# The Area Model

My nieces and I Skype weekly to chat about mostly math and sometimes about 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: NOTE:  Recreated for the blog post since I didn’t have the foresight to screen shot it at the time.

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: 36×17=612

Wow.  The rectangular array method stuck.  Notice the advancement in the array and the self-awarded smiley faces.  If that isn’t a testament to using the area model to do multiplication, I don’t know what is.

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.  [NOTE:  If a student wanted to multiply using a different method, that was fine.]  Take for example the multiplication problem (2x + 5)(3x – 7).  Here are two different ways to arrive at the answer: They way I learned to multiply binomials, and the way probably most mathematicians do it minus the arcs and maybe even minus the middle step since we can hold a lot of the steps in our heads. The way we as a department taught it this year to 220+ algebra one students.  It worked extremely well.

I absolutely loved teaching multiplication using what we collectively called “the box method”.  I know, I know, it’s not a box because a box is 3-D, but this was just our internal slang for the method.  All of my students were just young enough to have learned the area model for multiplication back in elementary school because that’s when the Common Core State Standards came in.  In addition, many students were concurrently in biology learning genetics and you guessed it, Punnet Squares.  When we moved to a binomial times a trinomial or even a trinomial times a trinomial, my students got those more labor-intensive problems correct without any help from me.  They even reverted to the area model for a monomial times a binomial, though I never demonstrated it that way.  I had always assumed that my students would think that drawing a rectangle to multiply would be too much work, but not one peep of complaint did I hear.  Caveat:  We taught factoring polynomials not by using the box but by “split the middle” which turns any trinomial into a grouping problem.  You can watch the technique here.

Ok, then back in January yet a third thing happened.  I was teaching calculus and we were figuring out the area underneath a curve geometrically, motivated by finding distance traveled from known velocities, building up 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, to determine the area of irregular figures.  When we then did average value, where I visualize any irregular shape squishing into a rectangle, there was the area model AGAIN.  My head was about to explode.  I mean, I had thought about it all before, but I hadn’t thought about it all at the same time across grades 4, 8 and 12.  Let’s just say, it was a lot of area for one month. Finding the area of an irregular shape is pretty much just summing up infinite rectangles, each of width nearly 0. And here is the rectangle with the same area as the irregular shape, and the same base (b – a).  The average value is the average height of the function.

That’s all for now.  Back to summer reading.

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