This post originally appeared on the blog Math Giraffe.
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:
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.
Try It (Download Files):
“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.”
“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.
Brigid has taught a range of middle and high school math courses, but her real passion is Geometry! She loves to teach proofs, and enjoys blending fun and rigor into math class. Discover creative and unique strategies for teaching by stopping by her blog. Or visit her on Pinterest, Facebook, Instagram, or at her TpT store Math Giraffe.