Tuesday, July 15, 2014

Intro to Proofs in Geometry

I wanted to blog about this a looooong loooong time ago but the school year got in the way along with moving across the country twice because of family health dramas (NY to California in the fall, now California to Oregon.  I know Cali to Oregon doesn't seem that far, but it is over 1,000 miles from San Diego to Portland.  California is freekishly big.)  So though I know posting lesson plans in the summer is kind of silly, I want to get it out of my system before I forget what I did.

The school in which I taught this past year was a high poverty school where 30% of our students had IEPs.  It is a charter school so it's pretty small meaning I got to work very closely with my students, colleagues and parents but we did lack funding and our students were weak in a lot of basic skills.  In fact, most of my geometry students this past year hadn't even passed algebra 1 yet.  The previous algebra teacher found them so lacking in basic skills that she gave the entire algebra 1 class "incompletes" because they didn't finish the algebra 1 curriculum.  My principal decided to enroll all these students in geometry anyway because she figured (rightly I think) that they needed a bit of a break from algebra and if they saw some algebra in a geometrical context it might make going back to algebra more meaningful (which it did.  Every time algebra popped up in geometry the students were actually excited because it was familiar and wasn't too difficult.  They really mastered equation solving, writing expressions and equations of lines by studying these topics through geometry.)  All of this meant that when proofs came up I was super freaked out.  I've always struggled with teaching them and I feel like I've done a very poor job in the past.  I put a lot of thought into how to build proofs into our curriculum this past year and I feel like what I did was relatively successful.  My students weren't scared of proofs for the first time in my teaching career.  When they came up, the students knew at least where to start and always attempted them.  So I want to lay down what I did just so I don't forget.

So here is the description of how the unit flowed.  It's definitely a more traditional approach to proofs and I stuck to two-column proofs.  I tried to transition the students to paragraph proofs, but their skills and confidence were too low; they liked the organized nature of two-column proofs.  First, I didn't nix the logic unit.  Even though logic is not in Common Core anymore, I think that the reasoning done in the logic unit helps prepare students for proofs.  

Logic Unit Lesson 1: Intro to conditional statements.

  • First I did Sam Shah's lesson introducing conditional statements.  I did the drawing activity and posted all their pictures on the wall.  It went really well- I was surprised at how much trouble some students had following the directions precisely.  A lot of them didn't know the geometric vocabulary (like what an isosceles triangle is) or were hesitant drawing so it was a great activity to do towards the beginning of the class.
  • Then I just did a mini-lecture on the notation of conditional statements, Euler diagrams and what a negation is.
  • Then I gave them this assignment: Logic Unit Lesson 1: Conditional Statements Intro 
Logic Unit Lesson 2: Manipulations of Conditional statements
Logic Unit Lesson 3: Word Proofs
  • First we did a syllogism activity  where I just cut the cards apart and had them put the syllogism in the correct order and a mini lecture on syllogisms
  • Then we did word proofs.  This is one of the most successful lessons I've ever taught on proofs.  I totally stole it from another blogger, and of course forgot to save their name in the name of the file I downloaded like I normally do.  When I figure out who made it I'll update this post.  I reformatted the file I stole from that other blogger and did the lesson in kind of a workshop style.  I did one or two of the word proofs on the board to demonstrate how to do it, then I gave them time to work on their own and then we compared answers.  I had them come up and show different solutions they'd discovered and we talked about the fact that there's more than one way to do a proof correctly.  Like I mentioned before, this was actually a pretty bright class but one lacking in discipline and both basic math and study skills.  I had to hold the kids back.  They were chomping at the bit to do more and more and more puzzles.  I made the last two pages of the lesson optional and almost all students did them anyway.  Whoever designed this lesson was brilliant because it really hooked students who are usually disengaged with math.  You can't not want to solve one of these puzzles when they're presented to you.  
  • There are two pages in the lesson to have the students make their own puzzles and switch papers with each other.  We had to skip this part because we ran out of time.
  • Here's the lesson: Logic Unit Lesson 3: Word Proofs
Logic Unit Lesson 4: Angle Proofs
  • This lesson was a little less fun than the last one, but it was very effective.  First we went over basic angle terminology: complementary and supplementary angles, vertical angles and linear pairs (they'd learned these before), and we also talked about the algebra properties of equality, transitivity and the substitution property.  
  • Then I had them do the lesson below in pairs.  It's structured exactly like the word proofs from the last lesson with 4 or so "rules" and space for them to use the rules to go from the given to the prove.  The students were able to stumble their way through these proofs without me doing any examples on the board based on what they did in the last lesson.  
  • Here's the lesson: Logic Unit Lesson 4: Angle Proofs
  • Finally we concluded with a Scrambled Proofs Activity
Here's where I made a mistake.  I did parallel lines and transversals as the next unit because I liked flowing from points, to lines, to parallel lines to triangles to polygons.  It seemed like the logical way to structure the course.  Also, congruent triangle proofs are so much richer and more interesting if the students already know their parallel lines and transversals angle relationships.  But if I were to do it again, I would do congruent triangles after the logic unit and then do parallel lines and transversals.  The proofs for parallel lines and transversals are a little more abstract and involve more vocabulary than congruent triangle proofs so trying to launch from the intro to proofs unit straight into parallel lines and transversals was too big a jump.  So here is how I built up proof using congruent triangles after I failed at teaching them proofs through parallel lines and transversal relationships.  

