Card Six: Cycle or Bounce?
Five cards from a series are laid on the table, and students must say what card 6 looks like. Each card carries three features: an outer shape, a number of small dots inside it, and a single small tab attached at one of four positions on the shape's edge, numbered 1 to 4 clockwise from the top.
Card 1: triangle, 5 dots, tab at 1. Card 2: square, 4 dots, tab at 2. Card 3: pentagon, 3 dots, tab at 3. Card 4: triangle, 2 dots, tab at 4. Card 5: square, 1 dot, tab at 1.
The outer shape is not in dispute: triangle, square, pentagon, repeating, so card 6 must be a pentagon. The argument is about the dots. One student says the count falls by one each time and, since it cannot fall below one, must bounce back to 2 on card 6. Another says the series is a cycle, so card 6 carries 5 dots. The printed answer choices offer only one of these. The tab pattern is agreed by everyone and is the only evidence that settles the question.
1. Which student is right about the dot count? Use the tab evidence and state the general principle you applied.
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The cycle reading is right, and the tab decides it, not the dots. The tab runs 1, 2, 3, 4, then 1 - after the fourth position it returns to the first rather than reversing to 3. That is a wrap-around cycle, and it is the only feature of the item that has run past the end of its range.
The dots run 5, 4, 3, 2, 1, so under the same rule type card 6 carries 5, not 2. The bounce reading requires a reflection rule - descend, then retrace - that no other feature uses, and it raises a question the cycle does not: why stop at one, and what determines the turning point?
General principle: when one feature of a series is ambiguous, the unambiguous features establish the rule type the item is built on, and the ambiguous feature is read to match. It is not that cycles are always preferred; it is that this item has already declared which kind of series it is.
2. Suppose the tab had shown positions 1, 2, 3, 4, 3. What would card 6's dot count be, and why?
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Two. A tab sequence of 1, 2, 3, 4, 3 reverses after the fourth position rather than wrapping, so the item's rule type is a bounce. Applied to the dots, 5, 4, 3, 2, 1 is a descent that turns around at one, so card 6 carries 2 and card 7 would carry 3.
The point of the variation is that the answer is not 'cycles are always right'. It is that the tab and the dots must be read under the same rule type: whatever the tab does at the boundary of its range, the dot count does at the boundary of its range.
A student who answers 5 in the original item and 2 here is reasoning from the evidence. A student who answers 5 both times is applying a fixed preference for cycling, which happens to be right once and wrong once - the classic way a partially learned rule hides itself.
3. Under the cycle reading, what dot counts do cards 6 to 10 show, and what single change to the five given cards would have removed the ambiguity without adding a card?
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Cards 6 to 10 show 5, 4, 3, 2, 1 - the same five values as cards 1 to 5, because the dot cycle has period five. The outer shape has period three, so the complete card pattern repeats only every fifteen cards; shape and dots realign only when both cycles return to their starting values at once. That is worth stating, because a student who spots the shape repeating every three cards may expect the whole card to repeat at that interval and will be surprised at card 6.
To remove the ambiguity without adding a card, change the starting dot count so a wrap is visible inside the five given cards. Counts of 3, 2, 1, 5, 4 would do it: the step from 1 to 5 is a wrap that no bounce can explain, so card 6 carries 3.
The general repair: make the disputed rule observable in the given evidence, not only in the answer key.