While generally speaking, i don't consider the US to be part of The Commonwealth countries, we do share:
language
common law-(US common law, in 49 states and at federal level is based on English common law
(2 point for knowing which state doesn't have most of its civil and criminal code based in english common law)
there is some commonality in government structure(sure we got rid of aristocracy and royalty, but we still have '2 houses' for government)
and
our current system of measurements:yards/inches, pints/gallons, quarts/pecks/bushels, rods/acre's etc are based on the old standards of the English.
(even the most widely practiced religions in US are ones that got started in UK!)

Its a bit of an exageration to say we are part of the commonwealth.. but we do still have a special relationship with UK..
(Websters says this about commonwealth)
commonwealth
• noun 1 an independent state or community, especially a democratic republic. 2 (the Commonwealth or in full the Commonwealth of Nations) an association consisting of the UK together with states that were previously part of the British Empire, and dependencies. 3 an aggregate or grouping of states or other bodies. 4 (the Commonwealth) the republican period of government in Britain between the execution of Charles I in 1649 and the Restoration of Charles II in 1660. 5 (the commonwealth) archaic the general good.


it really a matter of which date you pick for #2--as to whether or not we should be considered a member of the commonwealth.