Measuring Pi

After yesterday’s Family Math project I was thinking about a project with spheres so that we could talk about the area of the spherical caps from our printed shapes.  This morning i changed my mind and thought that a slightly more laid back project was in order, so we spent the morning trying to measure \pi.

The first thing that we talked about was basic definitions.  Just trying to set the stage for thinking about geometry, especially since I’ve not really spent much time talking about geometry with my younger son:

After getting through the definitions we started measuring.   We started with a can of chickpeas and then talked for a bit about why doing the same measurements with a cube would be different.

Now we moved on to some larger circles.  This required a larger area, so we moved to the garage.   Our first attempt here didn’t go so well as our estimate for \pi was about 3.5.  I wasn’t too disappointed, though, since learning that measurements don’t always produce what you expect is an important lesson.

Our last prop was a bicycle wheel.  This experiment required a little bit more room than our camera could handle, so we split it into three pieces.  We got an estimate for \pi that was a little low, but it the best estimate of the bunch.    After we finished the calculations on this one we talked through a few of the aspects of our measurements – is it easier to get an estimate for \pi with a large circle or a small circle, for example.

Definitely a fun little set of experiments.  Fun to see the boys rolling up their sleeves and taking a few measurements, too.

James Tanton’s geometry problem and 3D Printing

This past winter we saw this really amazing video (posted by Patrick Honner)  about how students were using the 3D printer at Brooklyn Tech:

Inspired by this video we decided to get our own 3D printer and see what we might be able to do with it.  The first lesson was easy – follow Laura Taalman’s 3D printing  blog:

From following her blog we saw incredible example after incredible example of how 3D printing can help open math doors that weren’t so easy to open before.  Using what we learned from Taalman’s blog, the first project we tried on our own was about the Prince Rupert Cube.  We wrote about some of the fun we were having here:

The second project that we did on our own came from a project that James Tanton posted on Twitter last week:

The problem itself is a little too advanced for the kids, we haven’t really discussed geometry at all,  but just trying to understand the shape of this object proved to be a fascinating exercise.  Again, using what I’ve learned from Taalman’s blog, I made a sequence of 5 shapes on Mathematica that help you build the shape up in your mind:



After a little bit of travel for everyone in the family this week we were finally able to sit down this morning and talk through Tanton’s problem.

In the first video I introduce the kids to the problem and talk through some of basic geometric ideas in it.  What is a tetrahedron?  What is surface area?  What does it mean that the area of each face is 1?  The nice thing about this problem is that the basic ideas are accessible to kids:

In the second video we take a quick look at a slightly easier problem – what if we were trying to find the points in a plane with a distance 1 from an equilateral triangle?  This just came up on the fly, but it turned out to be a really valuable detour since it gave us some nice insight into the 3D shape.    In fact, after reviewing this video I may come back to the 3D problem tomorrow to show how the pieces of the spheres come together since it is so similar to what we discussed in the 2D case.

Next we moved on to the 3D printed shapes that I made this week.  I think being able to hold these shapes in your hand really helps to get your head around Tanton’s problem.  As explained in the Brooklyn Tech video, learning how to write code that produces these shapes is a great math activity all by itself.  This would be a super fun project for students learning about linear algebra, 3D geometry, or even just learning to program in Mathematica.    I also think that walking through these 5 shapes we printed is a nice example of breaking complex problems down into simpler ones to help you get to the solution:

All in all, a really fun exercise and a neat example of how 3D printing can help understand some fun geometry problems.  Thanks to James Tanton for posting this great problem.

Jo Boaler’s exercise with snap cubes

[note:  sorry this is a little rushed – trying to get this published before heading up to Boston for the weekend]

Earlier this week I saw a new video posted from Stanford Math Education professor Jo Boaler:

The video (which hopefully you can click to in the above link) includes a Fawn Nguyen-like counting exercise that I thought would be fun to try with the boys.  For many similar exercises, check out Fawn’s amazing site:

We started off just talking through the problem.  In retrospect I should have set up the shapes first, but these are the things you don’t think about at 6:30 in the morning!  It was interesting to hear both of the kids try to explain the pattern in words.

I’d intended to proceed as Boaler does in her video, but my older son happened to notice hat the number of blocks in each step was a perfect square.  My younger son also picked up on that fact quickly, so we jumped into talking about the squares right away:

Talking about the squares led to a slight diversion to see if there was a different geometric way to see that 1 + 3 + 5 + 7 + . . .  (odd integers) always adds up to be a perfect square.  This is something that we’d talked about previously, but talking through that point one more time felt pretty natural here:

Finally, we returned to the original problem and looked for new ways to think about the patterns we saw there.  I thought that the last videos sort of got the kids anchored to thinking about the pattern in rows, so I switched to having the shapes all have the same color blocks.  Probably should have been doing that from the beginning.  Sorry for the slight goof up at the end of this video, but at least we caught the mistake!

