Prime Gaps and a James Tanton problem

Early this week James Tanton, a seemingly bottomless pit of great math ideas for kids, posted the following problem on Twitter:

A nice little twist on an old problem that inspired our Family Math talk for today.

The first thing that we did was do a quick review of prime numbers – what are primes, and can we list the first ten or so prime numbers?  After that we talked about the two types of “gaps” involving prime numbers.  One interesting, and still unsolved, question about prime numbers involves the number of “twin primes.”  So, are there infinitely many pairs of prime numbers like 3 and 5, or 11 and 13, that differ by 2.  An important step in answering this question was made last year by Tom Zhang of the University of New Hampshire who discovered (incredibly) that there are infinitely many prime numbers that are less that 70,000,000 apart from each other.   Not quite a difference of 2, but still amazing!

Almost the opposite question is the one posed by James Tanton – how large can the gap between consecutive prime numbers be?   That is the question that we’ll focus on for the rest of this talk:

Before moving on to answer the main question, though, in the last video my younger son mentioned that there are infinitely many prime numbers.   I thought it would be fun to show why that statement is true, so the next video walks through a simple proof that kids can understand.  I think (but have not verified) that this proof is attributed to Euclid.  In the course of this proof I also mention one reason why mathematicians do not like to consider the number 1 to be a prime number.

Finally we get around to discussing Tanton’s question.  We start by finding 1,000,000 consecutive non-prime integers and then move on to finding 1,000,000 consecutive odd prime numbers.  It was nice to see that my younger son was able to understand how to make the leap from all integers to just odd integers.

I think that there are lots of neat examples from number theory examples that kids will really enjoy.  The problem posed by James Tanton that we focused on today is a really fun problem to work through with kids.

Continuing with the Logistic Map

The last blog post talked about a fun project that I started with my older son this week which was inspired by some Steven Strogatz lecture videos.  See here:

We’ve continued studying the Logistic map this week and it has turned out to be as fun as I’d hoped.  Today we moved from the whiteboard to the computer to study the map more carefully.    Part of the fun hiding in this project for kids with a little bit of algebra background is the ability to talk about some basic transformations you need to build a simple graph on the screen.

We built our program on Khan Academy’s programming site since that’s the easiest way that I know how to share code.  Here’s a short talk about the program we made:

and here’s the link to the actual program itself.  We’ll play with this a little more next week (and hopefully improve the code a little).  A little spaghetti code notwithstanding, this was a really fun morning:

I think that playing around with the Logistic Map is a really fun math project for kids!

Steven Strogatz’s video lectures and Dynamical systems for kids

I’m so excited about the new project I’m working on with my son that I’m almost at a loss for words.  Yesterday I saw the following note on Twitter from Steven Strogatz:

Around 20 minutes in to the first lecture is a quote, or rather an “evangelical plea”, from Bob May stating something like –

“We should stop teaching only linear math to our college students and our graduate students and show them that once you allow systems to be non-linear all bets were off and you could discover all kinds of things.  It was time to stop lying to the students in the classrooms.”

This idea really struck me because it connected with a number of different things that I’ve heard over the last year  – Conrad Wolfram’s talk about computers and math comes to mind, for example  (see a link here: ).  Anyway, we were about to end the year talking about sequences and series, but spending a few weeks playing around with the logistic map suddenly seemed like a much better idea, so off we went.    I was also really excited to tackle this subject because I studied a little bit about the logistic map in high school in Mr. Waterman’s Enrichment Math class.  It is always doubly exciting to be able to pass along stuff I learned from Mr. Waterman to my kids.

The first thing I did when I got home from work yesterday was sit down with my son and introduce the concept.   Had I spent even two seconds thinking about what to do I probably would have started with an easier recurrence example – the Fibonacci numbers, say – but that idea didn’t occur to me until today.

