Contrapositives

As students walk in the room, they are given a page from the book, Alice Through the Looking Glass by Lewis Carroll. They read a passage about Tweedle Dee and Tweedle Dum. The author, Lewis Carroll, was a mathematician who specialized in the field of logic. Because of this, there are many references to logic within the text. In the passage, there are some conditional statements. The Do Now asks students to identify them and write the converse and inverse of each statement. The two statements are:

• If you think we’re wax-works, you ought to pay.
• If you think we’re alive, you ought to speak.

I give students about five minutes at the beginning of the class to read the short passage and write the indicated statements. As I circulate, if I see that students are having difficultly identifying the conditional statements, I ask the student to read the passage out loud. This makes it easier for the students to identify the conditional statements or have a discussion with another student about where the conditional statement appears in the text. 

In today's Mini-Lesson I introduce the term truth value. However, we do not go into too much depth (note: logic tables are not required in the Common Core). I also introduce the concept of a counter-example and we discuss how conditionals and their converses may have different truth values.

During the Mini-Lesson students will write the contrapositives of conditional statements. Students first identify the converse and inverse of a statement, then write its contrapositive. I usually get the question, “Which should we write first, the converse or the inverse?” This is a good time for students to investigate if there is a different outcome depending on the order they use to write the contrapositive.

In order to accomplish this I have half of the students in the class write the inverse of a statement and then write the converse of the inverse. The other half of the class writes the converse of a statement and then writes its inverse. Students are able to see that after using both of the methods, the same statement will be written. Since the contrapositive can be seen as the inverse of the converse or the converse of the inverse, it doesn't matter which way students think about writing it.

The activity for this lesson uses the same statements as in the activity from the previous lesson, “Converses and Inverses.”  I usually have the students rewrite the converses and inverses of the statements so they can see how the converse and inverse affect the contrapositive. I stop students after about 8 minutes and we go over the answers. Although students are working independently, I usually allow them to help each other figure out the answers.

At the end of this lesson, I have the students work independently on Contrapositives Summary Quiz. The quiz asks students to write the inverse, converse and contrapositive of a conditional statements. I collect the quiz at the end of the period and use it to assess student progress. 

I expect to see that, at this point, students are still mixing up the terms. It usually takes students some time to clarify the meanings of these terms and use them with consistency. Based on the assessment results, I will provide some clarification at the beginning of the next lesson.