My nieces and I Skype weekly to chat about mostly math and sometimes other things.  Here’s how one conversation went back in January:

Me:  What did you learn in math this week?

Niece:  Skeleton arrays.

Me:  Like for multiplying numbers?

Niece:  Yes.

Me:  Ok show me how you would multiply 12 and 23.

Squeak squeak squeak on her white board and a few minutes later she held this up to the camera so I could see the result:

I was astounded.  It took a little longer than the old method I had learned back in 1970-something, but it was so visually compelling, and she got the answer correct!  A few days ago I asked if she could multiply 36 and 17 using an array.  I wanted to see if, in the midst of summer vacation, she could remember the area model from the school year.  Plus, I wanted to see if she could do a more challenging two-digit multiplication problem.  This is what I got:

Back in January something else interesting happened too.  As our algebra one team met to discuss the topics we’d be covering for the next week, we all decided to teach polynomial multiplication by setting up an array instead of showing it via double and triple distribution.  For someone on the verge of teaching algebra and calculus for 30 years, this was a big concession on my part since I usually just distribute using the method I learned, also in 1970-something.  Take for example the multiplication problem (2x + 5)(3x – 7).  Here are two different ways to arrive at the answer:

 

 

 

Ok then back in January something else happened.  I was teaching calculus and we were figuring out the area underneath a curve geometrically and then building to the integral where one is taking infinitely many heights, f(x), and multiplying each of them by the infinitely small (almost zero) width, dx.

 

multiplying polynomials, high school

integrals, calculus

 

 

Special request I wrote for a colleague:  distributing and combining like terms circuit