# Showing that 1/40 is in the Cantor set is a great arithmetic exercise for kids

Yesterday we did a fun project on the Cantor set inspired by an amazing tweet from Zachary Schutzman:

That project is here:

Today we extended some of the ideas from that project by showing that the number 1/40 is in the Cantor set. Here’s how my son approached the problem – the idea he uses builds on the idea we talked about with the number 1/10 in yesterday’s project. I was happy to see that those ideas had stuck with him!

Now that we knew 1/40 was in the Cantor set, we talked about what other numbers of the same form must be in it. Although we don’t prove it (that’s what the paper in Schutzman’s tweet does), he’s now found all of the numbers with finite decimal expansions that are in the Cantor set

Finally, I wanted to go down a path relating these base 3 expansions to infinite series, but my son’s ideas took this last part in a slightly different direction. Which was fine and also fun. It really shows that kids can have fun exploring – and also have the capacity to have some great ideas about – infinite series.

These two projects have been really fun. I think the ideas about the Cantor set are great for kids to play around with!

# Exploring 0.99999…. = 1 using fractions and binary

Last week my younger son read chapter 2 of Jordan Ellenberg’s How not to be Wrong. In that chapter Ellenberg discusses the the nunber 0.999999…. and whether or not it equals 1.

We discussed his thoughts on that chapter here:

https://mikesmathpage.wordpress.com/2021/01/09/talking-through-chapter-2-of-jordan-ellenbergs-how-not-to-be-wrong-with-my-younger-son/

Today I thought it would be fun to approach the idea from the (slightly) different perspective of using fractions and binary.

We started with a review / refresher of how to write integers in binary since we haven’t talked about that in a while:

Then we talked about how you write fractions in binary including fun problem of writing 1/3 in binary:

Now I posed the question of how could we write 1 in binary – this part turned out to be the rare discussion that was as fun as I’d hoped it would be ðŸ™‚

Finally, having found an interesting way to write 1 in binary, we moved on to the question of how to write 1 in base 10:

This was a enjoyable project. The discussion of infinite series in How not to be Wrong is fascinating and accessible to a wide audience. Talking through the ideas in that chapter with my younger son has been really fun!

# Talking through Chapter 2 of Jordan Ellenberg’s How not to be Wrong with my younger son

For a math project this month I’m having my younger son (in 9th grade) read a chapter of Jordan Ellenberg’s How not to be Wrong each day. The book is terrific if you’ve not read it.

Yesterday my son read chapter 2 and today I asked him to pick three things that he thought were interesting or just caught his eye.

The first was the way the Greeks thought about pi:

The next thing he found interesting was Zeno’s paradox:

Finally, we talked about some of the really neat ideas about infinite series in the chapter:

I doubt that we’ll do a project on every chapter, but there are so many neat ideas in the book so I bet we’ll get to have at least 5 fun discussions. The book isn’t really aimed directly at kids, but I think most of the ideas are accessible. It’ll be fun to see what he thinks the interesting ideas are!

# Computing the area of a Koch snowflake

Yesterday my younger son and I explored the perimeter of the Koch snowflake. He was surprised that the perimeter was infinite, but he also thought that the area was finite. Today we calculated the area.

I started off by asking my son for a plan for how we should approach the problem:

In the last video we calculated the additional area that was added in each step of the process of creating the Koch snowflake. Here we look carefully at those amounts and take a guess at the pattern.

Now that we had the pattern, we added up all of the areas. I really enjoyed seeing my son’s approach to adding up the infinite series. I think this calculation is a great one to show to kids to sneak in a little arithmetic practice as well as introducing some simple ideas about sums of infinite series.

