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Four Related Conditional Statements

Run the Four Related Conditionals MicroSim Fullscreen

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Description

This diagram displays the four related conditional statements that arise from any conditional statement "If p, then q":

  • Original Conditional (p → q): The given statement
  • Converse (q → p): Switch the hypothesis and conclusion
  • Inverse (~p → ~q): Negate both the hypothesis and conclusion
  • Contrapositive (~q → ~p): Switch AND negate the hypothesis and conclusion

The diagram uses a concrete geometry example ("If it's a square, then it has 4 right angles") to illustrate each form and shows which are true and which are false.

Key logical relationships shown with arrows:

  • The Original and its Contrapositive are always logically equivalent (both true or both false)
  • The Converse and the Inverse are always logically equivalent to each other
  • Dashed arrows show how each form is derived from the original

Lesson Plan

Learning Objective

Students will be able to construct the converse, inverse, and contrapositive of a conditional statement and understand their logical relationships (Bloom's Taxonomy: Analyzing).

Warm-Up (5 minutes)

Ask students: "If it is raining, then the ground is wet." Is this true? What about: "If the ground is wet, then it is raining." Is that always true? Why or why not?

Direct Instruction (10 minutes)

  1. Display the diagram and walk through each quadrant
  2. Emphasize the pattern: how each form relates to the original
  3. Highlight the logical equivalence relationships (green and gray arrows)
  4. Discuss why the original and contrapositive always share the same truth value

Guided Practice (10 minutes)

Give students a new conditional: "If a shape is a triangle, then it has exactly 3 sides." Have them construct:

  • The converse
  • The inverse
  • The contrapositive
  • Determine which are true and which are false

Independent Practice (10 minutes)

Students create their own conditional statements from geometry vocabulary and construct all four related forms, determining truth values for each.

Assessment

  • Can students correctly form all four related statements?
  • Can students identify which pairs are logically equivalent?
  • Can students determine truth values using counterexamples?

References