To make the rocket:
Roll the cardstock snugly around the stiff tube, then tape along its length to make it into a tube. Slide the cardstock tube off, flatten one end, and tape it closed with packing tape. Optional now or after launching a few times: decorate/add rocket fins etc.
To make the launcher:
Pull the bike inner tube over the mouth of the plastic bottle and secure with duct tape. (Or push the PVC tubing inside the mouth of the bottle and secure with duct tape.)
Pull the other end of the inner tube over one end of the stiff tube and secure with duct tape. (Or push the other end of the PVC tubing inside the stiff tube and secure with duct tape.)
Take the rocket outside, or into a gym/hallway with a high ceiling.
Slide the cardstock rocket over the stiff tube, point away from people, then stomp on the water bottle.
To reinflate the water bottle, blow down the stiff tube, before putting the rocket on again for relaunch.
Replace plastic bottle once this one stops working - try a bigger bottle (e.g. 2 litre), although stiffer plastics tend to crack sooner.
Add fins to the tube to make it fly straighter and higher.
Build with PVC tubing: https://www.youtube.com/watch?v=za_2uj1mbyw
Discussion in terms of molecules:
When the plastic bottle is stepped on, the air is pushed out of the bottle and into the tubing next to it, creating an area of high pressure (i.e. the molecules of air are closer together). This high pressure region moves along the flexible tube (molecules always move from high to low pressure areas), and into the stiff tube. When the high pressure molecules exit the stiff tube, they hit the cardboard rocket and push it upwards. The pressure is great enough to exert a force that sends the rocket high into the air.
Some water bottles are hard for younger students to reinflate, so are not recommended. For smaller bottles, I found the thinner plastic water bottles (500 ml) are easy to inflate and can send the rocket high. The slightly larger, thicker plastic bottles (about 700ml, water or soda drinks) are much harder to inflate. 2 litre bottles work fine but crack open sooner than the thinner more flexible plastic of the small bottles.