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.) [i]Task Design in Mathematics Education: Proceedings of ICMI Study 22[/i], pp. 89-98, Oxford, U.K. [url]https://hal.archives-ouvertes.fr/hal-00834054v3[/url].

Table of Contents

  1. Duval's Model of Visualization and Reasoning in Geometry
    1. Duval's Model of Visualization and Reasoning in Geometry
  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
  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
  4. Soft Construction as Operative Apprehension to Foster Visualization and Reasoning
    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
  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)
  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

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)
[size=150][size=100][b]Operative apprehension[/b] in DGE: the insights revealed by operating on a pre-designed figure in the environment through [b]dragging[/b][/size][/size]
– Anthony OR 柯志明

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)

– Anthony OR 柯志明

2.2.

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2.3.

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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)

Circumcircle of a triangle (Robust Construction)
– Anthony OR 柯志明

3.2.

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3.3.

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3.4.

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3.5.

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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

– Anthony OR 柯志明

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
– Anthony OR 柯志明

5.2.

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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)

Triangle on Parallel Lines (Hint)
– Anthony OR 柯志明, GeoGebra Institute of Hong Kong

6.2.

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6.3.

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6.4.

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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:[br]Given are three parallel lines [math]a, b,[/math] and [math]c[/math]. Construct an equilateral triangle [math]ABC[/math] such that each of its vertices is on one of the lines.[br][br]Solution:[br]For any equilateral triangle a rotation with center one of the vertices and angle [math]60^{\circ} [/math] will map one of the remaining vertices onto the third vertex.[br]Let [math] B[/math] be an arbitrary point on line [math]b[/math]. [br][list][br][*]Drag the slider to the end to rotate line [math]c[/math] on angle [math]60^{\circ} [/math] around point [math]B[/math].[br][/list][br]If vertex [math]C[/math] is on line [math]c[/math], then the third vertex [math]A[/math] of the triangle should be on the image [math]c'[/math] of line [math]c[/math], but it also has to be on the third line [math]a[/math]. Therefore, vertex [math]A[/math] is the intersection point of the lines [math]c'[/math] and [math]a[/math]. This way we find two vertices and determine the side of the equilateral triangle.[br][list][br][*]Click on the Construction button to finish the construction.[br][/list][br]There will be two different solutions corresponding to the two possible directions of the rotation, at [math]60^{\circ} [/math] and [math] -60^{\circ} [/math] degrees. [br][list][br][*]Drag the gray points to change the positions of the lines. [br][*]What do you notice when line [math]b[/math] is above line [math]c[/math] or below line [math]a[/math]?[br][/list][br][br]A geometric construction using this transformation was first described by I. M. Yaglom, in [i]Geometric Transformations[/i] I, MAA, 1962, Chapter 2, Problem 18
Answer to the first challenge by Anthony OR - GGG 2015
– Irina Boyadzhiev

7.2.

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