Proof Presentations

Groups are assigned one of eight proofs to write and present. The proofs range from accessible (SSS or AAS then CPCTC) to genuinely difficult (Problem 8 requires decomposing given angles using Common Notion 3). Read the teacher notes for Problems 5 and 8 before assigning.

 

During the writing phase, resist the urge to answer questions directly. In Year 1, students relied too heavily on teacher support and the lesson did not produce the intended thinking. Redirect with questions: ‘What do you know from the given? What does that allow you to say?’ Or: ‘What would have to be true before you could write that step?’ A group that is genuinely stuck after sustained effort needs a hint, not an answer — the smallest possible push: ‘Have you thought about what the diagram gives you for free?’

 

During presentations, your role is facilitator, not judge. Let students ask the questions. Your job is to ensure questions get asked about every unjustified step — if the class misses one, you ask it. The most important moment in Year 1 came from a student who asked a gentle question that caused a presenting group to realize their argument was flawed. The class then completed the proof together. That is the culture this lesson is building: proof is a communal act of verification, not a performance.

 

If a group presents confidently but incorrectly and no one in the class challenges it — do not correct it yourself immediately. Ask the class: ‘Does everyone agree with this step? Can someone explain why it’s justified?’ Silence or agreement when a step is wrong is data.

 

Problem 5 — the proof that cannot be completed yet:

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Problem 5 asks students to prove KG ≅ MG given that GL bisects both KM and ∠KGM. This proof cannot be completed with the theorems currently in our system. To prove KG ≅ MG using CPCTC, students need to first prove two triangles congruent. The natural approach requires knowing that the angles at the bisection point are right angles — that GL is perpendicular to KM. That is true, but it is not yet proven in our system. It requires an intermediate theorem we have not established.

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In Year 1, a student conjectured: ‘If it bisects the angle and the side, then it must be perpendicular.’ This conjecture is correct. But it is an unproven proposition, and using it as if it were established is precisely the error we are training students to avoid. We can want it to be true. We can believe it is true. We cannot use it until it has been proven.

This connects directly to the Riemann Hypothesis: there are many results known to follow from it, but none of them can be accepted as proven until the Hypothesis itself is established. They hang on the truth value of an unproven antecedent. When this moment arrives in class, let the conversation develop. Ask: what would you need to prove first to unlock this problem? This is a preview of Unit 4.

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Teacher decision: you may leave Problem 5 out of the assignment, or assign it deliberately to a group you trust to engage seriously with the conversation it will generate.

 

Individual Follow-Up:

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After presentations, students individually write a proof for a problem a different group presented. They choose which problem to attempt based on their navigation instrument self-assessment. This is where students find out what they actually understand independently.

 

Circulate during the individual work just as you did during group work. The errors you see here are the errors to address at the start of Lesson 3.5c. Common patterns from Year 1: students correctly identified the shortcut but did not specify which parts were congruent and how each was established; students used alternate interior angles without citing a given of parallel lines; students wrote congruence statements with incorrect correspondence order.

 

Collect this work at the end of class. Before Lesson 3.5c, scan for the most common errors and plan a brief consolidation warm-up that addresses them without singling out individuals.