Sharing Numberphile’s “Amazing Graphs” video with kids

Last week Numberphile put out a fantastic video featuring Neil Sloane:

For today’s project we explored the sequence described in the first half of the video. Namely, the sequence that begins with a_1 = 1 and then continues with a_{n+1} depending on the greatest common divisor of n and a_{n}. See either the Numberphile video or the first video below for the full formula.

To introduce the boys to the sequence, I had them calculate the first 10 or so terms by hand:

Next we wrote (off camera) a Mathematica program to calculate many terms of the sequence, and studied what the graph of those terms looked like:

Finally, I asked the boys to watch the Numberphile video and the describe what they learned. They were both able to give a nice explanation of why the sequence eventually repeated:

I love math projects that allow kids to play with really interesting math and also sneak in some k-12 math practice. The first sequence in the Numberphile video is a perfect example of this kind of project!

Advertisements

Using Numberphile’s “Pancake Number” video with kids

Numberphile put out a great video on the so-called “pancake number” – see the 3rd video below. I watched it when it came out and thought it would be a fun project for the kids to explore.

Unfortunately I got mixed up on the procedure when we were doing the first two parts of the project this morning. So, at the beginning of the project we weren’t actually studying the “pancake numbers” but rather a variant. Not a big deal, though it required an explanation after we watched the video!

We started with an introduction to the sorting problem and studying the case of trying to put 5 elements in order using (what I thought were) the pancake sorting rules.

Next I asked the kids to explore the problem in general and they looked carefully at a few simple cases. One neat thing here is that one relationship between sorting N and N+1 blocks that they thought would be true turned out not to be true.
 

Now we move on to view the Numberphile video. Katie Steckles does such a great job here (and I love that the sheet of paper from the Graham’s number video is up on the wall behind her!).

 To wrap up we returned to the white board and played around with the correct rules. It was nice that they were able to find an example with 3 blocks that had a different result when we applied the right rules.

Definitely a great video by Numberphile and a fun project for kids. It is always really fun to play with an unsolved problem that is accessible to kids.

One more trip through Goldbach’s comet with the kids

We’ve now down a couple of projects on the latest Numberphile video on the Goldbach Conjecture:

Sharing Numberphile’s Goldbach Conjecture video with kids

Exploring the Goldbach Comet with kids

Following those projects I thought it would be neat to let the computer run and find the numbers that could be written as a sum of two primes in many different ways (specifically, in more ways than any number less than it). Looking at those results produced a nice surprise:

An unexpected surprise for me in the Goldbach Comet

A double surprise was that Numberphile had just (the day before) published a follow up Goldbach Conjecture video that talks a little bit about the idea that explains the pattern I was seeing:

Last night I walked the kids through some of the ideas. We first watched the end of the Numberphile video and then talked about it briefly.

Also, I was pretty under the weather yesterday, so sorry for the low energy from me in this project:

Next we moved on to looking at the Goldbach Comet and told them about the project I was looking at while they were up in New Hampshire hiking.

They noticed the same pattern that I saw and I showed them the prime factorizations of a few of the numbers on my list.

After we talked about the factoring, I wanted to show them another surprise – the Goldbach Comet looks surprisingly symmetric around the numbers that can be written as the sum of two primes in lots of ways.

Fianlly, we wrapped up the project by looking at the symmetry I mentioned above a bit more carefully. I’d like to explore this symmetry a bit more myself!

We’ve really had a fun set of projects on the Goldbach Conjecture. It is definitely accessible to kids and a great way to show them an unsolved problem in math!

Exploring the Goldbach Comet

My wife and kids are going hiking today and I was looking for any fairly light project to do with the boys before they left. This morning I thought playing around with the Goldbach Comet would be a fun idea. We learned about it last week in Numberphile’s Goldbach Conjecture video:

Our first project from that video is here:

Sharing Numberphile’s Goldbach Conjecture video with kids

Today’s project needs a little disclaimer . . . . Sometimes when I decide to try something at the last minute things actually work out ok. Today was much more stumbling around than usual, unfortunately. But we had fun exploring anyway.