All in all, a fun problem.  We’ve done many similar little project before, so this type of problem isn’t completely new for the kids – see here for just one example:

Definitely a fun morning before heading out on a quick trip.


Soccer Ball math

Five Triangles posted a neat picture of a standard soccer ball earlier this week:

Seemed like building the shape out of our Zometool set would be a fun exercise after a week of camping, so we gave it a shot this afternoon:

After introducing the problem we started building (off camera).  We’ve done a few other fun exercises with our Zometool set and actually just bought George Hart  and Henri Picciotto’s “Zome Geometry” so the kids are pretty familiar with building structures out of the Zome pieces.  The only trick for this little exercise is that you want to start with edges that are three times the normal length to make the truncation easier.   Also, once you have the icosahedron, it isn’t so obvious where the soccer ball is hiding:

Next we move to the truncation.  Since we started with side lengths that could be easily divided into three parts, truncating the icosahedron isn’t that hard.  It is, however, incredibly interesting to see the “soccer ball” shape emerge from the icosahedron.  The kids were surprised to see that “the pentagons made the hexagons.”  Here’s a peek from about half way through:


After we finished building we did a quick wrap up and talked about a few other questions that we could ask about our new shape – things like the number of edges, or number of pentagons.  I also asked them if they thought the Zometool shape was actually the same shape as the soccer ball and was surprised to hear that they thought it wasn’t.  We talked about that for a bit, too:


All in all a fun little geometry exercise.   Didn’t want to go into too much depth here since they just got back from a week of camping, but even without the depth they seemed to find all of the building to be really engaging.  Thanks to Five Triangles for the inspiration.


Area, Perimeter, and Fence Posts

A couple of years ago we stumbled on a pretty neat section of Khan Academy that talks about basic counting techniques:

I remember being surprised at how difficult it was for me to explain the concepts in this section to my older son.    Of course, I struggle to explain lots of math topics, but the struggle talking through these problems really stuck with me.  We even went outside to our deck and actually counted fence posts to try to help make the problems more real.

This week we ran across another “fence post” problem playing around with some old AMC 8 exams on the Art of Problem Solving site:

This problem gave me the idea for our Family Math topic for this weekend – area, perimeter, and fence posts.

As we often do, we started out with snap cubes.   After some informal talk about a few basic geometry concepts, we made a little shape out of 4 snap cubes and talked about area and perimeter.  I was happy that we were able to find  two completely different ways of calculating the perimeter of the shape:

Next we talked about fence posts.  Watching the video just now, I’m not thrilled with how well I explained the fence post idea, but hopefully the example with the line was helpful.  The main thing I was trying to show is that counting the fence posts isn’t always super easy.  With a straight line you get one answer and with a closed “loop” you get a different answer.    Not necessarily intuitive, but there is a nice relationship with the perimeter (at least for the simple shapes we are looking at today).

and having dealt with the cat distraction,  the dog gets into the act:

finally, can we finish up with out another distraction . . . . :

So, despite the distractions, this was a pretty fun project.  I remember the difficulty walking through problems like this a few years ago, and I can see that both kids remain challenged by this type of problem.  Hopefully playing around with the snap cubes and the Penrose tiles help them get a better feel for each of the three geometry concepts we were talking through today.

Pascal’s triangle and counting part 2

I’ve recently finished both Ed Frenkel’s “Love and Math” and Jordan Ellenberg’s  “How not to be Wrong.”  Both books are excellent and have given me lots of things to think about when it comes to communicating math ideas.  One theme that plays an important role in both books is that many different areas of math are tied together in unexpected and really interesting ways and these connections between seemingly unrelated areas of math can provide important insights in problems solving.

Pretty sure the first time I ran across this idea was reading Coxeter and Greitzer’s “Geometry Revisited” back in high school.  The section on projective geometry has a chart that helps you translate between “regular” geometry theorems and dual theorems in projective geometry.  Can’t say that these abstract connections made sense to me in high school, but with the nudging from Frenkel and Ellenberg I think I’m starting to get a better appreciation for the importance of communicating connections between different areas of math.

And that brings me to our little Family Math talk from yesterday about Pascal’s triangle:

Though I wasn’t really thinking about connecting different areas of math in this talk, we did talk about a lot of seemingly different ideas more or less by accident – Pascal’s triangle, counting blocks, binary numbers, and polynomials.  After finishing Ellenberg’s book last night I was thinking about how I could do a better job talking about relationships between different areas of math and realized I had a great example that was right in front of me and ready to go.

