We check "black". My kids are biracial. They know they're biracial. They know I'm white. If someone asks them about it in detail because mommy doesn't "match" (and, yes, it's happened because people are rude like that), they respond that they're both black and white and they like it like that because they get to be both. They also know that, when they are out, they are seen as black. If they want, they call themselves white. Doesn't change people's perspective, though. And it doesn't make them white. They simply aren't.
My husband is descended from a white confederate general. Damn sure doesn't make him white and he'd gag a little if somone suggested it did. Our last name is that of one of the largest plantation owners in the state. Probably has a little of that blood, too. Still doesn't make him white, and, looking at him, no one could guess any of his ancestors were white. He is very dark. He has cousins, however, who are lighter than our children. His mother, who is even a little darker than he is, has cousins who are lighter than me. And then there are the cousins who "passed". They decided they were white, and they disowned their black families and disappeared into a world where their darker-skinned cousins and siblings and parents weren't welcome. Because being white was easier. Others were white during they day so they could get a good job, and were black when they went back home in the evening, and the stress of this double life was incredible.
My kids couldn't actually pass, even if they wanted to, but I don't see anything shameful in acknowledging that they are not white. So what? That doesn't make them any less a part of me. And the reality is, they can call themselves white until the moon turns purple, but it doesn't make it true. What's my son going to say when he's pulled over for DWB? "Officer, you seem to have made a mistake. I'm actually white." 'Cuz that'll fly real well.
Obviously, I feel very strongly about this. I know the history of this region and I know the current reality, and my children have already been confronted with it. Again, my children's skin color doesn't make them any less a part of me, no matter what it's labeled. They aren't denying me. However, mislabeling them as "white" isn't going to change the very real impact of their race, and they've felt that already. The simple fact is, I can't pass my privilege on to them. It doesn't work like that. There are things they will face that aren't a part of my experience, and I can't change that with a different label.