Lesson 2.2 day 2
Converse, Inverse, Contrapositive, and Biconditionals
Focus: the four forms of a conditional, equivalence pairings, Venn diagram explanation, biconditionals, inverse error named
The thinking task asks students to write every variation of a conditional statement they can find by changing the order and negating parts, then test each one and look for a pattern. The statement to start with — ‘if a number is divisible by 4, then it is divisible by 2’ — is chosen deliberately because the original is true, the converse and inverse are false (counterexample: 6), and the contrapositive is true. Students who test carefully will discover the equivalence pairings before you name them.
The Venn diagram explanation of why the pairings hold is the intellectual heart of the consolidation. Draw it together. The p-circle sits entirely inside the q-circle. Anything outside q must be outside p — that is the contrapositive. Anything inside q may or may not be inside p — that is why the converse is not guaranteed. The inverse and converse fail for the same reason: they reverse the nesting direction. It is important to be precise here: the converse and inverse are not equivalent to the original conditional, but they are equivalent to each other. Students who understand one diagram understand why there are two equivalence pairs, not one.
Close by returning to the language precision examples from Lesson 2.2a. Students now have the vocabulary to say exactly what was wrong with inferring that rain means they stay inside. That error is the inverse error — assuming the inverse follows from the conditional. Give it that name. In Lesson 2.4 it will receive its formal name: denying the antecedent. For now the inverse error label is enough.
Consolidation: Guided Notes and the Venn Diagram
Build the four-form table together. For each row, ask students to supply or verify the example from the divisibility task before writing a geometric one. After the table, pause: why does the contrapositive always match the conditional? Let students speculate, then draw the Venn diagram together. The diagram makes the equivalence obvious rather than arbitrary. Make sure to name both equivalence pairs explicitly: the conditional and contrapositive are equivalent, and the converse and inverse are equivalent to each other — but neither the converse nor the inverse is equivalent to the original conditional.
Close with the biconditional: when both p→q and q→p are true, the two circles are identical — they occupy exactly the same space. Symbolic form: p↔q. The key observation worth making explicit: all definitions are biconditional. A definition must classify correctly in both directions — it includes everything that fits and excludes everything that does not. Students who remember this will have a test for the quality of their definitions throughout the course.