"We are cooking borsch."
Translation:Мы готовим борщ.
The remembering this is going to all be on you. It's hard to remember to take all of that into account but as for understanding:
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It's best if you create your own charts for cases 1) because you'll understand your own charts better and 2) online charts are rife with errors.
That said, if you want a chart to start with this chart, though I still use it, falls apart in places but is mostly ok with singular cases, pronouns, and most adjectival stuff. It's a decent start to figure out how you want to craft your own.
This is the best I can do to help you, next time try to be more specific on where you're falling down.
Interesting question. Нам is dative, so I (and I'm totally guessing) think the verb would have to change at the very least. Maybe to reflexive?
But then it seems like the meaning would be: The borsch is being prepared by us. Same basic meaning but maybe a different emphasis?
But, again, I could be reading too much into this and be totally wrong.