Sharing John Cook’s Fibonacci / Prime post with kids

Saw a neat post from John Cook about using a fun fact about the Fibonacci numbers to prove there are an infinite number of primes:

Infinite Primes via Fibonacci numbers by John Cook

Funny enough, we’ve played with the Fibonacci idea before thanks to Dave Radcliffe:

Dave Radcliffe’s Amazing Fibonacci GCD post

That project was way too long ago for the kids to remember, so today we started by just trying to understand what the Fibonacci identity means via a few examples:

Next we looked at the idea from Cook’s post that we need to understand to use the Fibonacci identity to prove that there are an infinite number of primes. The ideas are a little subtle, but I think the are accessible to kids with some short explanation:

We got hung up on one of the subtle points in the proof (that is pointed out in the first comment on Cook’s post). The idea is that we need to find a few extra prime numbers from the Fibonacci sequence since the 2nd Fibonacci number is 1. Again, this is a fairly subtle point, but I thought it was worth trying to work through it so that the boys understood the point.

Finally, we went upstairs to the computer to explore some of the results a bit more using Mathematica. Luckily Mathematica has both a Fibonacci[] function and a Prime[] function, so the computer exploration was fairly easy.

One thing that was nice here was that my older son was pretty focused on the idea that we might see different prime numbers in the Fibonacci list than we saw in the list of the first n primes. We saw quickly that his idea was, indeed, correct.

This project made me really happy ๐Ÿ™‚ If you are willing to take the Fibonacci GCD property for granted, Cook’s blog post is a great way to introduce kids to some of the basic ideas you need in mathematical proofs.

A fun project from Art Benjamin and the Museum of Math

Yesterday Art Benjamin gave a talk at the Museum of Math. One neat tweet from from the talk was this one:

It is a pretty neat problem and I thought it would make a fun project for the boys today. I didn’t show them the tweet, though, because I wanted to start by exploring the numbers with increasing digits:

 

Next we tried to figure out what was going on. My older son wanted to try to study the problem in general, but then my younger son noticed a few things that at least helped us understand why the sum should be divisible by 9.

 

For the third video we started looking at the problem in general. The computations here tripped up the boys a bit at first, but these computations are really important not just for this problem but for getting a full understanding of arithmetic in general.

 

For the last part of the project we looked at two things. First was returning to a specific example to make sure that we understood how borrowing and carrying worked. Next we applied what we learned to the slightly different way of multiplying by 9 -> multiplying by 10 first and then subtracting the number.

 

After the project I quickly explored Dave Radcliffe’s response to MoMath’s tweet:

It took a bit of thinking for the boys to see what “works in any base” meant, but they did figure it out.

I love this Benjamin’s problem – it makes a great project for kids!

Dave Radcliffe’s “unit fraction” tweet

Saw a neat tweet from Dave Radcliffe a few weeks ago:

I’d played around with it a bit on Mathematica and the code was still up on my computer screen when we were playing with base 3/2 yesterday, so the kids asked about it.

Radcliffe’s proof is a bit too difficult for kids, I think, but the general idea is still fun to explore. I stumbled through a few explanations throughout this project (forgetting to say the series should be finite, and saying “denominator” rather than “numerator” at one point), but hopefully the videos are still clear.

I started by explaining the problem and looking at a few simple examples:

Next we looked at how it could be possible for a finite sum of distinct numbers of the form 1 / (an integer) could add up to 100, or 1000, or some huge number:

Now that we understood a bit about the Harmonic series, we jumped to Mathematica. I sort of half explained / half skipped over the “greedy algorithm” procedure that Radcliffe uses in his paper. I thought seeing the results would explain the procedure a bit better.

We played around with adding up to 3 and then a couple of numbers that the boys picked.

After playing around with a sum adding up to 3, we tried 4 and the boys got a big surprise. We then tried 5 and couldn’t get to then end!

After we turned off the camera we played around with the sum going up to 5 a bit more sensibly and found that there are (from memory) 102 terms and “n” in the last 1/n term has 142,548 digits!

So, a little on the complicated side, but still a fun math fact (and computer project!) for kids to explore.

A great question from Dave Radcliffe

During a twitter conversation earlier in the week, Dave Radcliffe presented this question:

The question is a really deep and really challenging one for kids. Truthfully it is probably a little over the head of my kids, but I thought I’d give it a try anyway. I’ll revisit this one (hopefully!) several times over the course of this school year – although the question confused my kids a little bit, I really like it.