Congruent Triangles Lesson 1: Intro to Congruence
  • For this lesson I just did a standard lecture over what congruence is, the notation for congruence and examples of using congruence to find missing parts.  Nothing really exciting.  The only thing about this lesson that I like is my warm-up.  We define congruence as "identical in every way" and then I ask students if the two identical twins are congruent which leads to a great discussion of what "corresponding parts" means: 
  • Also, I always get a laugh out of my student by choosing another teacher who's the same height as me (at this school I used the principal) and I use that teacher to discuss the difference between the "equal" symbol and the "congruence" symbol.  Our heights can be equated, but if you accidentally use the congruence symbol you're saying I and this other teacher are identical in every way.  
  • Here's the assignment for that lesson: Congruent Triangle Lesson 1: Intro to Congruence
Congruent Triangles Lesson 2: Triangle Congruence Theorems
  • I used this cool illuminations app on congruent triangle theorems along with this worksheet that I wrote to introduce SSS, SAS, ASA and AAS.  I've done this activity twice, the first time I let the students pair up and use their own computers.  A lot of the students wouldn't or couldn't follow the directions so I spent the whole period frantically running around trouble shooting.  The more motivated students in the class were able to make the connections I wanted but the other students left class mostly confused.  So the second time, we did each activity first as a whole class with computers closed, then I let them open their laptops and play with the simulation to confirm the results for themselves. This worked much better.  All the students were successful on the homework without need for more instruction.  
  • Then I totally stole a few worksheets here and here from Math Teacher Mambo for the students to work on independently.  Here's my mash-up of her brilliance: Congruent Triangle Lesson 2: Congruent Triangle Theorems
Congruent Triangles Lesson 3: Using Theorems in Proofs
  • This is a lesson I created that I'm exceptionally proud of.  It was super boring though.  I realized that students weren't really getting what theorems were for or how to use them.  They were still struggling with problems like: if angle A and angle B are a linear pair and angle A measures 40 degrees, what is the measure of angle B.  They weren't thinking about what "linear pair" means and how to connect that meaning to the problem.  Even if they got this far they didn't understand that in a "proving" situation they needed to state how they know that angle A and angle B add to 180 (that they need to say: by def of linear pair or state the theorem).
  • So I made this worksheet: Congruent Triangles Lesson 3: Using Theorems to Make Deductions.  
  • In pairs I had them go through their notes and fill in the blanks for all the theorems we've covered.  This was the boring part.  I gave them a time limit though which helped keep them focused and this activity also reinforced the importance of taking notes.  I refused to tell them any answers.  If they didn't have it in their notes they needed to find someone who did.  
  • The "classwork" part of the lesson is where it all really paid off.  I did the first few examples with them- how to cite the correct theorem that had been used in each situation.  As soon as I started doing these problems on the board, a bunch of "ohhhh so that's why we needed those stupid theorems" exclamations went through the room.  It was especially rewarding to watch them do the last page where they have to think backwards- find the theorem that applies to the situation then figure out what deduction can be made.  
  • Students guarded the list of theorems they made as the first part of the lesson fiercely and insisted that I do a similar "fill in the blank theorem review" at the end of every unit.  
Congruent Triangles Unit Lesson 4: Proving Triangles Congruent.
  • This lesson went quite well although it's a very traditional lesson.  I just did a few example proofs with them and talked about how to set up a two-column proof table (I know... but these students liked structure.  I tried to show them a paragraph proof and their eyes all crossed and they started throwing paper.)  
  • I gave them this packet of problems from letspracticegeometry.com (there are a lot of typos in this worksheet though.  I would like to rewrite it) but without the first two pages.  
  • Then I gave them this proof template worksheet thing that I created.  It has the students choose which proofs they want to try, the harder proofs being worth more points.  They have to reach a certain number of points to get full credit.  
  • This small spin on a worksheet created a night and day difference in students' attitudes about proofs.  Every other time I've taught proofs students have been super whinny about them and would give up quickly.  But when I handed out the above assignment I saw at least half the students immediately turning to the last page to do the harder proofs.  A lot of them struggled on the proofs through a good chunk of the period without finishing and I kept suggesting they just go do more of the easier proofs, or build up to the harder ones but they said that they wanted to do hardest ones.  The fact that there was a choice between easy and hard involved made them want to prove to themselves that they could do the hard.  Students with less confidence started with the easy ones and were able to advance to the harder ones pretty smoothly.  Everyone was engaged and no one was complaining that I was making them do proofs.  
Congruent Triangles Unit Lesson 5: CPCTC theorem proofs
  • I taught this lesson the same as the last one.  Examples then a "choose your own problems" proof worksheet.  Here's the packet from  letspracticegeometry.com and here's the proof template worksheet I gave them.
  • I've had trouble in the past with students using CPCTC inappropriately so I put the following message up on the projector in giant letters and made them recite it in unison a few times.  I kept it up as they worked on the proofs and I didn't have students misusing CPCTC!  
To use CPCTC you MUST
FIRST: Prove triangles congruent
THEN: Say parts are congruent with CPCTC