In the first video we walked through the relation x_{n + 1} = 2R* x_n * (1 - x_n).   We compute a few iterations in the case where R = 2, and then finished off looking at a few points on the graph of y = 2 x * (1 - x).   I  admit that this might not be the most exciting start to the topic, but it does lay the foundation and also allowed me to double check that the math behind the quadratics and the iterations wasn’t too far over his head:

After last night’s basic introduction we spent about 30 minutes this morning diving into the geometry of the logistic map.   Seeing this connection between the geometry and the algebra in high school was absolutely amazing to me.  Prior to our little five minute film we spent time studying two equations  x_{n + 1} = (1/2) * x_n * (1 - x_n) and x_{n + 1} = 2 * x_n * (1 - x_n) and that allowed us to study one of the more complicated examples in the video:

Tomorrow we’ll look at some of the really baffling examples, though we got a little preview of that thanks to Alexander Bogomolny who saw some of my enthusiasm on twitter and alerted me to this section of his site:

For homework today my son read this page and played around with the applet – declaring it to be “awesome.”

Such a fun topic, and as I point out at the end of the second video, it really is cool to be able to introduce some relatively modern math to my son.    Don’t quite know where this is going to go in the next week, but it looks like we are going to have a really fun time no matter what direction we end up going!

Triangles in planes and spheres

Today our fun Family Math project was about geometry.  We did a little playing around with triangles in the plane and triangles on the sphere.   A more advanced version of this discussion would probably include some mention of Euclid’s 5th postulate.

Our first topic of discussion was parallel lines on a plane.  What does it mean to be parallel?  My youngest son sees parallel lines as lines that do not intersect and my oldest wants to define parallel in terms of the slope of the line.

After talking about parallel lines for a bit, we went on to talk about parallel lines and angles:

Next we go on to talk about triangles.  The point of this discussion is to see that the angles in a triangle can be rearranged to make the same angle as a straight line.   The main idea here is just the idea that we discussed in the last video:


Now we move on to some fun ideas about triangles.   Just using some of the  basic facts about angles that we talked about in the last movie + the Pythagorean theorem, we find the area of a equilateral triangle, and also some simple properties of an equilateral right triangle:

Finally the punch line – what happens if we try to extend some of these geometric ideas beyond the plane?  The easiest example to show is a sphere, and I illustrate a triangle with three right angles by drawing the picture on a softball.  I love that my youngest son’s reaction was that this triangle was impossible.  Ha, not impossible, you are looking at it right now!!


Feels like there are a lot of different directions to go introducing basic geometric ideas to young kids.  One unexplored idea here is to show a surface where a triangle’s angles add up to less than 180 degrees.   Maybe there’s a 3D printing / basic geometry project in the near future!


Using snap cubes to talk about the 4th dimension

Had a friend from college visiting for Memorial Day and thought it would be fun to do a video explaining the 4th dimension to all of the kids in the house this weekend.  This project didn’t go quite as well as I was hoping, but I think the idea here is fun.  Will probably try it again in a few months.

In the first video we walk through the concept of a zero dimensional object sliding in time.  Our model for a zero dimensional object is a snap cube.  We talk through how a zero dimensional object sliding in time can create a one dimensional object.  The concept may seem a little strange when you talk (or read) about it, but seeing the trail of the snap cube as it moves helps the idea make sense (I hope!).

One other thing that we’ll be keeping track of in each of the videos is the number of cubes we have at every stage of the sliding.  With a single sliding snap cube, counting the cubes is easy – we just get 1,2,3,4,5, . . .

Next we try to make a two dimensional object by paying careful attention the sliding zero dimensional object from the previous video.   We build a two dimensional object – sort of a triangle – out of the pieces that the sliding snap cube created in the last video.  In this section the number of cubes we need to build our object at each stage is 1,3,6,10,15, and etc:


Now we take the idea from the last video and apply again to make a three dimensional object.   This time we have to keep track of the shapes at every stage of the “sliding” in the last film and combine those shapes together as they slide in time.  The object we create this time around is a 3-dimensional pyramid.  The number of blocks at each stage is 1,4,10,20,35, and etc . . .

Now for the 4-D challenge.  We want to apply the same idea as in the previous two videos, but there’s a little snag.  We don’t have any dimensions left in our kitchen, so how are we going to put the 3D object together?  Unfortunately the 4D shape we are creating here is pretty hard to visualize, but we can at least understand what the slices look like – they are exactly the shapes from the prior video!  One neat thing is that even though it is difficult to understand the picture of the full shape, we actually can count the number of cubes at each stage – 1, 5,15,35, and etc.

