When benchmarks do not settle it, make the fractions easy to compare. You know how to make equivalent fractions, so rename them with the same denominator (or the same numerator).
Key idea
To compare $\dfrac{2}{3}$ and $\dfrac{3}{5}$, rename both with a common denominator, such as 15: $\dfrac{10}{15}$ and $\dfrac{9}{15}$. Now the pieces are the same size, so the one with more pieces is greater: $\dfrac{2}{3} > \dfrac{3}{5}$. Both fractions must refer to the same whole.
Why it works
Comparing thirds with fifths is like comparing 2 large slices with 3 small slices. Cutting both into fifteenths makes all the slices the same size; then you only count.
Worked example: One denominator is a multiple of the other
Compare $\dfrac{3}{4}$ and $\dfrac{5}{8}$.
- Rename fourths as eighths: $\dfrac{3}{4} = \dfrac{6}{8}$.
- 6 eighths is more than 5 eighths.
Answer: $\dfrac{3}{4} > \dfrac{5}{8}$
Worked example: Neither is a multiple
Compare $\dfrac{2}{3}$ and $\dfrac{3}{5}$.
- $3 \times 5 = 15$ works as a common denominator.
- $\dfrac{2}{3} = \dfrac{10}{15}$ and $\dfrac{3}{5} = \dfrac{9}{15}$.
- 10 fifteenths is more than 9 fifteenths.
Answer: $\dfrac{2}{3} > \dfrac{3}{5}$
Common mistake
Comparing only the numerators ("3 is more than 2, so $\dfrac{3}{5}$ is bigger") or only the denominators. Unlike pieces cannot be counted against each other. Rename first, then compare.