Unit 3
Proof and Congruence
Building Arguments That Must Be Accepted
In Unit 3, students stop treating proof as a side topic and start using it as the main event. Beginning with compass-and-straightedge constructions, they learn that “being able to build it” is the same as “being able to prove it exists.” From there, triangle congruence, CPCTC, and careful diagram reading become the raw materials for arguments where every step is earned. Read the full Unit 3 overview to see how the sequence builds genuine proof writers, not step‑copiers.
For the full story of how this unit is sequenced, what changed from Year 1, read the complete Unit 3 overview and navigation instrument below.
Unit 3 Lessons
Lesson 3.1 – Constructions as Proof
Students meet constructions as existence proofs, not drawings: if you can construct it with compass and straightedge, you have proved it exists. Through the dragon definition task, Euclid’s Proposition 1, and the “nullilateral triangle,” they see how precise definitions and postulates work together to make an argument that must be accepted.
Lesson 3.2 – What Makes Triangles the Same
Using manipulatives and counterexamples, students discover which shortcuts really guarantee triangle congruence (SSS, SAS, ASA, AAS) and why SSA and AAA fail. We also need to add HL. CPCTC is introduced as a consequence of congruence, with an emphasis on writing the full definition before using the abbreviation so every later step in a proof is justified.
Lesson 3.3 – Reading a Diagram
Students learn to read diagrams as sources of guaranteed information, not just tick marks. By listing everything a diagram alone ensures, then layering on given facts, they practice spotting shared sides, vertical angles, and other hidden structure. The class also formalizes a CPCTC “meaning first, abbreviation second” policy that will govern all later proofs.
Lesson 3.4 – Diagnostic Practice Day
This day checks whether students are actually ready to write proofs. Working individually through early rows of the navigation instrument, they apply congruence shortcuts and diagram reading in varied problems, then self-assess. The goal is a clear picture—for both teacher and students—of which tools are solid and which need shoring up before proof structure begins.
Lesson 3.5a – Proof Progressions
Students are introduced to proof as a growing argument rather than a fixed template. Using the Ray Allen story and a three-stage parallelogram reveal, they build flow-chart proofs where each box feeds the next. Two-column proofs are explicitly retired so attention stays on how statements chain together, not on filling a format.
Lesson 3.5b – Proof Presentations
In groups, students prepare and present flow-chart or paragraph proofs, then critique one another’s work. Class discussion focuses on whether each conclusion is truly earned from its premises and whether any assumptions slipped in without justification. Individual follow-up tasks ensure every student can revise and strengthen an argument based on feedback.
Lesson 3.5c – Scaffolded Proof Practice
Students choose their own entry point into proof practice—mild, medium, or spicy—so everyone works at a productive level of challenge. Within each track, tasks are scaffolded to fade support over time, pushing students to take more responsibility for structuring arguments while still insisting that every step be backed by a clear reason.
Lesson 3.6 – Review Day
This lesson consolidates the unit’s ideas before assessment. Through Stella’s Stunners, targeted practice sets, and an update to the navigation instrument, students revisit constructions, congruence, CPCTC, and proof formats. The tiered problem bank lets teachers steer students toward the specific proof moves and concepts they most need to reinforce.
Unit 3 Assessment
I’m happy to share my Unit 3 assessments and grading rubrics for teacher use, but I don’t want them publicly available on the internet. If you’d like a copy, please email me from your school account at jay@thinkingthroughgeometry.net and let me know which school and state you teach in.