Tag arithmetic

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

Saw a neat tweet from Gary Rubinstein yesterday:

This morning I thought it would be fun to look at the “Russian Peasant” multiplication video with the boys. Here’s Rubenstein’s video:

I had the boys watch the video twice and then we talked through an example. My older son went first. He had a fun description of the process: “It is like multiplying, but you aren’t actually multiplying the numbers.”

Next my older son worked through a problem. This problem was the same as the first one but the numbers were reversed. It isn’t at all obvious that the “Russian Peasant” process is commutative when you see it for the first time, so I thought it would be nice to check one example:

Next we moved to discussing why the process produces the correct answer. My older son had a nice idea -> let’s see what happens with powers of 2.

The last video looking at multiplication with a power of 2 gave the kids a glimpse of why the algorithm worked. In this video they looked at an example not involving powers of 2 (24 x 9) and figured out the main idea of the “Russian Peasant” multiplication process:

This was a really great project with the boys. It’ll be fun to work through Rubinstein’s videos over the next few months. I’m grateful that he’s shared the entire collection of ideas.

Playing with Three Sticks

I saw this tweet from Justin Aion at the end of July and immediately ordered the game:

When I returned from a trip to Scotland with some college friends the game was on the dining room table – yes!! Today we played.

In this blog post I’ll show how the game ships and two rounds of play (and we might not be playing exactly right) to show how fun and accessible this game is for kids.

First, the unboxing. The game comes out of the box nearly ready to play.

Here’s our first round of game play. I think we misunderstood one of the rules here, but you’ll still see that the game is pretty easy to play:

Here’s the 2nd round of play. I think we understood the rules better this time, which is good. You’ll also see how this game gets kids talking about both numbers and geometry:

Finally, here’s what the boys thought about the game:

I’m really happy that I saw Justin Aion’s tweet and now have this game in our collection. It is a great game for kids!

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.

Continuing our look at continued fractions

Yesterday we did revisited continued fractions:

A short continued fraction project for kids

Today I wanted to boys to explore a bit more. The plan was to explore one basic property together and then for them to play a bit on the computer individually.

Here’s the first part -> Looking at what happens when you compute the continued fraction for a rational number:

Next I had the boys go the computer and just play around.

Here’s what my younger son found. One thing that made me very happy was that he stumbled on to the Fibonacci numbers!

Here’s what my older son found. The neat thing for me was that he decided to explore what continued fractions looked like when you looked at multiples of a specific number.

So, a fun project overall. Continued fractions, I think, are a terrific advanced math topic to share with kids.

A short continued fraction project for kids

I woke up this morning to see another great discussion between Alexander Bogomolny and Nassim Taleb. The problem that started the discussion is here:

and the mathematical point that caught my eye was the question -> which positive integers are close to being integer multiples of \pi?

One possible approach to this question uses the idea of “continued fractions.” I learned about continued fractions from my high school math teacher, Mr. Waterman, who taught them using C. D. Olds’s book.

So, today I stared off by talking about irrational numbers and reviewing a simple proof that the square root of 2 is irrational:

Next we talked about why integer multiples of irrational numbers can never be integers. This I think is an obviously step for adults, but it took the kids a second to see the idea:

Now we moved on to talk about continued fractions. I’m not trying to go into any depth here, but rather just introduce the idea. I use my high school teacher’s procedure: split, flip, and rat 🙂

We work through a simple example with \sqrt{2} and also see that the first couple of fractions we see are good approximations to \sqrt{2}.

With that background work we went up to use Mathematica to explore different aspects of continued fractions quickly. One thing we did, in particular, was use the fractions we found to find multiples of \sqrt{2} that were nearly integers.

Finally, we wrapped up by using continued fractions to find good approximations to \pi, e and a few other numbers.

Definitely a fun project, and one that makes me especially happy because of the connection to Mr. Waterman. Hopefully the boys will want to play around with this idea a bit more tomorrow.

Sierpinski Numbers

I was trying (unsuccessfully) to track down a reference on the chaos game for Edmund Harriss and ran across an unsolved problem in math that I’d never heard of before -> the Sierpinski Numbers.

Turns out that Sierpinski proved in 1960 that there are infinitely many odd positive integers k for which the number:

k * 2^n + 1

is not prime for any positive integer n.

It turns out that the smallest known Sierpinski number is 78,557, though there are 4 smaller numbers for which no primes have been found, yet. Those numbers are 21181, 22699, 24737, 55459, and 67607.

There’s lots of info on the Sierpinski numbers on Wikipedia:

Wikipedia’s page on the Sierpinski numbers

Tonight I wanted to explain a bit about the Sierpinski numbers to the boys as a way to review modular arithmetic. I also thought it would be interesting to see how they thought you could attack a problem like this one – especially in the 1960s!

So, here’s how we got started – a bit of Sierpinski review and then an introduction to the theorem mentioned above. It isn’t the easiest thing for kids to understand, so I wanted to be extra sure they understood all of the parts:

Next we talked a bit about modular arithmetic and why it wasn’t too hard to see, for example, that lots of the number we were looking at were divisible by 3. The math work here is a great introductory modular arithmetic exercise for kids.

Next we went to Mathematica to explore the modular arithmetic a bit more. Once we had the idea with 3, it was a little easier to see why there were repeating patterns with the remainders mod 5. The fun part was that the boys were able to see that one out of every 4 numbers would be divisible by 5.

Finally, we looked at the problem a slightly different way and tried to see if it was easy or hard to see if 3 (or 5 or 7 or 9) was a Sierpinski number. Would we ever see primes?

This project was really fun – it is always neat to stumble on an unsolved problem that is accessible to kids. Also, I’d really love to know how Sierpinski’s proof went – sort of amazing that it took 8 years after the proof that there were infinitely many numbers with this property to find the first one!

