Using Brian Skinner’s terrific math joke for a lesson about logarithms

Yesterday I saw this fantastic post on twitter:

Since my older son leaned calculus last year, I thought it would be fun to run through the 9 equations with him, and then focus on the one about the logarithm of N!

Here are his thoughts on the equations:

Now we explored the one about the log of N! in a bit more depth – I was happy that after a few months off from calculus some of the main ideas still seem to have stuck around:

Finally, we went to Mathematica and explored the formula a bit more to see how good it was. We then wrapped up by looking at the Wikipedia page for Stirling’s approximation.

I’m glad to have gotten 2 days worth of laughs from Skinner’s post. Happy that it was also a fun starting point for a lesson, too 🙂

A neat logarithm example for kids thanks to Chanda Prescod-Weinstein and Bruce Macintosh

A saw a really neat twitter thread last week thanks to a re-tweet from Chanda Prescod-Weinstein:

The thread explained why thinking about the (astronomical) magnitude of an object moving through a telescope’s field of view is a little difficult. It was neat to learn that something I didn’t realize was hard is actually pretty hard (though it feels like basically everything in astronomy is like that!), but another thing that jumped off the page for me in that twitter thread was that it was an excellent example to show to kids learning about logarithms.

For reference, here’s the Wikipedia page we used in the project to learn about the concept of magnitude and also get a few examples:

Wikipedia’s page on Magnitude

My younger son (in 7th grade) is just learning about logarithms now and my older son (in 9th grade) has a bit more experience with them. We started by talking about the relative magnitude formula and working through a short calculation to show why the number 2.5 shows up in the formula:

Next we looked at the Wikipedia page linked above to get some examples of magnitudes of a few objects we recognize:

Now we talked through Bruce Macintosh’s twitter thread. I wanted to go through the thread carefully to make sure the kids had a basic understanding of the concepts he was discussing (arcseconds, for example). We talked about some of the calculations, but did not do any calculating ourselves in this part. One question for the kids here was why did Macintosh use a + sign in his formula when the Wikipedia page has a – sign in the formula?

Finally, we did the calculations and found the answer to the mystery of the + and – sign from the last video. Happily, we match the answers from Macintosh’s thread:

This project was really fun. It was a really happy accident that just as my younger son was learning about logs a neat (and “new to me”) example of where logs are used showed up in my twitter feed!

Sharing Grant Sanderson’s Fractal Dimension video with kids

A few weeks ago Grant Sanderson published this amazing video about fractal dimension:

I’ve had it in my mind to share this video with the boys, but the discussion of logarithms sort of scared me off. Last week, though, at the 4th and 5th grade Family Math night the Gosper curve fractals were super popular. That made me think that kids would find the idea of fractal dimension to be pretty interesting.

Here are the Gosper curves that Dan Anderson made for us:

Screen Shot 2016-04-05 at 4.54.02 PM

We’ve actually studied the Gosper curve several times before, so instead of just linking one project, here are all of them 🙂

A collection of our projects on the Gosper curve

So, today we started by watching Sanderson’s video. Here’s what the boys had to say about it:

At one point in the video Sanderson makes a comment that fractals have non-integer dimensions. I may have misunderstood his point, but I didn’t want to leave the boys with the idea that this statement was always true. So, we looked at a fractal with dimension exactly equal to 2:

Next we looked at the boundary of the Gosper island. I wanted to show that this boundary had a property that was a little bit strange. I introduced the idea with a square and a triangle to set the stage, them we moved to the fractals:

Finally – to clear up one possible bit of confusion, I looked at a non-fractal. For this shape we can see that the perimeter scaled by 3 and the area scaled by 7. Why is this situation different that what we saw with the Gosper Island?

Definitely a fun topic and I think Sanderson’s video makes the topic accessible to kids even if they don’t understand logarithms. I’m excited to find other fractal shapes to talk about now, too!

Linear vs. Non-Linear ideas in teaching math

Three tweets over the last two days have had me thinking about linear and non-linear ideas that arise when students are learning math.

