Complimentary counting,
(4) Permutations and combinations, and
(5) Correcting for over counting.

Here’s our quick talk through one solution to Taleb’s problem (and, again, this isn’t intended as a “discovery” exercise, rather we are just walking through my solution) :

To wrap up we returned to the idea of the balls spreading out completely -> a maximum of 1 ball per box. Both boys thought this case was pretty likely and were pretty surprised to find it was less likely than ending up with 3 or more balls in a box!

This problem is little bit on the advanced side for 8th and 6th graders to solve on their own, but they can still understand the ideas in the solution. Also, there are some fun surprises in this problem – the chance of the balls spreading out completely was much lower than they thought, for example – so I think despite being a bit advanced, it is a fun problem to share with kids.

A strange homework problem

One of my older son’s homework problems asked him to find 3 digit multiples of 7 whose digit sums were also multiples of 7. I was puzzled by this problem had it on my mind most of the day today.

I hoped that talking through it would help me understand what the math idea was behind the problem. Sadly no, but we still had a good talk.

Here’s the problem and the work my son did:

So – still quite puzzled about the problem – I decided to see if there was anything quirky that came up looking at a divisibility rule for 7 with 3 digit numbers. This gave us a nice opportunity to talk about modular arithmetic:

Finally, since I wasn’t making any progress seeing the point of the original problem, I had him talk about other divisibility rules that he knew:

So, a nice conversation, but I’m actually baffled. I’ll have to ask the author of the problem what he was trying to get at – I feel like I’m missing the point.

A second project from the Wrong but Useful podcast

Yesterday afternoon I was listening to rest of the latest (as of August 31, 2017) Wrong but Useful podcast. That podcast is here:

The Wrong but Useful podcast on Itunes

A little project we did from a “fun fact” mentioned in the first part of the podcast is here:

Exploring a fun number fact I heard on Wrong but Useful

The second half of the podcast was a really interesting discussion of math education. One thing that caught my attention was comparing math education to music education and the idea of having students do “math recitals.”

Another part that caught my attention was a problem used mainly to see the work of the students rather than the specific answers. That problem is roughly as follows:

Find two numbers that multiply to be 1,000,000 but have the property that neither is a multiple of 10.

Here’s how my younger son approached the problem – it was absolutely fascinating to me to see how he thought about it.

Here’s what my older son did. Much more in line with what I was expecting.

Fun little project – definitely check out the Wrong but Useful podcast if you like hearing about math and math education.

Exploring a fun number fact I heard on Wrong but Useful

I was listening to the latest episode of Wrong but Useful today:

The Wrong but Useful podcast on Itunes

During the podcast the following “fun fact” came up -> \ln(2)^5 \approx 0.16.

I thought exploring this fact would be a fun activity for the boys and spent the next 30 min daydreaming about how to turn it into a short project. I also wanted the project to be pretty light since today was the first day of school for them. Eventually I decided to explore various expressions of the form \ln(M)^N via continued fractions and see what popped up.

We started by looking at the approximation given in the podcast. During the course of the discussion we got to talk about the relationship between fractions and decimals:

Now we looked at some powers of \ln(3) until the phone rang. We found a neat relationship with the 5th power. This relationship was also mentioned in the podcast.

While I was on the phone I asked the boys to explore a little bit. Here’s what they showed me when I got back.

Oh, wait – EEEk – I just noticed writing this up that we counted back incorrectly in this video. Whoops! Here’s the number we thought we were exploring -> \ln(12)^{15} is very nearly equal to 850,454 + 19,118 / 28207.   The next approximation that is better is 850,454 + 33,481,089 / 49,398,529.

You can see in the pic below that the 19,118/28,207 is accurate to 12 decimal places!

Sorry for this mixup.

Continued Fraction

Next they showed me one more good approximations that they found -> \ln(8)^{18} is nearly an integer. After that I tried to show them one I found but we ran into a small technical problem, so no need to watch the rest of the video after we finish with \ln(8)^{18}.

