Learn German: Wir haben halb Asien besucht, ohne das Filmmaterial zu verlieren. - We visited half of Asia without losing the footage.

Normally, yes! But ganz (whole) and halb (half) are exceptions. When they come directly before a geographical name that doesn't use a definite article (like most continents, countries, and cities), they drop their adjective endings. So it is simply halb Asien or ganz Europa.

In English we say "half of Asia". Why is there no translation for "of" in the German sentence?

German treats halb and ganz as standard adjectives that directly modify the place name, rather than building a prepositional phrase with von (of). You simply put the uninflected adjective right before the noun.

Why does the second half of the sentence use ohne ... zu instead of ohne dass?

You use ohne ... zu (without doing something) when the subject is the same in both clauses. Since "we" are the ones visiting Asia and "we" are the ones who didn't lose the footage, you can use the shorter infinitive phrase. If the subjects were different, you would be forced to use ohne dass.

Where exactly does zu have to go in the phrase ohne das Filmmaterial zu verlieren?