There are a number of excellent resources for kids and for homeschool families who wish to study Dr. King in greater depth. Education World has a comprehensive list of resources primarily intended for classroom settings, but that can be adapted for use by homeschoolers, including timelines, audio and video, quizzes, writing, role-playing, and reading.
This example activity is meant to occur in a classroom setting with a group of children. So many homeschool families are ethnically and racially diverse already. There is a good chance you are part of a mixed race family, or have adopted children of other races and cultures. Larger homeschooling families can adapt this activity according to their number and ages of their kids, or several families can get together and try this. You can choose to divide the children up any way you like. The point is to give them an opportunity to experience discrimination.
Not that homeschoolers do not often experience a form of discrimination anyway, but it cannot compare to the racial discrimination that occurred when I was growing up.
From Education World:
FamilyEducation.com also has a list of resources and activities that families can do together.
Citizenship/role playing. This common activity is used in classrooms everywhere -- but it's one worth repeating from time to time! The activity helps students understand the concept of "discrimination." For this activity, divide the class into two or more groups. Some teachers divide students by eye or hair color; some invite students to select and wear badges of different colors (purple, green, and other colors that are not related to skin color); and others isolate students whose first names begin with the letter "b," (or whichever letter is the most common first letter of students' names in the class). For a class period or for an entire school day, one group of students (for example, the kids who have blond hair, those wearing orange badges, or the ones whose names start with "b") are favored above all others. Those students receive special treats or special privileges, and they are complimented often. Students who aren't in the "favored" group, on the other hand, are ignored, left out of discussions, and otherwise discriminated against. At the end of the period, students discuss their feelings. How did it feel to be treated unfairly, to be discriminated against? Invite students to talk about times when they felt they were judged or treated unfairly. How does this "experiment" relate to the life of Martin Luther King, Jr.? (Source: Kidsphere listserv)