The adjective in Arabic comes after the noun.
However, if you see the subject noun defined with AL while the adjective following it is not, then this adjective is said to be "predicative" (i.e. telling information about the noun, which is the room here). Hence, "is" is needed to complete the sentence in English.
If the adjective follows the noun in definition (as well as number and gender of course), then it is an attributive adjective, i.e. no need for "is" in English (the big room).
This is just a ... let's say a thumb rule ... to get around Duolingo.
The big room = الغرفة الكبيرة
Notice the addition of the definition article (AL) to the adjective كبيرة to make it in-line with the defined noun الغرفة - this is an attributive adjective; An adjective that attaches to the noun as a form of addressing, in the same manner as the BIG room.
Remove that (AL) and you are back to a predicative adjective; An adjective that tells information about the status of the noun, as in English the room IS big.
Attributive adjectives in Arabic mimic the noun they follow or describe in its aspects: plural/singular, gender, definition. While predicative adjectives do not follow the noun in definition (they come in the sentence without AL). Generally speaking, in nominal sentences (sentences in Arabic that start with a noun and typically contain no verbs like the one above), must have the subject defined (with AL) and the predicative undefined.
Typically when we write down, we don't put any markers on the definite articles. So, I'm not sure why Duolingo uses marks on AL here.
Anyway, I'm a bit confused about your remark but generally, the circle (Sukún) over لـ is for a stable letter (meaning a consonant without any vowel, just "L" here). The little stroke or line on the first letter, Alif, is fatHa and it simply means (A). It's not necessary to add it anyway (and there is another mark which resembles small صـ and is used in Duolingo, it doesn't change anything but it's common in Quranic orthography).
A big room: غرفة كبيرة (ğurfatun kabírah).
THE big room: الغرفة الكبيرة (al-ğurfatu al-kabírah).
The room is big: الغرفة كبيرة (al-ğurfatu kabiırah).
Each one of the above phrases/sentences has its own logic in the language. The first and the second are just a noun+adjective compounds (the first is indefinite, and the second is defined).
The third one, though, is a full sentence. It is a sentence that tells information or makes a statement for the listener or the reader (it has a subject, "the room", a verb, "is", and a predicate/final statement/result, "big").
Each of the three has its equivalent in Arabic and the main difference, if you noticed, where the definite article (AL) is placed.
Just for the record, "a room is big" cannot exist in Arabic and must be forced into "the room is big" because nominal sentences (sentences starting with nouns) cannot begin with indefinite nouns like "a room"; Except in very few situations.