It is legal to home school in all 50 states. Each state has its own laws pertaining to homeschooling. Some states, like Michigan, do not have any requirements and other states like California have extensive regulations.
In states that mandate core subjects for home school families; math, science, language arts, and history are required. For a ninth-grade student, these subjects should be taught at a high school level. The exact curriculum should be determined by a student's ability.
The best way to find out what is required for home schooling a ninth grader in your state is to contact your state's department of education for the exact information.
Ninth-grade students are high school students. Every high school student should take the PSAT/NMSQ, and either the SAT or ACT. Although these tests are not taken until the sophomore or junior years, students should start preparing for them in the ninth grade.
Some states, like Arkansas, require ninth-grade students take a norm-referenced standardized test like Stanford, TerraNova, or Iowa Test of Basic Skills.
Credits for ninth-grade students are calculated by semester. If a course takes half a year or 16 weeks to complete, a half credit is issued. For courses that take 36 weeks or a full school year to complete, one full credit is issued.
One way to determine the courses your ninth-grade home schooler should take is to look at college requirements. Take the required courses and divide them by four, one-year groupings. Start with lowest level courses for the ninth grade.