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The role of proof in mathematics
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The role of proof in mathematics


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    • 1. The Role of Proof in Mathematics
    • 2. The Role of Proof in Mathematics
    • 3. Proof in Mathematics
      Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of characters.
      Vladimir Arnold
    • 4. Standards on Proof
      Instructional programs that should enable students to:
      • develop and evaluate mathematical arguments and proofs
      • 5. select and use various types of reasoning and methods of proof
      • 6. By the end of middle school, students should be able to understand and produce mathematical proofs
    • 7. Proof
      • Convincing demonstration that a math statement is true.
      • 8. To explain.
      • 9. Informal and formal.
      • 10. Logic
      • 11. No single correct answer
    • Proofs
      Often proofs are constructed by working backwards. For example:
      Starting with the desired conclusion T, you could say, "If I could prove statement A, then using previously proved theorem B, I could conclude that T is true." This reduces your proof to proving statement A, then saying at the end of that proof, "Using Theorem B, T is true."
      Often there are many possibilities for A (and B).
      The trick is to pick one you can prove!
    • 12. Three Forms of Formal Proof
      • Synthetic Geometry
      • 13. Analytic Geometry
      • 14. Transformational Geometry
    • 15. Synthetic Geometry
      • A system illustrated by proving geometric relationships based on the use of a rational sequence of definitions, postulates, and theorems
      • 16. 19th Century
      • 17. Pure geometry
      • 18. Logical Arguments
    • 19. The most common proof – The Pythagorean Theorem
    • 20. Grade 7 – MathematicsFinding the value of (a-b)2 (Geometrical Proof)
    • 21.
    • 22. Analytic Geometry
      • Also known as coordinate geometry or Cartesian geometry
      • 23. Algebra
      • 24. Graphing Technology
      • 25. Computations
    • Analytic GeometryCartesian Geometry
      Also known as coordinate geometry-graphing
    • 26.
    • 27. Transformational Geometry
      • 20th Century
      • 28. Graphics technology
      • 29. MIRA
      • 30. Plane mirror
      • 31. Is a method for studying geometry that illustrates congruence and similarity by use of transformations
      • 32. Therefore a transformational proof is a proof that employs the use of transformation
    • Transformation Proof
      An isometry is a transformation of the plane that preserves length.  For example, if the sides of an original pre-image triangle measure 3, 4, and 5, and the sides of its image after a transformation measure 3, 4, and 5, the transformation preserved length.               A direct isometry preserves orientation or order - the letters on the diagram go in the same clockwise or counterclockwise direction on the figure and its image.             A non-direct or opposite isometry changes the order (such as clockwise changes to counterclockwise).
    • 33. Transformational Proof
    • 34.
    • 35.
    • 36.