Students have a lot of trouble understanding and mastering negative numbers and how to perform the basic operations on them. Part of the problem, I have noticed, is that students use their ingrained "regular" multiplication skill when trying to find an answer to a negative number math problem. For example, when asked, "What is negative 5 multiplied by 7?" one of the most common wrong answers is 35. Of course the correct answer is negative 35. A negative number, multiplied by a positive number is a negative number, but the force of habit is strong, especially on test day. The question then was, "How do you undo a bad habit and expand it so that it can handle the larger set of numbers?" The answer that I came up with was a "super multiplication table". I expanded the normal, positive numbers only grid of 10 x 10, to a 20 x 20 positive and negative number table. Now the answer to any multiplication question from -10 by -10 to 10 by 10 can be looked up directly on the table, just like that table in your old peachy folder, but more powerful. When I show students this table their memories are instantly rolled back and I can remind that negative number as in fact a totally different world. A super addition table could also be created. I am not saying that I don't also teach "the rules" i.e. "A positive times a negative is a negative", but the table really re-enforces it and show the results and implications of the rules. I hope that maybe you can use this idea in your own math tutoring. Send me a message if you have an even better idea.