Mount Rushmore represents the first 150 of the history of America. It is important because it shows America’s most prominent presidents, George Washington, Thomas Jefferson, Theodore Roosevelt, and Abraham Lincoln.
The given statement is an example of a phrase, specifically, a participial phrase. The element that is necessary to complete the sentence is an independent clause. A participial phrase consists of words with the -ing or -en form plus its modifiers. This kind of phrase always functions as an adjective. The phrase "Beginning in the fall of 1957" is describing a certain noun, which is found in the independent clause.