However, you as could see before the articles change from casus to casus.
Do you remember what your question is when you use the Nominativ? "Wer tut etwas?" The answers would be: "DER Mann tut etwas."/"DIE Frau tut etwas."/"DAS Tier tut etwas.
The question for Akkusativ is: "Wen sehe ich?" The answers would be: "Ich sehe DEN Mann."/"Ich sehe DIE Frau."/"Ich sehe DAS Tier." One handy thing about the Neutrum is that the article does not change from Nominativ to Akkusativ.
When you use the Genitiv it is: "Wessen Ball ist das?" The answers would be: "Das ist der Ball DES Mannes."/"Das ist der Ball DER Frau."/"Das ist der Ball DES Tieres."
To use the Dativ the question is: "To whom does ball belong?" The answers would be: "Der Ball gehört DEM Mann."/"Der Ball gehört DER Frau."/"Der Ball gehört DEM Tier."
Again, I am afraid it is about memorising the articles. However, do not be frightened now! It is easier than it looks.
Do you understand how it works? Great, so can you create some examples on your own?