So, we started with some simple Mathematica code to explore the number of ways to write an even integer as the sum of two primes:

I gave the boys a challenge of finding the largest even number that can be written as the sum of two primes in 6 different ways. Then we played around a bit more – stumbling around aimlessly . . . .

Finally we used a program from the Wolfram Demonstrations Project to play around with the Goldbach Comet. That project we used is here:

The Goldbach Coment on the Wolfram Demonstrations Project site

I mainly used the code here to ask the kids what they thought they were seeing.

So, a fun project despite the numerous stumbles. I’d actually never heard of the Goldbach Comet prior to the Numberphile video. It was neat to play with.

Sharing Numberphile’s Goldbach Conjecture video with kids

Numberphile released a really nice video about the Goldbach Conjecture today:

I thought it would make an excellent project with the boys even though some of the ideas involving logarithms might be over their head. So, we watched the movie and then talked about some of the ideas that caught their eye.

Next we moved on to the individual ideas. The first one was the chart that David Eisenbud made at the beginning of the video. Drawing and then filling in this chart is a nice little arithmetic activity for a kid in elementary school.

Next we talked about logarithms. I started with an idea I learned from Jordan Ellenberg’s book “How Not to be Wrong” – the “flogarithm”. That idea is to oversimplify the logarithm by defining it to be the number of digits in the number. That simple (and genius) idea really opens the door to kids thinking about logarithms.

With that short introduction I explained what the natural logarithm was and moved on to some of the properties of primes that Eisenbud mentioned in the video (after fumbling with the calculator on my phone for a minute . . . .).

(Also, I noticed watching the video just now that I forgot to divide by 2 at one point – sorry about that.)

Finally, we checked a specific example – how many ways were there to take two primes and add up to 50? This part is about as far away from the complexity of logarithms as you can get – just some nice arithmetic practice for kids.

To warp up I asked them if they knew any other unsolved problems about primes. My older son mentioned something about twin primes. I showed the boys a simple argument (fortunately quite similar to the one Eisenbud gave in the movie for why there are lots of ways two primes can add to be a given even number) for why there ought to be infinitely many twin primes.

I think that kids are going to be naturally curious about primes. The Goldbach conjecture is one of the few unsolved problems that kids can understand. It was fun to share this video with the boys tonight.

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

Explaining how 1 + 2 + 3 + . . . can possibly equal -1/12 to a kid

When I did the my biographies for my kids last week my older son said that the thing in math that he’s see but that he does not believe is this equality:

1 + 2 + 3 + 4 + . . . . = -1/12

This sum was made popular by a Numberphile video a couple of years ago (which now has over 4 million views!):

 

there have also been several good follow ups. For example this video with Ed Frenkel which was also produced by Numberphile:

 

and this video by Mathologer which is absolutely excellent:

 

I spent some time today trying to think about how to discuss this series with my older son. I’m glad that he is bothered by the result – it is obviously very very strange. Obviously I can’t go into the details about the Riemann Zeta function with him, but I still think there’s some what to help him make some sense of the series. So, I spent the day reviewing some ideas in G. H. Hardy’s book “Divergent Series.” Here are a few passages that caught my eye:

(a) Book Cover

I don’t remember where I heard about this book. My best guess is that it was mentioned in Jordan Ellenberg’s “How Not to be Wrong” in the section about Grandi’s series. Unfortunately I only have the audiobook version of “How not to be Wrong” and don’t know how to search it!

(b) first passage

The remark beginning at “It is plain . . . ” caught my attention.  This is right at the beginning of the book – section 1.3.   The statement:

“it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition.”

also felt very powerful to me.

(c) second passage

The continuation of the previous page is also important – the point about Cauchy was definitely mentioned in “How not to be Wrong” as well.

(d) third passage

 

For the third passage we have to go much later in the book – nearly to the end, in fact.  The passage here – 13.10.11, in particular – shows the strange result.  Not in a Numberphile video, or some other internet video, but in a math textbook by G. H. Hardy:

(e) fourth passage

Finally – and this really is just about the last page of the book – section 13.17 provides a word of caution and an example of what can go wrong playing around with these divergent infinite series.

 

So, I’m going to spend the next few days and maybe even the next few weeks thinking about how to share some sort of idea about this strange series with my son.  I’ll welcome any suggestions!