To teach fractions as operators, an algorithm approach can be used.
If the denominator of the fraction is 2, we have to make sure the test number is even. So 1/2 of 4 is 2.
Dividing by 2 and factoring 2 are the same if the number is even. If the number is not even, then 1/2 as an operator on naturals to naturals is not defined for that number.
This key point is not typically part of teaching fractions as operators. We first define a fraction as an operator on natural numbers, but only on natural numbers such that the fraction applied to them produces a natural number.
So we multiply the test number by the numerator and then try to divide by the denominator is one path. Or we reduce the fraction first and then require the test number is a multiple of the reduced denominator.
That for a given fraction, an infinite number of naturals are not in its domain is an important point to teach.
The fraction as a set of ordered pairs helps teach this.
The fraction 1/2 has the ordered pairs (0,0), (2,1),(4,2), etc. The odd numbers are never the first number, so they are not in the domain.
This helps teach the ordered pair and function concept.
Fractions as numbers are abstracted from fractions as operators. As we define the sum of fractions as operators and their product, we realize that fractions as operators behave like numbers. This leads us to the rule for adding two fractions with different denominators and the rule for dividing by a fraction.
We then define an abstract number class as a pair of numbers with the same rules as we developed for fractions as operators. This number class has new properties over the natural numbers. For example, between any fraction, there is another fraction. This can be shown for fractions as operators first.
Fractions as numbers are an abstraction and a new beginning in a sense. This is a key point to teach, as well as to learn.
Fractions are not there already, they are an invention. Using the invention we can model point sets in geometry and in physical space. We invent fractions as a number to model geometry and objects in space.