This is Home School Legal Defense Association's analysis of NY law.
NY law is kind of fussy, but people tend to make it more difficult than it is. I don't live in NY, but I have friends who do. Here are some things they have suggested (as well as the dates for filing the NOI and IHIP):
1. You file your initial Notice of Intent (NOI) by July 1.
2. You file your Individualized Home Instruction Plan (IHIP) by August 15.
3. Notice that the IHIP can be "Plan of instruction." This means you don't have to list all the textbooks or other materials that you're using. You can describe what you intend to do, using lots of words without being specific: teach the required subjects according to the children's abilities, using a variety of materials and methods, including but not limited to textbooks, community resources, multi-media.
4. You have to choose dates for your quarterly reports. Choose dates which don't coincide with major holidays such as Christmas.
5. Your quarterly reports can be your IHIP, just reworded a bit: "Orkie has shown appropriate progress in [list subject]." You have to include the number of hours. Divide the required hours for the year by 4, and list each quarter's hours on that quarterly report.
My strong recommendation is that you join HSLDA, as they're the only ones who can give you actual legal advice.