How to explain the commutativity of multiplication to middle school students?

Solution 1:

First do it with integers: make a rectangular array of dots, then turn the rectangle ninety degrees. Now instead of an array of $n$ rows, each with $m$ dots, it's an array of $m$ rows, each with $n$ dots. But the number of dots didn't change, only the way they were arranged.

More generally, a rectangle with real-length sides doesn't change its area when you rotate it a quarter-turn.

Solution 2:

One option, which my teacher did with us (8th grade, so may not work so well with 6th graders), is to show the class a system where multiplication (or perhaps even addition) is not commutative, like matricies or 3D vectors. After showing them that that type of multiplication is not commutative, it becomes much easier to understand that there is something special about multiplication on the reals.

The entire rest of the field axioms can be handled the same way, if you can just show systems that don't have all the properties that they take for granted.

• commutativity of addition
• distributivity of multiplication over addition
• associativity of addition/multiplication
• existence of identity/zero
• existence of an equivalence relation
• existence of an order relation
• trichotomy (for any $a,b$, exactly one of $a < b$, $a = b$, or $a> b$)
• transience of equality
• substitution ($a=b \oplus c \wedge c=d \to a=b \oplus d$)

Although most middle schoolers might not be willing to learn this sort of stuff.