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What does "for the rest of the double period" mean?
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The school day is divided up into periods, perhaps 45 minutes each. Normally, a different subject is studied in each one. Sometimes the same subject is studied in two consecutive periods; this is a double period.
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Rover_KE
context
It means a double lesson in a secondary school. Instead of having a lesson for the usual time, 45 minutes (referred to as a period), it would be for double that time.
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 Blue Jay's reply was promoted to an answer.