Use a Stated If-Then Rule

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A fictional geometry rule says: If a quadrilateral is a square, it is a rectangle. Figure 2 is a square. What follows from this rule? Candidate answers: (A) Every quadrilateral is a square (B) Every rectangle is a square (C) Figure cannot have equal sides (D) Figure is a rectangle.
• Independent check inputs: A fictional geometry rule says: If a quadrilateral is a square, it is a rectangle. Figure 3 is a square. What follows from this rule? Candidate answers: (A) Every quadrilateral is a square (B) Every rectangle is a square (C) Figure cannot have equal sides (D) Figure is a rectangle.

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. An implication allows its conclusion when its hypothesis holds. Applying a known classification rule to a supplied figure is different from asserting its reverse or extending it to unrelated shapes.

Here is the method on the supplied teaching case. The premise matches the stated sufficient condition, so the conclusion follows. The statement does not establish the reverse implication or make all quadrilaterals rectangles.

Track the exact direction of the rule. Knowing that squares are rectangles does not make every rectangle a square; a conclusion is justified only when the supplied premise matches the hypothesis.

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: A fictional geometry rule says: If a quadrilateral is a square, it is a rectangle. Figure 9 is a square. What follows from this rule? Candidate answers: (A) Every quadrilateral is a square (B) Every rectangle is a square (C) Figure cannot have equal sides (D) Figure is a rectangle. Show the mathematical reason, not only the selected answer. Expected result: Figure is a rectangle. The premise matches the stated sufficient condition, so the conclusion follows. The statement does not establish the reverse implication or make all quadrilaterals rectangles.

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