Tag number theory

How a kid approaches a challenging problem

We stumbled on this problem in the book my older son is studying over the summer:

A game involves flipping a fair coin up to 10 times. For each “head” you get 1 point, but if you ever get two “tails” in a row the game ends and you get no points.

(i) What is the probability of finishing the game with a positive score?

(ii) What is the expected win when you play this game?

The problem gave my son some trouble. It took a few days for us to get to working through the problem as a project, but we finally talked through it last night.

Here’s how the conversation went:

(1) First I introduced the problem and my son talked about what he knew. There is a mistake in this part of the project that carries all the way through until the end. The number of winning sequences with 5 “heads” is 6 rather than 2. Sorry for not catching this mistake live.

(2) Next we tried to tackle the part where my son was stuck. His thinking here is a great example of how a kid struggling with a tough math problem thinks.

(3) Now that we made progress on one of the tough cases, we tackled the other two:

(4) Now that we had all of the cases worked out, we moved on to trying to answer the original questions in the problem. He got a little stuck for a minute here, but was able to work through the difficulty. This part, too, is a nice example about how a kid thinks through a tough math problem.

(5) Now we wrote a little Mathematica program to check our answers. We noticed that we were slightly off and found the mistake in the 5 heads case after this video.

I really like this problem. There’s even a secret way that the Fibonacci numbers are hiding in it. I haven’t shown that solution to my son yet, though.

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Revisiting Graham’s number

We had only a short time this morning for a project. At this point I should know better than to rush things, but I don’t!

Based on some twitter conversations this week I thought it would be fun to revisit talking about Graham’s number. We’ve done several projects on Graham’s number in the past, but not in at least a few years.

To get started, I asked the boys what they remembered:

Next we talked about one of the simple properties of Graham’s number (and power towers) -> they get big really quickly!

Here we talked about why 3^3^3^3^3 is already as about as large as one of the usual “large” properties listed for Graham’s number. Namely, you couldn’t write down all the digits of this number if you put 1 digit in each Planck volume of the universe:

Next we talked through how to find the last couple of digits of Graham’s number. This part of the project is something that I thought would go quickly, but didn’t at all. Still, it is pretty amazing that you can find the last few digits even though there’s next to nothing you can say about Graham’s number.

If you search for “Graham’s Number” in my blog or in google, you’ll find some other ideas that are fun to explore with kids. I highly recommend Evelyn Lamb’s article, too:

Evelyn Lamb’s amazing article about Graham’s number

A quick look at remainders

My older son was learning about the polynomial remainder theorem yesterday and then the Theorem of the Day twitter account tweeted about the theorem:

I took it as a sign that we should review remainders. My younger son doesn’t have a lot of experience with polynomials, so I wanted the main focus of today’s project to be on remainders when dividing integers. Here’s how we got started:

Next we looked at remainders in different bases to see what was the same and what was different:

Now we looked at the relationship between divisibility rules and remainders

Two wrap up, we looked at polynomials. Obviously this part is not meant to be comprehensive as my younger son isn’t that familiar with polynomials. What I was trying to do here was just give a simple overview of the remainder theorem for polynomials, and show that it wasn’t really that different than what we’d just looked at for numbers.

It was definitely a fun surprise to see the polynomial remainder theorem show up in two totally different places yesterday. Hopefully this review of remainders today was a nice exercise for the kids and helped my older son see a connection between division with integers and division with polynomials.

Using Mathologer’s “Golden Ratio Spiral” video with kids

Mathologer recently published a terrific video about the Golden Ratio and Infinite descent:

As usual, this video is absolutely terrific and I was excited to share it with the boys. Here are their reactions after seeing the video this morning:

My younger son thought the discussion about the Golden Spiral was interesting, so we spent the first part of the project today talking about golden rectangles, the golden ratio, and the golden spiral:

My older son was interested in ideas about irrational numbers and why the spirals were infinitely long for irrational numbers. We explored that idea for using a rectangle with aspect ration of \sqrt{2}.

