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 and then continues with depending on the greatest common divisor of and . 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!

Today I was looking for a project with my son and flipped open to the chapter on quadratic reciprocity. It had a few introductory ideas that I thought would be fun to share with my younger son.

We first looked at Wilson’s Theorem:

After Wilson’s theorem, we moved on to talking about perfect squares mod a prime. After a fairly long discussion here my son noticed that half of the non-zero number mod a prime are perfect squares:

Finally, I asked him to make a mod 11 multiplication table and we talked through some of the patterns in the table – including that the non-zero numbers had multiplicative inverses:

It was a really fun discussion today. I know next to nothing about number theory, but I really would like to use Weissman’s book more to explore some advanced ideas with the boys.

For today’s project I tried a little challenge – I had my younger son read the first chapter in the wonderful book Count like and Egyptian and then we talked about what he learned. Here’s the book and and link to Evelyn Lamb’s review of it:

The goal for today was more for him to talk through what he learned as opposed to getting the math details right. We definitely had a few stumbles, but it was still fun and the multiplication and division ideas are really neat.

Here’s the introduction to the book and the arithmetic ideas:

Next I asked my son to talk through a few multiplication problems:

I saw a really neat tweet from James Tanton yesterday:

I've started a new project on quadratics – simple videos of me chatting in my home and matching PDF essays (perhaps better). The first three are up, all about the story of COMPLETING THE SQUARE. What do folk think? https://t.co/NxsCdZnMRR@cheesemonkeysf @mythemathics @k8nowak

The video in the “setting the scene” link is terrific and I had my younger son watch it this morning. I can’t figure out how to get the video to embed, so here’s a direct link to the page it is on:

One reason that I was extra excited to see Tanton’s video is that by total coincidence I’d used essentially the same idea with my older son to explain why a negative times a negative is a positive:

But I’d never gone through the same ideas with my younger son – at least not that I can remember. So, for our Family Math project today I had him watch Tanton’s video and then we talked about it.

Here’s what he thought about the video and area models in general – I was really happy to hear that my son liked Tanton’s area models and thought they were a really great way to think about multiplication and quadratics:

Next I had my son walk through the negative / positive area ideas that Tanton used to talk about multiplication. He did a really nice job replicating Tanton’s process. I think this is a great way for kids to think about multiplication:

These problems from yesterday’s math contest looked like they would make a fun project, so had the boys work through the first 6 this morning.

Here’s problem #1 – this problem lets kids get in some nice arithmetic practice:

Here’s problem #2 – the challenge here is to turn a repeating decimal into a fraction:

Here’s problem #3 – this is a “last digit” problem and provides a nice opportunity to review some introductory ideas in number theory. The boys were a bit rusty on this topic, but did manage to work through the problem to the end:

Problem #4 is a neat problem about sums, so some good arithmetic practice and also a nice opportunity to remember some basic ideas about sums:

Next up is the classic math contest problem about finding the number of zeros at the end of a large factorial. My older son knew how to solve this problem quickly, so I let my younger son puzzle through it. The ideas in this problem are really nice introductory ideas about prime numbers:

The last problem gave the boys some trouble. BUT, by happy coincidence I’m about to start covering partial fractions with my older son, so the timing for this problem was lucky. It was interesting to see the approach they took initially. When they were stuck I had the spend some time thinking about what was making the problem difficult for them.

Yesterday my younger son and I did a fun project on continued fractions and the quadratic formula. He really seemed to enjoy it so we stayed on the same topic today.

I started by asking him to recall the relationship that we talked about yesterday and then to make up his own (repeating) continued fraction:

After he’d chosen the continued fraction to study, we looked at the first few approximations to get a feel for what the

Finally he used the quadratic formula to solve for the value of the new continued fraction – it turned out to be !

My son had a 1/2 day of school today due to the snow storm. Instead of having him work through problems form Art of Problem Solving’s Algebra book, I thought it would be fun to do a quadratic formula project since we had more time than usual.

I was a little brain dead as the spelling in the project will show, but still this was a nice project showing a neat application of the quadratic formula.

We started by looking at a common continued fraction and seeing how the Fibonacci numbers emerged:

Next we tried to see how to find the exact value of this continued fraction – here is where the quadratic formula made a surprise appearance:

Finally, we tried to decide which of the two roots of our quadratic equation were likely to be the value of the continued fraction. We had a slight detour here when my son thought that was less than 1, but we got back on track after that:

It was definitely fun to show my son how the quadratic formula can appear outside of the problems in his textbook ðŸ™‚

Flipping through it last night I ran into an easy to state theorem that seemed like something the boys would enjoy exploring:

Wilson’s Theorem -> If is a prime number, then

The proof is a bit advanced for the kids, but I thought it would still be fun to play around with the ideas. There is an absolutely delightful proof involving Fermat’s Little Theorem on the Wikipedia page about Wilson’s Theorem if you are familiar with polynomials and modular arithmetic:

Here’s how I introduced the theorem. We work through a few examples and the boys have some nice initial thoughts. One happy breakthrough that the boys made here is that they were able to see two ideas:

(i) For a prime , is never a multiple of , and

(ii) There were ways to pair the numbers in (at least in simple cases) to see that the product was

Next we tried to extend the pairing idea the boys had found in the first part of the project. Extending that idea was initially pretty difficult for my younger son, but by the end of this video we’d found how to do it for the 7! case:

Now we moved on to study 11! and 13! At this point the boys were beginning to be able to find the pairings fairly quickly.

To wrap up this morning, I asked the boys for ideas on why we always seemed to be able to pair the numbers in a way that made Wilson’s theorem work. They had some really nice ideas:

(i) For odd primes, we always have an even number of integers – so pairing is possible,

(ii) We always have a product 1 * (p-1) which gives us a -1.

Then we chatted a bit more about why you could always find the other pairs to produce products that were 1. The main focus of our conversation here was why it wouldn’t work for non-primes:

Definitely a fun project! There’s some great arithmetic practice for the boys and also a many opportunities to explore and experience fun introductory ideas about number theory.

Section 9.2 of Art of Problem Solving’s Introduction to Algebra is one of my favorite sections in any book that my kids have gone through. The section has the simple title – “Which is Greater?”

One question from that section that was giving my younger son some trouble today was this one:

Which is greater or

I decided our conversation about the problem would make a great Family Math talk, so we dove in – his first few strategies to try to solve the problem resulted in dead ends, unfortunately. By the end of the video, though, we had a strategy.

Now that we’d found that and are close together, we tried to use that idea to find out more information about the original numbers.

I found his idea of approximating at the end to be fascinating even if it wasn’t quite right. It was also interesting to me how difficult it was for him to see that the two numbers on the left hand side of the white board were each bigger than the two corresponding numbers on the right hand side of the board. It is such a natural argument for someone experienced in math, but, as always, it is nice to be reminded that arguments like that are not obvious to kids.

Today I managed to get around to discussing the problem with the boys. First I put the pieces on our whiteboard and explained the problem. Before diving into the solution, I asked them what they thought we’d need to do to solve it:

Next we move on to solving the problem. My older son had the idea of reducing the problem to 1 variable by calling the short side of one of the rectangles 1 and the long side x.

Then the boys found a nice way to solve for x. The algebra was a little confusing to my younger son, but he was able to understand it when my older son walked through it. I liked their solution a lot.

Now that we’d solved for the length of the long side, we went back and solved the original problem -> what portion of the original square is shaded. The final step is a nice exercise in algebra / arithmetic with irrational numbers.

Definitely a fun problem – thanks to Catriona Shearer for sharing it!