# Triangle and construction review 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. Every now and then, though, I decided to check in and see how things are going. Today the topic he was looking at when I checked in was constructions. The two questions on the table when I stopped by were:

(i) Construct the incircle of a triangle, and

(ii) Construct the circumcircle of a triangle

Here’s how things went.

(i) The incircle

He has the basic idea for how the construction works, but misses one important idea. That idea is that the tangents of the incircle are not the feet of the angle bisectors. My guess is that this is a fairly common point of confusion for kids learning this topic:

(ii) the circumcircle

Here his solution was completely correct. He had a tiny bit of trouble in the beginning figuring out to construct the perpendicular bisectors, but he worked through that trouble fairly quickly.

It is always interesting to hear the ideas that kids have when they talk through a mathematical process. For me it was especially nice to hear that most of the ideas he’s learning as he works through this geometry book are sinking in pretty well.

# A quick look at remainders

My older son was learning about the polynomial remainder theorem yesterday and then the Theorem of the Day twitter account tweeted about the theorem:

I took it as a sign that we should review remainders. My younger son doesn’t have a lot of experience with polynomials, so I wanted the main focus of today’s project to be on remainders when dividing integers. Here’s how we got started:

Next we looked at remainders in different bases to see what was the same and what was different:

Now we looked at the relationship between divisibility rules and remainders

Two wrap up, we looked at polynomials. Obviously this part is not meant to be comprehensive as my younger son isn’t that familiar with polynomials. What I was trying to do here was just give a simple overview of the remainder theorem for polynomials, and show that it wasn’t really that different than what we’d just looked at for numbers.

It was definitely a fun surprise to see the polynomial remainder theorem show up in two totally different places yesterday. Hopefully this review of remainders today was a nice exercise for the kids and helped my older son see a connection between division with integers and division with polynomials.

# Using Mathologer’s “Golden Ratio Spiral” video with kids 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 $\sqrt{2}$.

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.

# Playing with Polynomials We’ll be doing a little bit of review work in the Integrated CME Project III book. Today my son came across an interesting problem about trying to (sort of) match two polynomials. He came up with a nice solution this morning and we talked about the problem when he got home from school today.

The problem goes like this:

Find a polynomial that agrees with $x^3 - x$ at $x = 1, 2$, and 3, and has a value of 0 at x = 4.

Here’s my son talking through his solution:

After he finished his explanation, I showed him my solution to the problem:

To wrap up we went to Mathematical to look at both solutions and also so that I could show him a little surprise:

So, a nice start to this review project. It’ll be fun to work through the book over the summer.

# Connecting the Euclidean Algorithm with geometry and continued fractions 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!

# Project 2 from “An Illustrated Theory of Numbers” -> Playing with the Euclidean Algorithm with kids We are spending a few weeks working through this amazing 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.

# Part 2 sharing Mathologer’s “triangle squares” video with kids Yesterday we did a project inspired by Mathologer’s “triangle squares” video:

Here’s the project:

Using Mathologer’s triangular squares video with kids

Today we took a closer look at one of the proofs in the Mathologer’s video -> the infinite descent proof using pentagons that $\sqrt{5}$ is irrational:

Here are some thoughts from the boys on the figure and the proof. You can see from their comments that they understand some of the ideas, but not quite all of them.

Watching Mathologer’s video, I thought that the triangle proof about the irrationality of $\sqrt{3}$ and the proof of the irrationality of $\sqrt{2}$ using squares were something kids could grasp, but thought that the pentagon proof presented here was a bit more subtle. We may have to explore this one more carefully over the summer.

After discussing the proof a bit, I switched to something that I hoped was easier to understand. Here we talk about the different pairs of numbers that create fractions close to $\sqrt{5}$.

The boys were able to explain how to manipulate the pentagon diagram to produce the fraction 38/17 from the fraction 9/4 that we started with. From there the were able to also show that 161/72 was also a good approximation to $\sqrt{5}$:

Next we went to the computer to explore the numbers, and also to see how the same numbers appear in the continued fraction for $\sqrt{5}$.

In the last video we tried to do some of the continued fraction approximations in our head, but that wasn’t such a great idea. Here we finished the project by computing some of the fractions we found in the last video by hand.

I love Mathologer’s videos. It is amazing how many ways there are to use his videos with kids. Can’t wait to explore these “triangular squares” a bit more!

# Using Mathologer’s “Triangular Squares” video with kids Last month Mathologer published an incredible video on what he calls “Triangular Squares”:

I’ve been meaning to use this video for a project for the boys ever since I saw it. Today I finally got around to watching it with the boys.

Here are their initial thoughts after watching the video:

Now we went through some of the ideas. First I asked the boys to try to sketch Mathologer’s argument that $\sqrt{3}$ is irrational. Then I asked what proof they would have given for that fact without seeing the video:

Next we explored the irrationality proof for $\sqrt{2}$:

Finally, we did a bit of exploration of the seeming paradox mentioned at the end of the video. That paradox is essentially -> the argument used to show that $\sqrt{3}$ is irrational seems to also show that 3 times a triangular number can never be a triangular number. BUT, there are lots of examples showing that 3x a triangular number is a triangular number. What’s going on?

So, another terrific video from Mathologer. His ability to shed light on advanced math topics for the general public is incredible. I love using his videos to help my kids see amazing math ides from new and beautiful angles!

# Using “An Illustrated Theory of Numbers” with kids I got a great book in the mail yesterday:

My plan is to spend the next 5 to 6 weeks using this book with the boys. They’ve both worked really hard this year going through Art of Problem Solving’s Geometry and Precalculus books and I want to end the year on a fun (and amazing!) note.

Today we took a quick look at Chapter 0. Here’s are a few initial thoughts from the boys:

For the project, I had each of the boys pick two problems from the end of Chapter 0 to talk through.

The first problem that my older son picked was about regular polygons. This led to a really nice discussion about which regular polygons can fold up into the Platonic solids

The first problem my younger son picked was about Hasse Diagrams – here we had a nice discussion about factoring:

My older son’s second problem asked to prove this statement -> If $x$ divides $x^2 + 1$ then $altex x$ must be +1 or -1.

Finally, my younger son’s second problem asked how to represent this arithmetic identity as a “spiral”: 100 = 10 + 2*9 + 2*8 + . . . . + 2*2 + 2*1.

Honestly, I can’t wait to do more from this book. The end of the book gets into a few ideas that are probably a little too deep for kids, but there’s easily 4 weeks of material that we can enjoy as the school year comes to an end!

# What a kid learning math can look like – incircles and circumcircles

This problem gave my son a lot of trouble this morning. It is from one of the challenge problem sections in Art of Problem Solving’s Introduction to Geometry book:

A triangle has side lengths of 10, 0, and 12, find the lengths of the radius of the inscribed and circumscribed circles.

This problem looks like a pretty plain vanilla math contest problem, but he’s not studying for math contests. He drew an amazing picture of the situation: After talking about the problem a bit this morning we moved on to other things. When he got home from school we revisited it. Here are his thoughts on the incircle part of the problem:

Here are his thoughts on the circumcircle part of the problem:

Finally – and unfortunately interrupted by a phone call – here’s his explanation of how the project from today relates to a 3d printing project that we’d done previously

Definitely a fun problem to talk through, and really nice to see that he was able to explain the ideas in a problem that he’d really struggled with in the morning.