Figure Classification: Separating Constant From Variable Attributes
A nonverbal reasoning group is shown three stem figures, described here in words. Stem 1 is a trapezoid with exactly one pair of parallel sides, a small dot inside its largest angle, and two short hatch marks on one leg. Stem 2 is a differently shaped trapezoid with the dot inside its largest angle and three hatch marks. Stem 3 is a third trapezoid with the dot inside its largest angle and one hatch mark. The four answer options are: W, a trapezoid with the dot inside its largest angle and four hatch marks; X, a parallelogram with the dot inside its largest angle and two hatch marks; Y, a trapezoid with the dot inside its smallest angle and two hatch marks; Z, a trapezoid with no dot and two hatch marks. One student chooses X because 'it has four sides and a dot.' Another chooses Y because 'Y is a trapezoid and has two hatch marks, like the middle stem.' The teacher asks the group to list every attribute and mark it constant or variable across all three stems.
1. Which option belongs with the stems? Justify by testing each attribute across all three stem figures.
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W belongs. Constant attributes: the outer shape is a trapezoid with exactly one pair of parallel sides, and the dot is inside the largest angle. The number of hatch marks is variable (2, 3, 1), so hatch count cannot be the rule. W satisfies both constants.
X fails shape: a parallelogram has two pairs of parallel sides. Y fails dot placement: the dot is in the smallest angle. Z fails dot presence. A valid classification rule must hold for all three stems and for the correct option, and it must exclude each distractor for a specific reason.
2. Evaluate the two students' rules. What is the error in each, and what general procedure avoids it?
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The first student uses “four sides and a dot,” which is too broad: a parallelogram has four sides and can contain a dot, but it lacks the defining trapezoid property. The student has replaced a specific constant with a looser one. The second student uses hatch marks, which vary across the stems; matching one stem is not enough, because a rule must fit all three. The procedure is: list shape, dot position, hatch count, orientation, and fill; mark each constant or variable; discard variables; then test options against the constant set. Distractors are designed to share some but not all constants.
3. If a fourth stem were added with a trapezoid, dot in largest angle, and zero hatch marks, how would that affect the rule and the answer set?
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It would strengthen the conclusion that hatch marks are variable, since the counts would then be 2, 3, 1, 0. The constant attributes remain trapezoid and dot in largest angle, so W still belongs. The added stem would not make Z correct, because Z lacks the dot entirely. It would, however, make any rule based on “odd number of hatch marks” untenable. This is a useful check: when a new example varies an attribute further, the rule should not change; if it does, the attribute was not the rule.