Pascal’s Triangle and Powers of 11

The most difficult thing about teaching my kids, by far, has been that I have no experience at all teaching elementary math.  When a concept is difficult for either of them to understand, quite often I struggle to work out exactly what it is that they struggling to understanding.  But we muddle along.

One obvious consequence (and sometimes it is a “feature” and other times a “bug” !!) is that I have no idea at all about the accepted ways to teach some of the most basic subjects.  Fairly often I just let them figure it out the elementary stuff on their own.  Basic arithmetic is a good example of a subject where my older son came up with his own methods (which have nothing to do with borrowing and carrying).  Since his ideas were perfectly fine mathematically, I just ran with ran with them.   Here are three examples that illustrate his method:

(1)  356 + 672 = 900 + 120 + 8 = 1028,

(2)  532 – 384 = 200 – 50 – 2 = 148

(3) 25 * 13 = 200 + 60 + 50 + 15 = 325

I think the main disadvantage of this way of doing arithmetic is that it is a little slow, but there are at least two nice advantages.  First, the approach highlights place value and therefore made it very easy to talk about arithmetic in other bases.  Second, this method looks pretty similar to the way we normally do arithmetic in algebra, so learning to multiply polynomials, for example, was not particularly difficult.

Yesterday I got a nice surprise when we stumbled on a new problem where this arithmetic method added a lot of value. We were doing some review work and one of the questions was simply to find the cube root of 1,331.  My son told me that he new the cube root of 1,331 was 11 because 1,331 was a row of Pascal’s triangle.

I love opportunities to talk math that come out of the blue, and this was as good an opportunity as any.  I’m not actually sure where the connection between powers of 11 and Pascal’s triangle came from since we haven’t talked about Pascal’s triangle in a long time.  However, as luck would have it, we’d spent the last couple of months talking about polynomials, so we were primed for a fun discussion.  It turned out to be even better than I’d hoped!

The first question I asked him was what he thought 11^4 was.  He drew out Pascal’s triangle and said he thought that 11^4 would be 14,641.  Fine, but the next row of Pascal’s triangle is 1 5 10 10 5 1, so what would 11^5 be?    Since the method of finding powers of 11 using Pascal’s triangle now appears to break down, he proceeded to calculate:

14,641 * 11 = 146410 + 14,641 = 100,000 + 50,000 + 10,000 + 1,000 + 50 + 1 = 161,051.

My plan, of course, why the connection to Pascal’s triangle has disappeared, but his unusual method of addition meant that the coefficients of Pascal’s triangle were right there on the board!  Ha ha, the joke was on me.  We revisited the calculation this morning:

Following that discussion this morning, we spent a few minutes connecting polynomials to Pascal’s triangle and showing why the powers of 11 are hiding inside the triangle.  Definitely a fun and surprising weekend of math!!

For me moments like these have always been the best part of teaching.  As I’ve said many times, I’m glad that I have the time and flexibility to teach my kids.

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