I like you guys... you have picked some good books. Now I am not saying here what I think public schools should read, but what I would have my children read... and for me this is a practical question, as we homeschool our 6 children.
The Bible (Spiritual Growth)
The Federalist Papers (gov't)
History of the World By: J.M. Roberts (world history overview)
Greek and Latin (Is that a book?)

Open: depending on child's gifts
I am assuming that by the time they are in high school they have mastered pretty well the english language, and have good practical math skills. Now it is time to help them learn how to
think. How to take those practical skills and put them to use.