완전한 문장 + Complete sentences

The following is taken from an online chat with my friend Jin-Bum. I've included some translations in parentheses.

Me: I have a Korean question...which is the best word for "phrase"? 숙어 or 성구?
Jin: What kind of phrase do you mean? Give me some examples.
Me: Well, just "phrase." For my 초급 (beginner level) class, I've been trying to teach the importance of speaking in 완전한 문장, or complete sentences. So I've been giving them phrases to turn into complete sentences...for example, the phrase "where from" should become "Where are you from?" When I introduced the idea of "phrases" in the last class, some of the students didn't seem to understand. I know the word "expression" is translated 표현, but that's not what I want to give them. There's a difference between a correct 표현 (expression) and an incomplete 문장 (sentences).
Jin: What's 성구? You just wrote this and I don't think it's a right word.
Me: I got it from Naver's online dictionary, along with 숙어...흔하지 않는 것 같아요. (I think it may not be common.)
(A couple of minutes pass by)
Jin: Wow. It exists
Me: Yes...I guess what I'm trying to express is an INCOMPLETE phrase...they need to understand this.
Jin: But you know...Do not use 숙어 to explain this.
Me: Yes...How would you express "incomplete phrase" in Korean? Maybe the word "phrase" by itself can't be implied to be incomplete in Korean.
Jin: 숙어 is a set of 단어 (words).
Me: I think 숙어 is best translated "idiom"
Jin: In this case a lot of Korean grammar teachers are using 절...so like noun phrases are 명사절 and adjective phrases are 부사절, but you need to use 불완전한 문장 (incomplete sentence) to explain things you mentioned.
Me: 아...So 불완전한 is "incomplete"...that's perfect.
Jin: was that what you were trying to figure out? 불 means "in" or "un" in front of some adjectives or adverbs.
Me: 맞아요 (correct)...Many English words have implied meanings, but that's less common in Korean. In Korean it needs to be very specific. In English the word "phrase" is OK to describe an incomplete phrase, but in Korean I have to specify that it's incomplete. I think that's why sometimes there's confusion with my students...I recognize the 불 from 불규칙 (irregular), which I had to learn a few weeks ago to explain irregular verbs to another class.
Jin: But "what from" is not a phrase.
Me: In English it is.
Jin: Isn't it just an incomplete sentence?
Me: "Incomplete sentence" is correct, but "phrase" is also OK in this context. If you showed me a Korean verb and subject without the proper markers--for example, 저 수업 계획 짜다 (literally "me class plan put together")--I would understand this as a phrase. It's an incomplete sentence, but I still understand it as a phrase that needs proper markers and conjugation--so the 저 수업 계획 짜다 becomes 저는 수업 계획을 짭니다 ("I'm putting together my class/lesson plan"). So if I give some Koreans something like "where from," I usually introduce it as a phrase that needs to become a complete sentence like "Where are you from?"
Jin: Oh...then here is a point. Most Koreans understand "phrase" as a piece of sentence...so like a subordinated concept of sentence. What you just explained is a phrase can developed to a sentence. So it's just not a set or series of words. So when I saw "where from," I didn't see that as an arrangement of words--a sentence missing something it needs.
Me: That's a good point. Although I wouldn't explain it that way to a group of beginner-level students. But then again, I've noticed that many Koreans seem to understand English grammar rules better than native English speakers.
Jin: Hahaha