If I got a 1099 MISC because I won a contest that awarded me with a $1000 gift card to buy supplies for my public school classroom, do I need to claim that?

Yes, prizes are included as ordinary income on your tax return.   If you used the money to buy school supplies, you may qualify for a tax break.  

Where do I enter a 1099-MISC?

The primary tax break for teachers is the Educator Expense Deduction.  To qualify for the Educator Expense Deduction for a given year, you must meet three criteria:

1. You worked as a teacher, instructor, counselor, principal or aide for students in kindergarten through 12th grade.
2. You worked at least 900 hours at a school certified by a state to provide elementary or secondary education. This applies to public, private and religious schools.
3. You spent money on qualified educator expenses.

The first requirement prevents college or other post-secondary teachers from claiming the deduction, while the second means homeschooling parents can't take it.   It is in the Deduction and Credits section if you qualify.