"Our kitchen is big."
I understand our need of adding "家の" to "だいどころ" so it is more exact too "our kitchen", even "家の" doesnt really make that meaning, but "the house kitchen" literally.
It would sound more natural in japanese just saying "だいどころ" and get the extra information by context - which is one of the most important things to understand japanese.
As spanish, japanese doesnt usually especify the subject, you get it due to context. In these cases happens the same, you get the subject or these kind of information not by saying it, but context.
more about うち as a concept here:
It seems so. The IMDb has a page for a 2016 comedy film titled Watashitachi no haa haa:
Note that the meanings of "watashitachi" and "uchi" are a bit different.
"Watashitachi" can be translated as "we" or "us". It would refer to a group of people of which you are a member. For example, the IMDb says that the film is about a group of four female high school students who like the same rock band.
"Uchi", on the other hand, can be translated as "house", "one's house", "one's home", "one's family", "one's household".
If you are talking about "our kitchen" to mean "the kitchen belonging to our household", "uchi no" would be a better fit.
The うち here is a pronoun that actually comes from the kanji 内 (literally means "inside", but is used as a pronoun here for we/our) rather than 家. As other commentors have noted, it's a more informal, almost slang thing, so I'm not sure why duo has chosen to include it. Also, I'm given to understand that the Japanese people usually type in hiragana when they want to use the word as a pronoun, and reserve the kanji for when they actually want to mean "inside".
Does うち have to be connected to something like うちのだいどころ or can it stand by itself like うちは
大 and 小 are the kanji for the concepts big and small. Try looking them up in jisho and you will see the different ways they can be read (pronounced) and different words that are formed using them. 大きい and 小さい use those kanji to form the adjectives for big and small. A lot of adjectives end in い . To learn more, read up on い adjectives vs. な adjectives.