We found a nice little example problem in Art of Problem Solving’s Introduction to Geometry book today.

They give you a picture of an equilateral triangle with three circles of equal size inscribed in it and then ask you to find the length of the side of the triangle given the radius of the circles.

When I saw the problem I thought it would make for a fun example of problem solving in geometry. My son’s walk through the problem turned out to also be a great example of what learning math sometimes looks like:

Today I thought it would be fun to revisit large numbers a little bit, so we looked at a few really large numbers on Wolfram Alpha and talked informally about logarithms. The spirit here is hopefully along the lines of Jordan Ellenberg’s description of logarithms in How not to be Wrong – as I was thinking about what to talk about today, his discussion of the “flog-arithem” inspired me to try this project.

First up, exploring logarithms and factorials and seeing what patterns we could find. A few Fibonacci numbers showed up in the beginning, but that pattern didn’t continue – wouldn’t it be cool if the number of digits in n! was related to the nth Fibonacci number!?! We did see the connection between the number of digits and the base 10 logarithm, though.

Next up we started looking at some large factorials and then moved on to other large numbers. We also ended up stumbling on some interesting properties of the logarithm function sort of by accident. At the end we looked at a pretty neat problem: which was larger 3^3^3^3 or 100 billion factorial?

As I was writing this up, both Dan Anderson and Burnheart123 on twitter realized there was an easy way to estimate the number of digits of 100 billion factorial – wish I would have realized their point when I was talking with the boys:

In the last video, the kids asked me about logs with bases other than 10. That led to a fun discussion about logs with a few other bases and we eventually arrived at base e. One fun surprise in this discussion is that 100 billion factorial has roughly the same number of digits in binary as 3^3^3^3 does in base 10.

The last bit of our talk was about the relationship between logs and prime numbers. This is the part specifically inspired by Jordan Ellenberg’s discussion in How not to be Wrong. Even if we can’t go into any details that he does his book, it is neat to show the kids this surprising connection.

Also, sorry here – the camera seems to have cut off in the middle of the discussion. In the part that got cut off, we checked the formula for approximating the number of twin primes.

So, a fun little discussion today piggy backing off of yesterday’s discussion about patterns on an Othello board. I’m also really happy that I can share some of Ellenberg’s discussion / ideas about logs and primes with the boys (even if that sharing is very informal). Also happy to have stumbled on the fun question about 3^3^3^3 and 100 billion factorial.

For today’s Family Math project I put an Othello board down in front of the kids and asked them to come up with their own math question. It took a little bit of thinking on their end, but each kid came up with a really interesting question:

(1) Younger son: If you placed the tiles on the board and randomly, what is the probability that all of the tiles would have the white side facing up?

(2) Older son: If all of the tiles have white facing up, and select one tile at random 32 times in a row – what is the probability that at the end of this process you’ll have exactly 32 black tiles and 32 white tiles facing up?

Here’s that conversation:

We tackled my younger son’s problem first. The main problem solving ideas here were reducing the problem to a problem that was easier to solve and then a little bit of pattern recognition.

My older son’s problem was next. We spent a few minutes making sure that we were clear about the problem, and then began looking at the problem. I asked my older son for a guess at the probability and he came up with a very thoughtful explanation of why he expected the probability to be small. We then looked carefully at his explanation.

To wrap up, we went to Wolfram Alpha to calculate the number we’d found in the last video. We also spent a couple of minutes repeating the calculation for the Birthday Paradox problem – how many people do you need to have in a room in order to have a 50% chance that two people have the same birthday?

So, a really fun morning working through the questions the boys came up with. Amazing the fun you can have just looking at an Othello board.

My older son and I are currently studying how angles and circles interact. I though that a good way for him to review some of the basic properties would be to give a short, sort of mini lecture on three of the properties.

First up, what happens when the angle is on the circle:

Next – what happens when the angle is in the circle:

Finally – what happens when the angle is outside of the circle:

Continued fractions have really caught my younger son’s eye the last couple of days. The link below (plus the link inside of that link) give two examples of our recent discussions:

Today we were playing around online with continued fraction calculators and other sites discussing continued fractions and found quite a surprise – there is a generalized continued fraction for that looks very similar continued fraction for . I used this little coincidence to review some basics of calculating continued fractions:

Next up – a short discussion of the generalized continued fraction that we found for . I think that kids will always find it surprising that there are relatively simple ways to describe even though just about everything you hear about is that it has no pattern.

