Starting with an advantage: Imagine if you were practicing takedowns with an opponent your own size and level and he told you he would let you start in a full single position – no fighting, no level changing, no shooting needed – you get the position for free. I'm sure you'd be pretty happy about that because it would take out some of the most difficult elements of takedowns and let you start with half the task already completed. Well, that's pretty much what happens when you start in a good half guard position with an hook. You essentially start in a single position. No need for all the difficulties of hand fighting, level changing and shooting – you're already in on the leg and ready to go – that's a big advantage! You could also say similar things about getting to an opponents – you're half way there also – the only thing between you and your opponents back is his over hook/wizzer. Beat that and the back is yours. Next you're working from half guard with an under hook – remind yourself that you begin half way to a single leg and/or the back and take heart that by getting to half guard you've gotten half way to two great situations before the action has even begun!