The fact is, it's more specific than that. If push something exactly twice as hard, it goes exactly twice as far. If something is exactly half the mass of the first thing, it will go exactly twice as far, all other things being equal.
We looked at this several ways. I used a very light ball, and a sand filled ball that happened to be a similar size, and tried tossing them. This is good because it confirms their everyday understanding that, yeah, you have to use more force to throw a heavier ball. Naturally, any boys in the class will try to throw the heavy ball as far as possible, but, in my view, that's okay- they can feel the extra effort it takes to heave the extra mass.
Then we tried dropping the two balls in the sand box. This was actually a better experiment because you could easily see the dents left behind by the two balls (the heavier ball left a deeper dent because it fell with more force, while the acceleration of gravity was constant)
We also rolled real golf balls and foam golf balls (you could use ping pong balls) down an incline and saw that the more massive balls rolled farther. Again, this is somewhat more useful than throwing because the gravitational acceleration is constant when you use gravity.
It also explains why, in the Pinewood Derby, you are not allowed to have your car weigh more (have more mass) than a certain standard. It would be an unfair advantage because the heavier cars would always win.
Now, I did say these rules applied with all other things being equal... Such circumstances are pretty rare on Earth. A baseball hit with twice as much force won't actually go twice as far. Why not? One reason is that it will have gravity pulling on it for a longer time since it will be in the air a longer time. Another reason is that it has to travel through more air, and therefore more air resistance.
Ah, air resistance. That's pretty important when you're talking about rockets. How far can you throw a piece of paper?
Try throwing a plain flat sheet of paper as hard as you can. It's pretty hilarious, actually. One of the kids immediately suggested that we wad it up into a ball. That went much farther, even though it was the same mass as the flat sheet. Then I folded a sheet into a paper airplane, and that went clear across the room and pegged Zorg. Sorry, Zorg!
Puff Rockets from this printable pattern. They are designed to have a very low mass, as well as minimal air resistance, and they go remarkably far on a puff of air (provided by you!).
You can also experiment with flying them with and without the fin assembly. It really lets the kids discover for themselves why rockets have fins.