Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry...
Transcript of Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry...
![Page 1: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/1.jpg)
Discovering Geometry Conjectures and Proving with Geogebra
GeoGebra Dynamic Mathematics North American Conference
Miami University, Oxford Ohio August 3-4, 2013
Dr. Steve Armstrong, LeTourneau University
www.letu.edu/people/stevearmstrong
![Page 2: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/2.jpg)
Unlikely Proofs (which you may have seen)
• Proof by clever variable choice
![Page 3: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/3.jpg)
Unlikely Proofs (which you may have seen)
• Proof by plagiarism
“As we see on page 289,..."
![Page 4: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/4.jpg)
Unlikely Proofs (which you may have seen)
• Proof by mumbo-jumbo
3
0
...
![Page 5: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/5.jpg)
• Proof by definition
Unlikely Proofs (which you may have seen)
![Page 6: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/6.jpg)
• Proof by tessellation This proof works the same as the
last …
Unlikely Proofs (which you may have seen)
![Page 7: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/7.jpg)
Commentary by Barry Simon L.A. Times, 2/6/98
• Fewer classes require geometric proofs
• There is more whittling away of exposure to
– Logic
– Critical thinking
Caltech Department of Mathematics Executive Officer
Statements Reasons
1. 2. . . .
![Page 8: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/8.jpg)
From HomeSchool.net
• Formal proofs difficult
• Hard in public school
–Hard for home school parents
• Consider lack of proving experiences in low grades
![Page 9: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/9.jpg)
Van Hiele Model
• van Hiele theory speaks to this dilemma
• Levels of geometric thought
• Exposure to lower levels needed prior a course of rigorous proofs
Geometry and proof. Mathematics Teacher, 88(1), 48-54. ©1995 by the National Council of Teachers of Mathematics.
![Page 10: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/10.jpg)
Van Hiele Levels
http://proactiveplay.com/the-van-hieles-model-of-geometric-thinking/
![Page 11: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/11.jpg)
Van Hiele Levels
• Levels are sequential
• Not age dependent
• Advance with appropriate experiences
• Inappropriate experiences inhibit
![Page 12: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/12.jpg)
Level 0: Visualization
That’s a rectangle because it looks
like a door.
![Page 13: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/13.jpg)
Level 1: Analysis
That’s a rhombus. It has four equal sides
![Page 14: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/14.jpg)
![Page 15: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/15.jpg)
Level 2: Informal Analysis
That’s a square. It’s a rhombus, but it has some
extra properties.
![Page 16: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/16.jpg)
Level 3: Formal Deduction
Those two triangles in the parallelogram are congruent
by Side-side-side.
![Page 17: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/17.jpg)
Level 4: Rigor
Look what I figured out about triangles when you place them on a sphere
and not on a plane. And I can prove it, too.
![Page 18: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/18.jpg)
Van Hiele Levels
• Levels are sequential
• Not age dependent
• Advance with appropriate experiences
• Inappropriate experiences inhibit
![Page 19: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/19.jpg)
Use Geogebra Transition between Levels
http://proactiveplay.com/the-van-hieles-model-of-geometric-thinking/
![Page 20: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/20.jpg)
Use of Geogebra
• Involve students in mathematical discovery
See if you can figure out what happens when you move the corner of the quadrilateral around.
![Page 21: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/21.jpg)
Use Geogebra Transition between Levels
http://proactiveplay.com/the-van-hieles-model-of-geometric-thinking/
![Page 22: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/22.jpg)
Use of Geogebra
• Facilitate making and testing conjectures
• Perform measurements D is the midpoint of BC AD connects the center
of the circle and the midpoint of BC. What
relationships do you see? Try to prove it to a jury.
![Page 23: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/23.jpg)
Try a Discovery http://padlet.com/wall/GGB2013-Session112
1. Create a circle
2. Place three points anywhere on the circle
3. Create segments from two of the points to the center
4. Create segments from the same two points to the third point on the circle
5. Measure the two angles formed
6. Move the points, make a conjecture
![Page 24: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/24.jpg)
Use of Geogebra
• What was your conjecture?
• Can you prove it to a jury?
![Page 25: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/25.jpg)
Another Discovery http://padlet.com/wall/GGB2013-Session112
1. Create a circle and a tangent to the circle at B
2. Create two other points on the circle, C and D
3. Place a point E on the tangent
4. Measure the angles DCB and DBE
5. Make a conjecture … move the points around to see if it always holds
6. Try to prove your conjecture
![Page 26: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/26.jpg)
Another Discovery
• What was your conjecture?
• Can you prove it to a jury?
![Page 27: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/27.jpg)
Use of Geogebra
• Ultimate goal, students able to prove … but …
• What if …. Require description of how a student demonstrated the theorem using Geogebra
• This moves students along in the van Hiele levels
![Page 28: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/28.jpg)
Use of Geogebra
• Agree or disagree …
Agree? Disagree?
• • •
• • • •
Require only “demonstration” of
theorems and conjectures
![Page 29: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/29.jpg)
Demonstration Proof
• Can be used for counter-examples
– To prove a conjecture FALSE
![Page 30: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/30.jpg)
Course Recently Taught
• College Geometry
![Page 31: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/31.jpg)
Course Recently Taught
• Each chapter starts with “Activities”
– Use Geogebra to construct figures
– Make conjectures
– Try to prove
• Concepts then presented in chapter
• Formal proofs presented
• Chapter exercises include formal proofs
![Page 32: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/32.jpg)
Activities
1. Draw an arbitrary triangle, ABC.
a. Construct an equilateral triangle on each leg of ABC. The constructed triangles should remain equilateral when you drag the vertices of ABC to new positions. Calculate the areas of each equilateral triangle. Drag the vertices A, B, and C to see what happens when ABC is acute, right, or obtuse. What do you observe? Make a conjecture. © 2006 by Key College Publishing. All rights reserved. Published by Key College Publishing, an imprint of Key Curriculum Press.
![Page 33: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/33.jpg)
Activities (ctd.)
b. Repeat part a, using circles on each side instead of equilateral triangles. Construct a circle on each leg so that the leg of the triangle is a diameter of the circle. Calculate the area of each circle. What do you observe? What happens when ABC is acute, right, or obtuse? Make a conjecture.
c. Prove your conjectures from parts a and b
© 2006 by Key College Publishing. All rights reserved. Published by Key College Publishing, an imprint of Key Curriculum Press.
![Page 34: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/34.jpg)
Exercises
4. Using coordinates, write a detailed step-by-step proof that the set of points equidistant from two fixed points, A and B, is the perpendicular bisector of the segment AB.
© 2006 by Key College Publishing. All rights reserved. Published by Key College Publishing, an imprint of Key Curriculum Press.
![Page 35: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/35.jpg)
Summary
• Formal proofs difficult
• van Heile suggests levels
• Students need transitional experiences
• Use discovery, experimentation
• Use Geogebra
![Page 36: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/36.jpg)
Questions?
![Page 37: Discovering Geometry Conjectures and Proving with Geogebra · 2017-08-03 · Discovering Geometry Conjectures and Proving with Geogebra GeoGebra Dynamic Mathematics North American](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0a6e4f7e708231d42b9927/html5/thumbnails/37.jpg)
Discovering Geometry Conjectures and Proving with Geogebra
GeoGebra Dynamic Mathematics North American Conference
Miami University, Oxford Ohio August 3-4, 2013
Dr. Steve Armstrong, LeTourneau University
www.letu.edu/people/stevearmstrong