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Designing GeoGebra Tasks for Visualization and Reasoning

Anthony OR 柯志明, Jul 11, 2015

Presentation in GeoGebra Global Gathering 2015 Or, A.C.M. (2013). Designing tasks to foster operation apprehension for visualization and reasoning in dynamic geometry environment. In C. Margolinas (Ed.) Task Design in Mathematics Education: Proceedings of ICMI Study 22, pp. 89-98, Oxford, U.K. https://hal.archives-ouvertes.fr/hal-00834054v3.

Duval's Model of Visualization and Reasoning in Geometry
1. Duval's Model of Visualization and Reasoning in Geometry
Operative Apprehension: From Quadrilateral Dissection to Dudeney's Puzzle
1. Dudeney’s Haberdasher Puzzle (1905)
2. Dissecting Quadrilaterals
3. Cut Triangles into Rectangles & Dudeney's Puzzle Generalized
Robust and Soft Constructions
1. Circumcircle of a triangle (Robust Construction)
2. Robust and Soft Constructions (Healy 2000)
3. Circumcircle of a triangle (Soft Construction)
4. Robust vs. Soft Constructions (Laborde 2005)
5. Incircle of a triangle
Soft Construction as Operative Apprehension to Foster Visualization and Reasoning
1. Soft Construction as Operative Apprehension to Foster Visualization and Reasoning
Challenges
1. Equilateral Triangle on 3 Parallel Lines
2. Equilateral Triangle Inscribed in a Triangle
Solutions of Challenges
1. Triangle on Parallel Lines (Hint)
2. Triangle on Parallel Lines (Construction and Explanation)
3. Equilateral Triangle Inscribed in a Triangle (Hint)
4. Equilateral Triangle Inscribed in a Triangle (Solution)
Answers of Challenges by Irina Boyadzhiev
1. Answer to the first challenge by Anthony OR - GGG 2015
2. Answer to the second challenge by Anthony OR - GGG 2015

Information

1.

Duval's Model of Visualization and Reasoning in Geometry

1. Duval's Model of Visualization and Reasoning in Geometry

1.1.

Duval's Model of Visualization and Reasoning in Geometry

Operative Apprehension in Dynamic Geometry Environment (DGE)

Operative apprehension in DGE: the insights revealed by operating on a pre-designed figure in the environment through dragging

2.

Operative Apprehension: From Quadrilateral Dissection to Dudeney's Puzzle

1. Dudeney’s Haberdasher Puzzle (1905)
2. Dissecting Quadrilaterals
3. Cut Triangles into Rectangles & Dudeney's Puzzle Generalized

2.1.

Dudeney’s Haberdasher Puzzle (1905)

2.2.

2.3.

3.

Robust and Soft Constructions

1. Circumcircle of a triangle (Robust Construction)
2. Robust and Soft Constructions (Healy 2000)
3. Circumcircle of a triangle (Soft Construction)
4. Robust vs. Soft Constructions (Laborde 2005)
5. Incircle of a triangle

3.1.

Circumcircle of a triangle (Robust Construction)

3.2.

3.3.

3.4.

3.5.

4.

Soft Construction as Operative Apprehension to Foster Visualization and Reasoning

1. Soft Construction as Operative Apprehension to Foster Visualization and Reasoning

4.1.

Soft Construction as Operative Apprehension to Foster Visualization and Reasoning

5.

Challenges

1. Equilateral Triangle on 3 Parallel Lines
2. Equilateral Triangle Inscribed in a Triangle

5.1.

Equilateral Triangle on 3 Parallel Lines

Given 3 parallel lines. How to construct an equilateral triangle with the 3 vertices lying on the 3 lines?

Equilateral Triangle on 3 Parallel Lines

5.2.

6.

Solutions of Challenges

1. Triangle on Parallel Lines (Hint)
2. Triangle on Parallel Lines (Construction and Explanation)
3. Equilateral Triangle Inscribed in a Triangle (Hint)
4. Equilateral Triangle Inscribed in a Triangle (Solution)

6.1.

Triangle on Parallel Lines (Hint)

6.2.

6.3.

6.4.

7.

Answers of Challenges by Irina Boyadzhiev

1. Answer to the first challenge by Anthony OR - GGG 2015
2. Answer to the second challenge by Anthony OR - GGG 2015

7.1.

Answer to the first challenge by Anthony OR - GGG 2015

Problem: Given are three parallel lines

and

. Construct an equilateral triangle

such that each of its vertices is on one of the lines. Solution: For any equilateral triangle a rotation with center one of the vertices and angle

will map one of the remaining vertices onto the third vertex. Let

be an arbitrary point on line

.

Drag the slider to the end to rotate line on angle around point .

If vertex

is on line

, then the third vertex

of the triangle should be on the image

of line

, but it also has to be on the third line

. Therefore, vertex

is the intersection point of the lines

and

. This way we find two vertices and determine the side of the equilateral triangle.

Click on the Construction button to finish the construction.

There will be two different solutions corresponding to the two possible directions of the rotation, at

and

degrees.

Drag the gray points to change the positions of the lines.
What do you notice when line is above line or below line ?

A geometric construction using this transformation was first described by I. M. Yaglom, in Geometric Transformations I, MAA, 1962, Chapter 2, Problem 18

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Designing GeoGebra Tasks for Visualization and Reasoning

Table of Contents

Information

Duval's Model of Visualization and Reasoning in Geometry

Duval's Model of Visualization and Reasoning in Geometry

Operative Apprehension in Dynamic Geometry Environment (DGE)

Operative Apprehension: From Quadrilateral Dissection to Dudeney's Puzzle

Dudeney’s Haberdasher Puzzle (1905)

Robust and Soft Constructions

Circumcircle of a triangle (Robust Construction)

Circumcircle of a triangle (Robust Construction)

Soft Construction as Operative Apprehension to Foster Visualization and Reasoning

Soft Construction as Operative Apprehension to Foster Visualization and Reasoning

Challenges

Equilateral Triangle on 3 Parallel Lines

Equilateral Triangle on 3 Parallel Lines

Solutions of Challenges

Triangle on Parallel Lines (Hint)

Triangle on Parallel Lines (Hint)

Answers of Challenges by Irina Boyadzhiev

Answer to the first challenge by Anthony OR - GGG 2015

Answer to the first challenge by Anthony OR - GGG 2015