Subtracting positive and negative numbers

These videos show how to subtract a negative number, but they don't explain why.
When my kids learned about negative and positive numbers, I told them, "If you owed your sister $7.00, then you have negative $7.00."  (This is after they understand what "having" negative $7.00 means.)
"Then suppose your sister takes away 3 of the 7 you owe?  How much do you now owe her?"
Now you only owe her $4.00.
So (-7)-(-3)=-4
That is the same as saying (-7)+(+3)=-4
The videos teach to change both signs, but do not explain why you can change the subtraction sign and the negative sign that comes immediately after it.

Next, I tell them even if a problem is 5-9= the 5 is positive, and the 9 is negative.  When they were just learning this, I told them to circle each number with the sign directly in front of it.  Then they could easily see the 5 was positive and that the negative sign went with the 9.

Another thing I'd like to point out, is that in text books, the negative sign is usually written up higher than the regular addition and subtractions symbols, and is sometimes not in parentheses.  But when being written, it is good to place parentheses around numbers and their negative signs to keep them separated from the regular addition and subtraction symbols.

(1) Subtracting integers, animated number line
"The difference of -4 and -7 is 3."  This makes so much sense!  
If it is -7º where you live, and it is -4º where I live, the difference is 3º.
And if it is 4º where you live, and it is -2º where I live, the difference is 6º.
So 4 - (-2) is the same as 4 + (+2) and it equals 6.


(2) Subtracting Integers, YourTeacher.com
Remember, when subtracting, you are finding the difference between two numbers.


(3) Subtracting integers, subtracting 3 numbers
Anytime you change a subtraction sign to an addition sign, you must change the sign of the number immediately after it.


(4) Subtracting Integers, with dogs and cats