Finally, having build and sort of understood a 4 dimensional object, I wanted to show a neat connection this project has to Pascal’s triangle.  In every video we found an interesting sequence of numbers by counting the number of blocks needed to build our object.  Each of those sequences comes from a diagonal in Pascal’s  Triangle!  Pretty amazing that Pascal’s triangle tells us how to count blocks in 4 dimensional pyramids.   The kids even speculated that other diagonals count blocks in higher dimensions.  Pretty fun:

So, although this one didn’t go as well as I’d hoped, it was still really fun.  At least it was nice to end on a really cool note with the connection to Pascal’s triangle.  Will definitely try to improve on this one later.

Ed Frenkel, the square root of 2 and i

A few weeks ago, some thoughts on twitter from Michael Person inspired this talk with my kids:

Last weekend I picked up the audio book version of Ed Frenkel’s “Love and Math” and Frenkel’s discussion of \sqrt{2} and i made me want to revisit this conversation about properties of numbers.

We started with \sqrt{2}.   Their reaction to hearing that we were talking about \sqrt{2} was to talk about why it was irrational, and since they nearly remembered the proof from last time, this proof made for an instructive start to the conversation today.  It is always nice to review some of the ideas behind these simple proofs with them and watch their ability to make mathematical arguments develop.

Next we moved on to talking about i.  They remembered a few basic properties about i, though my older son still thinks that it is something that math people just made up.  I’m not terribly bothered by that for now, but the ideas in Frenkel’s book are giving me some new perspective on how to present some of these more advanced concepts to the boys.  Hopefully this new perspective is going to lead to a much better approach to teaching them math.  In any case, here’s what we said about i:

The next two videos are the main point of the talk today – in what ways are \sqrt{2} and i similar?  This question is a specific example of the broad question of symmetries in math that Frenkel discusses in his book.  I felt like the book walked up a couple of stairs and then hopped into an elevator to the top floor, though.  The ideas were inspiring, but I was left (i) wanting more and (ii) wanting to fill in a few more details.  One focus of these math conversations with my kids over the next few years will be spent on (ii).  I’ll work on (i) by finishing the audio book on a drive to and from Boston this weekend!

For today, though, let’s just stick with some similarities between \sqrt{2} and i that Frenkel highlights:

So, without digging too deep into the details, it looks like the set of numbers that we get by adding \sqrt{2} to the rational numbers has some nice, simple properties.  If we add or multiply, we seem to never leave the system.  Pretty neat.  i seems to have the same property.  Frenkel make the point that is we aren’t too bothered by \sqrt{2}, we shouldn’t be that bothered by i.  This is a nice point, obviously, and a fun idea to share with kids.  I really loved that my older son made the connection between i and x from algebra.  Only one step away from polynomial rings . . .  ha ha!

So, definitely on the theoretical side, but a definitely a fun morning.  Looking forward to plucking a few more ideas  out of “Love and Math”  to share with the boys.

21 people in and around women’s ultimate you should meet

Sorry, another one not about math with my kids – maybe I should start an ultimate blog.  Oh well not today . . .

In response to Skyd’s article –

and in an effort to elaborate a little on some thoughts I had in a FB conversation, here’s my list of 21 people in and around women’s ultimate that i think you should meet.  I gave myself an hour to write this so that it wouldn’t be too long and rambling.  Also just wanted to try to come up with some ideas off the top of my head.  Oh, and since Gwen, Matty, and Michelle are in the Skyd article, I’ll leave them off this list on purpose.  To the other 4000 people I leave off accidentally, sorry 🙂

I have not yet met all of these people, but I hope to.

(1) Robin Knowler – 10 years coaching one of the top programs in the country, so she’s got plenty to teach you  Go meet her and ask her how to be a better teammate / leader / coach / person / or whatever.  I’d pick “coach” from that list and then just listen.


(2) Lou Burruss – I first met him in 1997 when he would fly back from Seattle to coach the Carleton women.  Amazing dedication to the sport and hence one of the most successful coaches of all time.  Ask him about moving to set up the next pass or how to play a 2 handler zone O.  Also read “The Inner Game of Tennis” in advance of meeting him.


(3) Suzanne Fields – part of the first class inducted into the Ultimate Hall of Fame, and one of the speakers at this year’s induction.  I’m always a little nervous around legends, but if I would have had the courage to talk to her at this year’s induction I probably would have asked something silly like  if she could believe she was standing there watching Chris O’Cleary and Nancy Glass being inducted into the hall of fame.

