Math Giraffe
  • Start Here
  • Blog
  • Doodle Notes
  • Shop
  • Classroom Management & Ideas
  • Algebra
  • Geometry
  • Middle School
  • Inquiry Learning
  • Subscribe
  • About
  • Pre Algebra Doodle NoteBook
  • Finance Doodle NoteBook
  • Distance Math Activities
  • Start Here
  • Blog
  • Doodle Notes
  • Shop
  • Classroom Management & Ideas
  • Algebra
  • Geometry
  • Middle School
  • Inquiry Learning
  • Subscribe
  • About
  • Pre Algebra Doodle NoteBook
  • Finance Doodle NoteBook
  • Distance Math Activities
Search by typing & pressing enter

YOUR CART

Picture

3/10/2015 21 Comments

Introducing Two-Column Geometry Proofs: A Different Approach

A Better Approach to Introducing Two-Column Geometry Proofs
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.


Pin it

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.

Tweet: Strategy for teaching Geometry Proofs - http://ctt.ec/3Mk5b+ from @themathgiraffe

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.
Properties and Postulates for Geometry Proofs
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. 
Pin it
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.
Sequence for Unit and Lesson Planning for Geometry Proofs
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:
Teaching Substitution and Transitive Property in Two-Column Proof Writing
Pin it

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.
A Smoother Introduction to Two-Column Geometry Proofs
A New Step to Help Students with Geometry Proofs
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.
Algebra and Geometry Proofs Practice

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".
Congruent Versus Equal in Geometry Proofs
Pin it

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.
Pin it
Printable Posters for Geometry Proof Writing - Steps and Justifications
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!

Subscribe to the Math Giraffe email list

* indicates required

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. 

To Read Next:

21 Comments
Linda Mitcham
6/22/2015 04:03:32 am

Reply
DocRunning link
4/19/2016 10:45:13 pm

Thank you for this great post. I am teaching geometry (which I despised as a student) for the first time, and I appreciate the insights. Can't wait to try this.

Reply
Math Giraffe link
4/20/2016 02:29:06 pm

Hey Doc! :)
Im so glad you can use this. It makes a world of difference. I hope you love teaching geometry more than you enjoyed taking it! Good luck!
Thanks so much for commenting. :)
-Brigid

Reply
Charles Vochatzer
5/25/2017 08:46:40 pm

I too did not like (despised might work) Geometry in HS. Now, next school year (2017-18) I get to teach it. Yeah! I'll be looking into getting lots of help with this.

Reply
Math Giraffe link
5/28/2017 04:53:22 pm

Hi Charles,
I hope you learn to love it this time around! :)
Hopefully some of the resources here will help as you get started! Best of luck - let me know how I can help.
-Brigid

Terri
8/5/2017 05:23:11 pm

Also first year teaching geo in a long time with new text at a new school, ao looking forward to sharing and learning best practices!

Reply
Math Giraffe link
8/7/2017 02:52:08 pm

Hi Terri,
That is so great! I really appreciate hearing that you can use some of this :) Enjoy the Geometry! I hope you love it.
Have an awesome school year!
-Brigid

Reply
Marilyn McArthur
8/7/2018 11:50:29 am

Great ideas. I for the most part have the same students for Algebra I and Geometry - so do try to "prep" them with algebraic proofs during Algebra course. One thing I do require when doing proofs is have also number justification - to correspond with statement. When not using templates they often are hard to match up.

Reply
Becky Z
10/18/2018 09:48:44 pm

I have taught Geometry for the past 3 years, and started my career teaching Geometry from 1996 - 2003; so I am a veteran of the subject but still change up my technique every year that I teach it. This subtle interlude into proof writing is so clever, and something I haven't tried - I thank you and look forward to doing this lesson before simple Geometry proofs.

Reply
Math Giraffe link
10/22/2018 11:03:14 am

Hi Becky,
I hope that it works well for your students! It made such a difference for mine.
Thanks for commenting! Have an awesome week :)
-Brigid

Reply
Rebecca
2/12/2019 01:28:46 pm

Using substitution and transitive properties with the algebraic proofs makes so much sense when outlined above. Here's something I can never quite do a good job of explaining. The difference between when to use transitive and when to use substitution. Is one inclusive of the other? Meaning, any transitive property could also be considered substitution property? But not every substitution property is transitive? For example, in the proof just before Try It, could yo have used Substitution based on lines 3 and 5? THanks!

Reply
Math Giraffe link
2/12/2019 02:42:05 pm

Hi Rebecca,
Yes, thanks so much! This is a challenge, but I wrote up a post about this exact question :)
Here is that link - https://www.mathgiraffe.com/blog/teaching-substitution-vs-the-transitive-property
Hopefully that will help :) Have a great evening,
-Brigid

Reply
Chris Beggs
4/19/2019 09:54:33 pm

I agree with your "pet peeve" - I regard it as a "category error", segment AB is in the Geometry category, measure of segment AB is in the Arithmetic (or even Algebra) category.

