>I have also provided a precise verbal mapping of the instructions that reveal the logic of why the products of both -ve x - and - x -ve are positive.
Perhaps you can rationalize it - you can't prove it in the mathematical sense without making prior assumptions. And its not the only possible rule.
But there is a obvious connection of the infamous sign rule that receives strong support from the usual verbal, and logical, concept of negation. If I do not, not go to the store, the usual interpretation is I went to the store. In logic, if its not the case that ~P, I conclude P. This of course should be brought up in any lesson plan introducing* the infamous sign rule.
*Some people don't know what "introducing" means. I can't help them, but obviously, these topics are things we expect may a have a long association with the students career. The word "introducing" makes perfect sense. Courses are regularly titled "introducing ...". People who can honestly confuse this usage with introducing one's cousin, or see it inherently in conflict with "teaching" somehow, are truly gifted in a weird negative, or ~ sense. ~Congratulations to them.