With the passage of time I’d probably ask her if she, Kelly Waugh, Katherine Greenwald, and Katie Shields played Heather, Shannon, Mia, and Emily in a game of goaltimate, who would win?

(4) Chris O’Clearly – see above.  One of this year’s inductions into the Hall of Fame and another legend in the game.  Seemed like everyone who ever played for Ozone was there to cheer her on at the induction.  An amazing leader and player.  Ask her how to build a team.

(5) Nancy Glass.  Also one of this year’s inductees.  Another absolute legend and practically royalty in Chicago ultimate.   Ask her about the tension between getting the sport to the “next level” like the Olympics or something and building the sport through grass roots growth.
(6) Jenny Fey.  One of the best players of the last decade who just came off of a national championship with Scandal.  Ask her how she sees the field and if she likes handling or cutting better.  Also, do me a favor and figure out how to guard her because I’ve not been able to do that.

(7) Cara Crouch.  Two time World Games team member, 2005 Callahan winner, and endless giver back to the game:

Ask her about the difference between the 2009 and 2013 World Game teams.  Seems like the two teams had totally different vibes – what worked well and what would she have changed looking back?

(8) Dominique Fontenette – Stanford, Fury, Godiva, Brute Squad, World Games, Riot.  As respected a player as there ever has been. Ask her about the influence that Molly Goodwin had on her.  Sprout, too.  Also, ask her to teach you to pull:

(9) Rohre Titcomb – One of the greatest minds in the game.  I’ll never forget seeing her play for the first time – it left me speechless.  Ask her to come to Atlanta and play a round of disc golf with Chris O’Cleary, ’cause that would be amazing.

(10)  Alex Snyder –  Multiple time national and world champion.  One of the things I will always remember is how different the  2013 US World Games team played during the one game she missed.  Ask her what she learned about the game coaching Wisconsin.

(11) Robyn Wiseman – A great young leader.  Ask her what she learned taking over coaching Wisconsin from Alex.

(12) Enessa Janes – I was so happy to get the chance to meet her in person at the 2013 US Open.  Played the single greatest half of ultimate that I have ever seen.  Ask her about the 2008 finals.

(13) Katy Craley – National champion at Oregon and now a key player for Riot.  Ask her about the transition from college to club.  Ask her about giving back to the ultimate community in South America.

(14) Ren Caldwell – The trainer for everyone within 300 miles of Seattle, I assume.  Ask her about the difference between training college athletes and club athletes.

(15) Claire Chastain – 2013 Callahan winner / U23 world champion and one of the best players I’ve ever seen coming out of college.  Ask her how her mentors impacted her ultimate career.

(16) Peri Kurshan – leader on the field with Brute Squad and Godiva.  Off the field with USA ultimate.  Current Nightlock coach.  As her about the transition from playing club to coaching club, and about the similarities between what Brute Squad looked like originally and what Nightlock looks like now.

(17) Erika Swanson – amazing player on both coasts and on the US Beach worlds team.  Ask her about how she balanced playing top level club ultimate with MIT and Caltech educations.    Ask her about how to defend the top cutters.

(18) Samantha Salvia – I’ve never met her, but her story is incredible.  Ask her about transitioning from other sports to ultimate, and ask her to write some more!


(19) Blake Spitz – helped build Brute Squad up from scratch and eventually past Godiva.  Ask her how to develop young players on a club team.  Ask her how to compete and eventually win out against one of the biggest dynasties ultimate has ever seen.

(20) Lucy Barnes – Captained Harvard, Brute Squad and now lives in England.  Ask her how far European ultimate has come in the last 10 years.  Has the US come as far?

(21) Kyle Weisbrod – coaches UW Element and the US under 19 team.  Ask him about the difference between the high school scenes in Atlanta and Seattle.  How could another city copy what either of these cities has done.

Dan Meyer’s Money Duck problem part 2

Last week I wrote about an interesting expected value / probability question that Dan Meyer posed on his blog.  Dan’s blog post is here:

[Confab] Money Duck

My first post about the problem is here:

In the comment section on Dan’s blog I described an exercise that I thought would be fun to try out with kids:

Your friend Dan walks in with 5 money ducks. Each of the ducks has some money hidden inside – one has $1, one has $5, one has $10, one has $20, and one has $50. You do not know which duck has what amount of money, but Dan does.

