Which Shape Doesn't Belong: Sorting Quadrilaterals and Equal Shares
Ms. Ferraro gives each group four shapes cut from card stock: a square with 4-inch sides, a rectangle that is 6 inches by 3 inches, a rhombus with 4-inch sides and no right angles, and a trapezoid with exactly one pair of parallel sides. Each group must sort the shapes into at least two groups and defend the rule. One group puts the square alone, saying 'it's a square, not a rectangle.' Another puts the rhombus in a group labeled 'not a quadrilateral, because it leans.' A third sorts by 'has right angles' and 'doesn't,' which puts the square with the rectangle and the rhombus with the trapezoid. A fourth group says the trapezoid does not belong at all, 'because a quadrilateral needs sides that match up.' Ms. Ferraro has pattern blocks and a set of shape cards for the follow-up lesson.
1. Evaluate the first two groups' justifications. State whether each claim is mathematically correct and what the student needs to understand.
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'It's a square, not a rectangle.' The grouping may stand, but the justification is wrong. A rectangle is a quadrilateral with four right angles (and, as a consequence, opposite sides equal). A square has four right angles, so a square is a rectangle - one with the extra property that all four sides are equal. If the group's rule is 'all sides equal,' putting the square alone is a valid sort. If the rule is 'is a rectangle,' the square belongs with the rectangle. The student needs the inclusive idea: a shape can belong to a smaller category and the larger one at the same time.
'Not a quadrilateral, because it leans.' Simply false. Leaning describes how the shape is placed on the page, not a property of the shape. Rotate the rhombus and it sits square to the table with the same four sides and four angles. Test it physically: turn the card and ask whether the shape changed. What makes a quadrilateral is four straight sides, four vertices, and a closed figure - none of which rotation affects.
2. Is the third group's sort valid? And how do you respond to the fourth group's claim that the trapezoid doesn't belong?
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The third group's sort is valid. 'Has right angles' / 'doesn't' is applied consistently: the square and rectangle each have four right angles; the rhombus and trapezoid (as described) have none. The rule holds for every shape, so the sort holds - even though it does not tell you what the four shapes have in common. That is the point to press: a valid sort is not the same as a complete classification.
What the four have in common: all are closed, all have exactly four straight sides and four vertices. That is what makes each one a quadrilateral, including the trapezoid. The trapezoid having only one pair of parallel sides is not a disqualification; parallel sides were never part of the requirement. If the class wants a sort within quadrilaterals - one pair of parallel sides, two pairs, four right angles - that is a different, and better, question.
The fourth group's rule ('sides that match up') is not a definition of anything and should be replaced, not argued with. Ask them to state a rule that the square, rectangle, rhombus, and trapezoid all satisfy, then test it against a triangle and a pentagon.
3. The class then divides the 4-inch square into four equal shares. One student says the shares can't be equal 'because they look different.' Show two different ways to divide the square into four equal areas and explain how you would settle the claim.
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Two divisions:
- Four 1-inch-wide strips. Each strip is 1 × 4, so each has an area of 4 square inches; 4 + 4 + 4 + 4 = 16, the area of the 4-inch square.
- Both diagonals, giving four triangles. Each triangle is half of a 4 × 2 rectangle, so each has an area of 4 square inches - again four equal shares of 16.
(A 2 × 2 grid of smaller squares is a third, and the one most students produce.)
The shares in method 1 and method 2 are different shapes but the same area. That is the whole idea: equal shares means equal amounts of area, not matching shapes.
Settling the claim: do not argue - measure. Have the student count square inches inside each part using a grid transparency, or cut the square apart and compare the pieces by stacking and trimming. Then return to the language: 'four equal shares' says nothing about shape. Finish by asking whether four parts that look different could ever be unequal in area - they can, which is why counting or matching, not looking, is the test.