Download Conditional, Converse, Inverse, Contrapositive Name Lewis Carroll

Conditional, Converse, Inverse, Contrapositive
Name
Lewis Carroll, the author of Alice's Adventures in Wonderland and Through the Looking Glass, was
actually a mathematics teacher. As a hobby, Carroll wrote stories that contain amusing examples of
logic. His works reflect his passion for mathematics and can be wonderful interdisciplinary teaching tools.
Consider this quote from a
conversation which occurred during
the Mad Hatter’s Tea Party
"Then you should say what you mean." the March Hare went on.
"I do," Alice hastily replied; "at least -- at least I mean what I say -- that's the same thing, you know."
"Not the same thing a bit!" said the Hatter, "Why, you might just as well say that 'I see what I eat' is the same
thing as 'I eat what I see'!"
"You might just as well say," added the March Hare, "that 'I like what I get' is the same thing as 'I get what I
like'!"
"You might just as well say," added the Dormouse, who seemed to be talking in his sleep, "that 'I breathe when
I sleep' is the same thing as 'I sleep when I breathe'!"
"It is the same thing with you," said the Hatter, and here the conversation dropped, and the party sat silent for a
minute.
- What’s up with these statements?? Are they true?
- Can you rewrite any of them as “If - Then” statements?
Let’s look a little closer at what the Dormouse said. Discuss these with your group members. How are these different
from each other? What’s similar? Are some of these true? False? Which ones?
If I am sleeping, then I am breathing.
If I am breathing, then I am sleeping.
If I am sleeping, then I am breathing.
If I am not sleeping, then I am not breathing.
If I am sleeping, then I am breathing.
If I am not breathing, then I am not sleeping.
These have all been examples of conditional statements, converses, inverses, and contrapositives.
A conditional statement is
If it is raining, then there are clouds in the sky.
EX. Rewrite the conditional statement in the “if-then” form.
1. All birds have feathers
2. Two angles are supplementary if they are a linear pair
So let’s look at those statements from earlier. Which is which? List em at the right.
If I am sleeping, then I am breathing.
If I am breathing, then I am sleeping.
If I am sleeping, then I am breathing.
If I am not sleeping, then I am not breathing.
If I am sleeping, then I am breathing.
If I am not breathing, then I am not sleeping.
So can you come up with the definitions for each of the remaining terms? Give it a shot.
A converse is
Write the converse of the statement: If it is raining, then there are clouds in the sky.
An inverse is
Write the inverse of the statement: If it is raining, then there are clouds in the sky.
A contrapositive is
Write the contrapositive of the statement: If it is raining, then there are clouds in the sky.
Let’s practice some more. I’ll do this one with you.
Given: Guitar players are musicians
Write the if-then form of this conditional statement. Is it True or False?
Write the converse. Is it True or False?
Write the inverse. Is it True or False?
Write the contrapositive. Is it True or False?
Let’s try a few with Geometry. Talk about the conditional statement, then converse, inverse, and the contrapositive of
these two examples with your partner. Be sure to identify if they true or false.
1.
If 𝑚∠𝐴 = 99°, then ∠𝐴 is obtuse.
2.
If a polygon is equilateral then the polygon is regular
So when is a biconditional helpful?
1.
Let’s take this definition:
If two lines intersect to form a right angle, then they are perpendicular.
Write its converse.
What do notice?
2.
What about this one?
If the sides of a triangle are all congruent then the triangle is equilateral.
Write its converse.
What do you notice?
So. I ask again. When is a biconditional helpful?
Write the two definitions from above as biconditional statements.
1.
2.
So let’s put it all together.
A conditional statement is given. Write in words a) the inverse, b) the converse, c) the contrapositive, d)
and the biconditional of that conditional. Determine whether each is true or false.
If two angles are vertical angles then they are congruent. (The Vertical Angles Theorem)
BTW.
This is a really helpful website if you would like some additional practice:
http://www.regentsprep.org/Regents/math/geometry/GP2/PracRC.htm