Coordinate Proof Day

Focus: slope, distance, midpoint — classification arguments — sufficient vs. necessary — two systems, same truth

This lesson unites two proof systems. In Lessons 5.2 through 5.4, students proved quadrilateral properties using Euclidean tools. Today they prove the same facts using coordinate geometry: slope, distance, and midpoint. Same conclusions, different starting points.

This is a full board-work day. Groups work through problems in three tiers of increasing complexity. There are more problems than most groups will finish — that is intentional. The coordinate tools are recalls, not new content: slope and the parallel/perpendicular slope relationships are Algebra 1, midpoint and distance are Unit 1. The new work is using those tools to make classification arguments. The most common error: showing one pair of parallel sides and concluding ‘parallelogram.’ Students need both pairs. Make this a specific coaching point during circulation.

The framing for this lesson: coordinate proof and Euclidean proof are two different formal systems built on different foundations, yet they agree on every result in this unit. Students who have been with the course since Unit 2 will recognize this as a return to the axioms conversation. Name it that way.

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