He suggests a couple of games:

(1) You pick any of the ducks you want and get to keep the money inside of it. How much money do you expect to win playing this game one time?

(2) You pick any of the ducks you want and then Dan tells you the lowest amount of money remaining in one of the 4 ducks that you didn’t pick (without telling you which one of the ducks contains that lowest amount of money). He then lets you pick a new duck if you want. Would you rather play this game or game (1)? Why?

(3) Same as (2), but after your first selection Dan tells you the highest amount of money that remains in one of the four ducks that you didn’t select instead of the lowest. He then lets you pick a new duck if you want. Would you rather play this game or game (2)? Why?

Depending on how advanced the students are, you could ask for expected values in games (2) and (3), too, but even without calculating an exact number, the discussion about which games people would rather play would be interesting.


Today I finally got around to playing these three games with my kids.  Here’s how it went.

First, since the my original blog post about this question mentioned the similarity between Pepsi’s “Play for a Billion” game and the activity that many people commenting on Dan’s blog wanted to try out with the Money Ducks, I brought out a couple of the old props from the show to help introduce the problem:

Now, on to the game.  The first game will be a one-shot game where you roll a 10-sided die and you have an equal chance of winning $1, $5, $10, $20, or $50.  The question is – how much money do you think you’ll win when you play this game?  Interesting to hear the boys trying to reason out the expected value:

At the end of the last video my older son thought we could figure out the expected value by playing the game 100 times.  We didn’t want to actually do that, but we did go through that math and concluded the expected value for playing the game once is $17.20.   At the end of this video I tried to explain that you won less than $17.20 60% of the time and more than $17.20 40% of the time.  I was trying to make the point that just because $17.20 was the average amount of money that you won, that doesn’t necessarily mean that you win more than that half the time and less than that half the time.  I did not explain this point very well at all 😦

Next we moved on to the two more complicated games that I thought would be fun to play.  In this video we play the game where the prizes are hidden, but I give you a little extra information after you select one of the prizes.    After you pick, and before you learn what your prize is, I tell you the lowest amount of money that you didn’t pick and then see if you want to pick a new prize.  Although analyzing this game isn’t super complicated, I didn’t want to go into the details.  Instead we just tried to talk through whether or not they thought you’d win more money playing this game or the first game:

The final game we played is similar to the one in the previous video.  The only difference is that after you pick your prize, I reveal the highest amount of money remaining from the ducks you didn’t pick.  It was fun to hear how the boys talked through whether or not they would want to play this game or the prior one.   Again, a complete analysis of this game isn’t super complicated, but the goal here wasn’t to do a detailed analysis, but rather to simply talking through the ideas.

All in all, I think there are lots of fun and mathy conversations that you could have with kids about the Money Duck.  As I wrote in the prior blog post, I think the “how much would you pay for the Money Duck” question is a little more complicated than just expected value, but it is still a fun question.   Thanks to Dan for asking for thoughts about fun ways to play with the Money Duck – it was fun to think through.

A Fawn Nguyen inspired geometry problem

Last week Fawn Nguyen posted that she was going to so a fun Five Triangles problem with her class:

I typically love the problems posted by Five Triangles and their geometry problems, in particular, are consistently outstanding.  Too bad I’ve not really covered any geometry with either kid yet 😦

But, I have been working on fractions and decimals with my younger son and this problem had a really interesting infinite series hiding in it, so I though it would be fun to talk through with the boys even if I would have to skip over the interesting geometry.

I spent the first 5 minutes just introducing some basic concepts about triangles so that they could understand the problem.  The fact that we are dealing with an equilateral triangle here significantly simplifies the explanation because we can work with the medians rather than the angle bisectors.  Also, though we didn’t dwell on it, with an equilateral triangle it isn’t hard to believe that the medians intersect in a single point.