Neat little morning – very fun to show a surprising way to see some structure in . Also quite a surprise to see the seemingly small change that changes into .

I’m a big fan of continued fractions – especially the many different ways that you can use them to help kids learn elementary math. Right now I’m studying square roots with my younger son and he’s taken quite a liking to continued fractions, too. See yesterday’s project, for example:

I intended for the focus of that discussion to be the standard proof of why is irrational. Instead, though, a large part of the discussion was about how you could use the continued fraction for to prove that it was irrational.

Having not learned my lesson already, I asked my son to sketch a proof of why is irrational, and he went down the continued fraction path again.

Even though this project is pretty difficult and many of the parts are really over my son’s head, I think this was a useful exercise. I also think that it all pretty much stands on its own, so I’ll present the four steps below without much comment.

Following this project, my son asked me if we could study more about continued fractions this week rather than just studying the current chapter in our book about square roots. Something about this topic has really caught his attention!

The continued fraction calculator we are using in the last video is here:

I’m having a bit of a roller coaster ride through square roots with my younger son. Sometimes things that I think will be hard about square roots are easy for him, and sometimes things I think will be easy are hard. Today was the latter case as some initial discussions about the square root of 2 led to more confusion than clarity. So I decided to scrap the overall plans for today and just talk about about . The new goal was to see why it was not rational.

So, we started off by discussion why it wasn’t an integer:

Next we tried to see if the square root of 2 was a fraction. I intended to talk about the standard proof by contradiction here, but about 30 seconds in my son remembered our continued fraction approximation for . That surprise memory led to a quick review of the first couple of convergents in the continued fraction expansion. We saw some fractions that were nearly equal to but none of them were exactly equal.

He understood that if we could write as a fraction, the continued fraction expansion would eventually stop (it may be a stretch to say that he understands this, but he at least has the intuition that this fact would be true). Since the continued fraction expansion goes on forever, there must be no rational number that is exactly equal to .

Finally, we covered what I intended to cover in the last little talk – the usual proof by contradiction that is not rational. We end the conversation by mentioning some other numbers that are not rational – some for the same reason as and some for other reasons.

So a fun and unplanned discussion about the square root of 2. Hopefully these little side discussions end up building up his number sense a little and help him gain a better understanding of square roots.

Thought it would be a fun one to go through with the kids, and it turned out to be even better than I was expecting as the kids approached the problem completely differently than I did.

I started off with my younger son. We spent the first half of this year going slowly through Art of Problem Solving’s Introduction to Number Theory book, so he has seen problems similar to this one. Another reason that I thought this would be a great problem for him is that we just began talking about square roots last week. So, well-timed Wordplay blog!

His reasoning for how to make the number 256 out of 5’s and 7’s around 4:00 in the movie made me really happy.

Next up was my older son. The problem gave him quite a bit of trouble in the beginning, and in fact he told me at one point he was stuck. We’ve learned an extremely important lesson from James Tanton about what to do when you are stuck – try something.

When he tried something, he found the path to the solution. I think his struggle through this problem was really productive.

Next up we moved to a little “broken calculator” computer program I wrote on Khan Academy’s site to illustrate the problem:

Using this calculator we talked through an approach to solving the problem that didn’t require you to find 256 first. Sorry this part is broken into two videos, I accidentally turned off the camera when I was moving it – oops (and the noise in the background is our dishwasher).

Finally, we wrapped up by quickly talking through why you could make every positive integer other than 1. I was much more interested in having the kids think through the first part of this problem rather than this part, so that’s why we didn’t go into all of the details here. Still, they were pretty surprised to see (even informally) that you could make every positive integer other than one.

So, a great problem from the Wordplay blog led to some great math conversations with the kids. The only little issue was some internet problems at our house last night that meant I had to write this one up at 5:00 am – so sorry for the likely sloppy writing.

[quick post publication note – one of my former students thought that I was a little too loose with my language with the kids in this project. He thought, quite fairly, that I needed to be clearer about the fact that it wasn’t just one trial that we were looking at, but rather looking at many rounds and the expected number of flips for HTT to appear the first time and the expected number of flips for HTH to appear the first time.

