|Volume XVIII , issue 1 ( 2015 )||back|
|Von der Klassifikation der Vierecke zum System der Vierecke||1$-$15|
The currently available classification schemes of quadrilaterals do not include all possible types of quadrilaterals, but only an arbitrary subset. Additionally, students often have problems to understand these classifications. Therefore, this study aims at constructing a comprehensive and logically structured systematics of quadrilaterals. Herein the characteristics of its elements should determine its elements. The basis here are six quadrilaterals of first order, which are characterised by either two same angles, two sides of the same length (both either adjacent or opposed to each other), two parallel or orthogonally sides. Combination of these quadrilaterals with one of the other characters results in quadrilaterals of second order; further character combinations lead to quadrilaterals of third or fourth order, the latter including the square. This approach reveals that several, even highly ordered quadrilaterals, are yet unnamed. In addition to the square there is another quadrilateral of fourth order, the complex double-orthogonal three-sides-equal quadrilateral. The applicability of the newly found quadrilaterals still needs to be studied. The here presented systematics of quadrilaterals with its logical basis should probably be easier to teach than the classifications available to date.
Keywords: Classification of quadrilaterals
MSC Subject Classification: G44
MathEduc Subject Classification: 97G40
|On Catalan numbers||16$-$24|
In this article we introduce one of the most frequently encountered sequences in mathematics, the sequence of Catalan numbers. The numerous and various occurrences and applications in combinatorial problems as well as their relations with other famous numerical sequences make Catalan numbers very suitable for creative teaching of mathematics at all levels.
Keywords: Catalan numbers; binary trees; associahedron; permutations.
MSC Subject Classification: 97K20
MathEduc Subject Classification: K25
|A simple proof of the change of variable theorem for the Riemann integral||25$-$28|
In this note, we present a simple proof of the change of variable theorem for the Riemann integral. The proof gives an alternate approach of how a class material on this topic can sistematically be carried out through a simple, succinct, and convenient way.
Keywords: Change of variable theorem; Riemann integral.
MSC Subject Classification: 97I55
MathEduc Subject Classification: I50
|On-line compendium of triangle construction problems with automatically generated solutions||29$-$44|
In this paper an on-line compendium of solutions to a class of triangle construction problems using straightedge and compass is presented. It consists of: informal description of constructions in natural-language form, formal description of constructions, illustrations of generated constructions, as well as step-by-step animations of generated constructions. This collection was generated completely automatically, using the system for automated solving of construction problems ArgoTriCS. To our knowledge this is the first automatically generated compendium of construction problems which offers solutions to almost all solvable problems from one large corpus of problems.
Keywords: Triangle construction; automatically generated solutions.
MSC Subject Classification: 97P50
MathEduc Subject Classification: P55