teaching geometry using a computer - university of … · geometrija 1 planimetry •triangle...
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Teaching Geometry using a
Computer
Elvira Ripco Šipoš
Bolyai Grammar School
Senta
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Introduction
Matematical creativity and talented students
Geometrija 1
Geometrija 2
Geometrija 3
Apollonius’ problems
Is there geometry after Euclid?
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Steps of spirality
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Teaching method
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Matematical creativity and talented
students Jane Piirto
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Geometrija 1
Planimetry
•Triangle centers (H
orthocenter, T centroid, O
circumcenter, S incenter)
•Euler’s line
•Cognitive aim of teaching
mathematics
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Geometry 1
Planimetry
1. Visualization on computer
i) experiments on computer
ii) experiences
iii) conjecture
iv) formulation of the theorem
v) Deducitve proof
(affective end effective aim)
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Geometry 1
Planimetry
2. Construction on paper with
technical pencil, rulers, bow
• Construction
• Control
• Psichomotorical aim
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Geometry 1
Planimetry
3. Systematization and repetition (teorethical proof)
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Geometry 3
Analitical geometry Given three vertices of the triangle: A(3; 5), B(6; 2), C(−2;−1).
Find the equation of the Euler’s line.
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Geometry 3
Analitical geometry Given three vertices of the triangle: A(3; 5), B(6; 2), C(−2;−1).
Find the equation of the Euler’s line.
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Geometry 3
Analitical geometry
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Geometry 3
Analitical geometry
General form:
Given three vertices A(a1,a2),
B(b1, b2), C(c1, c2) of the
triangle ABC.
Find the equation of the
Euler’s line
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Geometry 1 examines :
•Isometric transformations
•Transformation of similarity
•Inversion
Apollonius’ problems
Poincaré’s model for non-Euclidean geometry
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Inversion
1. The inverse of a circle (not through
the center of inversion) is a circle.
2. The inverse of a circle through the
center of inversion is a line.
3. The inverse of a line (not through the
center of inversion) is a circle through
the center of inversion.
4. A circle orthogonal to the circle of
inversion is its own inverse.
5. A line through the center of inversion
is its own inverse.
6. Angles are preserved in inversion.
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Apollonius’ problems
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Playfair’s axiom: Given a line and a
point not on it, at most one parallel to
the given line can be drawn through the
point.
Bolyai János-Nikolai Lobachevsky’s
axiom:
Given a line and a point not on it, at
least two parallel to the given line can be
drawn through the point.
Non-Euclidean Geometry
Hyperbolic geometry
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Non-Euclidean Geometry
Spherical geometry
Senta
Latitude 45 55’ 45,48’’N
Longitude 20 5’ 10,18’’ E
New York
Latitude 41 8′ 44″ N
Longitude 73 59′ 42″ W
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Results and Comments
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Results and Comments
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