Lesson 6.5 — Fold Nets and Track Opposite Faces
立体图形问题：展开图、相邻面与相对面
Turn flat nets into mental solids, keep labels attached while a cube rotates, and use several visible views to determine hidden opposite faces.
Know the difference between adjacent and opposite faces
A cube has six faces. Each face touches four faces along edges and has exactly one face directly opposite it.
Adjacent faces
They share an entire edge. A face has four adjacent faces.
Opposite faces
They do not share an edge or a vertex. A face has one opposite face.
Check the face relationships
Fold the numbered net
The cube’s faces are assigned in opposite pairs: front/back are 2 and 4, left/right are 6 and 8, and top/bottom are 10 and 12. The net shows 12, 2, and an unknown face.
Why the question mark is 4
After the net folds with face 2 at the front, the far-right square closes the back. The back is the face opposite 2, so it carries 4.
Read the folded net
Build an adjacency map around one face
Once an opposite face is known, the four faces left after excluding the target and its opposite must be adjacent to the target.
Target face 2
Adjacent faces: 6, 8, 10, 12.
Opposite face: 4.
Opposite pairs partition the six faces into three pairs. Knowing two pairs automatically determines the third.
Complete the map for face 2
Answer for face 2, even if you selected another target in the explorer.
Recover cuboid dimensions from a net
A cuboid net repeats its three edge lengths. Read repeated widths and combined measurements before multiplying for volume.
3+4=7, so the rectangular base is 3×4.
The overall height is 8 and the attached flap extends 3, so the cuboid height is 8−3=5.
3×4×5=60.
| Length | Width | Height | Volume |
|---|---|---|---|
| 4 cm | 3 cm | 5 cm | 60 cm³ |
Recover the dimensions
Infer opposite colors from several cube views
One view rarely shows an opposite pair directly. Collect adjacency evidence from several orientations, then eliminate the one color a face never touches.
Each cube has exactly the same arrangement of six colors on its faces, one color per face. Only its orientation changes between views. Color names are provided so meaning does not depend on color alone.
Flower-count reference
Red: 1 · Yellow: 2 · Blue: 3 · White: 4 · Purple: 5 · Green: 6.
These counts belong to the faces and stay fixed when a cube rotates.
Yellow’s evidence
Yellow already touches four different colors. A cube face has only four adjacent faces, so the remaining color—White—must be opposite Yellow.
Yellow ↔ WhiteFinish the pair map
Purple touches Yellow, Blue, Red, and—because Yellow is opposite White—Purple is also adjacent to White. The only color left for Purple’s opposite is Green.
Purple ↔ GreenThe final unused pair is:
Red ↔ Blue| Color | Flower count | Opposite color |
|---|---|---|
| Yellow | 2 | White |
| Purple | 5 | Green |
| Red | 1 | Blue |
Record the opposite pairs
Apply the same opposite map to a new arrangement
The next row shows top colors Blue, Purple, Yellow, and Purple. The visible front faces are useful orientation clues, but the bottom can be found directly from the top once the opposite pairs are known.
Top → bottom colors
Blue → Red, Purple → Green, Yellow → White, Purple → Green.
Flower total
1+6+4+6=17 flowers.
Complete the practice arrangement
Decide whether six squares really form a cube net
A connected arrangement of six equal squares joined along whole edges is not automatically a cube net. After folding, the six faces must occupy six different directions without overlapping.
Valid-net test
Choose one square as the front. Fold neighboring squares around shared edges. A valid net reaches front, back, left, right, top, and bottom exactly once.
Why nets fail
Two squares may try to occupy the same face, or a long strip may wrap around and collide with itself.
Test the four nets
Nets and hidden-faces workshop
Use folding, adjacency evidence, dimension repetition, and opposite-pair reasoning.
In the numbered cube, the face opposite 2 is:
In the numbered cube, the face opposite 6 is:
The cuboid net in Mission 4 folds to a cuboid with volume (cm³):
Yellow’s opposite color is:
Purple’s opposite color is:
Bottom-flower total under tops Green, Red, Yellow, Blue:
Bottom-flower total under tops Blue, Purple, Yellow, Purple:
Which net option in Mission 8 is valid?
Optional puzzle — try before opening worked review
Place the numbers 1 through 8 at the cube’s vertices, using each once, so every face has the same sum of its four corner numbers. Each vertex belongs to three faces. First use that fact to find the required face sum, then sketch an arrangement and check all six faces.
This extra puzzle is not automatically graded. After checking the workshop, you can compare your arrangement with the example in worked review. The example is one valid arrangement, not the only allowed answer.
Exit ticket
Complete all five questions without opening the lesson hints. A perfect result unlocks your certificate.
Lesson summary
Complete the missions to build a reliable mental model of folding and hidden faces.
Certificate of mastery
Cube-Net and Hidden-Face Navigator
Awarded for accurately folding nets, tracking adjacent and opposite faces, reconstructing cuboid dimensions, and inferring hidden cube faces from multiple views.
Teaching notes
The rotation explorer, adjacency map, valid-net laboratory, feedback, and assessments are newly designed instructional scaffolds.