Problem
Slide a face to an equivalent net
Figures A, B and C are all nets of a cube. Each has an arrow drawn in two of its squares. Sliding a face elsewhere still folds into the same cube. Pick the one that folds into a different cube.
Reasoning · Grade 5-2 Nets of a Patterned Cube
Your answer
Watch out Different net shapes look like different cubes, but one may be a face slid over.
How to solve
Strategy Create a Physical Representation — Sliding faces around, the way the Example does, is a fine way to show that two nets ARE the same cube, but it is an awkward way to show that two nets are NOT — you can slide for a long time and prove nothing. So I turn the question around: cut out all three figures, fold each one into a cube, and then just look at how the two arrows sit on the finished cube. That gives one number for each net that no amount of turning can ever change, and comparing three numbers settles the question for good. Folding real paper (or rolling a cube across the drawing) is the honest way to find where each square lands, and a short table at the end lines the three answers up side by side.
1STEP 1
Read the rule the Example is teaching
The Example shows sliding a face keeps the cube.
A net is a cube taken apart; comparing nets by their outline is like comparing two shirts by how they are folded in the drawer. Once you know sliding a face changes the outline but not the cube, you know the outline is the wrong thing to look at.
6.G.A.4Draw A Diagram2STEP 2
Find something turning a cube can never change
Turning cannot change the angle between the arrows.
This is the same idea as rotating a shape on a page: a rotation moves everything but keeps every length and every angle. That makes 'the angle between the two arrows' a fingerprint of the cube that survives every turn.
8.G.A.1Visualize Spatial RelationshipsTurning a cube can never change which faces are opposite each other, so that pairing is the thing to compare.
Why?
A turn lays the cube onto a copy of itself, so faces that were opposite before are still opposite afterwards.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
The six faces split into exactly three opposite pairs, so listing those pairs describes the cube completely.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
3STEP 3
Fold net A and look at the two arrows
Folding A leaves the arrows aligned.
net A: (0,1)→bottom, (1,1)→right, (2,1)→top, (2,0)→front, (2,2)→back, (3,2)→left
Instead of imagining the fold, roll a small cube across the drawing, one square at a time, and mark the face that lands on each square. Every square gets a different face, which is also your check that the figure really is a net.
6.G.A.4Create A Physical Representation4STEP 4
Fold net B and look at the two arrows
Folding B also leaves them aligned.
net B: (0,0)→bottom, (0,1)→back, (1,1)→right, (1,2)→top, (1,3)→left, (2,3)→front
The two folds land the pair of arrows in different-looking places on the cube, but that does not matter: turn the B cube so its bottom-left edge sits where the A cube's bottom-back edge is and the two pictures fall exactly on top of each other.
6.G.A.4Create A Physical Representation5STEP 5
Fold net C and look at the two arrows
Folding C makes them meet at a right angle.
net C: (0,0)→bottom, (1,0)→right, (1,1)→back, (1,2)→left, (1,3)→front, (2,3)→top
Four squares in a row always close into a ring, and going down that row is going round the ring. So an arrow drawn up or down the row ends up running round the ring, while an arrow drawn across the row ends up pointing at one of the two lids. Those two directions are perpendicular, and that is the whole difference.
6.G.A.4Create A Physical Representation6STEP 6
Line the three cubes up and compare
So only C is a different cube.
A: 0° B: 0° C: 90°
Comparing three things is much easier once each one has been boiled down to a single number that cannot lie. The table also shows the answer is not a near miss: A and B agree perfectly and C is a full quarter turn away.
8.G.A.1Make A Systematic List7STEP 7
Cross-check with the Example's sliding method
Sliding faces gives the same answer.
Once two nets have the identical outline, each square is forced to land on the identical face of the cube, so the pictures on the squares can be compared straight off — no folding needed.
6.G.A.4Draw A DiagramAnswer
C
The answer has to be one single letter, and it is. The check that matters is that the two nets I claim agree really do agree and are not just similar: folded, both A and B put their two arrows on faces that share an edge with both arrows pointing the same way along that edge, so one cube can be turned onto the other. And the reason C fails is a fact about C alone, not a failure to find a slide: its four-in-a-row belt forces one arrow to run round the belt and the other to point at a lid, and those directions are perpendicular. A quarter turn of difference is not something a clever rearrangement can remove. As a last check, all three figures do fold into cubes — rolling a cube across each one covers six different faces — so the answer really is about the arrows and not about the outline.
Takeaway
Fold first, then look: two arrows painted on a cube keep the same angle forever, so a net whose arrows come out square to each other can never be the same cube as one whose arrows come out side by side.
- Read the rule the Example is teaching
- Find something turning a cube can never change
- Fold net A and look at the two arrows
- Fold net B and look at the two arrows
- Fold net C and look at the two arrows
- Line the three cubes up and compare
- Cross-check with the Example's sliding method