For subtracting, I like to talk about owing money. If you’re at the restaurant, and you have a bill for $45 (-45), and I come along and say, “Hey, let me pay that for you.”, I would take away that negative: -45 – (-45) = 0. You now owe nothing.
You can do similar things for multiplying. 3(5) means I give you three 5 dollar bills: net result, add 15 (+15) 3(-5) means I give you three IOU slips (each owing $5): net result, lose 15 (-15) -3(5) means I take away three of your 5 dollar bills: net result, lose 15 (-15) -3(-5) means I take away three IOU slips: net result, gain 15 (+15)
getting something good is good getting something bad is bad losing something good is bad losing something bad is good
I learned that last one from a student, and who knows where he picked it up.