Here’s my older son’s (started 7th grade today!) thoughts on Dave’s question:

Here’s my younger son’s thoughts – he’s in 5th grade. I took a little extra time at the beginning with him to work through some examples with numbers so that the abstract symbols wouldn’t be so confusing:

Finishing Dave Radcliffe’s polynomial activity

We’ve spent the last two days talking about a math activity that I saw in a Dave Radcliffe tweet:

Those two project are here:

Dave Radcliffe’s polynomial activity day 1

Dave Radcliffe’s polynomial activity day 2

My younger son had a few ideas that we didn’t get to finish, so this morning we talked through a few of them. He was exploring (using Mathematica) (1 + x + x^2)^n in mod 3. We started by looking at Radcliffe’s picture:

 

We wrapped up today’s project by trying to find the pattern in the row numbers that had all 1’s for coefficients. This idea was a little bit of a struggle yesterday, but today we did find the pattern:

 

So, a really fun computer math project for kids. It was really fun to use Dave’s pictures to motivate the project and to help the kids explore some of the patterns that they found playing around in Mathematica.

Dave Radcliffe’s polynomial activity part 2

Last week I saw some really neat tweets from Dave Radcliffe. For example:

Those tweets led to a fun project yesterday:

Dave Radcliffe’s polynomial activity day 1

Today I had each of the boys explore (1 + x + x^2)^n mod 2 and mod 3. This is a harder exploration to do by hand (and made harder because I was out this morning and they worked on it alone). Still, it was interesting to hear what they had to day.

My younger son chose the more complicated activity of looking at the powers mod 3. Here’s what he found:

 

We then went to the computer to check if any of the patterns he thought were there would continue. He had some ideas but unluckily none of them worked. We’ll play more later to see if we can crack the code on the patterns:

 

Next I talked to my older son. He looked at powers of the polynomial (1 + x + x^2) mod 2.

Here’s what he noticed:

 

He didn’t have any conjectures, so I showed him the picture that Dave Radcliffe tweeted and that led to him seeing some additional patterns in what he’d written down on the sheet of paper:

 

So, I’m glad I saw Dave’s tweets because this project is a great computer math exercise. Exploring powers of these polynomials would have been next to impossible without the computer help, but with the computer help we were able to explore a few patterns. It’ll be fun to try to find ways to explore the patterns a bit more and see what we can find.

Dave Radcliffe’s polynomial activity day 1

Saw this really fun tweet from Dave Radcliffe yesterday:

This looked like a fun project for kids, though it wasn’t obvious how to get started. It turns out that Mathematica has a handy function called PolynomialMod[] that tells you what a polynomial looks like modulo an integer – so that made life easier!

I decided that for today’s project we’d explore (1 + x)^n using Mathematica and see what patterns we could find. The introduction to today’s project involved introducing basic polynomial multiplication. Luckily, a natural way to multiply polynomials looks a lot like multiplying 2-digit numbers. I used that connection to introduce the project:

After the introduction I had the boys play on Mathematica and compute various powers of (1 + x)^n starting with (1 + x)^0. We got a little confused between Fibonacci numbers and Pascal’s triangle, but here is what they saw:

For the last part of the project today we used PolynomialMod[] to look at the various powers of (1 + x)^n in mod 2. I wanted to get them used to this Mathematica function to make it easier to explore (1 + x + x^2)^n mod 2 tomorrow. After they explored the powers of (1 + x)^n mod 2 up to n = 8, we talked about patterns in the numbers:

So, a fun little computer math project. It was fun to hear the kids talk about the patterns and also fun to talk about some basic ideas like polynomial multiplication and modular arithmetic. Definitely excited to explore some of the more complicated patters tomorrow.

A neat problem from 5 Triangles and Dave Radcliffe

Saw a neat exchange on twitter last night:

Though my kids are far too young (4th and 6th grade) to find this solution on their own, I thought that going through this solution with them would be a useful and fun exercise. Each step in the solution is something that they can understand and Dave’s approach is also a great lesson in problem solving.

So, instead of our typical morning projects this morning we talked through this problem.

Here’s the introduction to the problem and a few initial thoughts. Right off the bat the kids have some nice thoughts about prime numbers:

The boys had some good thoughts about simplifying the problem in the last section. We looked at a few other simple examples – 3 consecutive integers in which two are prime (this led to a nice discussion about twin primes).

Next we moved to the computer to take a look at Dave Radcliffe’s idea. Luckily Mathematica has a function – PrimePi[n] – that counts primes less than or equal to n. We wrote a little program to count the number of primes occurring in a list of 2015 consecutive integers.

In this part of the project we began to use this program to explore the number of primes in various lists of 2015 consecutive integers.

