Sentence improvement (idiom “over someone’s head”): pick the correct preposition in the fixed expression Original sentence: "For some days the new professor lectured ___ the heads of his pupils." Choose the option that completes the idiom correctly.

Concept
The idiom is go/lecture/fly “over (someone’s) head”, meaning the content is too difficult for the audience. With plural “pupils,” we say “over the heads of his pupils.”

Option analysis
over the heads of — correct idiom; plural noun matches “pupils.” ✅over the head of — singular “head” mismatches plural “pupils.”on the heads of — wrong preposition; changes meaning.through the heds of — spelling error and wrong preposition.

Final Answer
“… lectured over the heads of his pupils.”