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Hi everyone! I've been having problems in diagramming this sentence. Could you help me?

The first college radio station began broadcasting in 1920, from Union College, under the personal call letters of Wendell King, an African-American student at the school.

The prepostional phrases are:

from Union College, this is an adverbial P.P that goes below "began"

In 1920. This is an adverbial P.P that goes below "began"

Here is my doubt.

under the personal call letters = This is adverbial p.p but I don't know if it should go below "began" or below "broadcasting" It's hard. Do you think it works with "began" or with broadcast?

of Wendell King = This is adjectival and goes below "call letters"

at the school = adjectival and goes below "Wendel King"

Thanks in advance for you time! Emotion: smile
Comments  
My impression is that it would go under broadcasting, because it didn't begin under the call letters. I'm glad to know students still diagram sentences! I think that is the very best way, (and, I always thought a fun way) to learn grammar. It's a type of puzzle that I always enjoyed solving.

Caveat--I was the 7th grade sentence diagramming champion, but that was MANY moons ago, so you might want to check with others to make sure my impression is correct.
I agree with your opinion that diagramming sentence is just the best way to learn grammar! It really makes the student think and evaluate what they're using in a language.
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Thank you so much for stopping by an answering my question!
I HAVE CHECKED MY GRAMMAR BOOKS BEFORE POSTING

(1) I, too, love diagramming (the Reed-Kellogg system).

(a) I think that it really helps people to better understand the

relationships of words and word groups. Very sad that few students

study Reed-Kellogg nowadays.

(2) I am 99% certain that at the school modifies (belongs to/ goes

under) student. That is:

Wendell King was an African-American who was a student at the school.

Where was he a student? At the school. Thus, we should probably

classify it as an adjectival prepositional phrase.

*****

(3) I do not wish to give an opinion about what under the call

letters of Wendell King modifies. I am not sure, and this

website asks that posters be super careful before offering an answer.
I HAVE CHECKED MY GRAMMAR BOOKS BEFORE POSTING

(1) I have researched and thought about this matter some more. I

am now 80% sure that I have "the" answer: under the personal

call letters does, as Sam says, modify/belong to/come under

broadcasting.

(2) Here is my reasoning (right or wrong):

(a) If I say The first radio station began under the call letters

XYZ, that is "good" English, and I am pretty sure "everybody"

would agree that under the call letters XYZ modifies began.

(b) I suggest, however, that the sentence in (a) is only an

ellipted (shortened) version of:

The first radio station began BROADCASTING/TO BROADCAST

(programs) under the call letters XYZ.

(c) I feel that it is pretty clear that under the call letters XYZ

modifies broadcasting/to broadcast.

(d) Furthermore, my dictionary tells me (if I understood it correctly)

that the preposition under in your kind of sentence = with.

In other words:

The first college radio station began broadcasting under/with the

personal call letters of Wendell King.

*****

Sometimes using the passive is a helpful check:

The first radio station began to broadcast programs under the

call letters XYZ.

Programs were begun to be broadcast under the call letters

XYZ. (When you read this sentence out loud, notice how the

word "broadcast" is stressed. That is, pronounced more loudly than

the other words around it.)

(If my answer is wrong, I ask the moderator to delete this post.)
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I think it's not wrong! In fact, I had a similar reasoning! I really appreciate your help!
Thank you, jossx, for your kind note.