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# 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
• ### Transcript

• 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
• http://vimeo.com/3386641
• 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
• http://www.graspr.com/videos/Math-Proof-Negative-Times-Negative-Equals-Positive-3
• 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. http://mathoverflow.net/questions/8846/proofs-without-words
cut-the-knot.org
cartoonstock.com
Bibliography
www.nctm.org