Refute a Universal Geometric Claim

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Among candidate rectangles with listed side lengths including 2 and 3, which refutes “Every rectangle has four equal side lengths”? Candidate answers: (A) 2 by 2 (B) 2 by 3 (C) 3 by 3 (D) 4 by 4.
• Independent check inputs: Among candidate rectangles with listed side lengths including 3 and 4, which refutes “Every rectangle has four equal side lengths”? Candidate answers: (A) 3 by 3 (B) 3 by 4 (C) 4 by 4 (D) 5 by 5.

Learn the idea

Geometry arguments connect definitions, givens and established properties. A conclusion needs a chain of reasons; an attractive diagram may suggest a relation but does not establish every length, angle or universal claim. A universal claim fails if one permitted example satisfies its hypothesis but violates its conclusion. A nonsquare rectangle provides a specific shape rather than a vague disagreement.

Here is the method on the supplied teaching case. A 2-by-3 rectangle has four right angles and therefore satisfies the hypothesis, but unequal adjacent sides because 2≠3. One counterexample refutes a universal claim.

A counterexample must belong to the group named in the claim. Giving a nonrectangle would not test a statement about every rectangle; verify the hypothesis before checking the failed property.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Among candidate rectangles with listed side lengths including 9 and 10, which refutes “Every rectangle has four equal side lengths”? Candidate answers: (A) 10 by 10 (B) 11 by 11 (C) 9 by 10 (D) 9 by 9. Show the mathematical reason, not only the selected answer. Expected result: 9 by 10. A 9-by-10 rectangle has four right angles and therefore satisfies the hypothesis, but unequal adjacent sides because 9≠10. One counterexample refutes a universal claim.

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