RE: 1 × 0 = 0 BUT 1 ÷ 0 is undefined - school is the rule

There are quite a few fundamental reasons why division by 0 is undefined. The first is to consider the graph of the function y = 1/x.

If we consider the limit from the right hand side (positive real numbers) and the limit from the left hand side (negative real numbers), we observe that they go in opposite directions!

Now, what we really want to do is consider is the statement The limit of 1 / x as x approaches 0.

In mathematical terms, this statement is asking what the value of the function 1/x is as x has an arbitrarily small magnitude -- with magnitude being defined as the distance from 0 (and distance is always positive).

In order for a limit, L, to be defined for a function f(x) as x approaches some value a, the following statement must be valid: If for every e, there exists a d, such that |f(x) - L| < e, then |x - a| < d.

In other words, for all possible open sets that contains your limit L in the y-direction, you are required to find a corresponding open set in the x-direction such that when you plug-in each of the elements of the open set in the x-direction that their outputs lie in the open set from the y-direction, then you have a limit L.

Granted, it's tedious to do this for every possible value e (there's an infinite number of them!), so you use different techniques to prove this.

And in order to prove that there is a limit doesn't work (and a limit has to be finite), all you have to do is show that one e doesn't satisfy the condition above.

In addition, there are other reasons why you can multiply by 0, but can't divide by it -- even though you would think these are inverse operations. In fact, they are invertible, except for the element 0!

0 is actually quite special when it comes to fields, which the real numbers are.

In short, your daughter is being taught things correctly, even if the logic behind it is unclear or not completely explained. The reasoning is quite evident and requires lots more math.