Comparing fractions
Two fractions of the same whole, and you want to know which is more. When is a glance enough, and when do you have to rename them?
Where you are 3 inside
What this area is about
Comparing fractions means saying which of two is greater, or that they are equal. It only makes sense for fractions of the same whole: half of a big cup is more than half of a small one. The three children are three ways to decide, from quickest to most general.
How these fit together
Sharing a number. Same denominator or same numerator: when the pieces are the same size, more pieces is more; when the counts are the same and not 0, bigger pieces is more. So for whole numbers a, b and n \neq 0, \displaystyle \frac{a}{n} > \frac{b}{n} exactly when a > b.
Benchmarks. Comparing against a benchmark places each fraction against \displaystyle \frac{1}{2} or 1. If one is below \displaystyle \frac{1}{2} and the other above, nothing more is needed.
The general method. A common denominator renames both fractions in matching pieces, which always works, then compares the counts.
A path through it
Same denominator or numerator first, then benchmarks, then a common denominator. Here the order is fixed: each method is built from the one before it.
Afterwards you can compare and order rational numbers, negative ones included, and decide which of two ratios is larger.