You might explain that BEDMAS is not the whole story when it comes to the order of operations. There is an operation called negation. It reverses the sign on numerical quantifies. It gives the additive inverse of any number, i.e. for all $x \in R$, we have $x + (-x) = 0$.
Unfortunately, most textbooks use the same symbol for both subtraction and negation. ...
As to question 1: I do have a german speaking background, so please excuse the source actually is in german:
There is a really nice journal called Wurzel, the german word for "square root". In there, Attila Furdek has a section called "Schlaue Leute werden durch die Fehler von anderen klug" (trans. "Clever people become wise through the mistakes of others"). ...
I tutored a student who had a hard time understanding this, and the way that helped him to understand it was this:
Any time there is a negative sign on a number, we can read it as $(-1)$ . So $-5 \equiv (-1)5$ and $-3^2 \equiv (-1)3^2 = (-1)9 \equiv -9$.
In the case you mention then, $-1^2 \equiv (-1)1^2 = (-1)1 \equiv -1$
The key to helping him ...
If your students already understand that exponents precede multiplication, and that multiplying by $-1$ is the "negation" operator, then you should be able to convince them that $$-5^2 = -1*5^2 = -1*25 = -25$$ is a reasonable way to interpret this expression.
Obviously the correct mathematical answer is to show how the exponent rules actually work, and when they do not work.
So please don't accept this answer.
Anyway, the educational answer is to see that student is using the fact that $1^2 = (-1)^2$ in the critical middle step. Show them the corresponding fact for cubes, because it's crazy, and might expand ...
I like the presentation on the NCTM Math Forum/Dr. Math website:
We don't usually list unary operators in PEMDAS because they're
thought of as being implied by the rules for binary operations. You
can think of the minus sign as either subtraction
-3^2 = 0 - 3^2 = 0 - 9 = -9
-3^2 = -1 * 3^2 = -1 * 9 = -9
and in ...
If you're main concern is students' writing, you could take the route of avoiding the issue altogether and tell students that $-1^2$, regardless of what it should be equal to according to BEDMAS, is just bad notation and that it can be avoided. The purpose of writing mathematics, of writing anything really, is to communicate effectively to a reader. The ...