I have been a slow (okay, glacial) convert to arrays.
They appear early in the Common Core curriculum.
This is what they look like in third grade, where they
are used for basic multiplication skills.
Then, they move onto this in fourth grade:
They start out on grid paper, and, in the beginning,
students are encouraged to manually count squares.
The numbers being multiplied get larger, and are
broken down into tens and ones. Students are shown
that larger numbers can be broken down into components,
which can be multiplied and then added together.
Often, these components can be easier to work with
mentally. It's easier to think about 10x8 and 4x8 and
adding them (80+32) than it is to mentally multiply
Then they move away from arrays done to scale,
where you can count grid squares, to skeleton arrays
that require multiplication work. Again, larger numbers
are broken down into units of tens and ones, and those
pieces are added together. Is this shorter than doing
an algorithm? No. BUT, this does demonstrate the
varied way that we can pull apart and reassemble
multiplication problems, which is HUGE for algebra.
(I wrote in little number number sentences
to show where the product in each cell came from.)
Here's a purely algebraic version, to illustrate where
the values in each square come from.
Oh, look, distributive property and algebra.
Hey, we're in middle school now! 8th grade, perhaps?
We've got our factored quadratic, and we're going
into standard form. And, voila, it's work we've been
doing since 4th grade. Isn't that nifty??
This is where we would foil in old school math.
If foiling is your jam, then you don't need this, but
if you are a student who looses track of what's been
multiplied and what hasn't, this is an easier to follow
method. (And yes, it works in reverse. You can use this
method to factor a quadratic, but that's a different post.)
I'm not quite done. This method continues to
evolve. Take the following:
Oh yes, imaginary numbers! This is big kid, high school
math, with things I can't visualize! Three factors being
multiplied by three factors! But, our array can handle
it. And look at how pretty it is! I can see my pairs
that cancel so clearly!
The symmetry and cleanness of this method makes
my little dyslexic heart SO happy. SO HAPPY.
Now, look below at the method I learned in school.
You tell me this foiling method is better. That mess
above.. that is how mistakes are made. That is how
elements are lost and math students cry. This is where
the new method is faster, neater, kinder, and sweeter.
Do I always love arrays? No. Sometimes they are slow,
clunky square wheels that make me say, "Put more
math in your math!" to my students, BUT, it has a
purpose and it is a great, powerful tool to have.