I went through a longer list of what you can do: Segments: Congruent, can be parallel, can be perpendicular, can intersect, can be collinear ... Measures: Equal, can add, subtract, multiply, divide, cube, square root...

Then I emphasised that the "road" between the two categories is the definition of congruence - you can turn congruence into equality or vice versa; the segment addition principle is the "bridge" between them - it lives in both worlds.

Similar things then follow for angles and angle measures. Angles can be vertical, a linear pair, supplementary, complementary, alternate interior, ... but numbers can't be vertical etc.

For me, the more things in the lists, the clearer it becomes to the students as to what beongs in which "world".

Reply
Math Giraffe link
4/25/2019 12:11:01 pm

Hi Chris,
That is even better! Sounds really great.
I love those additions. Thank you so much for taking the time to share! :)
Have a great day,
-Brigid

Reply
Vicky
10/2/2019 11:52:27 pm

I am trying to download the Proofs Posters guide, but it will not send it to me because I already subscribe to your website. Is there anyway that you can resend those to me?

Reply
Math Giraffe link
10/3/2019 10:22:29 am

Hi Vicky,
No problem! Just send an email to brigid@mathgiraffe.com and I'll send them over ;)
Have a great day!
-Brigid

Reply
Kal C
5/24/2020 05:44:09 pm

Thanks for this. The style of proofs should be as logical as the proof itself. So I agree with your two-column poem structure, which is a systematic way to write a proof. It is essentially the same way that professional mathematics write their proofs, except in prose not in poem.

Just my one issue is in the way you have written the proof in your example. A proof is like a staircase. Your legs should move up the staircase one logical step at a time. So you start with: m = as the bottom step, and: = 3h is the top step. You climb up the staircase of the proof by filling in the steps in between one at a time. So I would structure it like this:

Given #1: g = 2h
Given #2: g + h = k
Given #3: k = m

Step #1: Start with: m =
Step #2: = k, using Given #3
Step #3: = g+h, using Given #2
Step #4: = (2h) + h, using Given #1
Step #5: = 3h, using transitivity

We started at the first step: m =, and ended at last step: = 3h. So m=3h, and the proof is done.

However, in your proof, you start at an intermediate step, which is like jumping steps or splitting your legs between two different steps.

This is just my two cents. Overall the article is appreciated. Thanks.

Reply
Kal C
5/24/2020 05:52:43 pm

Sorry, for Step #5, disregard "using transitivity". That does not apply there.

Reply
Math Giraffe link
5/30/2020 02:31:44 pm

Hi,
Thanks so much for chiming in! Yes, I think you are right - I appreciate your comment. :)
Thank you and have a great weekend!
-Brigid

Reply
Brandon
1/21/2021 09:47:31 am

Is there, or do you have a complete curriculum for Geometry? I do not like the textbook style which is the typical format.

Reply
Math Giraffe link
1/23/2021 11:24:49 am

Hi Brandon,
I don't have a full curriculum yet, but I do have a large set of resources, and the proof content in there is very comprehensive and will take your students all the way through the process of learning to write a proof with this method. It's very heavy on the proof and logic, but is not a full curriculum. The preview file here will give a look at what is included: https://www.teacherspayteachers.com/Product/High-School-Geometry-Super-Bundle-1948872
But then let me know what other questions you may have. Thanks, and have a great weekend,
-Brigid

Reply



Leave a Reply.


    Archives

    March 2022
    December 2021
    November 2021
    September 2021
    July 2021
    April 2021
    November 2020
    September 2020
    August 2020
    April 2020
    March 2020
    November 2019
    September 2019
    April 2019
    March 2019
    February 2019
    January 2019
    November 2018
    October 2018
    September 2018
    August 2018
    July 2018
    June 2018
    May 2018
    April 2018
    March 2018
    February 2018
    January 2018
    December 2017
    November 2017
    October 2017
    September 2017
    August 2017
    June 2017
    May 2017
    April 2017
    March 2017
    February 2017
    January 2017
    December 2016
    November 2016
    October 2016
    September 2016
    August 2016
    July 2016
    June 2016
    April 2016
    March 2016
    February 2016
    January 2016
    December 2015
    November 2015
    October 2015
    September 2015
    August 2015
    July 2015
    June 2015
    May 2015
    April 2015
    March 2015
    February 2015
    January 2015
    December 2014
    November 2014


    RSS Feed

    Related Posts Plugin for WordPress, Blogger...
    Click to set custom HTML
Proudly powered by Weebly
Photos used under Creative Commons from kellywritershouse, Robert-Herschede, University of the Fraser Valley, mrhayata, Iwan Gabovitch, nateOne, Franklin Park Library, rhymeswithsausage, US Department of Education, Edsel L, Larry1732, philwarren, peteselfchoose, Dean Hochman, BryonLippincott, AFS-USA Intercultural Programs, Nilsze, philosophygeek, VividImageInc, cantanima, dcysurfer / Dave Young