Reasoning · Grade 6-2 Surface Area of Cube Stacks
Problem
With k touching faces the change is 2k minus 6.
Every square inch of surface belongs to some cube face, so instead of measuring the whole solid you only have to audit the six faces of the one cube that moves. Each face is 1 square inch, so counting faces and counting square inches are the same job.
6.G.A.4Change Focus Count The ComplementRemoving a cube changes the surface by twice its touching faces minus six, so only three touching faces leaves it unchanged.
Each touching face was hidden from both sides at once, so removing the cube exposes the neighbour's face as well as losing its own.
The surface is all the exposed faces put together, so the change is what is newly exposed minus what has gone away.
So only k equal to 3 leaves it unchanged.
A negative answer here means 'the skin got smaller' and a positive one means 'it got bigger'; the sign is doing real work, which is exactly what signed numbers are for. The values also step up by 2 each time, because gaining one neighbour both hides one more of this cube's faces and uncovers one more of a neighbour's.
6.NS.C.5Look For A PatternThe red one has k of 1, so it drops by 4.
A cube perched alone on top is almost all outside already, so taking it off removes far more skin than it opens up. This is the same reason a lonely tower has so much surface: it is the worst-packed cube in the solid.
7.G.B.6Visualize Spatial RelationshipsThe blue one has k of 3, so it is unchanged.
Three faces out, three faces in — the swap is even. This is the case the whole page is about: a cube tucked into a corner with exactly three neighbours can be removed and the skin of the solid measures exactly the same as before, even though the solid is now one cube lighter.
7.G.B.6Visualize Spatial RelationshipsThe yellow one has k of 4, so it grows by 2.
A deeply buried cube is barely showing, so pulling it out digs a pit whose four walls are brand new skin. The more neighbours a cube has, the more the surface grows when it leaves.
7.G.B.6Visualize Spatial RelationshipsThe green one has k of 2, so it drops by 2.
This is the trap in the picture: green looks buried because it is on the bottom level, but its top is open and one neighbour is missing, so it is really only half attached. Remembering that the bottom face of a ground-level cube is part of the surface is what makes -2 come out instead of -1.
7.G.B.6Visualize Spatial RelationshipsThe four cases give −4, 0, +2, −2.
Laying the four cubes out in one table shows the whole idea at a glance: the answer never depended on the solid being 16 cubes, only on how tightly each coloured cube was wedged in.
5.MD.C.4Make A Systematic ListCount how many neighbours a cube has: 3 neighbours means you can pull it out and the surface area does not change at all.