Sweep the floor before pulling out the mop.
Mopping without sweeping is exactly like throwing out the baby with the bathwater. You get nicely mopped floor with streaks of dirt. Of course, you could clean that up by getting on your hands and knees and wiping it away with a sponge or rag. But in that case, you could spend all day on your hands and knees, forgoing both broom and mop.
The choice is yours. But only one way works well.