When I was doing a lot of patterning work with shotguns I used a 30" circle steel plate at 40 yards and just painted the plate with white paint between shots. I made notes of the patterns rather than keeping paper pattern sheets. Nowadays it is even simpler by just taking a digital photo of each pattern as you shoot and before painting over.

Strictly speaking I was not truely checking the pattern of each choke as to do this you need a larger plate or sheet and impose a 30" circle over the 'centre' of the pattern.

I am only interested in the pattern I get on a 30" circle with the point of aim using the front bead being the centre of that circle, not interested in a pattern that is off centre as unlike a rifle you cannot adjust the shotgun to shoot to a specific point of aim if it does not already place the centre of it's pattern to where the bead is aimed (well not easily).

It is quite an eye opener when patterning a shotgun especially if using a gun with interchangeable chokes. Mostly they do not just provide a denser and denser pattern as you go up in choke i.e. from cylinder/skeet to full. I found with my Miroku O/U gun and #7 shot the 1/4 choke gave the best pattern with the 3/4 virtually the same, perhaps a whisker tighter. I look for patterns that have no 'holes' that a clay target on the angle could get through. The half and full chokes were useless, a whole box of clays could fit through those patterns.