My eighth and ninth graders know how to use the associative and commutative properties of addition and multiplication, they just don't know what they're called. You know how to use them, too. Here, I'll show you:­

If someone gives you a bunch of numbers to add, like

6 + 8 + 1 + 4 + 2 =

I bet you look for groups of ten; the six plus the four, and the eight plus the two. So far you've got two groups of ten, which makes twenty, and then one more, so the answer is 21, right? It makes the problem easier to do it that way. What you're doing, to use the mathematical terms, is re-arranging the order of the numbers and grouping them together in the most convenient way. You're using the commutative and associative properties of addition, respectively.

When I introduce these properties to reluctant PreAlgebra and Algebra students, I don't even use numbers. Or letters. Zeeesh, variables are even more confusing. What I do is pass out magic markers and a worksheet full of little boxes with plus and times signs scattered throughout, and the occasional bracket or parentheses. Each kid gets three different colors, and then line by line, we fill in the boxes:

red + blue = blue + red

This is the essence of the commutative property of addition. As long as you're adding, you can switch the order of the numbers (variables, colors, etc.) without changing the answer.

red x green = green x red

It's true for multiplication, too. Notice that the first two letters of COMMUTATIVE are C and O, and the commutative property is all about the order of the numbers. It's like, if two friends are walking together, does it really matter which one goes through the doorway first?

We check, does this work for subtraction?

Is 10 - 5 the same as 5 - 10 ?

Nope, this won't work for subtraction. Won't work for division, either. Just addition and multiplication. It only works for the operations that make the answer bigger, not the operations where the answer is smaller. (We're going to assume whole number operations for the time being.)

Okay. Onward, to the associative property. The associative property has to do with grouping. Who do you associate with? Your group of friends. Let's look:

( red + blue ) + green = red + ( blue + green )

This is a little harder to see, but the order of the colors didn't switch, just the parentheses. If you say the colors out loud, it helps: "red plus blue plus green, equals red plus blue plus green." The only difference is, first the red and the blue were all chummy, and the green was left out, and then after the equal sign, the red was by itself, and the blue and the green were hanging out together.

At this point, a little class participation helps. I call for three volunteers, and have them stand together in front of the class:

( Rachel + Andrew ) + Julie = Rachel + ( Andrew + Julie )

At first, I ask Rachel and Andrew to stand really close together, with Julie off a bit by herself (there's a lot of giggling at this point; these are 13- and 14-year olds) and then Andrew leaves Rachel's side and goes and stands next to Julie. They don't switch places, they're still standing Rachel, Andrew, Julie, it's just that Andrew slides away from Rachel, over closer to Julie. Rachel looks a little miffed, and I play it up; "How would you feel if you were Rachel? One minute, Andrew is walking you to class and being all nice, and then when class is over he leaves you and walks off with Julie."

I turn to Rachel, and point to the word Associative written on the board. "If I were you, I might have a name I wanted to call Andrew."

There are some red faces and more giggling by now, but they've got it. We repeat with multiplication, different colors, different placement of the brackets:

green x ( red x blue ) = ( green x red ) x blue

This time, it's red who's being a cad.

Will this work for subtraction? Let's check:

12  -  ( 6 - 2 )  ?  ( 12 - 6 ) - 2
   12 -  4                  6 - 2

        8   is not equal to   4

It doesn’t work for division, either.


Now, I know I’ve probably horrified lots of real mathematicians who will tell you that I’ve incredibly over-simplified these properties. Here’s my rationale: I’m dealing with teenagers. They spend a lot of time thinking about relationships, but not number relationships. Most of them are in school because it’s where they see their friends, and in Algebra because their parents want them to take it. I’m trying to keep them awake and engaged and, with any luck, feeling like math isn’t such a foreign language. I’d like them to feel a sense of competence, and maybe even enjoyment. Sometimes, I feel like I succeed.

(Sometimes I fail miserably, but at least I've amused myself.)

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