A Meditation on the Distributive Property

April 17, 2020

But first, a quick announcement

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We’re still going, though, so you can still get in on the campaign, and let others know about it. We also will be rolling out STRETCH GOALS! You can check out the campaign to find our more.

Meditation on the Distributive Property

In other news, I’ve been thinking about the distributive property. It’s usually summarized algebraically as:

a x (b + c) = (a x b) + (a x c)

A more general formulation is the old “FOIL” rule you learned in algebra class:

(a + b)(c + d) = ac + ad + bc + bd. (First, Outside, Inside, Last. FOIL!)

It’s possible to develop an intuition for the algebra, but not everyone does. And students of all ages tend to mess up these rules and formulas. Wouldn’t it be nice if there were a visual intuition for all of this?
Let’s look at what happens when we introduce some good visuals. Like arrays, for instance.

It might just be me, but I can actually SEE that the two cards on the left equal the cards on the right. I’m just taking a 5 by 5 array and chopping it into two pieces with a vertical cut, right?

I can express this almost as clearly in words if I say what I see: 3 groups of 5 plus 2 groups of 5 is 5 groups of 5. Or equivalent, (3 x 5) + (2 x 5) = (3 + 2) x 5 = 5 x 5.

This is just the distributive property again! In fact, that’s all the distributive property really says: that a full array is the sum of the two smaller arrays you get when you cut it with a single cut.

What about FOILing then? Let’s take a look at those pieces, and see what happens if we cut them into even smaller pieces.

Here’s 3 x 5 seen with a horizontal cut, as (3 x 3) + (3 x 2).


And here’s a similar cut for 2 x 5.

Now let’s put all those small pieces together.

It’s pretty clear visually that the four cards on the right would add up to the card on the right. Right? I just made a horizontal and a vertical cut, and separated the four pieces.

If I were to write out an equation for this, it would be (3 + 2) x (3 + 2) = 3 x 3 [First] + 3 x 2 [Outside] + 2 x  3 [ Inside] + 2 x 2 [Last]. It’s FOIL! But this time, we don’t have to do any of the annoying memorizing. We just make the natural observation that when you cut a whole into pieces, the pieces add back up to the whole. In other words, we could have expressed those algebraic expressions visually this whole time.

This is what the savvy teacher knows. Visual argument tends to be much easier to intuitively grasp. And when students intuitively grasp what they’re doing, they do better in math, enjoy it more, and identify as “math people.” The way to prep students for understanding math is to use the right visuals all along.

There ends my meditation on the distributive property.

Now if you’re looking for a puzzle to see how well you (or a kid, perhaps) grasps all this, let me pose one for you. Find four cards (representing arrays) that add up to 5 x 5, but that are not the four cards from this picture.

Four cards add up to 5 x 5. Can you find four different cards (or arrays) that also add up to 5 x 5?

If your kid can do that, algebra is doing to be that much easier.

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