Problem
Shade both ranges on one number line, solid dots where an end is included and open dots where it is not.
Seeing the two bars on the same line turns 'both ranges' into the simple idea of where they overlap.
6.NS.C.6Draw A DiagramThe later start wins and 56 is shut out, so the overlap begins at 57.
The overlap can only be as wide as the stricter range allows, so I take the bigger of the two starting points.
6.NS.C.7Identify SubproblemsThe overlap of the two ranges begins at 57.
A number belongs to both ranges only if it clears both starting points, so the later of the two starts is what decides where the overlap begins.
The numbers in both ranges are exactly the stretch the two ranges share, with nothing outside either one counted in.
The second range leaves 56 out, so the first whole number that clears its start is the next one up.
Whole numbers step along one at a time with nothing in between, so the first one past 56 is 57.
The earlier end wins and 72 is allowed, so the overlap stops at 72.
Just like the start, the overlap is limited by the stricter end, so I take the smaller of the two ending points.
6.NS.C.7Identify SubproblemsA run of whole numbers counts as last minus first plus one: 72 - 57 + 1 = 16.
Subtracting gives the gap; adding one back counts the starting number too, so nothing is missed.
6.EE.B.8Identify SubproblemsWhen two ranges overlap, the shared part starts at the later beginning and stops at the earlier ending—then just count the whole numbers in between.