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

Unlikely Proofs (which you may have seen)

• Proof by clever variable choice

Unlikely Proofs (which you may have seen)

• Proof by plagiarism

“As we see on page 289,..."

Unlikely Proofs (which you may have seen)

• Proof by mumbo-jumbo

3

0

...

• Proof by definition

Unlikely Proofs (which you may have seen)

• Proof by tessellation This proof works the same as the

last …

Unlikely Proofs (which you may have seen)

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. . . .

From HomeSchool.net

• Formal proofs difficult

• Hard in public school

–Hard for home school parents

• Consider lack of proving experiences in low grades

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.

Van Hiele Levels

http://proactiveplay.com/the-van-hieles-model-of-geometric-thinking/

Van Hiele Levels

• Levels are sequential

• Not age dependent

• Advance with appropriate experiences

• Inappropriate experiences inhibit

Level 0: Visualization

That’s a rectangle because it looks

like a door.

Level 1: Analysis

That’s a rhombus. It has four equal sides

Level 2: Informal Analysis

That’s a square. It’s a rhombus, but it has some

extra properties.

Level 3: Formal Deduction

Those two triangles in the parallelogram are congruent

by Side-side-side.

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.

Van Hiele Levels

• Levels are sequential

• Not age dependent

• Advance with appropriate experiences

• Inappropriate experiences inhibit

Use Geogebra Transition between Levels

http://proactiveplay.com/the-van-hieles-model-of-geometric-thinking/

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.

Use Geogebra Transition between Levels

http://proactiveplay.com/the-van-hieles-model-of-geometric-thinking/

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.

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

Use of Geogebra

• What was your conjecture?

• Can you prove it to a jury?

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

Another Discovery

• What was your conjecture?

• Can you prove it to a jury?

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

Use of Geogebra

• Agree or disagree …

Agree? Disagree?

• • •

• • • •

Require only “demonstration” of

theorems and conjectures

Demonstration Proof

• Can be used for counter-examples

– To prove a conjecture FALSE

Course Recently Taught

• College Geometry

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

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.

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.

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.

Summary

• Formal proofs difficult

• van Heile suggests levels

• Students need transitional experiences

• Use discovery, experimentation

• Use Geogebra

Questions?

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