A Feast of Proofs: Justifying Triangle Congruence

1. Use the triangle congruence criteria SSS, SAS, ASA, AAS, and HL to show that 2 triangles are congruent to one another. 
2. Utilize congruent sides, congruent angles and properties of Geometry to create statements (specific statements) to show that two triangles are congruent. 
3. Explore multiple methods of creating a Geometric proof (flow chart, statement/reason table, paragraph) to where students create their own proof level argument.

I utilize the principles of Dutch Mathematician Van Hiele’s Level of Geometric thinking when building up to the rigor of proofs. Van Hiele asserts that everyone has a Geometric experience. A student at Level 0 is able to engage with Geometry with visualization. As students move up the stages of Geometric thinking, students begin to categorize and then formalize into logical arguments what we know as proofs. Below is a visual example of this formalization taken from a student’s final product.

As we can see, this student engaged with triangle congruence theorems at a visual level, and then was able to write down the general rule. The student was able to formalize the notion of triangle congruence by writing down specific pairs of sides and angles that are congruent. It is imperative that students move through the following sequence when learning proofs:

1. Visualization of SSS, SAS, ASA, AAS, and HL for triangle congruence theorems.
2. Matching congruent sides and congruent angles.
3. Connecting marked diagrams into statements.
4. Creating proof style arguments with information that is marked on a diagram.
5. Beginning to find information not marked in the diagram (i.e. triangles that have vertical angles that are not marked congruent).
6. Using triangle scenarios with not everything given to create proof like arguments.

It is imperative that each student masters each stage sequentially. Forcing a student who cannot yet read a diagram to write a formal proof will lead to undesirable outcomes. Rather, this student should be given exercises of reinforcement of vocabulary (i.e. becoming comfortable with vocab used in proofs or identifying triangle congruence criteria).

Additionally, proofs are more accessible to students when they can see examples of the thought processes, rather than being forced to immediately create their own. I propose a gradual release model where students who are struggling with vocab are forced to read a proof first, and then they express the argument in their own way (i.e. writing a paragraph expressing the logic communicated in a statement/reason proof). Then, the student can move into a flowchart proof or a formal statement and reason proof.

One common mistake is the teaching of triangle proofs where a teacher decides to teach each triangle congruence theorem, and then immediately forces students to write formal proofs on the same day. Instead, I recommend creating lessons leading up to the Feast of Proofs, that gradually build rigor (i.e. spend one day on Theorem recognition, spend one day on heavily guided proofs).

• I utilize the rubric on the back page of the Proof Poster Project as a mastery data point for proofs, where I look at a student’s ability to prioritize vocabulary, and a student’s ability to explain the process of proofs accurately. Advanced projects are able to detail common misconceptions about proofs and communicate common mistakes seen in proofs. 
• Mastery Assessments used at the end of proofs (done 1-2 class periods after proof project):
  • Standard Level Assessment
  • Honors Level Assessment
  • Grading Criteria Example

I have moved to assessing proofs, using both the project and the written mastery assessment, with the awareness that some students may not sit as well for a conventional mastery assessment, but may be able to articulate the tenets of proof based thinking (sometimes, rather creatively and innovatively) in a proof poster format.

Video Lessons/Notetaking Guides:

These videos should be used to give students worked out examples of proofs as well as the vocab foundations. In my flipped classroom, I have my students watch videos and take notes from the following videos: 

• Skill 4.5: Triangle Congruence Theorems – Around a 13 minute video that shows the triangle congruence Theorems (the thesis statements of triangle proofs). 
• Delta Math Proof Tips:  – Around a 30 minute video in which the first 15 minutes are spent reviewing key vocab that is commonly seen in proofs. The final 15 minutes walk through how to fill out proofs on our online math program Delta Math. 
  • Accommodations for Students who need help with note taking/potential modifications for in- person:
    • GUIDED NOTES [The Proof Playbook]: Those who have a classroom that is largely in-person, those with students that struggle with video note taking, or those who teach students with organizational issues, can use this document to help create guided notes. I have aligned these with the 2 videos mentioned above. I made this a single document with the rationale so that students can refer to this document for their entire time with proofs, and as they complete the Feast of Proofs Project.
    • Non flipped in-person classrooms might want to consider chunking the guided notes document into 2-3 parts to give students some direct instruction in the days leading up to the Feast of Proofs Project.

Feast of Proofs Project- The main documents that you will need for this individual lesson. 

• Feast of Proofs Project [STANDARD LEVEL]
• Feast of Proofs Project [HONORS LEVEL]
• Feast of Proofs Project Brainstorming and Expectations PPT
  • Feast of Proofs Project Brainstorming and Expectations PPT PEARDECK– PERFECT FOR VIRTUAL/HYBRID settings!
• DeltaMath – I highly recommend DeltaMath as an online math platform. There are tons of great differentiated triangle proofs that range from heavily guided flowchart proofs with all the information given, to networks of triangles that will challenge even your honors level students.
• Non-DeltaMath Proof Packet – A library of differentiated proofs in a paper format for students who do not like the Delta Math format (or if you don’t use Delta Math).

