# What learning math sometimes looks like – geometry edition

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:

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# Which is larger 3^3^3^3 or 100 billion factorial?

Yesterday we ended up encountering some large numbers when asked some questions about patterns on an Othello board:

Looking at patterns on an Othello board

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.

# Looking at patterns on an Othello board and also the Birthday Paradox

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.

# Some angle basics with circles

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:

# *Pi and the square root of 10

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:

A Continued Fraction Experiment

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 $\pi$ that looks very similar continued fraction for $\sqrt{10}$. 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 $\pi$. I think that kids will always find it surprising that there are relatively simple ways to describe $\pi$ even though just about everything you hear about $\pi$ is that it has no pattern.

Neat little morning – very fun to show a surprising way to see some structure in $\pi$. Also quite a surprise to see the seemingly small change that changes $\pi$ into $\sqrt{10}$.

# A continued fraction experiment

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:

A surprise square root of 2 discussion

I intended for the focus of that discussion to be the standard proof of why $\sqrt{2}$ is irrational. Instead, though, a large part of the discussion was about how you could use the continued fraction for $\sqrt{2}$ to prove that it was irrational.

Having not learned my lesson already, I asked my son to sketch a proof of why $\sqrt{5}$ 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:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html

# A surprise square root of 2 discussion

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 $\sqrt{2}$. 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 $\sqrt{2}$. 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 $\sqrt{2}$ but none of them were exactly equal.

He understood that if we could write $\sqrt{2}$ 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 $\sqrt{2}$.

Finally, we covered what I intended to cover in the last little talk – the usual proof by contradiction that $\sqrt{2}$ is not rational. We end the conversation by mentioning some other numbers that are not rational – some for the same reason as $\sqrt{2}$ 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.