Biconditionals

During the first two days of this unit, we have been learning vocabulary and skills related using mathematical logic to write an argument. Today's lesson is the third in a series that I use to prepare students to write geometric proofs.

As the students walk in the room, they are given statements based on content from Lewis Carroll’s books. They are instructed to write the inverse, converse, and contrapositive of the statements. After five minutes, we go over the statements as a class.  These statements will be re-examined during the Mini-Lesson. 

We begin the Mini-Lesson by discussing the definition of the term biconditional. When we talk about the notation iff for if-and-only-if, I explain to the students this is shorthand mathematical notation and is only used in this context. We then talk about the criteria for writing a biconditional: both the conditional and its converse must be true.   

Using Statement 2 from the Do Now, we write an example together. Then, using Statement 3, we look at a non-example. Although every student in the class may not be familiar with the details of the story Alice’s Adventures in Wonderland, there will be some who have read the book or seen a version of the movie. These students help to identify the truth-value of the statements and their converses. Using this information, all of the students can determine if a biconditional can be written.

Before the students work independently, we go over two more examples. I give the students five minutes to look at two_mathematical_statements they have seen previously. They determine if a biconditional can be written, and if it can, they write it. In both of the statements, it is helpful to change the wording a bit. I tell the students to say the statement out loud in order to hear if it makes sense. When I want the students to work quietly, I tell them, “Think out loud. Say the statement in your head as if you were saying it out loud and ask, ‘Does this make sense?’”

Students work independently or in pairs on this next activity.  On the worksheet, there are 10 true mathematical statements. The task requires students determine if a biconditional statement can be written and if so, they write the statement. When biconditional statements cannot be written, students are instructed to give a counter-example of the converse to explain why a biconditional can not be written.  Each statement reflects a concept, which students have studied before. As a result, this activity serves as a bridge from the logic lessons to the proof lessons that follow.

In the first five statements, I give the students a prompt to write the truth-value of converse. For questions 5 through 10, the students should be able to determine if a biconditional can be written without the truth-value prompt.

After about 20 minutes, we go over the questions. I call on different students to give their answers. After each student answers, I ask the other students to verify the answer. When biconditionals cannot be written, I ask a few students to give their counter-examples.