Rotated or Flipped?
A spatial item shows a figure built from four equal squares on a square grid: a vertical column of three squares with a fourth square joined to the right-hand side of the top square. The column is the long arm; the single square is the foot. Call it the Flag.
Four options use the same four squares, all sitting squarely on grid cells, none resized or tilted: three squares in a horizontal row with the fourth below the right-hand square; three squares in a vertical column with the fourth to the left of the bottom square; three squares in a vertical column with the fourth to the left of the top square; three squares in a horizontal row with the fourth above the left-hand square.
Every option is drawn with the same line weight and no shading, so only the positions of the squares carry information. The question asks which options can be produced by turning the Flag on the page. Turning is allowed in any amount; the page may not be flipped over.
1. Which option cannot be produced by turning the Flag, and what invariant proves it?
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Use the turn test. In the Flag, walk along the long arm from its free end (the bottom square) up to the corner square, then step onto the foot: you turn right. Rotation preserves that right turn; a reflection reverses it to a left turn. Check each option:
- Horizontal row, foot below the right end: free end is the left square, travel right, step down onto the foot - right turn. A rotation.
- Vertical column, foot left of the bottom square: free end is the top square, travel down, step left onto the foot - right turn. A rotation.
- Vertical column, foot left of the top square: free end is the bottom square, travel up, step left - left turn. Not a rotation.
- Horizontal row, foot above the left end: free end is the right square, travel left, step up - right turn. A rotation.
Answer: the third option, the column with the foot at the top pointing left.
2. A student argues the third option is a rotation because 'it is the same shape, just upside down.' Explain the flaw using a concrete landmark rather than a general statement about rotation.
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The phrase 'same shape' is doing the damage: the Flag and the third option are congruent - same four squares, same edge lengths, same outline - but congruence does not mean one can be turned into the other within the plane. That is exactly what a shape without mirror symmetry blocks.
'Upside down' has a precise meaning: a 180° turn. Apply 180° to the Flag and the foot ends up at the bottom pointing left, which is the second option, not the third. The third option is the Flag reflected across a vertical line - what the student would get by flipping the paper over, the one move the item forbids.
The concrete landmark is the right turn: arm-then-foot is a right turn in the Flag and a left turn in the third option. One landmark is enough to settle it.
3. Would placing a single black dot on one square of the Flag help a student answer, or is it redundant? Would it change the answer?
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Redundant for this item, but useful - and it would not change the answer.
The bare outline already fixes handedness, since the mirror image is not among the rotations; that is precisely why the third option is wrong. A dot would add a second, independent landmark that requires no mental turning: a dot on the foot square stays on the foot square under every rotation, so any option showing the dot on an arm square is eliminated at a glance.
It would become necessary if a later version used a figure that is unchanged by reflection. For such a figure, position alone cannot detect a flip, and a marked square has to carry the information instead. So the dot is an aid here and a requirement there - worth saying out loud when reviewing the item.