Oh boy, I get to wax philosophical again!
Morals, the set of behaviors and rules we govern with, derive from the fact we have agency, the ability to choose, a finite time on this earth and the need to interact with our fellow humans. The writers of the Declaration of Independence summed it up actually quite well with life, liberty and the pursuit of happiness and all men (and women) are created equal.
Actually, the writers of the declaration and constitution knew full well morals do not derive from religion. During the era, there was a lot of philosophy and just intellectual thinking happening (see Ben Franklin's Common Sense) or Jefferson or Hamilton. They were able to see that our morals derive from the fact we exist as humans and the fact we have to interact with each other. They very specifically wrote to not endorse a religion because of a push for the moralities of that religion. The founder's clearly saw the effects of the Church of England on England and wanted the ability to practice their own religion without state interference.
Over our existence morals evolve and change, but always come back to the our ability to make a choice and how we interact with others. If you believe in a God, then the intent of commandments or directives is to guide those morals.
What I find interesting with this notion of the forcing the 10 commandments in the classroom is Jesus Christ summed them up with "love God and love your neighbor" all other commandments are based on this notion. Basically, right there - right there, even Jesus Christ acknowledges morals are derived from our interactions with one another.
The hard part is we each of different needs and definitions of what life is, liberty and the pursuit of happiness. Consequently, we form structure, hierarchy and yes, governments to try to delineate what our interactions and consequences should be given certain circumstances.
So, are our morals derived from God? yes and no. There are plenty of successful societies with our perception of good morals that do not even believe in the 10 commandments or Christianity.
My recommendation - teach kids choices and consequences. Teach them we are all equal (something a lot of people in our country have a problem with) and that we respect each other despite our differences. The natural consequence of this would be the development of mature adults with morals that are probably not far off from the 10 commandments without forcing religion down someone's throat.