Problem
Transfer segments from cube to net
The same cube appears three times, each with three segments drawn on three faces. Beside each is a net of that cube. The three nets are all different shapes. Draw the three segments onto each net.
Reasoning · Grade 5-2 Nets of a Patterned Cube
Your answer
Watch out Getting the face right is not enough — the corners must line up too.
How to solve
Strategy Create a Physical Representation — Nobody can hold six faces in their head at once, so I do not try. I copy each net onto paper, cut it out, and label it one square at a time using a single rule: two squares that share a fold line share the two corner letters at the ends of that fold line. Starting from the square whose letters are already given, that rule spreads the labels over the whole net with no guessing. Then every red segment is easy — it is a diagonal of one named square, and I just join the two named corners. Folding the paper up at the end and checking that the red ends really meet on the cube is the proof that the labelling was right.
1STEP 1
Name the six faces of the cube
Name the cube's six faces.
top ABCD, bottom EFGH, front BCGF, back ADHE, left ABFE, right CDHG
Six short lists take a minute to write and remove all the guessing later. A cube only has six faces, so the list is short enough to be worth making every time.
6.G.A.4Make A Systematic List2STEP 2
State the one rule that labels a whole net
Fill each net square with its corner labels.
Rolling a cube across a table shows it: the edge the cube tips over never leaves the table, so its two endpoints are shared by the face that was down and the face that comes down next.
6.G.A.4Create A Physical RepresentationOnce one square of the net is named, every other square's face is decided by how the net folds.
Why?
Folding sends each square of the net to exactly one face of the cube, so the labelling has no freedom left once it starts.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Folding does not stretch anything, so a line drawn on a face keeps its position relative to that face's corners.
🧱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).
3STEP 3
Part (1): check the printed labels and locate the faces
In (1) the printed labels fix where each face sits.
A net that is already labelled is a free rehearsal. If your rule reproduces every letter that is printed there, you can trust it on the nets that are blank.
6.G.A.4Make A Systematic List4STEP 4
Part (1): draw the three segments
Draw (1)'s three segments corner to corner.
BD → R₁ (bottom-left to top-right), GD → R₂ (bottom-right to top-left), BG → lower square (top-right to bottom-left)
Because triangle BDG is closed on the cube, its three pieces must still join up on the net wherever two of the faces stay side by side — a quick way to see you have not put a diagonal in backwards.
4.G.A.1Draw A Diagram5STEP 5
Part (2): label the staircase net
Label the staircase net of (2) the same way.
L₁=ABCD, L₂=CDHG, M₃=ADHE, M₂=ABFE, M₁=BCGF, T=EFGH
You never have to picture the folded cube: at each fold line you already know two of the four letters, and the face list from step 1 supplies the other two, because only one face besides the one you are standing on contains that edge.
6.G.A.4Visualize Spatial Relationships6STEP 6
Part (2): draw the three segments
Draw (2)'s three segments.
AC → L₁ (top-left to bottom-right), CH → L₂ (bottom-left to top-right), FC → M₁ (top-right to bottom-left)
On the cube the three segments all touch C, but a net can only keep the faces together where they still share an edge; the other copy of C has been torn apart by the cut, which is exactly why the third segment looks stranded.
4.G.A.1Draw A Diagram7STEP 7
Part (3): label the net around the central square
For (3) start labelling from the central square.
centre=ABCD, above=CDHG, right=BCGF, far right=EFGH, below=ABFE, left of below=ADHE
Starting at the middle square is the easy way round: four of the five other squares touch it directly, so most of the net is one step away from what you were given.
6.G.A.4Visualize Spatial Relationships8STEP 8
Part (3): draw the three segments
Draw (3)'s three segments.
GD → square above (bottom-left to top-right), BD → centre (top-left to bottom-right), BE → square below (top-right to bottom-left)
The zig-zag is a good sign: segments that share a vertex on the cube must still touch on the net whenever their two faces are still joined, so three joined faces in a column give one unbroken path.
4.G.A.1Draw A Diagram9STEP 9
Fold each net up and check
Folding shows all three nets land correctly.
Any labelling mistake shows up as a red line that misses its partner by a whole edge when the paper closes up, so the fold test catches errors you would never spot by staring at the flat net.
6.G.A.4Create A Physical RepresentationAnswer
carry the corner labels onto the net, then join
Three separate checks agree. First, in each part the six squares of the net came out as the six different faces of the cube, with no face used twice and none missing — a labelling error would have shown up as a repeat. Second, every fold line carries the same pair of letters on both of the squares it separates. Third, segments that share a vertex on the solid still touch on the net wherever their two faces stayed joined: in (1) BD and GD meet at D across the fold between the first two squares of the row; in (2) AC and CH meet at C across the fold in the bottom row; in (3) all three link up into one zig-zag down the central column. And every drawn segment is a corner-to-corner diagonal of exactly one square, which is what a face diagonal has to become when the cube is opened out flat. The published answer key shows exactly these nine diagonals.
Takeaway
Two squares that share a fold line share the two letters at its ends — write those in, and the whole net labels itself, so every red line is just a corner-to-corner diagonal you can copy straight across.
- Name the six faces of the cube
- State the one rule that labels a whole net
- Part (1): check the printed labels and locate the faces
- Part (1): draw the three segments
- Part (2): label the staircase net
- Part (2): draw the three segments
- Part (3): label the net around the central square
- Part (3): draw the three segments
- Fold each net up and check