How to understand operations with negative numbers in a simple, intuitive way? I have to teach my $29$ year old brother math. He remembers basic arithmetic from school, but he always hated math, which is why I want to start him from the beginning with intuitive explanations for everything. 
I am thinking of starting off with the number line. Adding two positive numbers is easy enough to explain, because you can represent positive number as lengths; if you want to add $5$ to $7$, you take the compass and measure $5$. Then you put the compass down at $7$, and the result will be $12$. However, how would I explain addition with negative numbers? The simplest way I can think of is to represent numbers as vectors in $2D$. Positive numbers are vectors pointing right, and negative numbers are vectors pointing left. With this representation, when we add two numbers, we add their vectors tip-to-tail. And when we have the negative of a number, we keep the length but just reverse the direction of the arrow.
I am not very satisfied with the approach above, as I think it overcomplicates things a bit. I would very much prefer to be able to return to representing numbers as just lengths. Is there a better way to explain this?

The approach with my children was to use temperatures to introduce negative integers (it works better with Celsius degrees, since negative temperatures in Fahrenheit are uncommon). For instance,
the temperature is +2°C and we loose 3 degrees, to arrive to -1°C: children can visualise the difference on a thermometer. 
I have to confess that this method has some drawbacks. First, it works mainly in winter. Secondly, global warming may reduce the opportunities to have practical exercises with negative degrees. Thirdly, electronic thermometers are a killer for this method...
