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'gym' is short for 'gymnasium'.
'gymnasium' is a place in a school where students do sports. It is usually a big room.
'gym' is the casual abbreviation for 'gymnasium'.
'jungle gym' is slang for a place where little kids climb on things - like in this photo. This slang comes from the idea that monkeys climb trees in the jungle.
'gym' is the same sound, but a different spelling from 'Jim'. "Jim" is slang for the name "James" in English.
'gymnasium' in North American English has a different meaning from Europe. In Europe, "gymnasium" means a school of higher learning, like a high school or college.
"gymnasium" is a Latin word, so for the plural, you can say "gymnasia", but most Americans would say "gymnasiums".
