Quarter-turn cell symmetry
Check exact occupied-cell sets after quarter turns.
Adult planning suggestion: about 9–12 · Shape & spatial thinking
- Players
- 1–4 with adult support
- Time
- 15–25 minutes
- Materials
- 3×3 numbered coordinate grids
Prepare
On a 3×3 grid with columns A–C and rows 1–3, shade A2,B1,B3,C2. The rotation center is the middle of B2. Keep a fixed reference grid; physically turning the sheet must not turn the coordinate labels of the reference.
Play
A clockwise quarter turn maps column x,row y to new column y,new row 4−x, using A=1,B=2,C=3. Apply it to every shaded cell, not only a recognisable corner.
Worked example
A2 maps B3; B3 maps C2; C2 maps B1; B1 maps A2. The resulting set equals the original, so 90,180,270 and 360 degrees all return this pattern: rotational order 4.
Finish and check
A second pattern A1,C3 returns at 180 and 360 but not 90 or 270, so its order is 2. Compare complete cell sets. A shape merely overlapping itself in one or two cells has not returned exactly.
Adaptation and limits
Finish after listing the images of every occupied cell for both patterns. A full 360-degree turn counts once in the order, and zero degrees is not an additional count. This rotation task does not establish whether either pattern has a mirror axis.
Purpose, observation and provenance
Within this activity: Check exact occupied-cell sets after quarter turns.
Compare all image cells with the original set; count complete coincidences rather than partial overlaps.
Our rules, examples and printable diagrams are separately prepared. The earliest inventor of the broader mechanism is not established here.
Math Is Fun: Rotational Symmetry — Reference supports the stated concept; our fictional numbers, instructions and paper diagrams are independently prepared, not reproduced teaching pages.
Secondary practice: Cooperation and explaining a check.
For a class of 30, use ten groups of three. Prepare one player-material set per group and keep the host key with the adult. Rotate operator, recorder and checker after each worked example; a reader or scribe may help without choosing answers. Avoid public ranking or diagnostic labels.
Age bands are planning suggestions. Background references may address older learners and do not validate suitability for this age band. Goals describe observable actions, not measured long-term gains or institutional endorsement.
Free alternative: Copy the supplied grids, labels and cards onto reused paper. No specialist product is needed. Adults may prepare pieces or act as a scribe.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Rules to keep beside the materials
- On a 3×3 grid with columns A–C and rows 1–3, shade A2,B1,B3,C2. The rotation center is the middle of B2. Keep a fixed reference grid; physically turning the sheet must not turn the coordinate labels of the reference.
- A clockwise quarter turn maps column x,row y to new column y,new row 4−x, using A=1,B=2,C=3. Apply it to every shaded cell, not only a recognisable corner.
- A2 maps B3; B3 maps C2; C2 maps B1; B1 maps A2. The resulting set equals the original, so 90,180,270 and 360 degrees all return this pattern: rotational order 4.
- A second pattern A1,C3 returns at 180 and 360 but not 90 or 270, so its order is 2. Compare complete cell sets. A shape merely overlapping itself in one or two cells has not returned exactly.
- Finish after listing the images of every occupied cell for both patterns. A full 360-degree turn counts once in the order, and zero degrees is not an additional count. This rotation task does not establish whether either pattern has a mirror axis.
Quarter-turn cell symmetry — player materials
| Original cell | 90° | 180° | 270° | 360° |
|---|---|---|---|---|
| ________ | ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ | ________ |
Host check — keep separate
Cross pattern order 4; opposite corners order 2.
Examples are stipulated model cases, not measured experimental or learning outcomes.