You can prove by contradiction at the basketball court!

 Another Monday morning discussing different ways of performing mathematical proofs. 

I asked my students if they were able to think about the proof of the proposition we left at the end of the last class. I got a timid answer..." I tried"...so we started with this class activity.  The idea was to get them familiar with proof by contrapositive.  They were to write the contrapositive of the original statement and then we discussed it together on the board. 

Next stop...proof by contradiction.  I believe the way the textbook explained it (despite being formal and correct), was a bit complicated for newcomers to the world of mathematical proofs, so I proceeded to write in my own words the meaning of the proof by contradiction method. 

Bottom line: "assume the conclusion is false and get something that contradicts your hypothesis or assumptions" 

One of the most illustrative examples: Prove by contradiction that √2 is irrational. 

Questions I asked before and at the beginning of the proof:

• What is an irrational number?
• What is a rational number?
• What is an irreducible fraction?
• What can we say of an integer if its square is even? 
• Where is the contradiction in the argument we are building? 
• Can the proposition p and the negation of p be true at the same time? 

I asked them to prove a proposition in the subway. They should take a picture of the notebook so I can see the subway seats in the background. The alternative is proving the statement in any place/context outside the classroom, office, or home (not driving!)

One of the asked me..." could it be anywhere then?", and I replied, "could be at the basketball court!". I saw the biggest smile of the day!

https://www.brooklynbridgepark.org/things-to-do/basketball/