Motivating Complex numbers for kids

Saw this post on Twitter when I got up this morning:

Made me want to share my attempt to talk about complex numbers with my older son earlier this year.   Up until this discussion all we had really covered was basic definitions of complex numbers – hardly anything more than that i = \sqrt{-1}.  This series of discussion was motivated by my son asking me why the cube root of 343 had 7 as the only solution, but something like the square root of 9 had two solutions -> +3 and -3.

As with just about everything I do with my kids, this was all on the fly (except some details in the last talk).   I’m sure there are mistakes, but here’s what we did:

The first step was just talking about the question he’d posed:


The next discussion was an attempt to do an example similar to the one in the first video but not quite as complicated – the roots of x^4 - 1 = 0.  I probably should have started with this equation in retrospect.  The important step in this problem was looking at where two sets of roots we’ve found appear in the complex plane:

The next step was a detailed look at the geometry that we had looked at in the last video.  The focus here is the three solutions to x^3 - 1 = 0.  In particular, the three solutions look like they lie on our unit circle – do they?  The next thing would be to see if they do actually form an equilateral triangle.  Not sure why I skipped this second part:

The next thing we did was to circle back to the original question.  We had used a combination of factoring and the quadratic formula to find the solution to x^3 - 1 = 0, but we never bothered to check that all three of the numbers we found were actually solutions.  Here we crunch through that math just to get some experience calculating with complex numbers:

I wanted to end the series talking about a neat result – the fundamental theorem of algebra.  It was tricky to figure out how to cover this topic with a young kid, but I wanted to give it a shot.  Obviously I don’t get into any details of Gauss’s proof, but I’m always happy to talk about important math results.  This particular topic was actually more fun than usual since, to my son, Gauss is famous for adding up 1 + 2 + 3 + . . .  + 100 – and nothing else!! Ha ha:

I definitely enjoyed throwing this little sequence together.  Even though there was no direct follow up, every now and then I think it is fun to show my kids more advanced ideas that use some of the basic math we are covering.  Probably my favorite example along those lines – and one that turns out to be pretty closely connected to the sequence here – was a talk about regular pentagons.  A little longer than usual, but one of my favorite examples of how some basic math comes together for a neat result:

Penrose tiles and some simple 3D variations

This weekend we spent a few hours making shapes with Penrose tiles and playing around with shapes that can fill up 3 dimensional space.  This was another weekend Family Math project inspired by an Evelyn Lamb tweet:

Although what gave me the  idea for this project was the rhombic dodecahedron blog post that Lamb linked to, I thought that it would be easier to play around in 2D first.  Penrose tiles seemed like the obvious place to start.

It wasn’t surprising to see that someone had already posted a template for Penrose tiles on Thingiverse.  Actually, there’s probably more than one, but here’s the one that we used:

We printed 8 kites and 8 darts and the boys spent 20 minutes or so making various shapes with them before we got going with the movie:

Following that, we began talking about 3 dimensions.   The easiest idea of a shape that could fill up 3 space was a cube, so we stacked up some 2x2x2 Rubik’s cubes at the beginning.  Following that, we talked about an incredible pair of shapes that can form a special cube – Iwahiro’s “Apparently Impossible Cube” – that we’d found by coincidence on Thingiverse earlier in the week:

The boys had really enjoyed trying to solve Iwahiro’s puzzle (which may be more difficult to get apart than it is to put together!).

After this little 3D intro we moved on to what had inspired Evelyn Lamb’s tweet:  Laura Taalman’s (aka @mathgrrl ‘s) post about printing a rhombic dodecahedron:

We printed 4 of the small rhombic dodecahedrons and then spent about 30 minutes building the same shapes out of our Zometool set.  It was a little hard to get the printed shapes to stay stacked together, so the Zometool set was actually a big help.  Here’s our talk about these shapes:

Playing around with all of these shapes made for a really exciting afternoon of math.   It is so cool to see all of the different ways that the Penrose tiles can fit together, and it is amazing to be able hold these special 3 dimensional shapes and see first hand how they  can fit together to fill up 3 dimensions.  Definitely a fun day.

