I would be honest with the school ahead of time. You need to find out their policies on unexcused absenses. If your decision did, in fact, lead to your children having to stay back or go to summer school, they would have a long, long time to regret your decision. You really should find this out before you decide to go. Personally, I would never take my chidlren out for that long a time--and they wouldn't want to miss that much.
You can't count on the 12-14 teachers all giving yoru child a "packet". Seriously, I can't imagine the size of packets my oldest two would get in that situation! And my oldest would never have wanted to miss that much of math class, in particular--it would have taken her weeks to catch up. And she's in honors algebra, it's just not her best subject, and she has to work to maintain her place in the class.
I also think it's wrong to send the message to your children that a vacation is more important than school. Sure, vacations have value, and everyone trots out the tired argument that "you never know what's going to happen". Yeah, true enough--but your children aren't great students to begin with, and they are going to pay whatever price is to be paid fr this trip. Bottom line, that's not their burden to have to bear.