Describing a Shape Sequence in Words
A sequence of tiles is described in words. Each tile is a 3 by 3 grid. Tile 1: a black dot in the top-left cell; a small arrow pointing up. Tile 2: the dot moves one cell clockwise around the outer edge to the top-middle cell; the arrow rotates 90° clockwise to point right. Tile 3: the dot moves to the top-right cell; the arrow points down. Tile 4: the dot moves to the middle-right cell; the arrow points left. The sequence continues in the same way. A pupil says, ‘The dot is reflecting across the main diagonal, so tile 5 will have the dot in the top-left again.’ Another pupil says, ‘The dot is walking clockwise around the edge, one cell at a time, and the arrow is turning 90° clockwise each time.’ The teacher asks the group to test both rules against tiles 1 to 4, then predict tile 5 and tile 7. The class must also decide what happens when the dot reaches a corner.
1. Which rule fits all four tiles? Show the position and arrow direction for tile 5.
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Answer: The clockwise-walking rule fits; the reflection rule does not.
The dot positions are top-left, top-middle, top-right, middle-right. That is one cell clockwise around the outer edge each time. The next position is bottom-right.
The arrow directions are up, right, down, left. Each is 90° clockwise from the previous one. The next direction is up again.
So tile 5 has the dot in the bottom-right cell and the arrow pointing up.
The reflection rule fails immediately: reflecting top-left across the main diagonal leaves it at top-left, but tile 2 has the dot at top-middle. Reflecting top-middle would give middle-left, not top-right. So reflection is not the rule.
2. Predict tiles 6 and 7. Explain what happens at the corner and why.
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Answer: Tile 6: dot in the bottom-middle cell, arrow pointing right. Tile 7: dot in the bottom-left cell, arrow pointing down.
From tile 5, the dot is at bottom-right. Moving one cell clockwise around the edge goes to bottom-middle, then bottom-left. It does not skip the corner or jump to the middle-left immediately; the dot follows the outer edge cell by cell. The arrow continues rotating 90° clockwise: up → right → down.
So the full step is: tile 5 bottom-right/up, tile 6 bottom-middle/right, tile 7 bottom-left/down.
3. A third pupil claims the arrow is reflecting, not rotating. Give one observation that disproves this.
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Answer: If the arrow were reflecting in a fixed vertical line, an up arrow would stay pointing up, or an up arrow reflected in a horizontal line would point down. But the observed sequence is up → right → down → left, which is a quarter-turn clockwise each time.
A reflection changes left-right or up-down orientation in a mirror way; it does not produce the steady clockwise turn seen here. The arrow’s direction changes by 90° in the same rotational direction at every step, so rotation is the better rule.