Sharing Kelsey Houston-Edwards’s video about Pi and e with kids

Yesterday I a new video from Kelsey Houston-Edwards that just blew me away. At this point I don’t have the words to describe how much I admire her work. What she is doing to make challenging, high level math both accessible and fun for everyone is amazing.

The new video was about this question on Math Overflow from Erin Carmody:

If I exchange Infinitely many digits of Pi and E are the two resulting numbers transendental?

Before showing the boys Houston-Edwards’s video, I wanted to see what they thought about the question. So, we just dove in:

Next, I took a great warm up idea from Houston-Edwards’s video and asked the boys if they could find *any* two irrational numbers that you could use to swap digits and produce a rational number.

Now, with that little bit of prep work, we watched the new video:

After the video we talked about what we learned. I think just tiny bit of prep work we did really helped the boys get a lot more out of the video.

One of the fun little challenge questions from the video was to show that (assuming \pi and e differ in infinitely many digits, then you will produce uncountably many different numbers by swapping different digits. I didn’t expect that the boys would be able to construct this proof, so I gave them a sketch of how I thought about it (and hopefully my idea was right . . . . )

I think that kids will find the ideas in Houston-Edward’s new video to be fascinating. It is so fun (and sadly so rare) to be able to share ideas that are genuinely interesting to professional mathematicians with kids. As always, I can’t wait for next week’s PBS Infinite series video!

Sharing Numberphile’s Collatz Conjecture video with kids

Numberphile published a beautiful video on the Collatz Conjecture today. I thought it would make for a fantastic project with the kids tonight:

We have looked at the Collatz Conjecture before, so we aren’t starting from scratch here. Two of our prior projects are here:

Revisiting the Collatz Conjecture

the Collatz Conjecture and John Conway’s Amusical Variation

I started the project tonight by asking the kids what they thought was interesting about the video:

Next we tried to recreate the “tree” that was in the video. This exercise was a nice way to check that the kids understood what was going on in Numberphile’s video:

To wrap up I wanted to walk through one example of how the Collatz conjecture plays out. Somewhat unluckily, though, my son chose 31 as the starting point. 31 takes more than 100 steps to converge!

BUT, this video shows why I think the Collatz conjecture is such a fun math idea to share with kids – you can sneak in a lot of arithmetic practice 🙂

So, we gave up after maybe 30 steps in the last video and went to check how long it would take to converge using Mathematica. Someday I’ll learn that when I zoom in too far on Mathematica the video gets super fuzzy . . . but today was not that day 😦

I’m really grateful to Numberphile for their video – I think videos like it will really help show off the beauty of math to a large audience.Collatz.jpg

A challenging but super instructive inclusion / exclusion example

My son had a really interesting problem as part of the homework for an enrichment math program he’s in. I’m writing this post from the road so I don’t have the exact statement of the problem in front of me, but it went something like this:

You are going to make 7 digit numbers using the digits 1, 2, 3, . . . , 7 exactly once. How many of these numbers have no consecutive digits with common divisors?

So, for example 1,234,567 is a perfectly fine number, but 2,413,567 doesn’t work.

My son’s solution was nice, but complicated. He found the number of ways to separate the even numbers (there are 10) and then found the ways to fill in the odd numbers in each of those cases.

I couldn’t find an easier solution and wondered on Twitter if there was one. One response I got pointed me to a similar problem that was discussed on the Art of Problem Solving problem forum:

Looking through the thread I stumbled on a really clever inclusion / exclusion solution. Since we’ve been taking a closer look at inclusion / exclusion ideas I thought it would be fun to step through this solution with the boys. I think this a really instructive inclusion / exclusion example. One thing that was a little tough for the boys to understand was that the elements we were “excluding” were pairs of integers.

Also, just to be clear, I’m not expecting the boys to have a complete understanding of this solution. Rather, I just wanted to show them an inclusion / exclusion example that had some interesting twists.

So, we started by introducing the problem because my younger son hadn’t seen it before:

Next we dove in to the inclusion / exclusion solution. The “no restrictions” case is easy! Seeing the way to express the restrictions is pretty challenging. Once we understood that case we looked at subtracting away the cases with 1 restriction.

Next we looked at the 2 restriction case. Now things get really tricky – the fact that we have now have pairs of pairs of numbers is one bit of confusion. Another bit of confusion comes because one pair of pairs is not like the others.

Finally we looked at the case with 3 restrictions. This part, I think anyway, is really cool. The surprise is that several of the cases are impossible!

Despite being a very challenging problem, I love this problem as an inclusion / exclusion example for kids. No individual piece is beyond their reach and if you walk through the problem slowly everything is accessible to them.

Extending our arithmetic / geometry connection project to calculus

Yesterday we did a fun project connecting arithmetic and geometry:

Connecting Arithmetic and Geometry

While we were talking about the shapes my older son commented that one of the shapes looked like a pyramid. I thought it would be fun to make the shapes look even more like a pyramid and see what the kids thought.

We started by just talking about the shapes – the most interesting thing to me here was how challenging it was for them to compare the volumes of the shapes:

Because they were having a little bit of difficulty with the volumes I spent a little extra time on the idea. Things seemed to clear up a little bit, luckily:

Finally, I thought it would be interesting for the boys to see some of the math I used to create these shapes. Although this section goes on a little longer than I would have liked, I think this is a fun little introduction to functions and scaling even if we don’t define those ideas explicitly:

A fun little project. I think that some of the broad ideas from calculus are within the grasp of kids even if the underlying calculations probably aren’t. It was fun for me that a question from my older son led from us jumping from arithmetic to geometry to calculus 🙂