My students think PEMDAS means that all addition should be done before
any subtraction is done. How can I show and convince them that they
need to do both operations at the same time working from left to right?

To get the difference by the missing addend method we are not supposed
to change the given problem into subtraction of one subtrahend, but to
work out the problem as it is. We are not even to use regrouping.

Does the distributive property contradict the rules of order of
operations? When we have a(b + c) order of operations says we should
add first in the parentheses, then multiply, but the distributive
property says we can multiply first to get ab + ac, then add.

A question on how to subtract 6 1/8 - 3 1/3 leads to examples of
borrowing and regrouping in various contexts, including time, money,
liquid volume, and of course mixed numbers. An alternative approach of
producing an equivalent but easier problem is also presented.

We learned the rule that to subtract negative numbers you change to
addition and switch the sign on the number, so that -9 - (-3) becomes
-9 + (3). I want to know why we do that, not just that we have to do it.