Introducing Two-Column Geometry Proofs: A Different Approach

Leading into proof writing is my favorite part of teaching a Geometry course.  I really love developing the logic and process for the students.  However, I have noticed that there are a few key parts of the process that seem to be missing from the Geometry textbooks. 

I started developing a different approach, and it has made a world of difference!   I noticed that the real hangup for students comes up when suddenly they have to combine two previous lines in a proof (using substitution or the transitive property).  Most curriculum starts with algebra proofs so that students can just practice justifying each step.  They have students prove the solution to the equation (like show that x = 3). 

These just were not sufficient to prevent the overwhelm once the more difficult proofs showed up.

The Solution (There IS a Better Way!):

So, I added a stage of algebra proofs to fill in the gap that my students were really struggling with.  We worked with the typical algebra proofs that are in the book (where students just justify their steps when working with an equation), but then I led them into algebraic proofs that require the transitive property and substitution.  We did these for a while until the kids were comfortable with using these properties to combine equations from two previous lines.

My "in-between" proofs for transitioning include multiple given equations (like "Given that g = 2h, g + h = k, and k = m, Prove that m = 3h.") 

This way, the students can get accustomed to using those tricky combinations of previous lines BEFORE any geometry diagrams are introduced.  They are eased into the first Geometry proofs more smoothly.  This extra step helped so much.  I'll never start Segment and Angle Addition Postulates again until after we've practiced substitution and the transitive property with algebra proofs.

Sequencing the Transition:

After finishing my logic unit (conditional statements, deductive reasoning, etc.), I start (as most courses do) with the properties of equality and congruence.  I also make sure that everyone is confident with the definitions that we will be using (see the reference list in the download below).  I introduce a few basic postulates that will be used as justifications.  I spend time practicing with some fun worksheets for properties of equality and congruence and the basic postulates.

Then, when we start two-column proof writing, I have students justify basic Algebraic steps using Substitution and the Transitive Property to get the hang of it before ever introducing a diagram-based proof. 

Click the image to download the flowchart I use to organize my proof unit.  The PDF also includes templates for writing proofs and a list of properties, postulates, etc. that I use as a starting point for the justifications students may use.

The extra level of algebra proofs that incorporate substitutions and the transitive property are the key to this approach.

This addition made such a difference!  By the time the Geometry proofs with diagrams were introduced, the class already knew how to set up a two-column proof, develop new equations from the given statements, and combine two previous equations into a new one.  Check out this sample proof to see what I'm talking about:

Try It (Download Files):

Taking a couple of days to develop this thought process helped my students so much.  After practicing these proofs, they had no problem easing into the next level of proofs with Angle Addition Postulate and Segment Addition Postulate.  (Click here for a fun worksheet for practicing with these postulates.)  This made them ready for what used to be such a huge leap.  We avoided all the struggle that usually comes with introducing proofs.  They did not feel nearly as lost.

Try these algebra proofs in your own classroom.  You'll love the way this additional lesson leads your students into proof writing more smoothly.  This PDF includes a few examples that are half-sheet size.  They work really well as warm-ups.

Pet Peeve to Emphasize:

Here's the other piece the textbooks did not focus on very well - (This drives me nuts).  There is a difference between EQUAL and CONGRUENT.  This is a mistake I come across all the time when grading proofs.  I spend a lot of time emphasizing this before I let my students start writing their own proofs.  I make a big fuss over it.  I require that converting between the statements is an entire step in the proof, and subtract points if i see something like "<2 = <4" or "<1 + <2 = <3".

When we finally got into the good stuff, after watching me demonstrate a few proofs, a lot of kids would say things like...

“Ok I kinda get what you are doing, and each step makes sense, but you are just making it look easy.  It seems like you're just making it up."

or

"I understand some of where it is coming from, but there is just NO WAY I could come up with these steps myself and get from the beginning to the end on my own.”

Posters as a Guide When Stuck:

To help them organize the procedure and get "un-stuck" when they were unsure how to progress to the next step, I developed a series of steps for them.  Some kids really depended on this, and some thought that it didn’t help much.  For students who do need that structure, this chart is on their desk at ALL TIMES for a month straight.

Another group of students seemed to need a reference list of what kinds of things can be used as justifications.  Proofs are so different from anything that has been done before in their math classes.  Each student seems to get stuck on a different part of the process.  I found that having a reference sheet helped them a lot.

DOWNLOAD POSTERS for FREE:   Printable versions of these two pages are included in an email that I send out to subscribers.  If you would like to have the 8.5x11 posters for your students, subscribe to the Math Giraffe email list, and they will be sent straight to your inbox!

If you like this sequence and structure for introducing proofs, you may also want to check out my full proof unit or one of these practice packs.

These related resources are available for teaching proofs (Click images to link to products.)

I hope that the downloads I've included will help you organize your proof lessons and get you started with incorporating the new level of algebra proofs.  This approach has made a world of difference for me!  Let me know how it goes for your students. 

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