At the end of the last section my younger son noticed the point that Dave Radcliffe had made last night -> as you go down the list the number of primes changes by +1, 0, or -1, but no other number.

In this part of the project we discussed why the changes were never greater than 1 and also how this property might help us solve the original question.

Finally we discussed how we could find a long list of consecutive integers with no primes. Unfortunately we were running a little longer than I expected so this part was a little more rushed than I would have liked. Still, though, they seemed to understand the idea.

So, I think walking through this problem with kids is a fantastic exercise. There are lots of interesting mathematical ideas from arithmetic and from number theory kids might find fascinating. Also, the idea that the proof shows a list of 2015 integers with exactly 15 primes exists without actually finding it is also an amazing idea (and likely one that is brand new to kids).

Finally, Dave Radcliffe’s idea to look at lists of 2015 consecutive integers to see how many primes they have is a fantastic problem solving idea. Seeing how a simple idea like Dave’s changes an almost unapproachable problem into a one that is now much easier to understand is an important example for kids to see.

Definitely a fun project.

Dave Radcliffe’s neat “linearity of expectation” tweet

Learned something from a Dave Radcliffe last week:

The link in his tweet goes here:

https://twitter.com/notch/status/675273039991345154%5Bembed%5D

I had to dig a little deeper to see what Dave meant about linearity of expectation, but when I did I found an incredible solution to the problem in just a couple of tweets:

Screen Shot 2015-12-15 at 7.04.05 PM

Prior to this series of tweets from Dave, probably the most interesting “linearity of expectation” example that I’d seen was a new-to-me proof of the Buffon Needle problem in Jordan Ellenberg’s How not to be Wrong.

Although I don’t have the exact reference in Ellenberg’s book (I have the audio book version),ย  Lior Pachter has more or less the same neat explanation on his blog. It is amazing to me that changing the needle to a circle solves the problem in a snap!

Buffon’s Needle Problem on Lior Pachter’s blog

Anyway, a couple of points about problem in Dave’s tweets.ย  First, it wouldn’t have occurred to me to use linearity of expectation to attack this problem, and that is definitely my bad.ย  Dave’s example really showed me the power of the approach.ย  Second, my intuition for an approximate answer to the question was off by miles!ย  It actually took a while for me to understand how the number could be as high as 488ย  (I mean, you’ve got a 50% chance to win after about 500,000 tries, and, really, how many different people could have won 10 times by then . . . .), but I’m glad that Dave’s tweet made me think about this problem – I definitely needed the intuition adjustment!

Pretty incredible what you can learn on Twitter ๐Ÿ™‚

An Abstract Algebra question from John Golden

Saw this question from John Golden on Twitter a few minutes ago:

My immediate idea was two prior Zometool projects that we’ve done that touch on rotation groups, but they require a Zometool set.

My two next thoughts were a bit more technical – Galois Theory and Elliptic curves. On reflection, though, I feel like both are pretty tough tasks for one class.

So my next idea related to three things I’ve seen on Twitter recently.

(1) Start by watching the first 10 minutes or so of this wonderful public lecture by Jacob Lurie from last year’s Breakthrough Prize:

In the first part of the talk he discusses rings and touches on Emmy Noether’s work on the subject in the early 1900s.

Here’s how I used this video with my kids last week (we did not explicitly dive into abstract algebra, but we did talk about clock arithmetic):

Using Jacob Lurie’s Breakthrough Prize Lecture to inspire kids

(2) Next check out this video linked by Steven Strogatz last week:

In this video you learn about a few incredible ideas related to abstract algebra. For example, when you adjoin i to the integers, you get new primes, but you still have unique factorization. However, when you adjoin the square root of 5, you lose unique factorization. These ideas are just one step removed from what Lurie touched on in his lecture.

Oh, and the punchline of this video about the square root of 163 is pretty amazing!

(sorry not TeX-ing this, I’m writing in a hurry)

So, even just stopping with the ideas in this video you’ve got some neat facts that are pretty accessible (and cool!).

(3) Finally, if you have time, take a look at this “new to me” proof that e is irrational that Dave Radcliffe tweeted about last week:

https://twitter.com/daveinstpaul/status/669205374034034688%5Bembed%5D

Essentially this proof looks at numbers of the form A + B*e where A and B are integers. This set of numbers isn’t a ring, but it is at least another example of expanding a number system. For a one day lecture it seems close enough to what’s going on in part (2) above to keep the class flowing. Plus, it is sort of fun to see this proof that e is irrational.

It is also easy to skip of the first two parts take longer than expected.

Anyway, that’s my “pondering this Twitter question for 20 minutes” idea.