If this is a test for "low ability students", it is likely a bad idea to come across with mathematics as something abstract and an end in itself. The prerequisite for higher mathematics is to be fully comfortable with basics (say, arithmetics up to integration).
Hence the usual approaches, and in particular rigour axiomatics, don't lead to the desired educational results. It is better to introduce mathematics as tool, even if it hurts ones understanding of "mathematical purity" and so on. For these people, mathematics is a tool at best, and should be related to applications.
So where negative numbers actually come from? It is probably not by counting things. You do not pile up coins and obtain -5 dollars, as much as you do not measure length and obtain -23.5 meters.
A feeling for negative numbers is better conceived if you introduce a sense of direction and a pivot point.
For example, take a look at the thermometer. You have a pivot point "degree 0" ( in Celsius, I am continental European), and temperature compares with this point. The sign of the temperature tells you whether below or above the pivot, and the magnitude how much. Equally important accountancy. Having a surplus of money or being in debt can be conceived as a sense of balance, around the pivot "0 dollars".
Second, A sense of direction can be perceived if we measure relative positions and movement. Say, on a street you step backward and forward. This can be perceived as moves in positive or negative directions, and in this case, you do not even have a fixed pivot.
Third, I would like to give a perspective partly inspired by geometry. If you have positive and negative numbers, you have an orientation. The sign of the thermometer or your account status are arbitrary. You could measure coldness and poverty instead of warmth and prosperity.
Maybe these three points are helpful for mathematical didactics. Actually I know think that bigger or smaller is indeed the wrong naming. You can have more debts, you can go farther backwards, temperature can be more freezing, comparativly. In so far, "bigger" and "smaller" is a convention that does not contain the full meaning or applicability of negative numbers.