So, today I wanted to show how all of the different ideas in yesterday’s talk could be come together to help us think about a counting problem that seems to be much more complicated.   The problem involves counting blocks again, but this time we have two copies of each block.   Here’s the introduction:

After the quick little introduction we dive right into the problem and try to count the number of ways we can pick some small sets of blocks.  Hopefully these relatively simple examples help illustrate the new problem as well as some of the complexity that arises when you have multiple copies of the same block.   From some of the simple examples we construct a new Pascal-like triangle that is a little bit different from what we saw yesterday:

Next we started looking for patterns.  The first one that the kids saw was that the rows in our new triangle add up to be powers of 3 rather than the powers of 2 we saw in Pascal’s triangle.  That pattern presents a nice opportunity to show how each of the subsets correspond to a number in base 3 (though I give this point about 1 second of screen time . . . . whoops!).    The next pattern we talk about is the relationship between one row and the next one in this triangle.  This relationship allows us to speculate on the numbers in the next row rather than trying to find all of these numbers by counting blocks:

We left off in the last video thinking that we found a relationship between rows in our new triangle.   The next thing I wanted to show was how the connection between rows of Pascal’s triangle and powers of the polynomial (1 + x) could help us understand how the rows in our new triangle were related to each other.  This part is surely the most difficult part of kids, but I thought it would be fun to talk about anyway.   I only saw this connection between polynomials and block counting in college, so I’m hoping that introducing connections like this really early on will help the boys think of polynomials as more than just abstract math symbols on the whiteboard!

Finally, we end up on Wolfram Alpha multiplying out polynomials.  Sure enough, we are able to replicate the numbers that we found in our triangle.  We even find the number of 5 block subsets from 5 pairs of 2 blocks – something that would have been super hard for us to count directly.  Sorry the computer screen is hard to see on this one – I published this video in HD to hopefully make the small numbers on the screen easier to see.

So that’s my first try at presenting a problem that has a fun and surprising solution that draws from several different areas of math.  Having seen the emphasis that both Frenkel and Ellenberg put on connections in mathematics, I’m going to try a little harder to emphasize these connections in these little math projects with the kids.


Pascal’s triangle and some fun counting for kids

Last night I asked my younger son what he’d like to learn about in today’s Family Math and his answer was Pascal’s triangle.  Since we just started our little summer project on counting and probability, this was a timely suggestion.

We started with a simple review of how you create the triangle and also talked about some simple patterns.  This video went a little over 5 minutes because my younger son noticed an interesting pattern with prime numbers that took an extra minute to explain:

The next part of today’s talk was relating the numbers in Pascal’s triangle to ways we can pick groups of objects.  We illustrated our groups with snap cubes.  After a little introduction to ways to choose groups from sets of two and three objects, we show that the main identity in Pascal’s triangle – that two adjacent numbers in a row add together to get the numbers in the next row – can be understood in terms of selecting groups.   I’m not sure how clear the explanation was, but I hope it made sense:

The next step was to show one way that picking groups of blocks can help us understand why the rows in Pascal’s triangle always add up to be a power of 2.   This fact is a little easier to understand that the example in the last video (as long as you know binary).  In retrospect, I should have done this identity first.

Finally, we wrap up back at the whiteboard talking about why we see powers of 11 in Pascal’s triangle.  We actually did an entire Family Math about this fact a while ago:

I was pretty happy that my son remembered the powers of 11 in the first video, so I was really happy to be able to do go over this idea again.  We revisited why those powers show up (which involves a short discussion of polynomials) and then use the same idea to compute a few other similar computations.  (and ugh!  sorry we went off of the bottom of the screen at the end 😦  )

All in all a fun morning talking about Pascal’s triangle.  It is always fun to revisit old topics and dive a little deeper than we did previously!


Fawn Nguyen’s influence

A few weeks ago I asked the boys what math they wanted to do for the summer.  They decided on a summer project that they could work on together rather than the separate algebra / prealgebra math we did during the school year.    Not the easiest task in the world given the 2.5 year age difference, but I ended up settling on a slow walk through Art of Problem Solving’s “Introduction to Counting and Probability.”    Some parts of the book might be pretty challenging (and skipped for now!), but it looks like we’ll have a fun summer of basic combinatorics.

Book Pic

I got a nice surprise right off the bat and it showed quite directly the value of all of the work we’ve been doing with Fawn Nguyen’s material.  Particularly the visual patterns:

The first section in the book discusses counting lists of numbers.  One of the slightly complicated problems was counting the numbers in this list:

3 2/3, 4 1/3, 5, 5 2/3, . . . , 26 1/3, 27

My older son’s solution had a really nice grouping strategy that surprised me and clearly showed Fawn Nguyen’s influence:

Definitely going to be a fun summer!

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:


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:

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:

(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!