Finally, I got the technical glitch fixed and showed them that \ln(11)^2 is approximately 5 3/4. The next better approximation is 5 + 1,907 / 2,543

So, a fun little number fact to study. Sorry for the bits of the project that went wrong, but hope the idea is still useful!

Sharing Gary Rubinstein’s Perfect Number video with my 6th grader

Saw a neat tweet from Gary Rubinstein earlier in the week:

Our first project with his videos used his “Russian Peasant” multiplication video. That project is here:

Using Gary Rubinstein’s “Russian Peasant” video with kids

Today my older son is away at camp, so I was working with my younger son alone. I asked him to pick another one of Rubinstein’s videos and he picked the one on perfect numbers.

After watching the video I sat down with him to do a project – there was enough in Rubinstein’s video to easily fill three short videos, but we did just one. It was absolutely incredible to see how much my son took out of the perfect number video. There’s a fun and totally unexpected and unplanned connection with the Russian Peasant video at the end, too:

I love it when the projects go this well 🙂

Two AMC8 problems that gave the boys a bit of trouble

I had a couple of things going on today and just asked the kids to work through an AMC 8 rather than doing a longer project. Each had one problem that gave them some trouble, so we turned those problems into a short discussion.

Here’s the first problem – this one gave my younger son some trouble – it is #21 from the 1992 AMC 8:

AMC 8 first problem

 

Here’s our discussion of the problem:

 

Here’s the 2nd problem – it is problem #24 from the 1999 AMC 8.

AMC 8 2nd problem

There’s some questionable advice from me and also some terrible camera work, but it was a nice discussion!

I like using the AMC problem to help the kids see a wide variety of accessible mathematical ideas. Despite being in a bit of a rush today, this was a fun project.

Going through an IMO problem with kids

Last week I saw this problem on the IMO and thought that the solution was accessible to kids:

 

IMO

The problem is problem #1 from the 2017 IMO, just to be clear.

My kids were away at camp during the week, but today we had a chance to talk through the problem. We started by reading it and thinking about some simple ideas for approaching it:

The boys thought we should begin by looking at what happens when you start with 2. Turns out to be a good way to get going – here’s what we found:

In the last video we landed on the idea that looking at the starting integer in mod 3 was a good idea. The case we happened to be looking at was the 2 mod 3 case and we found that there would never be any repetition in this case. Now we moved on to the 0 mod 3 case. One neat thing about this problem is that kids can see what is going on in this case even though the precise formulation of the idea is probably just out of reach:

Finally, we looked at the 1 mod 3 case. Unfortunately I got a little careless at the end and my attempt to simply the solution for kids got a little to simple. I corrected the error when I noticed the mistake while writing up the video.

The error was not being clear that when you have a perfect square that is congruent to 1 mod 3, the square root can be either 1 or 2 mod 3. The argument we go through in the video is essentially the correct argument with this clarification.

It is pretty unusual for an IMO problem to be accessible to kids. It was fun to show them that this problem that looks very complicated (and was designed to challenge some of the top math students in the world!) is actually a problem they can understand.

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 🙂

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

Exploring induction and the pentagonal numbers

Yesterday we did a fun project based on this tweet by James Tanton:

That project is here:

Exploring a neat problem from James Tanton

During the project yesterday we touched on mathematical induction and also on the pengatonal numbers. Today I wanted to revisit those ideas with slightly more depth.

We started with a quick review of yesterday’s project:

Now we looked at a mathematical induction proof. The example here is:

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

(the nearly camera ran out of batteries, that’s why this part is split into two videos)

Here’s the 2nd part of the induction proof after solving the battery problem:

To wrap up the project we went to the living room to build some shapes with our Zometool set. The Zome shapes really helped the boys make the connection between the numbers and geometry.

The boys really liked this project. In fact, my younger son spent the 30 min after we finished making the decagonal numbers 🙂