In the order I saw them:

(1) A linear approach to understanding fractional exponents:

(2) A student mistaking a non-linear idea in trig for a linear one:

https://twitter.com/crstn85/status/693976757314129921

(3) A second example of mistaking a non-linear process (functions / logarithms) for a linear one:

Something in the example in the first tweet left me uneasy and I couldn’t quite put my finger on what it was. Seeing the next two tweets, though, helped clear the fog – at least a little.

The first example takes a non-linear process – a geometric series – and uses it to illustrate how to understand a linear process – adding exponents. BUT, it is maybe a little surprising, especially to students, that the linear / non-linear relationship works so nicely in this situation. As the next two examples from students show, a non-linear idea doesn’t always simplify so easily. I worry that the first example subtly plants the idea that everything is linear and could lead to the type of misconceptions illustrated in the 2nd two tweets.

Still trying to think through all of this, though.

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.

Primes, Logs, and showing some modern math to kids

Earlier in the week I saw a nice exchange on twitter between Michael Pershan and Justin Lanier.   Sorry for the cumbersome tweet linking, but here’s the part of the exchange that caught my attention:

After seeing this exchange I wanted to come up with a project that (i) gave the boys a peek at some modern math, and (ii) showed a connection to something that they already understood.  Prime numbers were on my mind since my younger son is working through Art of Problem Solving’s Introduction to Number Theory, and some amazing recent papers from a Terry Tao led team and James Maynard also have been in the back of my mind, too,.  So the topic I chose was a walk through how mathematicians have come to understand prime numbers.

We began by looking at two results known in ancient Greece.  The first result we look at is Euclid’s proof that the number of prime numbers is infinite.  I have talked about this result with both of the kids previously and they are familiar with the sketch of the proof.  A key detail that I take for granted is that every integer can be factored into prime numbers in a unique way, but you have to start somewhere in an overview.

The second idea from ancient Greece that we discuss is the construction of prime numbers via the Sieve of Eratosthenes. This topic is also familiar to both of the boys from their Introduction to Number Theory book.

So, 2500 years ago we know that there are an infinite number of prime numbers and we knew how to construct them.  Even if the way of constructing them is not super efficient, these two facts together are a pretty good starting point for kids to learn more about primes.

Now fast forward to the 1800s.  In this century mathematicians began to think about  how to approximate the number of primes less than a given integer.  The eventual result was the so-called Prime Number theorem proven by Jacques Hadamard and Charles Jean de la Vallee-Poussin in 1896. With this work mathematicians extended Euclid’s idea that the number of primes is infinite to be able to say roughly how the prime numbers are distributed in the integers.

To get our arms around the work here we need to have a basic understanding of the logarithm function.  For simplicity I used the “flogarithm” idea from Jordan Ellenberg’s book How not to be Wrong.  That idea is simply that the number of digits of a given number is a good enough approximation to the logarithm for purposes of thinking about these prime number theorems.  I do also mention the difference between log base 10 and log base e, but this is not a point I wanted to dwell on.

Next we moved into the first half of the 1900s.  The idea that mathematicians began to think about during this time was how to approximate number of prime numbers that divide into a given number.   For purposes of these theorems you count repeated primes numbers multiple times (so 18 = 2*3*3 has three prime divisors).  In the early 1900’s Hardy and Ramanujan found that the expected number of prime divisors of a given integer n was about log (log n).  In the mid 1900’s Erdos and Kac improved this estimate with a beautiful theorem describing the distribution of the number of prime divisors of the integers.  Instead of discussing the normal distribution used in the theorem, though, I used the binomial distribution from Pascal’s triangle as a picture because it was more familiar to the kids.

One piece of particularly fun math from this video is at the end when we talk about the incredible and bizarre connection between the prime numbers and random numbers.  Thanks to Eratosthenes we have an explicit way to construct all of the prime numbers – so the prime numbers are not random at all.  However,  the beautiful description of the number of prime divisors of a given number from  the Erdos-Kac theorem arises from ideas about random numbers.  How can ideas from random numbers work so well to describe numbers that are not random at all? I think that the answer to this question remains a mystery.

Finally we jump to the developments in understanding prime numbers that have been made in the last two years.  The first thing we discuss is Zhang’s paper about prime gaps and the subsequent work that was done around the world after the publication of his paper.  Next we discuss an incredible coincidence from this year when a group led by Terry Tao proved another theorem about prime gaps one day ahead of the Canadian mathematician James Maynard announcing a different proof of the same theorem.   Incredibly, this problem about prime numbers  that had two different solutions posted one day apart had been unsolved for the previous 75 years!

