I tried to teach my son multiplication using a rectangle. (e.g. 3cm * 4cm = 12cm^2). Now I have 12 little squares. But how do I explain where the "little square" came from?
My best guess: If I cut a little square in two, I can get a rectangle 2cm long * 1/2cm deep. Then I cut that rectangle in two, I get 4cm long * 1/4 cm deep. I keep going and say I get a rectangle 100,000cm long by 1/100,000 cm deep.
This is very long and very thin. And I deduce, that the limit would be a line. And so I conclude that the area of a line = 1. (units become irrelevant, infinity takes care of that).
I also like this to explain why the integral of y=0x+1 (for x from 0 to 1) is 1. Note: this should be a 1 unit square, and integrals are an area but we say the answer is 1 not 1 unit^2.
My question: Do you agree? (if not, why not?)