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!

A fun fact about the Cantor set and a great arithmetic exercise for kids!

Yesterday I saw an amazing tweet about the Cantor set:

The amazing paper posted by Zachary Schutzman was in response to this question posed by Jordan Ellenberg:

I thought explaining some of the ideas about the Cantor set to my younger son and then having him play around with some fractions in base 3 would make a pretty fun project. So we tried it out tonight.

First we talked a bit about the Cantor set and he shared some initial thoughts:

Next I asked him to try to compute 1/4 in base 3. I always like projects like these with kids as they sneak in a little extra practice with fractions. Here’s his work:

Finally, I asked him to compute 1/10 in base 3 using an idea I mentioned at the end of the last video. After he did that, I asked him to find a few other fractions of the form k/10 that must also be in the Cantor set.

This was definitely a fun project. The math ideas here are slightly tricky, but hopefully the work here shows that the are accessible (and interesting!) to kids.

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!

Calculating the perimeter of the Koch snowflake is a great arithmetic exercise for kids

This morning I asked my son to flip through Martin Gardner’s The Colossal Book of Mathematics and pick out a chapter he thought would be interesting to talk through. One of the chapters had a discussion of the Koch snowflake that caught his eye.

Here’s that chapter and why he thought it was interesting:

For today we decided to explore the perimeter to see if we could figure out why it was infinite. For starters we tried to calculate the perimeter of the first four iterations. He had a little trouble with the 4th, but I think that trouble shows why this is such a great arithmetic exercise for kids:

Next we went back to look more carefully at the 4th step to make sure that we had the right number. With this review we found the correct perimeter.

For the last step we found the pattern for how the perimeter changed at each step. This was, unfortunately, slightly rushed as we were about to run out of memory in the camera. But still, I thought my son gave a nice explanation of why the perimeter eventually went to infinity.

Having the kids talk through a neat problem shared by Tim Gowers

Last week Tim Gowers shared a great math problem on Twitter – here’s my retweet of it (again to help avoid the temptation to get hints in the original thread:

If you’ve not seen the problem before I’d definitely suggest spending some time thinking about it – it is really a terrific problem. The videos below give the solution, so fair warnng . . .

I’d talked about it a bit with my younger son on a car ride back and for to his (outdoor) karate class earlier this week. My older son hadn’t seen the problem until this morning. A discovery that my younger son had made in the car earlier in the week helped the boys solve the problem today, but even with that prior discovery the discussion was still really great.

Here’s how I introduced the problem – you’ll see that some of the elements in the statement of the problem that are pretty standard for mathematicians are a little confusing to the boys. This introduction clears up a bit of the confusion:

With the definition of a “repetitive” number now clear, we checked if 1/7th was a “repetitive” number – the boys were pretty sure that it was, though explaining exactly why that was true in a 100% precise way was a little challenging:

Now my younger son gave his explanation for why he thought $\pi$ was repetitive:

At the end of the last video the boys were starting to think that all numbers were repetitive. In this last video they finished the solution to the problem:

I really like this problem and think it is a great way for kids to have a fun – and non-computational – mathematical exploration. As the videos show, some of the ideas can be a little difficult for kids to make precise, but I think that’s just another nice reason to explore this problem with them!

Writing fractions in binary

I saw an interesting tweet last wee that got me thinking about fractions:

Today as sort of a unusual way to play around with fractions I thought it would be fun to try to write some fractions in binary. It has been a while since we talked about binary, though, so I had my son tell me what he knew about binary first:

Next we moved on to writing fractions in binary – we started with some simple cases:

Finally, we tried to write 1/3 in binary. This video shows what a kid thinking through a math problem can look like, and also shows why I thought this exercise would be a nice fraction review:

Sharing a new result about the Cantor set with kids

Earlier in the week I saw a tweet announcing a new (and really cool!) result about the Cantor set:

The new result is that any number in the intrrval [0,1] can be written as the product $x^2 * y$ where $x$ and $y$ are members of the Cantor set.

After reading the paper, I thought that it would be really fun to try to share some of the ideas with kids. The two ideas I wanted to highlight in the project today were (i) the geometric ideas in the construction of the Cantor set, and (ii) the interpretation of the Cantor set in base 3.

I started with a question about base 3 -> how do you write 1/2 in base 3?

Now we looked at constructing the Cantor set by removing intervals. The boys had lots of interesting ideas about what was going on

Next we looked at the incredible property that you can make any number in the interval [0,2] by adding two numbers in the Cantor set. This ideas here were a little harder for my younger son to understand than I was expecting, so I ended up breaking the discussion into two parts.

I think the ideas here are fun for kids to think through – how do I pick a number from one set and a second number (possibly from a different set) to add up to a specific number.

Here’s part 1:

and part 2:

Finally, we took a peek at the result from the paper -> how does multiplication work? This was also a fun discussion. The ideas necessary to see why you can find three numbers from the Cantor set that multiply to any number in [0,1] are obviously way out of reach for kids. However, seeing why the multiplication problem is difficult is within reach.

It is always a real treat to find math that is interesting to mathematicians to share with kids. I think talking through some of the ideas related to this new result about the Cantor set makes for an amazing math project for kids!
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Sharing Kendra Lockman’s Desmos activity with my son

I saw this tweet from Kendra Lockman yesterday:

It looked like a fun activity to try, so we spent 20 minutes this morning going through it. It was nice to year what my son son thinking about fractions throughout the activity. The first 4 videos below show his work and the last is some quick thoughts from him on the morning:

Part 1:

Part 2:

Part 3:

Part 4:

Here’s his summary of the activity:

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.Â Â  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!

What a kid learning math can look like – fraction confusion

My younger son is currently in the review section on percents in his algebra book. Last night he chose a fairly standard problem on percents for our movie. The arithmetic with fractions tripped him up a little, though. The first video shows his struggle:

After we finished the problem I decided to propose an alternate solution just to get him a second round of fraction practice. His work here was really good, but the fraction arithmetic at the end still also gave him a tiny bit of trouble:

You never know what’s going to give a kid difficulty and it is interesting to watch them try to work through these unexpected struggles.