Converse and Contrapositive –The most important variations of a conditional statement are the converse and the
contrapositive. Unfortunately, these two sound similar, and students often confuse them. Emphasize the converse
and contrapositive in this lesson. Ask the students to compare and contrast them.
Converse and Biconditional –The converse of a true statement is not necessarily true! The important concept of
implication is prevalent in Geometry and all of mathematics. It takes some time for students to completely understand
the direction of the implication. Daily life examples where the converse is obviously not true is a good place to
start. The students will spend considerable time deciding what theorems have true converses (are bicondtional) in
subsequent lessons.
Key Exercise: What is the converse of the following statement? Is the converse of this true statement also true?
If it is raining, there are clouds in the sky.
Answer: The converse is: If there are clouds in the sky, it is raining. This statement is obviously false.
Practice, Practice, Practice –Students are going to need a lot of practice working with conditional statements. It
is fun to have the students write and share conditional statements that meet certain conditions. For example, have
them write a statement that is true, but that has an inverse that is false. There will be some creative, funny answers
that will help all the members of the class remember the material.
Deductive Reasoning
Inductive or Deductive Reasoning –Students frequently struggle with the uses of inductive and deductive rea-
soning. With a little work and practice they can memorize the definition and see which form of reasoning is being
used in a particular example. It is harder for them to see the strengths and weaknesses of each type of thinking, and
understand how inductive and deductive reasoning work together to form conclusions.
Recognizing Reasoning in Action– Use situations that the students are familiar with where either inductive or
deductive reasoning is being used to familiarize them with the different types of logic. The side by side comparison
of the two types of thinking will cement the students’ understanding of the concepts. It would also be beneficial to
have the students write their own examples.
Key Exercise: Is inductive or deductive reasoning being used in the following paragraph? Why did you come to this
conclusion?
- The rules of Checkers state that a piece will be crowned when it reaches the last row of the opponent’s side of the
board. Susan jumped Tony’s piece and landed in the last row, so Tony put a crown on her piece.
Answer: This is an example of detachment, a form of deductive reasoning. The conclusion follows from an agreed
upon rule.
- For the last three days a boy has walked by Ana’s house at 5 pm with a cute puppy. Today Ana decides to take
her little sister outside at 5 pm to show her the dog.
Answer: Ana used inductive reasoning. She is assuming that the pattern she observed will continue.
Which is Better?– Students quickly conclude that inductive reasoning is much easier, but often miss that deductive
reasoning is more sure and frequently provides some insight into the answer of that important question, “Why?”.
Additional Exercises:
- What went wrong in this example of inductive reasoning?
Teresa learned in class that John Glenn (the first American to orbit Earth) had to eat out of squeeze tubes, and her
mom says the food served in airplanes is not very good. She just had a yummy pizza for lunch. She sees a pattern.
Food gets better as one approaches the center of the earth. Therefore the food in a submarine must be delicious!
Answer: She carried the pattern too far.
2.2. Reasoning and Proof