You could think of integers as pairs of natural numbers $(a,b)$ modulo the equivalence $$(a,b) \sim (c,d)\quad \mathrm{if}\quad a+d = b+c.$$ (In other words, you think of $(a,b)$ secretely as $a-b$.) Addition is just entrywise addition of natural numbers: $$(a,b)+(c,d) = (a+c,b+d).$$ Negative numbers are numbers of the form $(0,a)$. Now the equation $-7 -3 = -10$ is $$(0,7) + (0,3) = (0,10).$$ In summary, you can restrict yourself to adding natural numbers, provided that you consider pairs.
I'm not sure this helps in your friend's case, though.