Last night I got an interesting comment on twitter in response to my Younger son suggesting that we write the numbers in a circle – a suggestion that we didn’t pursue:

At the beginning, there was a suggestion to write the numbers in a circle, then draw lines between the pairs. It is worth making a prediction about how that will look. The result may surprise you.

So, today we revisited the problem and wrote the numbers in a circle:

Next I asked them to try to find another set of numbers that would lead us to be able to pair all of the numbers together with the sum of each pair being a square. The discussion here was fascinating and they eventually found

This problem definitely made for a fun weekend. Thanks to Michael Pershan for sharing the problem originally and to Rod Bogart for encouraging us to look at the problem again using my younger son’s idea.

Yesterday I returned from a trip and the boys returned from camp, so we were together again for the first time in two weeks. I also happened to see this tweet from Michael Persian:

Shared this fantastic problem with counselors today.

This problem seemed like a nice one to use to get back in to our math project routine.

Here’s the introduction to the problem and the full approach the boys used to work through it the firs time:

When they solved the problem the first time around, they started by pairing 16 and 9. I asked them to write down their original pairs but to go through the problem a second time without starting with 16 and 9 and see if the choices really were forced. Here’s how that went:

This is a really nice problem for kids. It is easy to understand, so kids can jump right into it. There’s also lots of different ways to approach it. Definitely a fun way to get back into our math projects.

My older son is working thorugh the Integrated CME Project Mathematics III book this summer. Last week he came across a pretty interesting problem in the first chapter of the book.

That chapter is about polynomials and the question was to find a polynomial with integer coefficients having a root of . The follow up to that question was to find a polynomial with integer coefficients having a root of .

His original solution to the problem as actually terrific. His first thought was to guess that the solution would be a quadratic with second root . That didn’t work but it gave him some new ideas and he found his way to the solution.

Following his solution, we talked about several different ways to solve the problem. Earlier this week we revisited the problem – I wanted to make sure the ideas hadn’t slipped out of his mind.

Here’s how he approached the first part:

Here’s the second part:

Finally, we went to Mathematica to check that the polynomials that he found do, indeed, have the correct numbers as roots.

I like this problem a lot. It is a great way for kids learning algebra to see polynomials in a slightly different light. They also learn that solutions with square roots are not automatically associated with quadratics!

Mathologer recently published a terrific video about the Golden Ratio and Infinite descent:

As usual, this video is absolutely terrific and I was excited to share it with the boys. Here are their reactions after seeing the video this morning:

My younger son thought the discussion about the Golden Spiral was interesting, so we spent the first part of the project today talking about golden rectangles, the golden ratio, and the golden spiral:

My older son was interested in ideas about irrational numbers and why the spirals were infinitely long for irrational numbers. We explored that idea for using a rectangle with aspect ration of .

Unfortunately I did a terrible job explaining the ideas here. Luckily we were reviewing ideas from Mathologer’s video rather than seeing these ideas for the first time. I’ll definitely have to revisit these ideas with the boys later.

We are slowly working through this amazing number theory book:

Tonight my older son was out at a viola lesson, so I was looking for a project on the Euclidean Algorithm to do with my younger son. I decided to show him how the Euclidean Algorithm is connected to geometry and to continued fractions.

First, though, we reviewed the Euclidean Algorithm:

Next we looked at a geometric version of the arithmetic problem that we just did:

Finally, we looked at a connection with continued fractions

Exploring the Euclidean Algorithm is such a great topic for kids. There are so many interesting connections and so many interesting math ideas that are accessible to kids. Can’t wait to explore more with this new book!

Currently we are looking at the second on the Euclidean Algorithm, and last night I had a chance to talk through some of the ideas with my older son.

Here are his initial thoughts on the Euclidean Algorithm after reading through a few pages of chapter 1. We worked through the example of finding the greatest common divisor of 85 and 133:

Next we moved on to trying to solve the Diophantine equation 133*x + 85*y = 1. We had already looked at this equation on Mathematica, but had not discussed how to use the ideas from the Euclidean algorithm to solve it.

In this video you’ll see how my son begins to think through some of the ideas about how the Euclidean algorithm helps you solve this equation.

