Should schools be divided between the rich and the poor?

Today, I had an opportunity to visit seven very different high schools.  One high school looked like a university campus.  White pillars were set in front of a marble facade.  When I walked on the school grounds during lunch, the students sat quietly eating their lunch, read books or studying for tests.  When I walked down the halls, I saw many different signs that proclaimed "Robotics Club will meet Thursday after school", or encouraged students to joined the chess club.  Finally, above the main doors were flags that represented the universities that facility graduated from, for all students to see.

Then,  I visited an alternative high school and what a difference it was... The school was clean, but very bare and very antiseptic.  No signs for clubs, no words of encouragements to the students, no display case with trophies, or no university flags.  Just a building to get kids through high school...

Later on the long drive home, I thought about how sad the disparity was between these schools in the same district. Why does one school have robotics and chess club, when the other school does not? Do the parents of these children know about this disparity between these schools?  I say no.  But what I think happens is parents "see" their students' school and believe ALL schools are like their students' school.  But the truth is all schools are very different.

Would parents accept this?  Would Americans accept this?  Should public schools be based upon the local tax system?  Should the resources of a school be limited to what the local neighborhoods can bear? Should schools in the same school district be divided between the "rich" and the "poor"?  What do you think?