Physics and Math

By this time more than 1 million people have seen the video of Andrei Linde being told his prediction about gravity waves and inflation had been proven correct.  It is one of the most heart warming scenes in science that you will ever see:

I was lucky enough to be in the audience in 1992 when George Smoot and the COBE team released the first detailed map of the cosmic microwave background radiation.  The excitement from that announcement inspired me to do a little research in theoretical physics for my undergraduate thesis and also to study mathematical physics in graduate school.   As much as I really love math, I try to not just do  “math for math’s sake” stuff with the boys, and have really enjoyed sharing fun results in physics to motivate various aspects of math that we’ve studied.  It isn’t always easy to do this with younger kids, but we have found a few really neat examples.

A few of my favorites:

(1) Comparing the relative sizes of the Earth and the Sun using a marble, a yoga ball, and a soccer field:


(2) Talking about one neat little property of the Higgs boson I read in a Frank Wilczek article:


(3) Finally one of my absolute all time favorites – how fast is the Sun moving around the center of the galaxy?  Sorry that this one is a little long, but one super fun thing in this problem is that the number 1 quintillion comes up!


I really love having the opportunity to share these ideas with the boys.  As they are both struggling a little this week with some new math concepts we are covering, reviewing these old videos reminds me of why I love teaching them.  Hoping for a few fun physics examples in 2014!


Evelyn Lamb put some interesting thoughts about notation on Twitter last week starting with this tweet:


Lamb’s sequence of tweets got me thinking about some of my own battles with notation over the years.  I remember, for example,  struggling mightily with the notation in Hartshorne’s Algebraic Geometry in graduate school.  So, with the thoughts about notation in kicking around in my mind, I had an interesting week watching both of the boys have their own struggles with math notation.  I hope 20 years from now their memories of the Art of Problem Solving books don’t match my memories of Hartshorne!  (note post writing:  I’m surprised that another immediate reaction to Lamb’s post didn’t involve learning Latex!!)

First up – some of my older son’s struggles.  We began a new chapter about exponentials and logarithms this week.  Unfortunately a work trip to London next week is going to mean putting this chapter on hold for a bit, but I was still really excited to dive into this new subject.    We spent the first few days going through several different examples and eventually came to problem 19.4 in  Art of Problem Solving’s Introduction to Algebra book:

Solve the equation:  2^{(16^x)}  =  16^{(2^x)}

The difficulty my son had with this problem came in simplifying the right hand expression.  His instincts told him that  (2^4)^{(2^x)} was equal to 2^{8^x} instead of 2^{2^{(x + 2)}}.  It was a little surprising to me that the introduction of powers in the powers would cause as much difficulty as it did for him.  We ended up spending two full days on this problem – making various charts of numbers and doing several different versions of the problem.  He didn’t have too much difficult with examples which had only numbers, but problems that included variables with powers in the powers were really a struggle for him.   After a few days  he seemed to get a little more comfortable and was  able to work though this problem without too much help:

At essentially the same time I was in the review section of chapter 5 of Art of Problem Solving’s Prealgebra book with my younger son.  We were reviewing some simple algebraic expressions and eventually landed on a problem with an expression similar to this one:

\frac{4x + 3}{2}

The algebraic expressions without fractions hadn’t given him too much trouble, but the introduction of fractions in the expressions really caused him to struggle.    His instincts told him that since the 4 could cancel with the 2, but the 3 could not cancel with the 2, the expression would simplify to be \frac{2x + 3}{2}.  As with the discussion with my older son above, what seemed to me to be a minor change – in this case the inclusion of fractions – caused quite a lot of difficulty.  Working through the difficulty with the fraction notation took a couple of days.  We worked through some examples without variables and then worked on some progressively more difficult examples which included variables.  Eventually we tried out a little “find the error” problem which he seemed to understand pretty well:


One of the nice things about working through all of this math with the boys is that we aren’t really in any hurry.   If something isn’t making sense we have the opportunity to take a day or two (or longer) to make sure the concepts are sinking in.   Lamb’s twitter post about difficulties that her college students were having with notation definitely brought back memories of  my own struggles in graduate school.   Maybe she just planted the idea in my head, but it was interesting to observe the notation-related struggles that my kids were having as their own math work.  I wonder if you ever get to a level of math where that struggle goes away?

A really neat problem that Gauss solved

Last night I pulled an old favorite off the shelf:


I love just flipping through this book because it has so many interesting problems and stories.  The one I ran across last night was a really neat problem that Gauss solved at age 23.  What makes it especially neat is that almost all of the ideas (right up until the end) are accessible to anyone with just a little bit of background in algebra and geometry.  I thought it would be a fun problem to talk through with my older son this morning.

The problem is pretty easy to understand:  If you pick an integer at random, what is the expected number of ways to be able to write that integer as a sum of perfect squares?

For example (and clarity) there are 8 ways to write 5 as a sum of perfect squares:

  1. 5 = (1)^2 + (2)^2
  2. 5 = (-1)^2 + (2)^2
  3. 5 = (1)^2 + (-2)^2
  4. 5 = (-1)^2 + (-2)^2
  5. 5 = (2)^2 + (1)^2
  6. 5 = (-2)^2 + (1)^2
  7. 5 = (2)^2 + (-1)^2
  8. 5 = (-2)^2 + (-1)^2

With that introduction here’s our talk through the problem:

After the introduction we moved on to talk about one bit of  basic number theory that comes into play in this problem – the sum of two perfect squares can’t have a remainder of 3 when divided by 4:

Now we get to the meat of the problem – what is the average number of ways to write an integer as a sum of perfect squares.  The next part of the discussion was just explaining this problem in a way that is (hopefully) accessible to kids.  The notation may make the problem seem difficult, but hopefully the words make the problem easier to understand:

On to Gauss’s solution.  We look at a circle in the plane with a slightly different equation than we usually see.  I spent a minute or two making sure that the notation didn’t trip up the understanding of the problem (despite tripping me up for a second!).  Then we just talk through the geometry:

Finally Gauss’s clever geometric observation – the number of points inside the circle is a decent approximation to the area of the circle.  I skipped some of the analysis details to keep the problem moving, but those details aren’t that difficult.  He essentially drew a slightly smaller circle and a slightly smaller circle and showed that the number of points always is always in between the value of the area of each of these circles.  That allows him to show that the average value we are looking for converges to \pi.   I just show that the average value we are looking for looks a lot like \pi.  Good enough for now.

I love being able to show connections between two seemingly different areas of math.  This example shows an amazing connections between arithmetic and geometry.  I also was happy to see that an amazing discovery by Gauss could be explained to kids.  Definitely a fun morning.

Perfect brackets

I’ve been meaning to write about perfect brackets for a while but haven’t been able to figure out what I wanted to say until today.  This morning I woke up to find my son playing a game he calls “dice racing.”  The game involves rolling two 6-sided dice and keeping track of the totals.  The first total you get 42 times “wins.”  You also win since you get a nice little binomial distribution picture when you finish.  This game seemed like a nice lead into a discussion of brackets.  The initial focus for the bracket discussion was exploring some simple 2 and 4 team tournaments:

Next was a 16 team tournament.  The main point here was simply to try to understand how many games you’d have in general.   From there we moved on to talk about the probability of getting selecting a perfect bracket  in a 64 team tournament (assuming that all games were 50/50 coin flips):

Next, with the camera off, we talked through the probabilities of picking a perfect bracket if your chance of picking each game was equal but different from 50%.  In addition to the 9 quintillion number that comes up when you have a 50% chance of picking each game, we looked at what happens when you have a 60%, 70%, and 80% chance of picking the outcome of each game correctly.  The numbers you deal with in this problem are all so large that they are a little difficult for kids to understand, that’s why I wanted to spend time on this part of the perfect bracket problem.  The other thing that I think kids will find surprising is how much the numbers change when you change the chance of picking each game right.  Raising numbers to the 63rd power is not intuitive!

