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That is true. You can call TRS and they will give you the precise day. Or check with your school district.
A half year is precisely that: half of your contracted days. If you work a standard ~190 day contract, your "half a year" won't hit until 95 work days are worked - which for many is sometime in late January.
TRS says half a year is a year. You can quit in December or January and have the same amount coming in as you would leaving in May or June. Something to think about if you are that miserable.