How Does A Hot Air Ballon Rise?

I'm not sure what grade level you are working at so please pardon me if this explanation seems overly simplified:

Hot air balloons utilize the "displacement" effect.  When air is heated, the air molecules move faster and spread further apart - thus becoming less dense.  The air on the outside of the balloon remains at the same temperature and density.  This causes the trapped air in the balloon to rise to a place where the air density is equal.  It's also why a balloon doesn't float out into space - as the air outside the balloon becomes less dense than the air inside the balloon rises no higher.

Here's a good article on air density: www.usatoday.com

A hot air balloon is filled with hot air, which is less dense than the cold air surrounding the balloon.  The balloon then rises until it is at equilibrium with the cooler air outside the balloon.  If the balloon pilot wishes to rise even further, he just puts more hot air into the balloon.

Because it's full of hot air. Hot air rises, cold air sinks.

Hot air is pushed into the balloon causing it to rise since hot air is lighter than cold  or cooler air.

Because the particles are buoyant because particles move fast when the air is hot which means the air is less dense causing it to float.

The air is hot. As a matter of fact, hot air rises as it is less dense. Colder air is denser. The hot air inside the balloon is less dense and so it enables the balloon to rise.

Hot air balloons rise because, air has greater density than inside the hot air balloons. It is because heat increases the energy in every molecules that drive another air molecules away

Because lift is pushing it upwards and the low pressure is also pushing it up