# Exploring the generalized binomial theorem with my younger son My younger son is reading Conway and Guy’s The Book of Numbers right now and one of the early sections in the book on the binomial theorem caught his attention. We talked about the binomial theorem for a bit and then I showed him a few examples of the generalized binomial theorem and he was really interested.

Today we talked about the ideas a bit more, starting with a reminder of what the theorem says:

Now we used the ideas from the last video to take a look at an approximation to $\sqrt{2}$.

Next we looked at an approximation that turns out to be one step more complicated – $\sqrt{3}$

Finally, we went back to Mathematica to look at how good the approximation from the last video was:

It was really fun showing my son some of these advanced ideas. I’m excited to explore these ideas with a few more approximations tomorrow.

# Continuing our continued fraction exploration in Mathematica Yesterday we played around with continued fractions and showed that the square root of 2 is irrational:

https://mikesmathpage.wordpress.com/2021/02/13/my-favorite-proof-that-the-square-root-of-2-is-irrational-continued-fractions/

Today we explored continued fractions a bit more using Mathematica. I started by showing my son the relatively simple commands at taking a closer look at the continued fraction for the square root of 2:

Now we explored a few other continued fractions for other square roots and looked for a few patterns – he did notice that there always seemed to be a repeating pattern:

Next we looked at pi and found a few, fun surprises:

Finally, we looked at e. We only had about 2 min of recording time left, so this last part was, unfortunately, a little rushed:

The last few days exploring continued fractions has been really fun – hoping to do a few more projects over February break studying them.

# My favorite proof that the square root of 2 is irrational -> continued fractions Last week my younger son and I did two fun projects studying the proof in this tweet from Lior Patcher:

This weekend I thought it would be fun to explore my favorite proof – the approach using continued fractions.

We’ve talked about continued fractions before, but probably not for a few years, so I started the project today by asking my son what he remembered about them:

Before moving on to the square root of 2, we talked about why rational numbers would always have finite continued fractions:

Now we calculated the continued fraction for the square root of 2 – it has a pretty fun surprise:

Finally, and this part was just for fun, I showed him the neat little mathematical trick for quickly calculating the convergents. We looked at the first few fractions that were good approximations to the square root of 2.

# Playing around with Ed Southall’s tweet about triangles with angles near 45 degrees I saw a neat tweet from Ed Southall when I got up this morning:

I thought trying to find other triangles with near 45 degree angles would make for a great project, so I introduced the idea to the boys and asked them how they thought we could find other triangles with this property:

My younger son went first – here we explore a triangle with side lengths 99, 100, and $100 \sqrt{2}$:

My older son noticed that Ed’s triangle was a right triangle with sides whose legs different in length by 1 unit. We were going to search for other right triangles like that (with integer sides), but he noticed that a 3-4-5 triangle had that property. So we looked to see how close the angles in that triangle were to 45 degrees:

Finally, I showed them how you could used continued fractions to find triangles with angles that are really close to 45 degrees. They were surprised that we could find a triangle that was smaller than the one we looked at in the 2nd video with angles that were much closer to 45 degrees:

This was a really fun exercise – I think it is a great way to review some basic ideas from geometry and trigonometry with kids.

# Continued Fractions and the quadratic formula day 2 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 $(3 + \sqrt{17}) / 2$!

# A fun connection between quadratic equations and continued fractions My younger son is beginning to study quadratic equations in Art of Problem Solving’s Introduction to Algebra book. So far he’s essentially only seen quadratic equations that factor over the integers. For today’s project I wanted to show him that there are simple equations with fairly complicated (compared to integers!) roots.

We started with a problem similar to ones that he’s already seen:

Next I showed him a type of equations that he’s not see before and we spent 5 min talking about his ideas of how you could solve it:

Finally, for the specific equation we were looking at, I showed him how we could use continued fractions to solve it. As a bonus he remembered the connection between the Fibonacci numbers and the golden ratio and that got us to the exact solution!

# 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.

# 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!

# A fun math surprise with a 72 degree angle. We’ve been talking a lot about 72 degree angles recently. Yesterday’s project was about a question our friend Paula Beardell Krieg asked:

Paula Beardell Krieg’s 72 degree question

In that project we learned that a right triangle with angles 72 and 18 (pictured below) Is nearly the same as a right triangle with sides of 1, 3, and $\sqrt{10}$ Today I wanted to show the boys a neat surprise that I stumbled on almost by accident. The continued fraction expansion for the cosine of the two large (~72 degree angles) are remarkable similar and lead to the “discovery” of a 3rd nearly identical triangle.

We got started by reviewing a bit about 72 degree angles:

Now we did a quick review of continued fractions and the “split, flip, and rat” method that my high school teacher, Mr. Waterman, taught me. Then we looked at the continued fraction for $1 / \sqrt{10}$:

Now we looked at the reverse process -> given a continued fraction, how do we figure out what number it represents? Solving this problem for the infinite continued fraction we have here is a challenging problem for kids. One nice thing here was that my kids knew that they could do it if the continued fraction had finite length – that made it easier to show them how to deal with the infinitely long part.

Finally, we went to the computer to see the fun surprise:

Here’s that 3rd triangle: I love the surprise that the continued fractions for the cosine of the (roughly) 72 degree angles that we were looking at are so similar. It is always really fun to be able to share neat math connections like this with kids.

# Exploring a fun number fact I heard on Wrong but Useful I was listening to the latest episode of Wrong but Useful today:

The Wrong but Useful podcast on Itunes

During the podcast the following “fun fact” came up -> $\ln(2)^5 \approx 0.16$.

I thought exploring this fact would be a fun activity for the boys and spent the next 30 min daydreaming about how to turn it into a short project. I also wanted the project to be pretty light since today was the first day of school for them. Eventually I decided to explore various expressions of the form $\ln(M)^N$ via continued fractions and see what popped up.

We started by looking at the approximation given in the podcast. During the course of the discussion we got to talk about the relationship between fractions and decimals:

Now we looked at some powers of $\ln(3)$ until the phone rang. We found a neat relationship with the 5th power. This relationship was also mentioned in the podcast.

While I was on the phone I asked the boys to explore a little bit. Here’s what they showed me when I got back.

Oh, wait – EEEk – I just noticed writing this up that we counted back incorrectly in this video. Whoops! Here’s the number we thought we were exploring -> $\ln(12)^{15}$ is very nearly equal to 850,454 + 19,118 / 28207.   The next approximation that is better is 850,454 + 33,481,089 / 49,398,529.

You can see in the pic below that the 19,118/28,207 is accurate to 12 decimal places!

Sorry for this mixup. Next they showed me one more good approximations that they found -> $\ln(8)^{18}$ is nearly an integer. After that I tried to show them one I found but we ran into a small technical problem, so no need to watch the rest of the video after we finish with $\ln(8)^{18}$.

Finally, I got the technical glitch fixed and showed them that $\ln(11)^2$ is approximately 5 3/4. The next better approximation is 5 + 1,907 / 2,543

So, a fun little number fact to study. Sorry for the bits of the project that went wrong, but hope the idea is still useful!