A fun conversation about complex numbers from a problem that didn’t look so interesting at first

Last night my younger son was working in Art of Problem Solving’s Precalculus book and came across this problem:

Following the fun conversation we had last night, I thought it would be good to revisit the problem tonight for a project. We started by talking through the problem and reviewing the first solution that came to his mind:

Next we moved to a second solution – this one involving geometric series. Here we also talked through a solution that didn’t come up last night.

Finally, we talked through a solution involving trig. As I mentioned in the twitter thread, this was a fun one for me because on a stand alone basis, calculating Cos(36) – Cos(72) is a pretty difficult problem:

When I first saw the problem here, I have to admit that I thought it looked pretty dull. Turned out to be a terrific problem to illustrate a variety of different ways to talk about complex numbers! Definitely a fun surprise.

Talking about medians

My younger son has been studying Art of Problem Solving’s Introduction to Geometry book this year. He’s been doing most of the work on his own, but I check in every now and then.

Today he was working in the section about medians, so I thought I’d ask him what he’s learned.

Here were his initial thoughts.

In the last video he was struggling to remember how to prove that the 6 small triangles formed when you draw in the medians of a triangle all have equal area. Here I gave him a hint and he was able to finish off the proof:

We wrapped up with a short discussion about the lengths of the medians.

Overall a fun discussion. I’m a big fan of the Art of Problem Solving books and am happy that my son is enjoying working through their geometry book.

Working through a tough geometry problem with my younger son

My younger son is working through Art of Problem Solving’s Introduction to Geometry book this school year. He works for about 30 min ever day and seems to really enjoy the challenge problems. Today the last challenge problem from the chapter on similar triangles gave him some difficulty.

I could tell he was having trouble understanding the problem and I asked him what was wrong. His answer was interesting -> they didn’t give any side lengths.

The problem is a pretty good challenge problem for kids learning geometry, but I thought talking through it tonight would make a good project. So, here’s the problem and a short introduction to what was giving him trouble:

In the next part of the project we began to solve the problem. There are (I think) two critical ideas -> (i) finding all of the similar triangles, and (ii) finding the parts in the diagram which have the same length.

It takes a few minutes for my son to find all of the relationships, but he does get there. Despite being a little confused, his thought process is really nice to hear.

Now we moved to the last side to see if we could find another relationship that would simplify the equation that we are hoping will be equal to 1.

After we had that last relationship he was able to see how the expressions in the equations corresponded to various side lengths in the picture. From there he was able to see why the sum was indeed 1.

I like this problem a lot and am happy that my son wanted to struggle with it.

Starting AoPS’s Precalculus book this year

Since we stopped home schooling about 2 years ago I’ve mostly been doing projects with my kids rather than covering new content. This year I wanted to get back into content and have decided to do a slow walk through Art of Problem Solving’s Precalculus book with my older son.

Today this problem gave him a little trouble:

Find the domain and range of f(x) = 1 / (1 + \frac{1}{x})

We talked about the problem when he got home from school tonight. Here’s what he thought about the domain:

Next we talked about the range which is a more complicated problem (at least I think so anyway):

So, I’m happy that he was able walk through this explanation after school today. WE spent a while talking about it this morning and I was hoping that the ideas wouldn’t slip out of his mind during the day.

What learning math can look like – arithmetic and factorials

Once in a while the way one of my kids solves a problem catches me by surprise. In today’s case I got an example of my younger son seeing a common math contest problem for the first time. When you’ve seen a problem 1,000 times you forgot what it is like to have not seen the problem previously.

I love his solution – especially because it shows how a kid can think about a pretty complicated problem involving numbers that you can’t really write down:


Building number sense using ideas from number theory

I love using Art of Problem Solving’s “Introduction to Number Theory” book as a way to help my younger son build number sense. We went through the book together a few years ago and he’s going through it on his own now.

Yesterday he worked on a section discussing perfect, deficient, and abundant numbers. These concepts are something that a kid can understand, but also come into play in unsolved problems in number theory. For a 4th grader I think the fun is in understanding the new ideas rather than their connection to the Riemann hypothesis:

Superabundant numbers on Wikipedia

For me, though, the ideas are a sneaky way to build numbers sense.

So, is 60 perfect, abundant, or deficient?

Here’s my son explaining what those terms mean:

Here’s his answer to the question:

An interesting introduction to completing the square

My older son started a new section in Art of Problem Solving’s Introduction to Algebra book today. The chapter begins to look at quadratics in a bit more depth. Given the question my son picked for today’s movie I assume that one of the topics will be completing the square. It is always interesting to see how a kid approaches an advanced topic before really knowing much about that topic.

The question asks you to find the minimum value of x^2 + 10x - 7. Here’s his work:


After he finished his work I gave a really basic introduction to completing the square using this problem as an example:


A great complex number question from Art of Problem Solving

My son picked a great problem from Art of Problem Solving’s Introduction to Algebra to talk through today:

A lot of people on Twitter thought the problem was fun and many people commented on the difference between a geometric and an algebraic approach.

A geometric approach is beyond my son’s understanding right now, but the algebraic approach has so many great lessons going for it – many more than you think if you aren’t generally around kids working through math problems:


After we were done I talked through some of the geometry for just a little bit – I don’t think introducing the geometry solely through this problem is such a great idea, but I wanted to try anyway. Here’s how it went:


Finally, as an extra shout out to AoPS – here’s one of my favorite math videos of all time – Richard Rusczyk illustrating how powerful a geometric approach to complex numbers can be with a stunning solution to an old math contest problem:

A geometric look at x^3 – y^3

Yesterday my older son started the section about factoring sums and differences of cubes in Art of Problem Solving’s Introduction to Algebra book. After he did some problems we talked about a geometric way to understand how to factor x^3 - 1.


In the evening we looked at x^3 - y^3:


It is neat to see these algebraic identities appear geometrically. The next thing we are going to look at is a geometric way to understand how x^3 + 1 factors.

Differences of squares and cubes

My older son is working his way through Art of Problem Solving’s Algebra book and has come to a section about factoring sums and differences of cubes.  This topic is new to him and I thought we’d work through a few introductory examples with numbers before diving in today.

There are a couple of surprises, I think.  First, although x^2 - y^2 is easy to factor, x^2 + y^2 is not.  Second, when you move up to cubes it turns out that x^3 - y^3 is reasonably easy to factor after you play around for a bit, and x^3 + y^3 is, too.

I wanted to show him a bit about what was going on before he dove in this morning.

Here are our short talks:

(1) We started by talking about x^2 - 1


(2) From there we moved on to x^2 + 1 and found a lot of primes


(3) Next up was x^3 - 1 which we were able to factor with a little work.


(4) Finally, we looked at x^3 + 1 which actually did factor in a very similar way to x^3 - 1!


So, hopefully a useful introduction. I’d like to do a few more projects over the course of the week to help give some different perspectives on factoring differences of squares and cubes equations.