Triangles, Congruency and Proofs Oh My!

There are few words that a teacher can utter in a Geometry class that inspire fear students at just the mention of the word.  The big ones are of course, test, quiz, exam and maybe "you have to write a paper".  In my class, however, these words compete with one more - proofs.  Literally, as soon as I mention the word I get
hit with a tidal wave of sound and complaints that all have one common theme - they hate proofs even though they have never written one yet. 

So this year, I set out on a mission to teach my students how to write proofs without the fear. I planned out what I thought was the best way to introduce them and how I thought I could transition them to writing Geometric proofs.  My game plan started out with a discussion on logic and how to try to communicate what you thought to be true with a logically sound argument.  Given that they are high school students I knew that I need to started it out a little fun so I shared with them my favorite logic, but not logic, but logic movie clip! The Battle of Wits from the Princess Bride.  The students had a blast watching it and it spawned an awesome discussion on logic, arguments, and how to support your point.

Cut, Match and Paste

From there my game plan moved us on to the most non-threatening version of proofs that I could think of - Algebraic Proofs!  By the time students reach Geometry they have taken (at least in my district) two years of Algebra and therefore can solve pretty well a multi-step linear equation.  After teaching Algebraic Proofs I have my students do a cut, match and paste group activity.  I have designed the activity to be easily differentiated depending on student level and needs.  Different levels include asking students to fill-in missing pieces, cut apart scrambled proofs and reassemble them or to write the entire proof from scratch provided only the given and the prove statement. With my class, the majority did the cut and paste aspect as I am trying to focus on hands-on activities this year.  This really helped to focus the students how to support each statement with a reason.

Cut, Match and Paste

The next step in my game plan revolved around Methods of Proving Triangles Congruent.  These types of proofs have a "formula" to them more than any other type of Geometric Proof and therefore are the easiest starting place.  I spent a couple of days teaching the students about the five main methods (SSS, SAS, ASA, AAS and HL) and what they look like.  Once I felt that they had a foundational knowledge, I again put them in groups and had them do a second cut, paste and match activity on those methods.  The students are given the five main headings plus "more than one method" and 24 sets of triangles.  They need to identify the methods being used, plus draw conclusions about vertical angles and shared sides to deduce other methods as well. The discussion that the groups had during this activity was wonderful, full of mathematical facts and had all of the students engaged and learning! 

Missing Reasons

I felt that my students were close to being able to write proofs and feel confident about it, but I wanted to take one more lesser step along the way so I gave them a set of proofs that had all of the statements, but were missing the reasons.  I prepared two versions of this activity.  The first has all of the reasons (plus a couple of extra) in a box at the bottom of each proof.  This is designed to help the students who need to see the possibilities.  I also created a version without the "reason box" for those more advanced students.  My students really enjoyed doing this and I found that some students choose to do the activity without the reason box!

Proof Practice

The last step on our journey through proofs was to write the entire thing without anything besides the given, diagram and prove statement.  I was much less nervous about doing this than I had been in past years because I felt that my students were really prepared and had the foundational knowledge to be successful!  The best part, when I handed it out, the only complaints I got were the normal
"I don't want homework" versus the "I hate proofs, proofs are hard, this stinks".  When I graded it, the majority of the proofs were correct and well written!  I count this as win! :) 

Our next step...Geometric Proofs on Segments and Angle Addition.  Hopefully, this journey will help that task to be much less painful than previous years!