Sorry for not being clearer about that point in this post, but I’m glad to have people pointing out ways to make these posts better.]

Saw this incredibly interesting probability problem on Twitter yesterday:

If you click through to the Peter Donnelly Ted Talk video, you’ll see a discussion of the problem starting around 4:00.

I had a hard time getting my arms around Donnelly’s explanation of the answer to the problem, and ended up spending a couple of hours last night thinking of how to explain this problem to the boys. Eventually I decided a computer-based approach might be the most enlightening way since that would allow us to talk informally about the math and then see the results relatively quickly. It ended up being one of the more interesting projects that we’ve done in a while.

We started with a quick explanation of the problem followed by some neat discussion from the kids. It was really interesting to me to hear their thoughts about probability.

Next up we tried a little “coin flipping.” Except that we did some dice rolling because it is hard to see coin flips on camera. Even numbers were “heads” and odd numbers were “tails.” It took 5 rolls to get a HTH sequence and it took only 3 rolls to get HTT. The boys understood that one time looking for each sequence wasn’t enough to tell us what the expected value was, and suggested instead that we’d need an infinite number of trials – ha ha.

Because of the random luck of getting HTT in 3 rolls, I decided to take a little diversion to talk about the probability of that happening. Although this diversion made the project a little longer, I’m glad that we took a closer look at this situation since computing the probability of getting HTT in 3 rolls was a little bit confusion to the boys. The mathematical idea we are talking about here is independent events.

Next we switched over to a little computer program I wrote last night. I know that some people have strong feelings about Khan Academy, but their computer program interface makes it easy to share programs like this one – feel free to play around with it:

The first part of the video below is just me explaining the computer program. After a couple of minutes we begin using the program to analyze the coin flipping game. We start by looking at how often we get a match in round 3. From the calculations we did in the previous video we expect to find a match 12.5% of the time. That’s about what we find. We also find that we get a match on round 4 about 1/8th of the time.

Now we took a second diversion to take a closer look at getting a match on round 4. Again this exercise proved to be slightly confusing at fist, but the boys seemed to catch on fairly quickly. Their understanding led to a great discussion of what could happen on round 5. The reason round 5 is important is that’s the first time you see a difference between the two sequences:

Finally we returned to our computer program to look at what happens in round 5. We found that both sequences behaved more or less how our conversation from the prior video predicted (despite my incorrect calculation of 75% of 12.5 . . . sheesh).

After that we looked at how many flips we expect to have to do to find a match to each of our sequences. Unfortunately and super unluckily our camera ran out of storage in the middle of this discussion, so I had to break this discussion into two videos. Sorry about that 😦

So, a super fun project and a super interesting math problem. Thanks to Bob Lochel and Math Curmudgeon for sharing it!

Today we used the shape below as a basis for our Family Math talk:

To start off I put the original shape in front of the kids and asked them to come up with a question that we could ask about it. My older son’s question was about the area of the shape and my younger son’s question was about the number of struts in the shape.

We approached the question about the number of struts after we recreated the top of the shape. My younger son’s two approaches to counting the struts shows how great the Zometool set is for facilitating math conversations with kids. Though the question itself is not particularly complicated, it gets my son thinking about (i) systematic ways of counting and (ii) the beginnings of the inclusion / exclusion principle as his second way of counting over counts some of the struts.

Now we came to the question about the area of the ring. My older son also found two ways of approaching this problem. Both ways use the area formula for an equilateral triangle that he’s recently learned from our Introduction to Geometry book, though you could have this conversation without knowing that formula by simply assuming that the area of these small triangles was 1. The math ideas in this conversation are ideas like (i) breaking a difficult problem up into pieces that you already know how to do, (ii) some basic ideas about symmetry, and (iii) by coincidence, a little bit of the inclusion / exclusion idea from the last question.

Last up was my question – what is the ratio of the area of the big hexagon to the area of the little hexagon in the picture?

Each kid gave an answer to this question using a different idea. At first, my older son calculated the areas the same way that he did in the last video. At the end he was able to answer the question just by counting the smaller equilateral triangles that lived inside of both shapes.

My younger son used ideas of scaling to discuss the relationship between the two areas.

So, a super fun project inspired by a little creative playing around with the Zometool set. Pretty amazing to see the possibilities for exciting math conversations that come from the shapes kids create!