DeltaMath Proof Implementation Guidance:

This document provides an overview of what DeltaMath offers in terms of proofs and how I scaffold up to the Feast of Proofs project as I teach proofs in my class.

Proof Poster Best Practices/Exemplars:

I made this PowerPoint to celebrate particularly excellent attributes of proof posters/projects found in my Fall 2021 Geometry classes. These can provide you with an example of particularly creative/best practice approaches of students to build what your future Geometry scholars can create.

1. Bellwork (7 minutes):

Students should practice explaining how to solve a multi-step equation from Algebra class on a whiteboard. They should think about this individually and show their thinking on their whiteboards. I like having students show this thinking on a whiteboard because it helps me and the students understand what their inherent thinking style is. I choose an Algebra problem because most students in Geometry have taken and passed Algebra before and are much more used to working with an equation. After students are done, they should consider what their thinking style is.

2. Discussion Questions (10 minutes):

You might want to do a think, pair, share format for each question. In a virtual format, one could implement this by using the breakout room feature of Zoom and allowing students to discuss the questions in breakout rooms, and then as a whole class. Another way that this could be done is for students to respond in a Pear Deck, and then warm call and showcase exemplary responses as a class. The discussion questions on the PPT are pretty open-ended, but get students thinking about exemplar work and some of the features of effective proof style thinking that are seen in quality proof posters. I encourage maybe one minute of individual thinking, one minute of partner discussion and calling on 2-3 students for a whole-class discussion.

3. Project Expectations (10 minutes):

Walk students through the rubric and the different options for the project. Encourage students to think about the difference between the 3 point and the 4 point column of the rubric as that will encourage students to brainstorm about rigor.

• Idea for collaboration: Create or use 3 exemplar products that score between the 2 and 3 levels of the rubric that have a selection of strengths and weaknesses. Divide the room into sections or into breakout rooms. Each group is assigned an exemplar product to grade against the rubric. Have students arrive at what they would score each project and cold call or warm call on individual students to justify what score they would give.

4. Feast of Proofs Work Time (50 minutes):

Students should be able to finish at least ⅔ sections of the proof poster by the end of a block schedule period. It will be important to check in with students one-on-one regarding their progress and their interpretations of the directions. I would recommend for students, who are more likely to struggle, to have some exemplar projects on hand for students to look at.

Potential Accommodations for ELL and Resource Students: 

• Encourage electronic formats like a PPT or a word document for students who have issues with fine motor skills and handwriting. I often find that for these students the task of writing things down is burdensome, but if they are able to screenshot an example and then write a few bullet points to summarize, they can have the benefit of learning by example.
• For students that have skill gaps, language issues, or issues following directions, having exemplar products out for students to interact with works out best. I like to show these students exemplar products in a variety of formats and ask them which one they prefer. I allow them to look at and interact with the exemplar product while they create a product of their own in the same format. If you are using push-in supports, my push-in support/resource department did the following to support: 
  • Utilize the theorem organizer where students have the written definition and picture. Students can connect the reading of the diagram with the triangle congruence theorem and written description.
    

• I recommend some of the guided proofs from the non-Delta Math proof document. My push-in teachers/resource departments found that focusing on 2-3 of these proofs and engaging the students in a sort of discussion over them taught them to really see the structure of proof-thinking while having some of the support of the given information.
  

5. Closure (5 minutes):

Consider doing a quick Google Form or a survey to give students a checkpoint on their progress, a time to take questions/concerns, and a game plan towards turning in a completed proof poster project during the next class. 

6. Presentations Formats (things to try the next day):

Teaching during the pandemic has brought on certain social-emotional issues. The old model of calling all students up front to present in front of the whole class might make certain students feel uncomfortable in being called out. Therefore, it is better to put students in a position where they are presenting their ideas in a low pressure/collaborative situation. Here are a few ideas on how one could organize presentations in the time of pandemic:

• Whole class presentations: If you are going the whole class route in person, I recommend doing the project in groups of 2-4 instead of individually, and to use larger sized poster papers. This creates a more time-efficient and collaborative experience. You could even create a peer grading form to ensure engagement and purposeful feedback.
• “A (Google Classroom) Library of Proof Posters”: If you are doing the virtual format, I recommend taking the projects and creating a class library of projects on Google Classroom for students to interact with and study from. If you want to add to this assignment, you could create a Peer Grading Google Form where students can write feedback on the “Proof Posters” they saw (I recommend having students choose 2 projects if you go this route). Teachers can forward students’ feedback received from peers regarding the proof posters via email or through the online platform used by your school. 
• MasterTeacher: This is a creative and fun way of doing presentations that I have come up with during the pandemic. It’s called MasterTeacher because it’s structured like the game show “MasterChef.” It is my preferred method of presentations for in-class projects. 
  • ALL students participate in the “Team Challenge”- where students present their poster to their table partner. They have 5 minutes to present their poster. 
  • Upon the completion of the “Team Challenge,” students may opt into the classroom MasterTeacher competition. This is voluntary. Each MasterTeacher contestant is given a time limit of 5 minutes to present to the class. They are subject to questions of the peers and the teacher at the end of the presentation. 
  • When all presentations are complete, students vote on who will be crowned the _____ period Masterteacher (i.e. the B2 Geometry MasterTeacher). The winner could pose in front of a trophy and earn a triangular-shaped treat.

