"I am drinking my juice."
Translation:Ich trinke meinen Saft.
It's not in dative, it's in accusative. Look at the table here: http://en.wiktionary.org/wiki/mein
In this case, Saft is masculine and is in accusative, so the form to use is "meinen". To know if something is in accusative, just look for the subject of the sentence in passive: The juice is being drink by me.
Alternative, a more abstract explanation is "the subject which receives the action of the verb". It's usually the answer to the question "What (is/are SUBJECT) (VERB)?", in this case "What are you drinking?" (Answer: The juice).
This brings up one of my frustrations with Duolingo and, in a very different way, Rosetta Stone. How best to teach these grammatical details to adults? Yes... I get that children don't learn the cases by explanation. That's usually the cop-out answer given by those that advocate this style of learning.
Truth is, adults have the benefit of understanding rationale and can be taught how to construct something without having to acquire the knowledge tangentially.
It takes children (with vastly more supple minds) several years to wire their brains to grammar. And they often do it incorrectly due to primacy of learning if they aren't taught properly.
Duolingo could benefit greatly by having a few simple instructional videos that help with these concepts. Something along the lines of the more traditional classroom briefs on these topics.
please look at the chart in the following link http://deutsch.lingolia.com/en/grammar/pronouns/possessive-pronouns.
mein is for neutral, meine is for feminine/plural, and meinen is for masculine.