As a homeschooler in Mn, there are certain State requirements that must be met to be "in compliance". 

You are required to turn in a reporting form by Oct 1 each year to your school district (no required form, but there is required information that must be included- so you can send a basic letter with the info if you want to, to your superintendent's office).

You must also do an annual test with a Nationally Norm-referenced and Standardized Test.

And you must provide instruction in the following areas (and be able to prove it if the district audits you): communication skills (reading, writing and speaking), math, science, social studies, health & phy-ed. There is no requirement to cover every topic every day or even every week. There is also no requirement to keep detailed calendars or lesson plans.

So on this page you will find links to articles that address these requirements and provide your options within the requirements.

Each state has somewhat different requirements, so if you are moving from MN, be sure to do your homework concerning homeschooling in your new state. And for those who might be abroad with missionary assignments, you need to follow the homeschooling requirements in whatever state you are maintaining a permanent address (which I believe is also true for your tax filings). So if MN is your permanent address, you will still need to file homeschooling paperwork by MN deadlines, and you can still claim your expenses on your MN income tax filing.

And here's a MN technicality that causes families great confusion- Fall reporting and testing (the homeschooling law in general) only applies to children who are already 7 yrs old or older on Oct 1 of that fall, and those who have not yet turned 17 by Oct 1. That means your 5 or 6 yr old, or a child who turns 7 on Oct 2nd of that year does not have to be included in your paperwork filing or documentation or testing. Yes, you can homeschool those children, but you don't have to include them in state filings. You can also just keep them home and NOT do any homeschooling and just let them play and explore and mature a bit. 5 and 6 yr olds aren't required to do anything academic at all! I'm not kidding and yes, I can show you the state statute to back me up. The exception here is if your child attended school for Kindergarten and then you took them out-- You need to file paperwork because the child is on the school's census and they need to know why they aren't returning to school. 

In Mn, you can deduct most of your homeschooling expenses on your State Income Tax filing. You'll find an article here on how that works, too.