With that basic introduction out of the way, now we could spend a little time talking through the problem.  The first challenge is to find the radius of the second circle.  My older son had a one geometric idea that was going to be a little more difficult to work through than I was hoping for, but then my younger son noticed that we could draw in a new line segment that would make finding the radius of our new circle pretty easy.  From there we moved on to talking about the infinite series that is hiding in this problem:

From our picture in the last video we were able to see that 2/3 + 2/9 + 2/27 + . . .  = 1.  Now we try to see if there’s a way to sum up that series without appealing to the geometry.  This particular problem is pretty similar to converting repeating decimals to fractions which is what my younger son and I have been talking about for the last week.   I really loved the various ideas that the kids threw out here:

Finally, we wrap up by showing how to sum up the above series by using base 3.  We start by talking about why .9999… = 1 in base 10 and move on to show how the same argument shows that .22222…. = 1 in base 3.  Luckily the series we are looking at is easy to write in base 3.  Fun!!

So, yet another thanks to Fawn Nguyen for alerting me to a really great problem.  Though not quite the point of the problem as originally posed, I love the connection between arithmetic and geometry hiding in this problem.  It was really fun to talk through with the boys.


Michael Pershan’s Exponential post (part 2 /3 )

Despite an extra day to try to think things through I remain confused about my own approach to teaching / talking about exponential functions.   I’m actually struggling to even understand what the struggle is.  After all, running across this fun little connection between \pi and e as a college freshman is what convinced me to major in math:


As I mentioned in the first post in this series, my approach to teaching exponentials has not been nearly as formal as my approach to teaching arithmetic.  We began by talking about powers where I introduced exponents as essentially a time saver.  Probably like just about everyone who has ever talked about exponents,  one of the early conversations was about the zeroth power and negative powers.  That talk about exponents with my younger son remains one of my favorite math  conversations he and I have ever had:

Feels as though you are almost forced to introduce integer exponents early one if you want to talk about place value or different bases or other similar topics, but the path to exponentials from here just isn’t that satisfying to me.  You’ll have to introduce fractional powers and then define non-rational powers by some sort of limit process (at least if you want to approach things formally).  If you are going to bed at night fearing a Grant Wiggins-like “conceptual understanding of exponential functions” exam, you probably won’t like this path at all.

There are fun topics, though, so I’m not suggesting that integer or rational exponents are a waste of time.  Two of my favorite topics here have been finding a formula for the Fibonacci numbers while we were studying quadratic equations:

and talking about Graham’s number, which is one of the most fun math activities that we’ve ever done, and probably as much fun and excitement with integer powers as you are ever going to have.  It took me a week to figure out how to put this one together (and I stopped after a week, because there was no way I was going to figure out how to do it anyway!)

With all this background I’m kind of surprised that I can’t really think of a nice, easy transition from exponents to exponential functions.   As I was riding home last night I tried to keep a look out for anything I saw that I naturally thought of as being associated with an exponential function – something / anything that kids might see occasionally in their life.  I couldn’t find a single thing which made me a little sad.  Maybe I’m just not being creative enough.

In yesterday’s post I mentioned a few things from finance and probability where exponential functions appear pretty naturally, but those are well outside the realm of things that kids see or worry about.   What I didn’t mention was a different field where exponentials play an incredibly important role – physics.  Representing waves in the form e^{i * \omega t} is pretty convenient, to say the least, but again is way outside of what might be reasonable examples for kids.

So, I’m lost.  A non-formal approach starting with integer exponents does let you talk about some really interesting problems, but doesn’t really seem to set you up too well to move to exponential functions in general.  I’m a little frustrated at my inability to find  any great (or even reasonable) natural exponential examples to share with kids.  And, to top it all off,  starting with a formal approach like defining e^x as the limit as n approaches infinity of (1 + x/n)^n just seems stupid.

The perplexing thing is that both e^x and ln(x) play such incredibly important roles in math.  You’d think that there would at least be a few easy examples you could talk through with kids to introduce / motivate these ideas.  I mentioned in yesterday’s post that I gave one formal approach a try.    That was in response to a question I saw about logs on twitter when I happened to be talking about them with my older son.  That question motivated me to throw together a fun overview of some of the areas in math where logs are part of important results.  I wasn’t expecting my son to get much of anything out of it other than to see some really amazing math involving prime numbers.  That blog post is here:

Tomorrow, or over the weekend, I’ll try to come out of the fog and write about what I’d like my kids to learn about logs and exponentials.  Hopefully I’ll have it all figured out by then.  Ha!