There is no escape...believe it or not. Mathematics is everywhere.
Well even though I'm not a teacher, and I even agree with what you said, I need to disapprove the second method (walking).
The best way I think is to go with my example, it is a basic principle that even a 5 year old would understand. It requires no previos knowlage, and you don't have to walk a mile to understand it. Certanlly, as a student- I appreciate correct methods that will produce best possible results in small amount of time.
For example, if (-1) * (-1 ) would be ( -1 ), if that was a convention, ( " The enemy of my enemy is my enemy " ) would be in contradiction to all things known to man, even politics doesn't work that way :). And try explaining distribution law, using your mentioned " walk " method... Or you coud try to explaing in what cases, (-1)^n would result in 1, or in (-1).
Another thought-observe the following pattern
-5 x 3 = -15 = (-5)+ (-5 )+ (-5)
-5 x 2 = -10 = -15 +5
-5 x 1= -5 = -10 + 5
-5 x 0 = 0 =-5 + 0
-5 x -1 = 5 = 0 + 5
-5 x -2 = 10 = 5 +5
using fundamental def. of multiplication as addition we can conclude product of two negatives is positive
You could use basic reasoning and explain this :
But we all know it is a convention ( -1 ) * (-1) = 1 ...