Place 5 leaves of mint in the bottom of a large glass or in a 16-ounce mason jar. Add simple syrup. We suggest you start with 1 tablespoon, then add more as desired later, until you know how much you want to use each time.
If you have a muddler, use that to break up the mint and mix the two together. (If not, skip this step.)
Squeeze juice from the lime half into a separate container. Add half of the juice now.
Add rum, or skip this step if making a virgin Mojito.
Add ice to fill the glass. (If you did not use a muddler, we recommend shaking up the glass at this point. You will want to have something to cover the top of the glass, or transfer it to a mason jar with a lid.)
Add club soda to fill the glass. Stir.
Pour the remaining lime juice on top. Garnish with the sprig of mint. (Add more simple syrup, if desired.)