Today’s Family Math was really fun to work through.  While I was uploading the videos onto the computer my younger son came in to ask a question about logarithms (using the number of digits definition).  He said that he thought that no matter how big a number was taking the log of that number, then the log of the result, and so on would always get you down to 1.  Once you got to one, though, you’d be stuck since 1 has 1 digit.   Nice to see him thinking about these ideas 🙂

So, thanks to Justin Lanier and Michael Person for the inspiration for this Family Math.  It was definitely a fun morning.

Half way through Jordan Ellenberg’s book “How not to be Wrong”

I picked up an audiobook copy of “How not to be Wrong” for the drive to and from Boston this weekend.  I’m about half way through it at this point and have really enjoyed it.  It is a great book if you are interested in getting a look at the world through the eyes of a mathematician.

Among his achievements, Ellenberg was one of the top students in math competitions the year I graduated from high school and was also one of the top students in the Putnam exam during his time at Harvard.  Our senior year of college, his Harvard team beat my MIT team in a way similar to how a #1 seed beats a #16 seed in the NCAA basketball tournament.  He’s gone on to have a really interesting career in academic math, writing, and probably 20 other things that I don’t really know anything about 🙂

So far the book offers fascinating insights into math education, scientific and political proclamations about probability, and an interesting range of more mathy topics such as Zhang’s breakthrough on prime numbers from last year, and a great discussion on the group of MIT students that figured out how to win one of the Massachusetts lottery games.

Several more comprehensive reviews of the book by Cathy O’neil and Evelyn Lamb are here:

How Not To Be Wrong by Jordan Ellenberg

and here:
http://blogs.scientificamerican.com/roots-of-unity/2014/05/31/how-not-to-be-wrong-book-review/

 

Some of the fun takeaways for me so far include:

(1) His take on the math education wars explains my own position better than I could have myself.

(2) I wish that I would have seen his discussion of Zhang’s prime number breakthrough before I did this little talk with my kids last week:

https://mikesmathpage.wordpress.com/2014/05/31/prime-gaps-and-a-james-tanton-problem/

Ellenberg’s explanation of the problem, Zhang’s solution, and why the problem is interesting to begin with is as clear and understandable as I’ve seen.  Also, his back of the envelope explanation of logs is really great.

(3) The part about the Massachusetts lottery and why the MIT team filled out their forms by hand is really incredible.  I’m actually still in the middle of that one, but having heard the story from a few other people loosely connected to the group that figured out this hole in the lottery game, Ellenberg’s telling of the tale ties a lot of loose ends together for me.

(4) He talks through a proof of the Buffon needle problem that I’d never seen before.  I’d previously know that problem to be only a neat high school calculus example, but this other proof  makes a clever geometry / probability argument and is remarkably similar to this problem that Guass solved:

https://mikesmathpage.wordpress.com/2014/03/23/a-really-neat-problem-that-gauss-solved

(5) Finally, probably the most interesting bit to me personally was the section on how humans guess numbers in a way that is totally non-random.  In the two big billion dollar games that I’ve been involved in – Pepsi’s “Play for a Billion” in 2003 and 2004, and the Quicken Billion dollar bracket game in 2014 – reviewing the guesses by the contestants after the game was over (1000 numbers from 000000 to 999999 for the Pepsi game, and NCAA brackets for the Quicken game) it was almost shocking how the guesses of the contestants clustered.

I’m sure I’ll have many more fun takeaways from the book once I finish it.  Even half way through it, though, I’d recommend this book without any hesitation to anyone looking for a fun book about how math comes into play in all sorts of different situations.  While that topic may seem like  a pretty heavy one, Ellenberg is a gifted writer and the book is a really enjoyable read.    In a way I’d put it on par with Roger Lowenstein’s “When Genius Failed” or Bethany McLean’s “Smartest Guys in the Room” and “All the Devils are Here” – potentially difficult topics turned into relatively easy and informative reads (and basically must reads)  by great writers.

Go pick up your copy today!