Unfortunately I did a terrible job explaining the ideas here. Luckily we were reviewing ideas from Mathologer’s video rather than seeing these ideas for the first time. I’ll definitely have to revisit these ideas with the boys later.

Connecting the Euclidean Algorithm with geometry and continued fractions

We are slowly working through this amazing number theory book:

Screen Shot 2018-05-18 at 12.32.33 PM

Tonight my older son was out at a viola lesson, so I was looking for a project on the Euclidean Algorithm to do with my younger son. I decided to show him how the Euclidean Algorithm is connected to geometry and to continued fractions.

First, though, we reviewed the Euclidean Algorithm:

Next we looked at a geometric version of the arithmetic problem that we just did:

Finally, we looked at a connection with continued fractions

Exploring the Euclidean Algorithm is such a great topic for kids. There are so many interesting connections and so many interesting math ideas that are accessible to kids. Can’t wait to explore more with this new book!

Project 2 from “An Illustrated Theory of Numbers” -> Playing with the Euclidean Algorithm with kids

We are spending a few weeks working through this amazing book:

Currently we are looking at the second on the Euclidean Algorithm, and last night I had a chance to talk through some of the ideas with my older son.

Here are his initial thoughts on the Euclidean Algorithm after reading through a few pages of chapter 1. We worked through the example of finding the greatest common divisor of 85 and 133:

Next we moved on to trying to solve the Diophantine equation 133*x + 85*y = 1. We had already looked at this equation on Mathematica, but had not discussed how to use the ideas from the Euclidean algorithm to solve it.

In this video you’ll see how my son begins to think through some of the ideas about how the Euclidean algorithm helps you solve this equation.

By the end of the last video my son had found some ideas that would help him solve the equation 133*x + 85*y = 1. In this video we finish up the computation and (luckily!) find a solution that was different than then one Mathematica found.

Comparing those two solutions helps to show why there are infinitely many solutions.

I’m on the road today, but hope to be able to talk through some of the ideas from the Euclidean Algorithm with my younger son tonight. The topic is a great one for kids – there are lots of neat math ideas to think about (and to review!). Hopefully we’ll get to explore some of the connections from geometry, too.

Using Mathologer’s “Triangular Squares” video with kids

Last month Mathologer published an incredible video on what he calls “Triangular Squares”:

I’ve been meaning to use this video for a project for the boys ever since I saw it. Today I finally got around to watching it with the boys.

Here are their initial thoughts after watching the video:

Now we went through some of the ideas. First I asked the boys to try to sketch Mathologer’s argument that \sqrt{3} is irrational. Then I asked what proof they would have given for that fact without seeing the video:

Next we explored the irrationality proof for \sqrt{2}:

Finally, we did a bit of exploration of the seeming paradox mentioned at the end of the video. That paradox is essentially -> the argument used to show that \sqrt{3} is irrational seems to also show that 3 times a triangular number can never be a triangular number. BUT, there are lots of examples showing that 3x a triangular number is a triangular number. What’s going on?

So, another terrific video from Mathologer. His ability to shed light on advanced math topics for the general public is incredible. I love using his videos to help my kids see amazing math ides from new and beautiful angles!

Using “An Illustrated Theory of Numbers” with kids

I got a great book in the mail yesterday:

My plan is to spend the next 5 to 6 weeks using this book with the boys. They’ve both worked really hard this year going through Art of Problem Solving’s Geometry and Precalculus books and I want to end the year on a fun (and amazing!) note.

Today we took a quick look at Chapter 0. Here’s are a few initial thoughts from the boys:

For the project, I had each of the boys pick two problems from the end of Chapter 0 to talk through.

The first problem that my older son picked was about regular polygons. This led to a really nice discussion about which regular polygons can fold up into the Platonic solids

The first problem my younger son picked was about Hasse Diagrams – here we had a nice discussion about factoring:

My older son’s second problem asked to prove this statement -> If x divides x^2 + 1 then $altex x$ must be +1 or -1.