# Sharing ideas about herd immunity from Carl Bergstrom, Natalie Dean, and Tim Gowers with kids

This morning I saw a nice twitter thread about herd immunity from Tim Gowers. In that thread I learned about a NYT opinion article written by Carl Bergstrom and Natalie Dean. Here’s Gowers’s twitter thread which has a link to the article:

I thought that both the article and the twitter thread would be interesting reads for the boys this morning. We started with the article – here are a few things they found interesting:

[before diving in – our regular camera stopped working, so I filed this project with my phone. Sorry that the film quality is poor]

After talking about the article a bit, we dove into exponential growth. I think they’d understood the exponential growth ideas in the article at a high level, but going a little deeper really did help them understand the ideas about growth rates better. It was particularly interesting to hear them talk about what happens when 1 person infects 1.5 other people on average:

Next I had them read through Tim Gowers’s twitter thread (while I learned how to download videos from my phone to iMovie ðŸ™‚ ). They looked at the thread for about 10 min – here are their initial thoughts:

Finally, we took a close look at the infinite series that Gowers used in his twitter thread. My older son was already pretty familiar with infinite geometric series, but my younger son is not as used to them. Here we talked through the ideas behind the general formula for the sum. My younger son had some good ideas for how to sum the series, so this turned out to be a really worthwhile discussion:

This project was really fun. I’m glad that so many scientists and mathematicians are sharing their ideas with the public. I’m especially thankful for ideas that are presented so clearly that they can be understood by middle and high school kids.

# An introductory talk about power series with my son and a surprise (to me) misconception

I’m wrapping up sequences and series this week with my son and the final topic is Taylor Series. We’d had a few discussions here and there about power series, but it all comes together this week. Looking through some old problems form the BC calculus test, I found a nice one from 2011 that I wanted to use to introduce the idea of error terms.

I intended for the first three parts to be review, but one interesting misconception came up – so the talk was more than just review.

Here’s the introduction to the problem and my son’s work on the first part of the problem. This problem asks you to write down the usual series for $\sin(x)$ and then write down the series for $\sin(x^2)$

The next question asks you to write the series for $\cos(x)$ and then write the Taylor Series around x = 0 for the function $\cos(x) + \sin(x^2)$.

Here my son wrote the series for the 2nd function in a way that surprised me:

Once we wrote the correct series for the 2nd part of the last question, we moved on to part (c) of the problem -> find the 6th derivative of the function above evaluated at x = 0:

Finally, we looked at the last part of the problem. The question is about the error in a Taylor series approximation. I’d hoped to use this question to introduce ideas about error terms in Taylor Series, but unfortunately I completely butchered the discussion. Oh well – we’ll be covering the ideas here in a much more detailed way later this week:

# Working through a series problem from the 2012 BC calculus exam

We are about to start the section on power series, but since I haven’t blogged about our calculus work in a while I wanted to do a blog post about testing convergence of series. I chose this problem from the 2012 BC calculus exam:

Here is my son’s work on the first series:

Here’s his work on the 2nd series:

Here’s his work on the 3rd series:

This has been a fun topic to cover. I’m excited to start on power series tomorrow!

# A nice problem about primes for kids from James Tanton

Saw a really cool tweet from James Tanton today:

Tonight I sat down with the boys to make sure they understood the problem. They noticed that half the numbers would have no powers of two – good start! After that observation they started down the path to solving the problem really quickly. In fact, my younger son thought that we might have a geometric series.

Since we covered a few ideas pretty quickly in the last video, so I stared this part of the project by asking them to give me a more detailed explanation for how they got the 1/2, 1/4, 1/8, . . . pattern in the last video. It turned out to be a little harder for them to give precise arguments, but they did manage to hit the main points which was nice.

At the end of this video my older son was able to write down the series that we needed to add up to solve the problem.

Now that we had the series, we had to figure out how to add it up. My guess was that they’d never seen a series like this, but my older son had a really cool idea almost immediately – rewrite the series!

The boys were able to sum the series in this new form – so yay!

At the end of of the last video my younger son said that he was surprised that the “expected value” wasn’t zero since zero was the most likely value. In this part of the video we talked a bit more about what “expected value” meant.

Once we had that I asked what I meant to be a quick question -> is the expected number of 3’s higher or lower. It turned out to be a longer conversation than I expected, though, because my older son was actually able to write down the answer!

Definitely a fun problem. I think it is fun for kids to see how to add up a series like the one in this project. I also think it is fun for kids to explore some of the basic ideas about primes that pop up in this problem.

As an aside, one other place where I’ve seen the series that came up here is in this post from Patrick Honner:

Proof Without Words: Two Dimensional Geometric Series

His “proof without words” for the sum is this picture – can you see how it works?

