I have a friend with dyscalculia and was teaching her some some mathematics (namely, solving a linear equation, simplifying certain expressions, and what (affine linear) functions are).
She understood solving equations of the form $ax + b = 0$ by first adding $-b$ to both sides and then diving by $a$. Dealing with negative $a$ and with expression $b - b$ was something of a problem, but I hope she figured it out, also.
Adding slightly more complexity created more problems. For example: $2x + 3 = -7$. We subtract 3 from both sides, getting $2x = -7 - 3$. She has great trouble seeing that $-7-3 = -10$.
How to communicate and teach the concepts here? I tried using the thermometer analogy, explaining how $a - a = a + (-a) = 0$ and, somewhat poorly, that $-7 - 3 = - (7 + 3) = -10$. How to justify the last attempt in a useful way? What other models or intuitions are there for understanding the negative numbers and particularly summing them?