"I could be bounded in a nutshell, and count myself a king of infinite space" - Hamlet, Act 2, Scene 2
Lesson
Where was the stuff that makes up your body before you were born? Where was it before the earth was made? Where was it before then? Eventually, if we go back far enough, before the first stars, before planets or anything else that we see in the universe, the universe was a big cloud of invisible gas and was completely dark, before then, it was filled with a glowing white hot cloud that you couldn't see through. Before then it was a tiny dot, but even if you were in that tiny dot, the universe would look infinitely large, even though outside of it only looked like a tiny dot.
Alternative approach:
About 14 billion years ago everything you see in the sky and on the earth, everything in the universe existed in a tiny point smaller than the smallest speck of dust you have ever seen. For some reason (we don't know why), it started expanding. It's not that everything was in a point in space, space itself was in that point. Also, time started at that point. Some people think that if you started now and asked somebody and kept asking "what's before that," that eventually you'd get to the Big Bang, and you could say not only that there's nothing before that, that there isn't even a before.
Because everything in the universe was crammed into such a small space, it was so hot that a special state of matter above gas happened called plasma. For about 400,000 years the entire universe was a filled with a big glowing, white-hot fog as it expanded. After 400,000 years the fog dissipated, and the earliest air was formed. These gases were invisible though, so the entire universe was completely black, there were no stars, galaxies, or planets, just a lot of invisible air floating through space. Eventually some of that gas started to pull together because of gravity, and enough gas came together that they lit on fire and became the very first stars.
The universe is expanding faster than the speed of light. Because of this, when we look at very far away galaxies, we are looking at galaxies that are very young, we are basically looking back into time. [Show them a picture of the Hubble Deep Field image with the really dull red ones in back and explain that these are some of the earliest galaxies ever]. The more powerful our telescopes, the farther back we can see, except we can see some light from the Big Bang itself called the Cosmic Microwave Background radiation [show them a map of the COBE]. This light is a color that we can't see called microwave, this is the same light we use to cook our food in microwave ovens.
However, because both time and space are expanding, space can be infinite (we have infinite galaxies, planets, and suns) within a ball, because time is different in different places in that ball. So even when the universe was the size of a room from the outside, if we were in that room-sized universe it would still look infinite to us.
Some scientists think that there are other universes besides ours, some of the math predicts that there are.
Additional activity: Blow up balloon, draw points on it representing stars and galaxies and show how they all move away from each other when the balloon is expanding, explain that that's how we found out about the Big Bang.
Recommended reading for adults
Our Mathematical Universe, Mag Tegmark (first, Big Bang-relevant parts).
Future
I don't quite get the inflation-era stuff and need to brush up on that when my kids get older. Also, I don't quite understand how the volume of the interior of an inflating sphere can be infinite while the sphere is finite from the outside. It was explained in the Tegmark book above but it's one of those things I'll need to set aside some time to really focus on.
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