Finally, my younger son’s second problem asked how to represent this arithmetic identity as a “spiral”: 100 = 10 + 2*9 + 2*8 + . . . . + 2*2 + 2*1.

Honestly, I can’t wait to do more from this book. The end of the book gets into a few ideas that are probably a little too deep for kids, but there’s easily 4 weeks of material that we can enjoy as the school year comes to an end!

Sharing some number theory with kids thanks to Jim Propp’s “Who knows two?” blog post

Jim Propp published a terrific essay last week:

Here’s a direct link in case the Twitter link has problems:

Who knows two? by Jim Propp

Yesterday we did a fun project about card shuffling using the ideas from Propp’s post:

Sharing a card shuffling idea from Jim Propp’s “Who knows two?” essay with kids

Today we did a second project for kids based on some ideas from Propp’s post. The topic today was “primitive roots”. Unfortunately this isn’t a topic that I know well and I messed up one explanation in the first video below. Oh well . . . still a really neat idea to share with kids.

So, I started by introducing the concept of primitive roots by reminding them of the 8 card and 52 card shuffles we looked at yesterday (pay no attention to my explanation about powers and mods at the end. It will become clear in the next video that I goofed up that explanation . . . . ):

Now we looked at some examples of primitive roots with small numbers. These simple examples give a nice way for kids to get a little bit of arithmetic practice and also help them see the main ideas in the problem that we are studying.

After working through these smaller examples, we moved to the computer to continue studying the problem. My older son noticed that the examples that seemed take the longest time to work were primes, but not all primes took a long time. That’s exactly the math idea we are looking at here.

Next we made a small change to the program to study all of the odd numbers up to 1,000 all at once. After correcting a little bug we found that the numbers we were looking for were indeed all primes.

We wrapped up be talking about what was known and what wasn’t known about these primitive roots. I was happy that my older son seemed to be particularly interested in this problem.

Definitely a fun project. It is always fun to find unsolved problems that are accessible to kids (and lots of them seem to come from number theory!). We will definitely have to do some follow up projects to explore the ideas here in a bit more detail.

Sharing a card shuffling idea from Jim Propp’s “Who knows two?” essay with kids

Jim Propp published a terrific essay last week:

Here’s a direct link in case the Twitter link has problems:

Who knows two? by Jim Propp

One of the topics covered in the essay is a special type of card shuffle called the Faro shuffle. We have done a few projects on card shuffling projects previously, so I thought the kids would be interested in learning about the Faro shuffle. Here are our prior card shuffling projects:

Card Shuffling and Shannon Entropy

Chard Shuffling and Shannon Entropy part 2

Revisiting card shuffling after seeing a talk by Persi Diaconis

I started the project by asking the kids what they knew about cards. They remembered some of the shuffling projects and then introducing the idea of the Faro shuffle.

My younger son thought he saw a connection with pi, which was a fun surprise.

We continued studying the Faro shuffle with 8 cards and looked for patterns in the card numbers and positions. The boys noticed some neat patterns and were able to predict when we’d return to the original order of cards!

Next we looked at the paths taken by individual cards. My older son thought that there might be a connection with modular arithmetic (!!!) and the boys were able to find the pattern. I’d hoped that finding the pattern here would be within their reach, so it was a really nice moment when he brought up modular arithmetic.

Finally, we wrapped up by talking about how to extend the ideas to a 52 card deck and calculated how many Faro shuffles we’d need to get back to where we started.

I think that kids will find the idea of the Faro shuffle to be fascinating. Simply exploring the number patterns is a really interesting project, and there’s lots of really interesting math connected to the idea. I’m really thankful that Jim Propp takes the time to produce these incredible essays each month. They are a fantastic (and accessible) way to explore lots of fun mathematical ideas.