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GBK Geometry

GBK Geometry. Jordan Johnson. Note. I was out Monday. On Monday, Asg #9 was due The class worked on Asg #10, and on the test analysis and corrections. Today’s plan. Greeting Hand in Test Analysis & Corrections Practice Quiz: Definitions Check Asg #10

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GBK Geometry

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  1. GBK Geometry Jordan Johnson

  2. Note • I was out Monday. On Monday, • Asg #9 was due • The class worked on Asg #10, and on the test analysis and corrections.

  3. Today’s plan • Greeting • Hand in Test Analysis & Corrections • Practice Quiz: Definitions • Check Asg #10 • Lesson: Relationships & Transitivity • Break • Lesson: Indirect Proof • Homework / Questions • Clean-up

  4. Test Analysis • Hand in your test analysis & corrections: • Staple them to the test. • Top: Test analysis • Middle: Corrections • Bottom: Original test • Take out Asg #10.

  5. Practice Quiz: Definitions Consider:An animal is a horse iff it has hooves. • Write the two conditional statements that, taken together, are equivalent to this statement. • How are they related? • This definition is obviously not good enough.One of the two conditional statements is wrong; which one?

  6. Solutions • An animal is a horse iff it has hooves. • If an animal is a horse, it has hooves. • If an animal has hooves, it is a horse. • The two statements are converses. • The second statement is the flawed one.

  7. Relationships • Think about how • numbers can be related to each other • people can be related to each other • words can be related to each other • Examples: • One number can be greater than another. • Someone can be someone else’s brother. • Two words can sound identical. • List (on paper) as many relationships as you can, in three minutes.

  8. Transitivity:A Pattern in Relationships • Some relationships are transitive: • If x > y and y > z, then x > z. • If person x is person y’s brother, and y is person z’s brother,then x is z’s brother. • If one word sounds identical to another word, andthe second word sounds identical to a third word,then the first word sounds identical to the third word. • Equality and “>” are others: • If x = y and y = z, then x = z. • Some relationships are not transitive.For example, “” is not: • If x  y and y  z, then x  z.

  9. Transitivity Examples • Transitive relations: • If x = y and y = z, then x = z. • If x< y and y < z, then x < z. • If x y and y  z, then x  z.

  10. Relationships • Look at the relationships you wrote earlier. • Which of those relationships are transitive? • Which aren’t?

  11. Stretch Break

  12. 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.

  13. 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:

  14. 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.

  15. 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.

  16. 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?

  17. Indirect proof in Sudoku 6

  18. Work • Have Asg #10 out for me to check now. • For Day 3 (Thu/Fri): Asg #11 • Exercises 16 and (B+) 20:Write the entire proof. Box/underline your answers. • Exercise 21 (B+) and Set III (A+) require one or more complete sentences. • For Monday: Asg #12

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

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