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GBK Geometry

This lesson provides an introduction to indirect proof techniques in geometry, focusing on how to demonstrate the impossibility of a statement by assuming its opposite. Students will review key concepts and engage in practical exercises, including the classic checkerboard tiling problem. Through a series of group activities and a poker proof example, students will learn to identify contradictions that arise from false assumptions. The lesson concludes with reminders for clean-up and preparation for upcoming assignments and tests.

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GBK Geometry

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  1. GBK Geometry Jordan Johnson

  2. Today’s plan • Greeting • Unit 1 Review / Practice Quiz • Lesson: Indirect Proof • Homework / Questions • Clean-up

  3. Card Tricks • For this exercise, speak only when called on. • Split into two groups: • one against the front wall • one against the back wall • One volunteer from each group, step forward.

  4. Indirect Proof Prove X. Original problem: i.e., prove that the opposite leads to a contradiction Prove the opposite of X is impossible. New problem:

  5. Example How did we prove the checker-board tiling was impossible? • Suppose we could tile the board. • If we can, then every domino covers a white and a black square. • Thus we’d always cover the same number of white and black squares. • There are 30 white & 32 black squares, contradicting statement #3.

  6. Indirect Proof / Contradiction • Alternative to direct proof: • Start by assuming the opposite of the conclusion you want, and prove that it leads to a contradiction. • If you want to prove (a c)… • Assume that a holds • Assume that (not c) holds • See if you can come up with a contradiction.

  7. Practice problem – A Poker Proof • Theorem: • If I hold this hand,nobody holds a royal flush. • What’s our initial assumption? • What’s our reasoning? • What contradiction do we reach?

  8. Work • Write up today’s lesson in your journal. • For tomorrow: Unit 1 Test Analysis • For Wednesday:Asg #10 • For Thursday/Friday: • Asg #11 • Week 6 Journal

  9. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!

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