Different Denominators

$\frac{1}{3}+\frac{1}{5}$ cannot be added as it stands. Thirds and fifths are different sizes, and you can only count pieces that are the same size. So first rename both fractions with the same denominator.

The new denominator must be a multiple of 3 and of 5, so that both fractions can be renamed. 15 works: $\frac{1}{3}$ is $\frac{5}{15}$, top and bottom times 5, and $\frac{1}{5}$ is $\frac{3}{15}$, top and bottom times 3. Now the pieces match: $\frac{5}{15}+\frac{3}{15}$ is $\frac{8}{15}$.

On the board, the renamed line $\frac{5}{15}+\frac{3}{15}$ is accepted straight away, because renaming never changes a value. Any common multiple of the denominators will do, and multiplying the two denominators always gives one.