Download - The role of proof in mathematics
![Page 1: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/1.jpg)
The Role of Proof in Mathematics
![Page 2: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/2.jpg)
The Role of Proof in Mathematics
![Page 3: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/3.jpg)
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
Proof in Mathematics
![Page 4: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/4.jpg)
Standards on Proof
Instructional programs that should enable students to:
develop and evaluate mathematical arguments and proofs
select and use various types of reasoning and methods of proof
By the end of middle school, students should be able to understand and produce mathematical proofs
![Page 6: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/6.jpg)
Proof
Convincing demonstration that a math statement is true.
To explain.
Informal and formal.
Logic
No single correct answer
![Page 7: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/7.jpg)
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!
![Page 8: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/8.jpg)
Three Forms of Formal Proof
Synthetic Geometry
Analytic Geometry
Transformational Geometry
![Page 9: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/9.jpg)
http://www.graspr.com/videos/Math-Proof-Negative-Times-Negative-Equals-Positive-3
![Page 10: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/10.jpg)
Synthetic Geometry
A system illustrated by proving geometric relationships based on the use of a rational sequence of definitions, postulates, and theorems
19th Century
Pure geometry
Logical Arguments
![Page 11: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/11.jpg)
![Page 12: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/12.jpg)
The most common proof – The Pythagorean Theorem
![Page 13: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/13.jpg)
Find the value of (a-b)2
Suppose AC = AF = A unitsAB = AG = b units\ BC = AC - AB = (a - b) unitsGF = AF - AG = (a - b) unitsHI = BC = (a - b) units and HE = GE = (a - b) units\ Area of the square HIDE = HI * HE= (a - b) * (a - b) = (a - b)2 sq. units ..........IArea of square HIDE = Area of square ACDF - Area of rectangle ACIG - Area of rectangle GHEF= AC * AF - AC * AG - GH * GF= a.a - a.b - (a-b).b= a2 - ab - b(a-b)= a2 - ab - ab + b2
= a2 - 2ab + b2 ..........II\ From I and II Þ (a - b)2 = a2 - 2ab + b2
Grade 7 – MathematicsFinding the value of (a-b)2 (Geometrical Proof)
![Page 14: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/14.jpg)
![Page 15: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/15.jpg)
Analytic Geometry
Also known as coordinate geometry or Cartesian geometry
Algebra
Graphing Technology
Computations
![Page 16: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/16.jpg)
Analytic GeometryCartesian Geometry
Also known as coordinate geometry-graphing
![Page 17: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/17.jpg)
![Page 18: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/18.jpg)
Transformational Geometry
20th Century
Graphics technology
MIRA
Plane mirror
Is a method for studying geometry that illustrates congruence and similarity by use of transformations
Therefore a transformational proof is a proof that employs the use of transformation
![Page 19: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/19.jpg)
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).
Transformation Proof
![Page 20: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/20.jpg)
Transformational Proof
![Page 21: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/21.jpg)
![Page 22: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/22.jpg)
![Page 23: The role of proof in mathematics](https://reader035.vdocuments.us/reader035/viewer/2022070318/55746b39d8b42a146f8b521b/html5/thumbnails/23.jpg)
http://mathoverflow.net/questions/8846/proofs-without-words
cut-the-knot.org
cartoonstock.com
Bibliography
www.nctm.org