Back at our summer convention all the girls were using blue dresses.
Back at our graduation party all the boys got together to sang her favorite song and most of them ended up crying like children.
Does these phrases make any sense?
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Yes, the use of "back at" in these sentences is natural and grammatically correct.
In these sentences, "Back" indicates that the scene is in the past. This is redundant, because the sentences' verbs also use the past tense. In formal writing, similar sentences usually start with "At" instead of "Back at".
In these sentences, "at" is appropriate because a particular event is named. "At" refers to a particular time, instead of a lengthy time range. If you needed to discuss a longer time period, you would use "Back when" or "When":
Back when I was growing up, cats ate grasshoppers.
When I was growing up, cats ate grasshoppers.