Croquet is the perfect game for schools.  (Assuming the mallets don’t become weapons).

Adults and children can compete as equals. It is a game of calm, rational, strategic filth.  A game where unsportsman like conduct and generally stuffing up your opponents is actively encouraged.

A half decent set of equipment costs from £30 for a cheap set  to £120 for a more robust set up.

The absolute joy of watching a child give their toes a firm rap with a large wooden mallet is absolutely free.

Set up a lunchtime league and I guarantee you that your most competitive boys will be all over it.  There’s even a National Association.