After worming my pet I noticed lots of worms in his stool for several days after. Is this normal?

I recently got a new puppy. I noticed he had worms after having to clean up after another housebreaking accident. I gave him worm medicine and noticed over the next several day every time he went to the "bathroom" his stools are full of worms. Is this normal and if so how long will this last? Do I need to re worm him? And also I have a cat that seems to have a real problem with worms even after worming her. What do I do?