Available online
Description
This programme explains the idea of classification using equivalence relations using examples from geometry.
This programme explains the idea of classification using equivalence relations using examples from geometry.
| Module code and title: | M101, Mathematics: a foundation course |
|---|---|
| Item code: | M101; 29; 1986 |
| First transmission date: | 14-09-1986 |
| Published: | 1986 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:45 |
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| Producer: | David Saunders |
| Contributors: | Bob Coates; Allan I.,1936-2013 Solomon; Bob Margolis |
| Publisher: | BBC Open University |
| Keyword(s): | Area; Euclidean congruences; Geometry; Hyperbolas; Matrix transformations; Triangles |
| Footage description: | This programme explains the idea of classification using equivalence relations (introduced in section 2 of the unit) using examples from geometry. The programme starts by posing a problem involving the areas of two triangles; although there is a straightforward but lengthy way of solving the problems using calculation, the ideas about to be introduced provide a much simpler solution. These ideas are explained by looking first at the symmetries of a given equilateral triangle, and asking how the points of the plane can be classified using these symmetries. A computer animation shows that in general there are sets containing six equivalent points, where each point is related to any other by either a reflection or a rotation which preserves the triangle. The programme explains that such a relation is characterised by the properties of being symmetric, reflexive, and transitive, and is called an equivalence relation. Students are then reminded of a problem involving matrices from the television section of the text, and are shown the strategy for proving that the relation so defined is actually an equivalence relation; computer animation demonstrates that the corresponding sets of equivalent points are concentric circles. It is explained that the matrix transformations - rotations - are special cases of Euclidean congruencies, and in fact a rotation can be used to give a simple solution to the problem from the beginning of the programme. Finally, students are posed a further problem involving matrix transformations which give rise to hyperbolas rather than circles. |
| Master spool number: | HOU5543 |
| Production number: | FOUM258K |
| Videofinder number: | 2491 |
| Available to public: | no |
