Proof By Contradiction Class Activity

Call two students to the front of the classroom and show them and the class one $10 and two $1 bills. Ask the pair to close their eyes and to raise a hand over and behind their head where it cannot be seen (by the student herself). Place a $1 bill in each raised hand and put the $10 bill away. Before instructing them to open their eyes, explain to them the situation (not the part about which bill they were given) and ask them to tell the class when they know whether they are holding a $1 or $10 bill. When they open their eyes, each is able to see the other’s bill but not their own.

Once they open their eyes, there is usually a long period of time during which both students grow progressively more awkward or giggly as they stand there unable to tell whether they have the $10 bill or the leftover $1 bill. If they say they know which bill they have, ask them for their reasoning. The claim that you would not give them a $10 bill is not deemed a solid proof! Ultimately (this can take a few minutes of squirming and some eventual reassurance that they can and will figure out the problem), one student announces that if they had the $10 bill, their classmate would instantly announce that they have one of the $1 bills. Since the classmate seems confused, then the proof-giver must also have a $1 bill. Write out their reasoning parallel to a list of the steps of a proof by contradiction:

Your Claim.	I have a single.
Assume the opposite of what you are trying to prove.	If I had the ten...
Show that a contradiction is generated by this assumption.	Then my classmate would know that he had a single, which he doesn’t seem to know.
Conclude that the assumption, in leading to a contradiction, was false.	Therefore, I cannot have the ten.

The idea for this activity comes from:

Harold Jacobs (1987). Geometry (Instructor's manual). New York, N.Y.: W. H. Freeman.