That's right. When you get paid a fee for a service, you can claim your related expenses directly against that income. Against tutoring income you can deduct mileage driving to the student, supplies, internet access, cell phone, and any other expense "ordinary" and "necessary" in connection to earning that income. You report your income and related expenses to the IRS on a Schedule C as part of your federal tax return.
Obviously, if your income is only $1,000, you wouldn't deduct 100% of your cell phone and internet access for the year. Even so, I don't think you would have any problem coming up with more than $600 of expenses, which would be sufficient for you to not file a tax form for this year.
So should you report the $1,000 in income, pay $141 in self-employment taxes, and contribute $930 to a Roth IRA? Depends on your financial situation. If you have the $1,071 to spare, and you think the government won't change the Roth IRA rules down the road, then it might be worth considering. (How's that for a non-answer?)