# Sharing Kelsey Houston-Edwards’s Infinity video with kids

The latest PBS Infinite Series video came out this week:

This is the 4th video in an incredible series from Kelsey Houston-Edwards. Our projects on the first 3 are here:

Sharing Kelsey Houston-Edward’s [higher dimensional spheres] video with kids

Sharing Kelsey Houston-Edward’s Philosophy of Math video with kids

Sharing Kelsey Houston-Edward’s Pigeonhole Principle with kids

I had the boys watch the new video together and started today’s project by asking them what they thought was interesting.

After hearing what the kids found interesting, we dove into the idea of bijections. We talked a bit about how a bijection has to work both ways using the bus idea from the video.

After the bus example we moved on to the example of the bijection between the points in an interval and points on the real line.

We finished up by talking about the bijection between the national numbers and the positive even integers.

Since we’ve done several prior projects where infinity played some role, the next thing I asked the kids was for some thing that they already knew about infinity – both things that they thought made sense and things that they thought didn’t make sense. The discussion and examples here were amazing – “no one knows what infinity divided by infinity is” ðŸ™‚

Finally, we wrapped up the project talking about why the infinity associated with the real numbers is larger than the infinity associated with the natural numbers.

I thought this would be a fun way to end the project since it was one of the key ideas in Houston-Edwards’s video:

So, another really fun project from the new set of math videos from PBS Infinite Series. I love this new series – can’t wait for the next one!

# Talking about infinite series with kids

When we did our “math biographies” project a few weeks ago I asked the kids to tell me about something in math that they heard was true but that they do not believe is true.Â  My younger son mentioned the seemingly strange property of the Koch Snowflake having an infinite perimeter but a finite area, and my older son brought up this series:

1 + 2 + 3 + . . . .Â  = -1/12

The “math biographies” project is here:

Math Biographies for my kids

Our two projects about the Koch Snowflake are here:

Exploring the perimeter of the Koch snowflake

Talking through the area of the Koch Snowflake with kids

And a little bit of my thoughts leading up to the project today are here (including the 4 screen shots of passages in G. H. Hardy’s book “Divergent Series”):

Explaining how 1 + 2 + 3 + . . . can possibly equal -1/12 to a kid

I think that explaining to kids how the series 1 + 2 + 3 + . . . can possibly equal -1/12 is practically impossible. You might actually be able to explain the idea to high school calculus students – and it would be fun to try – but, anyway, today’s project doesn’t quite go all the way to an explanation.

Instead what I tried to do was show some other strange results with infinite series that would seem to not have a sum and then walked the boys through some of the philosophical ideas in Hardy’s book.

We started with two infinite series that they were already familiar with:

(1) 1 + 1/2 + 1/4 + 1/8 + . . . = 2, and

(2) 0.99999999….. = 1.

Listening to what both kids have to say about these two series, I’m actually pretty happy with how they understand them.

For the second part of the project I extended the idea in the “proof” that the boys used to show that 0.99999…. = 1 and applied it to the series

9 + 90 + 900 + 9000 + . . . .

They were surprised to see that using exactly the same idea that they had just used, this series “sums” to -1.

I loved that my younger son immediately recognized that this strange number:

….99999.999999…. has to equal zero!

Now me moved on to looking at geometric series and finding a close formula for the series:

$1 + x + x^2 + x^3 + \ldots$

Looking at the formula we found, we see (possibly surprising) explanation for the results that we’ve seen previously in this project.

I love the questions that the kids had during this part of the project.

Finally we looked at some of the passages in G. H. Hardy’s “Divergent Series” book. I wanted to go through these passages so that (i) the boys could see how a professional mathematician like Hardy thinks about these seemingly strange results, and (ii) more simply, so that the boys could see these strange results in writing.

Some of the ideas are probably a little too philosophical for the boys to understand, but I thought it would still be important for them to see them.

So, that’s my best effort ðŸ™‚ Maybe several years from now when my older son is learning Calculus, we’ll revisit the topic. It was fun to think about how to share these ideas with kids, and especially nice to hear my younger son say at the end of the last video that this project was “pretty fun.”