A "core" is like a lecture. That's the part where a professor stands at the front of the room, talks for an hour, and you write down notes.
A "tutorial" is when a small group of students from the class, usually 15-30, get together with a Teaching Assistant and discuss readings, take up homework, talk about assignments, write tests, etc. What you do in your tutorial depends on your class.
When signing up for a class, both the core and the tutorial are mandatory.
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