Opportunities for Collaboration: 

Feast of proofs  can be done in groups of 2 or even 4 to promote more dialogue and discussion on proofs.

• Consider ability grouping (Honors students should be paired with other Honors students, higher standard Geometry students can be paired with lower standard Geometry students, average standard students can be paired with standard Geometry students).
• Groups of 2 poster projects would be similar to the individual version of this assignment, except for the fact that it is done with a collaboration of 2 people instead.
• If you are considering a class-wide oral presentation component, a group of 4 might make sense and you can upgrade the poster size from 11×17 to a larger poster size. The students can then present key talking points and the general flow of a completed proof to the class.

Accommodating Struggling Learners (IEP, EL, Below Grade Level):

• For direct instruction, move to the Guided Notes [The Proof Playbook] document to go along with the video. This will help the students have their information all in one place and give them a place to start from for the Feast of Proofs Project.
• I recommend that these students use and summarize information from Proof Playbook to meet the vocab requirement of The Feast of Proofs.
• I recommend that these students use the more guided proofs from the Option 1 or Option 2 sections of the Non-Delta Math proofs. 
• I find that having examples of completed projects readily available for students to examine closely as they start work will help give these students more guidance and see some explanation strategies that they might find most helpful for them.
• Standard Level tests can be modified for exceptional learners by adding word banks for proofs or more “fill in the blank” type of questions (i.e. Line segment AB is congruent to _____________ )

Connections to Other Content/Bigger Picture:

1. Bodily Kinesthetic Learning: 

• Constructions are an excellent place to start building proof style thinking for a bodily kinesthetic learner. For instance,  you could have students copy a triangle that is drawn on a page with a compass, ruler and/or protractor. It is also fun to restrict the tools that a student can use (i.e. copy a triangle with only a ruler and no compass/protractor).
• Sites like Desmos and GeoGeBra are good places to do virtual constructions. Delta Math has an excellent selection of constructions but does not yet include a copying triangle construction problem. However, I like to use things like the perpendicular bisector construction earlier in the course.
• When doing constructions, I like bringing in a picture of The School of Athens to discuss an original Geometry classroom from Ancient Greece. My school is a classical high school and it fits right into my School’s classical mission. The picture below shows that the notion of congruence has been around for thousands of years and has been explored for thousands of years.
  
  

2. Pattern Making

• I used to have inductive/deductive reasoning skills in the course, but have cut out those explicit skills. However, problems that are similar to the dot problems in the last page of this document are nice examples of ways to observe patterns and generalize. They begin to engage the student at an individual level.
• One of my favorite problems to use to teach pattern making is the Handshake Problem. This problem often generates some stimulating discussion.

3. Logic/Deductive Reasoning

• Logic puzzles are an excellent way to hook students into the proof unit. A bell work or a whiteboard warmup of a logic puzzle often is a good place to hook students into the logical/proof-type of thinking. There are tons of examples in the logic puzzle magazines, and on the logic puzzle websites.
• One of the crazes right now is the game Wordle. A fun bell work to do at some point in the proof unit is to assign students the daily Wordle (there is also Nerdle (with numbers) and Noodle). Students can then volunteer to discuss their strategy in figuring out the Wordle. Your non-math/verbally oriented people might actually articulate some of the most detailed examples of deductive reasoning/proof type of thinking.

4. Triangles in Art: 

• In my summer school classes (on Zoom), I invited a local artist to discuss how he used Geometry in their art. He told us about how he found the triangle to be the easiest shape to work with in his designs and how they helped his sculptures withstand the laws of Physics. If you know (of) a local artist, I recommend bringing them to the classroom to discuss how they use triangles in their artwork.
• During the past few years, when we have had more time (perhaps a week or two before Final Exam Review week) my colleagues and I have used a Line Design Project.

From the Indiana Geometry Math Content Connectors

G.T.2: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G.LP.1: Understand and describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms and postulates, methods of reasoning, and theorems). Understand the differences among supporting evidence, counterexamples, and actual proofs.

G.LP.4: Develop geometric proofs, including direct proofs, indirect proofs, proofs by contradiction and proofs involving coordinate geometry, using two- column, paragraphs, and flow charts formats.