Question: I can't remember when and where I had this discussion, but I remember being corrected when I was speaking by a stranger saying that it is never correct to say *give me half of this*; instead, the grammatically correct phrase would be *give me one half of this*. I've never been a pro at where numbers fit in with the English language, so maybe someone here could shed some light on this.

Answer: 
*Comment posted as answer - as requested*

In idiomatic usage, you would seldom say 'one half of this'. You might say 'give me one half-pound pack of sugar' but the hyphen shows that it is a different construct. You might say 'give me one third of that' (as opposed to 'two thirds of that'), but with halves, the alternatives are none and all. However, even with thirds, it would be more usual to say 'a third' than 'one third'. So, whoever 'corrected' you was actually misleading you.
You don't need the 'one' in expressions like 'give me half a cookie'. Where you do need the 'one' is in when units of measurement are involved, like "please give me one half pound of sugar". You can use 'a' instead of 'one', but leaving out any determiner is wrong.

Correct:

> 
> Give me one half pound of sugar.  
> 
>  Give me a half pound of sugar.  
> 
>  Give me half a pound of sugar. 
> 
> 
> 

Wrong:

> 
> \*Give me half pound of sugar.  
> 
>  \*Give me one half a pound of sugar.  
> 
>  ?Give me a half a pound of sugar. 
> 
> 
> 

(You do hear the last quite a bit in the U.S., although not in England; this came up in [another question](https://english.stackexchange.com/questions/43294/a-half-a-cup-of-something) here.)