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

14x8.

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.

Then, we start to move away from the visual, to

the number driven math:

(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.