CPCTC says that:
IF two triangles are congruent THEN their corresponding parts are congruent.
Prove the "IF" first, Only then can you use the "THEN"
  • Again, through this class period students were working on proofs without complaint and without giving up.  If they started to have trouble they could persevere or choose a new problem and this flexibility eliminated a lot of the griping I've experienced in the past with proof practice.  Boring but effective.  

WHEW.  That's all.  I just wanted to catalog what I'd done because this was my most successful proof teaching experience so far.  It still needs a lot of work though.  And I know that under Common Core, it may not even be relevant anymore because I didn't work in any proving congruence with transformations.  


  1. Thanks so much for this- you were very specific and thorough. I feel like I could follow along with everything you were saying and it gave me some new resources for my proof unit. Great job!

  2. Are all the word proofs solvable in the worksheet for 2-3? I cannot figure out 8 and 10

  3. Did you get the work sheet for Lesson 3 from this website:


  4. I know this is an old post, but is there any chance you have an answer key to the Word Proofs activity? I just want to make sure I'm doing them correctly before I give it to my students!

  5. I would like to include your include your Exponent Cards and Going on a Vacation activity, that have been uploaded to Squibd, in a sixth grade extension packet for teachers within my district. Squibd said to contact you directly. Please feel free to email me at sjtisler@gbaps.org. Thank you for your excellent blog!