So why avoid negative denominators?
I considered this question early in my teaching career.
I found two reasons why negative denominators are undesirable.
. . Forgive me for resorting to baby-talk.
(1) Naming the fraction
We want a specific share of some pie.
The denominator indicates the size of the slices.
. . We have divided the pie into 4 equal parts.
. . Each part is called a "fourth".
The numerator indicates the number of slices we want.
. . We want 3 of those parts.
. . Hence, we want three fourths.
It says "Cut the pie into -4 equal parts, and take 3 slices."
And we find that we can't do that.
We can cut a pie into two equal parts.
We can even cut a pie into one equal part (although it sounds silly).
We don't know how to cut a pie into zero equal parts.
And we certainly don't know how to cut a pie into negative-four equal parts.
 Common denominator?
The LCD seems to be -12.
This takes more work and includes some added risk.
We may change any two of the signs.