By the end of the last video my son had found some ideas that would help him solve the equation 133*x + 85*y = 1. In this video we finish up the computation and (luckily!) find a solution that was different than then one Mathematica found.

Comparing those two solutions helps to show why there are infinitely many solutions.

I’m on the road today, but hope to be able to talk through some of the ideas from the Euclidean Algorithm with my younger son tonight. The topic is a great one for kids – there are lots of neat math ideas to think about (and to review!). Hopefully we’ll get to explore some of the connections from geometry, too.

What is a Number, that a Mind may know it? (And do minds fully know numbers after all?) My latest Mathematical Enchantments essay is "Who Knows Two?" https://t.co/IAdip3bAQQpic.twitter.com/b6pkny1ccP

Today we did a second project for kids based on some ideas from Propp’s post. The topic today was “primitive roots”. Unfortunately this isn’t a topic that I know well and I messed up one explanation in the first video below. Oh well . . . still a really neat idea to share with kids.

So, I started by introducing the concept of primitive roots by reminding them of the 8 card and 52 card shuffles we looked at yesterday (pay no attention to my explanation about powers and mods at the end. It will become clear in the next video that I goofed up that explanation . . . . ):

Now we looked at some examples of primitive roots with small numbers. These simple examples give a nice way for kids to get a little bit of arithmetic practice and also help them see the main ideas in the problem that we are studying.

After working through these smaller examples, we moved to the computer to continue studying the problem. My older son noticed that the examples that seemed take the longest time to work were primes, but not all primes took a long time. That’s exactly the math idea we are looking at here.

Next we made a small change to the program to study all of the odd numbers up to 1,000 all at once. After correcting a little bug we found that the numbers we were looking for were indeed all primes.

We wrapped up be talking about what was known and what wasn’t known about these primitive roots. I was happy that my older son seemed to be particularly interested in this problem.

Definitely a fun project. It is always fun to find unsolved problems that are accessible to kids (and lots of them seem to come from number theory!). We will definitely have to do some follow up projects to explore the ideas here in a bit more detail.

Here is a dartboard. You have an unlimited number of darts to throw. Every dart you throw will hit the board. What is the highest score you cannot get under 100? #MTBoSpic.twitter.com/J3OWZHVQW2

We had a hard time finding the volume of the pyramid and tetrahedron by filling them with water because, despite our best efforts with tape, our shapes were not even close to water tight. They were definitely “popcorn tight” though, so we *had* to try out this activity.

Kathy was nice enough to share the handout she used, so designing today’s project was a piece of cake:

I don't know if it helps, but here you go. Originally we did these with 3×5 cards but feel like the bigger cards are easier to work with. https://t.co/51QB3ACRbS

So, I had the boys make the shape’s prior to filming – we started the project with a quick discussion of the construction of the shapes. Then we talked about their volume.

My older son thought the volumes would be roughly the same. My younger son thought the one with the rectangular base would have the largest volume.

Next we tried to calculate the area of the base of each prism. Rather than using graph paper as the handout suggested, we found the area of each base by measuring. That gave us a chance for a little arithmetic and geometry practice, too.

Next we went to the kitchen scale to measure the change in weight when we filled the shapes with popcorn kernels. We found *very roughly* the relationship we were expecting, which was nice!

Finally, we revisited the pyramid and the tetrahedron project and looked at the two different volumes using popcorn. We found the ratio of the volumes was roughly 1.96 rather than the 1.7 to 1.8 ratio we found using water.

This is such a great project and I’m super happy that Kathy Henderson shared it yesterday. Working through the project you get to play with ideas from arithmetic and geometry. With a larger group you probably also get to discuss why everyone (presumably) found slightly different volumes.

So, a fun project that was relatively easy to implement. What a great start to the weekend ðŸ™‚

For Pi day today we explored the amazing near integer

I started by showing the boys the numbers as well as just how close it was to being an integer. I measured the closeness both in terms of the decimal expansion and in terms of the continued fraction expansion of the number:

Next I asked the boys to each take a turn finding another number relating to that was either nearly an integer or nearly a rational number. It turned out – especially with my younger son – to be a really nice way to discuss properties of powers of numbers.

The number my younger son found was

The number my older son found was

So – obviously just for fun – but still a neat way to talk about numbers and continued fractions. And a pretty fun number at the start, too ðŸ™‚