EUCLIDEAN GEOMETRY
(also called Parabolic geometry) is flat “2D” geometry
Axioms
(1) Things that equal the same thing also equal each other.
(2) If equal parts are added to equal parts, the wholes are also equal.
(3) If equal parts are subtracted from equal parts, the remainders are also equal.
(4) Things that are the same are equal.
(5) A whole is greater than a part.
Euclid’s postulates
(1) A straight line can be drawn from any one point to any other point.
(2) A finite line can be extended continuously as a straight line.
(3) A circle can be described with any point as its centre and any distance as its
radius.
(4) All right angles are equal to one another.
(5) If a transversal falls on two lines in such a way that the interior angles on one
side of the transversal are less than two right triangles, then the lines meet on
the side where the angles are less than two right triangles.
Euclid’s fifth postulate is also known as the parallel postulate.
Role of the fifth postulate in the development of non-euclidean geometries
For 2,000 years following Euclid, mathematicians attempted either to prove the fifth
postulate as a theorem (based on the other postulates) or to modify it in various ways. Two
scientists later independently published a description of a geometry that, except for
the parallel postulate, satisfied all of Euclid’s postulates and common notions. Essentially,
in the 19th century it was shown that the fifth postulate is independent of the other
postulates. It is possible to build a theory of geometry where the fifth postulate is not true.
It is this geometry that is called hyperbolic geometry, a non-euclidean geometry.
Eg.
Non-Euclidean Geometry
A theory of geometry where the fifth postulate is not true.
Non-euclidean geometries include:
• hyperbolic
elliptical/spherical
• projective geometry
, Platonic solids
Using manipulatives
• Manipulatives or concrete apparatus are teaching and learning aids
• They make it easier for learners to construct/measure various configurations/geometric shapes
• Learners can see and touch the real-life shapes
• Learners can be able to use the concrete material as models in order to describe and understand
abstract concepts in geometry.
• Teaching and learning occur as a process that moves from the concrete to the abstract.
(also called Parabolic geometry) is flat “2D” geometry
Axioms
(1) Things that equal the same thing also equal each other.
(2) If equal parts are added to equal parts, the wholes are also equal.
(3) If equal parts are subtracted from equal parts, the remainders are also equal.
(4) Things that are the same are equal.
(5) A whole is greater than a part.
Euclid’s postulates
(1) A straight line can be drawn from any one point to any other point.
(2) A finite line can be extended continuously as a straight line.
(3) A circle can be described with any point as its centre and any distance as its
radius.
(4) All right angles are equal to one another.
(5) If a transversal falls on two lines in such a way that the interior angles on one
side of the transversal are less than two right triangles, then the lines meet on
the side where the angles are less than two right triangles.
Euclid’s fifth postulate is also known as the parallel postulate.
Role of the fifth postulate in the development of non-euclidean geometries
For 2,000 years following Euclid, mathematicians attempted either to prove the fifth
postulate as a theorem (based on the other postulates) or to modify it in various ways. Two
scientists later independently published a description of a geometry that, except for
the parallel postulate, satisfied all of Euclid’s postulates and common notions. Essentially,
in the 19th century it was shown that the fifth postulate is independent of the other
postulates. It is possible to build a theory of geometry where the fifth postulate is not true.
It is this geometry that is called hyperbolic geometry, a non-euclidean geometry.
Eg.
Non-Euclidean Geometry
A theory of geometry where the fifth postulate is not true.
Non-euclidean geometries include:
• hyperbolic
elliptical/spherical
• projective geometry
, Platonic solids
Using manipulatives
• Manipulatives or concrete apparatus are teaching and learning aids
• They make it easier for learners to construct/measure various configurations/geometric shapes
• Learners can see and touch the real-life shapes
• Learners can be able to use the concrete material as models in order to describe and understand
abstract concepts in geometry.
• Teaching and learning occur as a process that moves from the concrete to the abstract.
