I just taught a class on closed properties (ex. the set of Natural Numbers is closed under addition).  Unfortunately, not having a lot of time to plan, I just ran through the notes, and way over the kids’ heads.  Upon further thought, I would’ve explained it like this:

A set of numbers is like a classroom.  (this illustration works best for a finite set of numbers)

For example, I have a set {-1, 0, 1}.  Pretend that the {} are little doors into the room.  Now, we’re going to perform all the operations on these numbers.  If we come up with an answer that is still in this room, the doors stayed CLOSED (we didn’t have to open them to find the solution).

Let’s try it:

• Addition: -1 +1 = 0; -1+ -1 = -2 **-2 wasn’t already in the room, therefore we had to OPEN the door to let it in (not closed!)
• Subtraction: 0 – -1 = 1; 1-1 = 0; -1 – -1 = 0; 1 – -1 = 2 **2 wasn’t already in the room, therefore we had to OPEN the door to let it in (not closed!)
• Multiplication: 0 x 0 = 0; 1 x 1 = 1; -1 x -1 = 1; -1 x 0 = 0; 1 x 0 = 0; 1 x -1 = -1 ** because I tried every possible combination of numbers, and my solutions were always already in the room, my door stayed CLOSED
• Division: 0 / 1 = 0; 1 / 0 = does not exist **this answer obviously isn’t in the room, therefore we had to OPEN the door to let it in (not closed!)