### Mathematical Realizations

NOTE: S stands for students, it was not just one voice speaking but many different voices

Me: Look at the the two tables, one with a fixed perimeter and the other with a fixed area.  What do you notice?

S: The numbers on the right table are changing.  There is 9 in between, and then 7, and then 5, and then 3, and then 1.

S: Yeah, it is like a pattern.  But then the other side is ummm (calculating) 34, 10, 4, and 2 in between, so there is no pattern.

S: I see something weird.

M: Is it okay we change the subject and look at N's weird thing?

S: Yes.

M: Go ahead please.

S:  when the area is 36, the perimeter is 24, on the left table, and when the area is 24 the perimeter is 36 on the right one.  They are flipped.

Ss: Cool.  Why is that Mr. D?

M: Well, what do you think?  Look hard at the chart, you may see something special about those two shapes.

S:  Oh!  They are squares!  The others are rectangles and they don't flip.

S:  And look at this, in the area table, the square is the smallest number, but in the perimeter table it is the biggest.

M:  Wonderful observation!  Everybody think about that.

S:  (long pause) I don't get it

M: Imagine you are a farmer, and you're building a fence for you pigs.

S:  Well, the square will give us the biggest area, so the pigs get more room.

S:  It also uses the least amount of fence, so it is cheaper and and we can build it faster.

S:  That's why so many buildings are squares!  You get more space and use less wood!

What a great moment!  I had no idea that was coming, I was just simply hoping they would notice that when one value is kept constant, the other can vary.  It is moments like this that makes this job the best job in the world!