The role of proof in mathematics

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  • http://vimeo.com/3386641
  • http://www.graspr.com/videos/Math-Proof-Negative-Times-Negative-Equals-Positive-3
  • The role of proof in mathematics

    1. 1. The Role of Proof in Mathematics<br />
    2. 2. The Role of Proof in Mathematics<br />
    3. 3. Proof in Mathematics<br />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.<br /> Vladimir Arnold<br />
    4. 4. Standards on Proof<br />Instructional programs that should enable students to:<br /><ul><li>develop and evaluate mathematical arguments and proofs
    5. 5. select and use various types of reasoning and methods of proof
    6. 6. By the end of middle school, students should be able to understand and produce mathematical proofs</li></li></ul><li>http://vimeo.com/3386641<br />
    7. 7. Proof<br /><ul><li>Convincing demonstration that a math statement is true.
    8. 8. To explain.
    9. 9. Informal and formal.
    10. 10. Logic
    11. 11. No single correct answer</li></li></ul><li>Proofs<br />Often proofs are constructed by working backwards. For example:<br />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." <br />Often there are many possibilities for A (and B). <br />The trick is to pick one you can prove!<br />
    12. 12. Three Forms of Formal Proof<br /><ul><li>Synthetic Geometry
    13. 13. Analytic Geometry
    14. 14. Transformational Geometry</li></li></ul><li>http://www.graspr.com/videos/Math-Proof-Negative-Times-Negative-Equals-Positive-3<br />
    15. 15. Synthetic Geometry<br /><ul><li>A system illustrated by proving geometric relationships based on the use of a rational sequence of definitions, postulates, and theorems
    16. 16. 19th Century
    17. 17. Pure geometry
    18. 18. Logical Arguments</li></li></ul><li>
    19. 19. The most common proof – The Pythagorean Theorem<br />
    20. 20. Grade 7 – MathematicsFinding the value of (a-b)2 (Geometrical Proof)<br />
    21. 21.
    22. 22. Analytic Geometry<br /><ul><li>Also known as coordinate geometry or Cartesian geometry
    23. 23. Algebra
    24. 24. Graphing Technology
    25. 25. Computations</li></li></ul><li>Analytic GeometryCartesian Geometry<br />Also known as coordinate geometry-graphing<br />
    26. 26.
    27. 27. Transformational Geometry<br /><ul><li>20th Century
    28. 28. Graphics technology
    29. 29. MIRA
    30. 30. Plane mirror
    31. 31. Is a method for studying geometry that illustrates congruence and similarity by use of transformations
    32. 32. Therefore a transformational proof is a proof that employs the use of transformation</li></li></ul><li>Transformation Proof<br />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).<br />
    33. 33. Transformational Proof<br />
    34. 34.
    35. 35.
    36. 36. http://mathoverflow.net/questions/8846/proofs-without-words<br />cut-the-knot.org<br />cartoonstock.com<br />Bibliography<br />www.nctm.org<br />

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