"This is your half."
Translation:Das ist deine Hälfte.
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The word "halb" works together with a noun, you have to write it and right after use a noun. "hälfte" can be written without the need of a noun. For example:
Ich habe ein halbes Hühnchen
Ich will ein halbes Brot
Das ist die Hälfte, die dir gehört
Ich habe die Hälfte des Hühnchens gegessen.
"Dies" is something very close to you, while "das" would be more along the lines of "das dort" (that there). But we don't really use "dies" anymore except in books or things like that. You can always use "das" and will be understood perfectly - it sounds a lot more natural, too. Depending on where you live it's also ok to say "Dis" (we don't write it though, so it looks kinda weird here^^;) it's something between "dies" and "das". I live in the north of Germany and use it all the time. It's pretty colloquial, though.
Think of the drop-down list as a very brief dictionary entry. "Halb" is a possible translation of "half", but it doesn't work in this sentence, because it's an adjective, not a noun. A dictionary will typically give you several possible meanings for a word, and it's up to you to figure out which one to use in the sentence you are constructing.
"Das" is used as an "article", a definite article or it can mean, that, this but something pointing to an object where as "Dass" is used as a conjugation, to connect to words. When you are using "dass" as a conjugation to connect to words, the verb stays at end. kyky gives the example underneath this message: