Tag arithmetic

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.

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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 🙂

Sharing Stephen Wolfram’s MoMath talk with kids

I saw an amazing tweet from Stephen Wolfram today:

Based on the blog post, his talk at MoMath must have been incredible!

I decided to try out one of his explorations with the boys tonight. We did the first few parts by hand and the last part using Mathematica and the code from Wolfram’s blog post.

The process we studied works as follows:

(1) Pick an integer to start with and pick a number n to multiply by in step (3),

(2) Cycle the digits of the number to the left. A few examples will make the process clear:

123 goes to 231
402 goes to 024, or simply 24
111 would stay 111

(3) Multiply the new number by n and then add 1.

(4) Return to step (2) with the new number.
The video below shows how our exploration began. Our initial integer was 12 and we multiplied by 1 at each step (so, starting easy, though I picked 12 at random so I really didn’t know what was going to happen):

Now we moved to a slightly more complicated example -> the same process as in the first part but we’ll be working in binary rather than in base 10.

We started with the number 6 (110 in binary) and multiplied by 2 at each step. Once again we found a fun surprise:

To get one more round of practice in before moving upstairs to the computer we looked at the same situation as in part 2, but this time starting with 1 and looking at several cases – multiplying by 1, by 2, and by 3:

Finally, we went to the computer to explore the process in many different situations. We used code from Wolfram’s blog post to recreate the work from the MoMath talk:

What I *love* about this project is that the exploration works really well with kids on the whiteboard and on the computer. The whiteboard exploration gave us a great opportunity for a little practice with arithmetic, with binary, and with algorithms. We also saw some really fun surprises!

The computer exploration is obviously fantastic, too. I’m so grateful that Stephen Wolfram shared the ideas from his talk!

A fun project on the Arecibo Message inspired by a Holly Krieger Tweet

Saw this neat Tweet from Holly Krieger earlier today:

After reading the post I was super excited to go through it with the boys when they got home from school.

So, we read the post after dinner and then made a code out of snap cubes. Here’s what the boys thought of the post:

and here’s our secret message!

We had a lot of fun with this project. It looks like something that could be pretty fun with a group, too, so I’m thinking about using it for 4th and 5th grade Family Math night at my younger son’s school next month.

Sharing a “visual pattern” triangular number identity with kids

Saw a fun tweet last night from Matt Enlow:

Here’s the underlying tweet since it doesn’t show up in wordpress:

Shortly after seeing the tweet my younger son and I were playing Othello. The combination gave me the idea for today’s project.

We started by talking about the triangular numbers and why consecutive triangular numbers might sum up to be a perfect square. My older son’s idea of how to think about triangular numbers was computational rather than geometric.

Now we moved to the Othello board and looked at the geometry. My younger son found two different geometric ideas which was fun.

Finally, I gave the kids a challenge to try to find another geometric version of the identity. This question was a bit more challenging that I intended it to be, but we eventually got there and even saw how our new picture related to the sum formula that my older son used in the first video:

Writing 1/5 in binary

I’ve spent the last couple of days talking about binary with my younger son. We were inspired a bit by Kelsey Houston-Edwards’s latest PBS Infinite Series video on binary. It has been a fun little review.

Tonight we talked about how to write 1/5 in binary. I didn’t really know how the conversation would go, but it ended up being a nice little arithmetic review.

We started talking about the problem and he settled on the idea that we needed to find a number that would equal to 1 when we multiplied by 5. That got us going on the arithmetic review since that idea works in any base.

Now we had to figure out now to divide 1.000000000…. by 101 in binary. This long division problem gave us an opportunity to talk about subtraction (and borrowing) in binary:

The last step was multiplying the number we thought was 1/5 by 101. Once again this was a great opportunity to review some basic ideas about arithmetic and multiplication.

So, an unexpectedly fun project! We learned what 1/5 was in binary and had a nice review of subtraction, division, and multiplication along the way 🙂