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.