Next we looked at a few other probabilities – 40%, 90%, 69% and 71%.  The first two numbers came from the boys, the second two came from me.  The size of the numbers caused my older son to wonder what would happen if you picked a different bracket every second.   It turns out that if you had a 52% chance of picking each game right and picked one bracket per second it would take about 14 billion years (or about the age of the universe) to pick a perfect bracket!

Finally, some fun numbers from this year’s tournament.  ESPN has close to 11 million people selecting brackets on their website.  They also publish stats at the end of every round.   That allowed us to watch and see what happened to the number of perfect brackets at every stage.  There were a couple of really interesting surprises.  The surprising math behind these specific stats is a little over their heads, but still fun to talk about.

The Mercer upset over Duke was a surprise for two reasons.   First, obviously, any time a 15 seed beats a 2 seed it is a surprise.   Second, only about 2.5% of the 11 million people who entered brackets in the ESPN contest had selected Mercer, but more than 4 times the percent of the people who had perfect brackets at the end of the first day had picked that upset.  Pretty amazing.  But the bigger surprise came next.

Two games later  #10 seed Stanford beat #7 seed New Mexico (the game in between was a 1 vs 16 seed game).  At the time of this game there were about 2,000 remaining perfect brackets in the ESPN contest and when Stanford won the number of perfect brackets dropped to about 60.  Roughly 97% of the perfect brackets were eliminated even though about 40% of the 11 million people in the overall contest had picked Stanford.  Based on the overall numbers you would have expected the number of perfect brackets to go from 2000 to 800, not 2000 to 60.  I have no explanation for why the people with perfect brackets up to that point did not like Stanford, but I’d love to know!

For me, studying perfect brackets is a great way to show kids some fun math.  With so many people enjoying the basketball tournament, I’m sure that many kids will find it interesting to talk about this math.  For me personally, the math behind the perfect brackets is one of the most interesting “real world” math problems I’ve ever worked on.  I hope to be able to write more about it soon.

A fun and unusual absolute value example

Last week I wrote about some of the fun we were having with our new 3D printer:

The main example I used was our print of the shape for “Prince Rupert Cube” problem.  We’d learned how to make simple shapes like this one from playing around on Laura Taalman’s outrageously amazing 3D printing blog and from watching  a Wolfram 3D printing video (both linked in the blog post above).  Here’s a picture of a cube passing through another cube that is (or was!) exactly the same size:


I wanted the original blog post to include the Mathematica code we’d used to construct this Prince Rupert cube shape, but I just couldn’t find the right way to talk through it.    Since the main purpose of the post was to talk about some of the things we were learning, the details of the code seemed secondary anyway and I just left it out.

BUT, a few pieces of the code were actually pretty neat and writing about it was still kicking around in the back of my mind this week.  Then a new blog post on absolute value from Fawn Nguyen today brought it right back to the front of my mind!

So, here’s one of the pieces of the code that was really neat.

In order to create the Prince Rupert cube in Mathematica  the first thing we needed to do was to draw a solid cube.   Honestly I wasn’t sure how to do that (in no way am I a Mathematica expert), but luckily drawing a cube was one of the examples in the Wolfram 3D printing video from post I linked above.  Lucky break for us!

