Reasoning · Grade 5-2 Nets of a Patterned Cube
Problem
Pair the net's three opposite faces.
Skipping one square along a straight strip always lands on the opposite face, and two flaps hanging off the same square always end up back to back. Those two little rules pair off all six faces without folding anything.
6.G.A.4Make A Systematic ListA and B show opposite patterns together, so out.
Faces that never touch cannot both be facing you at once. Testing that takes a glance and here it removes half the candidates before any real folding starts.
6.G.A.4Eliminate PossibilitiesPairing alone leaves two.
A right glove and a left glove have the same parts joined in the same way, yet no amount of turning makes one into the other. Three patterns round a corner behave the same way, so their turning direction has to be checked.
6.G.A.4Visualize Spatial RelationshipsRead the order round one corner of the net.
Folding the paper outwards bends the three squares up around the corner but never flips any of them over, so the direction you read them in stays the same. That is why the loop can be read on the flat net before you fold.
6.G.A.4Create A Physical RepresentationThe order the three patterns run round a corner can be read on the flat net and never changes when it is folded.
Folding bends the squares up around the corner without turning any of them over, so the direction they read in survives.
A cube whose corner reads the three patterns the other way round is refuted at once, because turning can never reverse a loop.
D's order matches the net.
Once you have the loop from the net, checking a picture is just reading three patterns round its front corner and seeing whether they come in the same order.
6.G.A.4Visualize Spatial RelationshipsC's order runs backwards, so out.
Two loops that use the same three things but run in opposite directions can never be matched by turning the cube — that is exactly the difference between a shape and its mirror image.
6.G.A.4Eliminate PossibilitiesFolding the net confirms D.
Paper never lies. Two minutes with scissors turns a reasoning answer into something you can hold, and it is the surest way to catch a mirror-image slip.
6.G.A.4Create A Physical RepresentationOpposite faces can never show at once — that knocks out two cubes; for the last two, check which way the three patterns go round their shared corner, because a mirror image can never be turned to match.