Explaining something to the half I'm a private tutor in my free time, teaching some basic high school mathematics and I've often been asked: ''Why is something to the half equal to the root of that something?''.
And I'm having problems explaining it. I have an idea of why in my head but obviously this idea is not strong enough, as I can't explain it properly. Can anyone lay it out?

Well if algebra does not work for you, then here is a weird approach. Give them a stick of length (say $9 (=3^2)m$) and measurement tape. And assuming your student know the meaning of $a^2$, and ask them if length of the stick is equal to some number $a^2$, by using the measurement tape find $a$. The change the length and rerun this experiment with a different stick length. 
After all this, tell them $a$ is the root of stick length as squaring the root will give me the length. I know it is a funny experiment but this is something they can feel about!!
