Hoping is the present-participle form of hope ("to wish for"). Hopping is the present-participle form of hop (like a bunny).
- Jimmy left for school early the next morning, hoping to avoid Benny and his buddies.
- Kolya was hopping along like a chicken trying to catch a worm.
- Hoping that no one would recognize her, Lori went hopping across the stage like a kangaroo.
(a) She saw him _____ along the pier.
(b) She was _____ that he wouldn't trip.