Here’s how the Wolfram folks made a cube in just two lines.  The first line defines the cube, the second line makes the 3D plot:

cube[{x_, y_, z_}, r_] := (Abs[x] < r && Abs[y] < r && Abs[z] < r)

RegionPlot3D[(cube[{x, y, z}, 1], {x, -1.5, 1.5}, {y, -1.5, 1.5}, {z, -1.5, 1.5}, Mesh -> None, PlotPoints -> 50]

That code draws this nice little picture:


I thought the definition of the cube in the first line was neat example for math students because of the interesting (and unexpected to me at least) use of absolute value.  The function “cube[]” takes two inputs.  The first is a point (x,y,z) in 3 space, and the second is a distance r.  The function tells you that if the point (x,y,z) satisfies all  three of these conditions:

  1. the absolute value of the x coordinate is less than the distance r,
  2. the absolute value of the y coordinate is less than the distance r, and
  3. the absolute value of the z coordinate is less than the distance,

then the point (x,y,z) is in the cube.  If any of the conditions are not satisfied, then it tells you that the point (x,y,z) is not in the cube.

I think this function naturally leads to a lot of interesting absolute value questions.  Such as:

Why do these three conditions define a cube in the first place?

Where is the center of this cube?

What are the coordinates for the corners of this cube?

What if I wanted to make a box whose width and height were equal, but whose length was twice as big, how would I do that?

How could i have defined this cube without using absolute value?

There are probably many other natural questions, too.  Definitely a fun and non-standard example for absolute value and one of the neat ways that 3D printing makes you think through some interesting math.

Happy Square Root of 10 day!!

\pi day was a just a few days ago.  Maybe I was paying more attention this year, but it seemed like there was quite a backlash against it this year.  Oh well, we had fun with it anyway, and no one eating the pie seemed to object!


But yes, some of the objections about \pi day are valid and inspire fun little tweets like these:

Actually, wait, that’s a good idea – let’s celebrate \sqrt{10} day, too!!

First, the square root of 10 is irrational:

That was fun and full of giggles, what’s next?  How about representing \sqrt{10} in an easier way with continued fractions:

Maybe we can make every day a fun math day!!

Pi Day

I have to head in to work early tomorrow so I was looking for a little fun math  to do tonight with the boys.  Numberphile came to the rescue when they posted some of their  \pi related videos on Twitter.  Their Buffon Needle video caught my eye:

As usual, a really nice job by Numberphile.

This problem is one of my favorites from high school calculus and I thought it would be fun to try to do the experiment with the boys. We watched the first couple of minutes to get the hang of what we were supposed to do and then went off to throw some straws as construction paper.

There are, I’m sure, 100’s of fun little math related activities that people will do tomorrow for \pi day.  This seems like an especially fun one to do with kids since it just involves counting.  The appearance of \pi in this  problem shows one neat connection between counting and geometry and also shows that \pi isn’t just about pies 🙂

Steven Strogatz on Word Problems

Patrick Honner’s interview with Steven Strogatz in Math Horizons is a great (and, sadly, way too short) read:

[update:  Steven Strogatz sends the following link for the complete interview:

Click to access strogatz_feb14.pdf



Sitting outside of both the worlds of education and academia but teaching my own kids, I try to spend as much time as I can reading about what others have to say about math education.  This part of the interview really stayed with me:

“[Strogatz] It just struck me as unbelievable that we’re doing word problems in one night’s homework. Students should spend at least two to three weeks on word problems. They’re hard! Every old-fashioned word problem is being thrown at her in one night.”

Funny enough I just started word problems with my youngest son last week and had practically the same reaction.  Right now we are in week two and you can already see that he has quite a bit more comfort in approaching simple word problems than he did ~10 days ago:

MathProblems120: from March 4th:

and MathProblems127 from today:

I’ll freely admit that I know nothing about how kids normally adapt to word problems, but I’m happy that he’s starting to think though the problem before diving in.  I’m definitely looking forward to at least another week of these word problems.

So my own anecdotal experience leaves me in agreement with Strogatz – one day of word problems just seems wrong.