# Using Richard Green’s post about complexity of a number with kids

Saw a another great Google+ post from Richard Green today:

It is so great to see ideas from research mathematicians that you can use with kids. We’ve actually used a couple of Green’s posts before:

Another great piece of math to share with kids from Richard Green

Using a Richard Green Google+ post to talk about geometry with my son

Tonight I was looking for a quick little project since I’ll be on the road for work tomorrow, and talking about the complexity of a few numbers seemed like it would be a lot of fun.

(oh, and sorry for the poor lighting, we were working on the floor of the study while my wife was watching football . . . )

I started with my younger son – he caught on to the idea of complexity quickly and even formed a nice little conjecture about the complexity of prime numbers. The conjecture turned out to be false, but it helped him understand a new way to compute complexity. He even wondered about how you’d compute the complexity of big numbers.

My older son also found computing the complexity of a number to be interesting. We ended up talking for about 8 minutes, so I split the video into two pieces.

He also caught on to the idea fairly quickly and chose to compute the complexity for 10 first. After that he tried 13 and, similar to my younger son, he thought you’d need 13 1’s because 13 is prime. However, with a little more thinking he found a way with 9 1’s and then eventually 8.

In the second half of the conversation I asked him to come up with a few other things to explore. He thought it would be tough to come up with a general formula. I decided to show him one thing that might be a little surprising – if a > b, the complexity of a isn’t necessarily greater than the complexity of b.

After that he wondered what the smallest possible value of the complexity for a two digit number. He had some ideas that led him to believe the smallest complexity would be 7.

To wrap up, we guessed at the value of the complexity for 100.

So, a great post from Richard Green that isn’t too hard to use with kids. It is so fun to be able to share ideas from math research with kids. This idea from Green was especially fun because it got the boys thinking about numbers in ways that are a little different than usual. Amazing what you can do (and what is unsolved!) with just addition and multiplication with 1’s ðŸ™‚

# Another great piece of math to share with kids from Richard Green

Saw this really cool post from Richard Green over the weekend:

I love unsolved problems that kids can understand! In this case what really jumped off the page was that there were so many different directions to go when sharing this problem with kids. I picked the first three ideas that came to mind and used them for a fun little project with the boys this afternoon.

Sorry that this one goes a little longer than usual, but you’ll see the kids remain totally engaged (and fascinated) all the way through. So much fun!

I started by simply sharing Green’s google+ post with them:

The first project based on this unsolved problem that I thought would be interesting to kids was looking to see if they could find numbers that were written with just 0’s and 1’s in base 2 and in base 3. To a mathematician this probably doesn’t seem to be that interesting of a problem, but the kids found it to be pretty neat. They were really excited when they discovered the pattern!

The second project I thought would be interesting took about 10 minutes. The idea in this part of the project is to see if we can find a pattern in the way to convert numbers from base 2 to base 4. It took a while for the kids to see the pattern, but they were really happy when they found it. Again, the connection here probably isn’t really that surprising to mathematicians, but it is amazing to watch kids see it for the first time:

The last project that I thought the kids would find interesting was finding the probability that a number written in base n would have just 1’s and 0’s. To simplify the project we just looked at 3 digit numbers. The kids had some really great ideas here and we got to explore a couple of different ideas and patterns.

At the end of the second video in this part we returned to talking about the ideas in the original problem.

As I said at the beginning, I love sharing unsolved math problems that kids can understand. The really nice thing about the problem Richard Green shared is that there are lots of neat properties of base number arithmetic that are closely connected to this problem. Talking through some of these properties is a fun way for kids to explore math, and maybe even get a tiny little glimpse of mathematical research. Definitely a fun afternoon ðŸ™‚

# Using a Richard Green google+ post to talk about geometry with my son

Patrick Honner shared an amazing post from Richard Green yesterday:

The post caught my attention for a couple of different reasons. First, the result is absolutely amazing, and following Green’s summary I clicked through and skimmed the original paper. It was really cool to see all 111 tilings. Second, the mention of the Paul Monsky result about triangles in a square was fascinating. Monsky was (and still is) a professor at Brandeis when I was in graduate school and he was always incredibly generous with his time and ideas. I found the Monsky paper with a quick google search and his proof is amazing (though pretty technical and not really something that you could share with kids).

Lastly, though, Green’s post intrigued me because I’m just finishing up a section about angles with my younger son and it sure seemed as though there was a project for kids hiding somewhere in this post. I tried one project idea with my younger son this morning.

The first thing we did was look at Green’s opening paragraph – “It is easy to cut an equilateral triangle into four smaller equilateral triangles . . . ” Perfect, let’s talk about that! Right away we get to have a fun little conversation about triangles and counting.

Following my son’s idea of how to chop up an equilateral triangle into smaller triangles, I had him build the object he described out of our Zometool set and I built an example that used a (slightly) different idea. He sees a pattern in the number of triangles that goes 1, 4, 16, 64, . . . . When you include the triangle that I showed him you get a different pattern 1, 4, 9, 16 . . . . So, we get a couple of nice patterns to talk through.

We also talk briefly about the Monsky result at the end of this video.

After that brief introduction, we moved to the end of Green’s post and I had my son talk about some of the shapes he saw. It is always fun to hear ideas that kids have about math, and these tilings are so cool that I’m sure that kids will have all sorts of really fun things to say about them.

Finally, let’s talk about some angles. We used the shape that caught his attention and then tried to calculate what some of the angles in the tiling. The first angles that he noticed were the right angles, and then the octagon at the center of the tiling caught his attention – what are the angles in that octagon?

After finding the angles in the central octagon, we went looking for one last set of angles to calculate, and my son chose the angles in one of the hexagons. This calculation is a tiny bit more difficult because not all of the angles are the same. I love hearing his ideas about how to find these angles, and also his surprise that two of the angles are actually right angles!

So, a super fun geometry project based on Richard Green’s post. It isn’t that often that you can use ideas from current math research in conversations with kids, but the ideas in Green’s post were just too good to pass up. Thanks to Patrick Honner for sharing the post yesterday and thanks to Richard Green for pointing out and explaining this amazing geometry paper.