Volume XIX , issue 2 ( 2016 )back
The connection between order grades and congruence theorems through the example of systems of triangles57$-$67
Joachim T. Haug, Carolin Haug
Ludwig-Maximilians-Universität München, Biozentrum - Department Biologie II, Gro{ß}haderner Str. 2, 82152 Planegg-Martinsried, Deutschland


Different types of polygons are treated differently, or at least the treatments are differently focused. For triangles, congruence theorems seem to be considered to be important, while for quadrilaterals classification seems to be one of the most important aspects. Recently, we suggested a systematic treatment for quadrilaterals to substitute the often rather arbitrary-appearing common classificatory systems. For this system, we suggested certain characteristics, which lead to subsequently higher ordered quadrilaterals. This approach is generalised here and expanded to be applied also to triangles. The number of ordered triangles is significantly lower than that of quadrilaterals: there are only four, two triangles of first order and two of second order. We furthermore show that the resulting system of triangles is directly linked to the triangle congruence theorems. With this approach, we try to bridge the different ways how triangles and quadrilaterals are treated and aim at a more coherent understanding of geometry.

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Keywords: Classification of triangles

MSC Subject Classification: 97G40

MathEduc Subject Classification: G44

Strict monotonicity of nonnegative strictly concave function vanishing at the origin68$-$75
Yuanhong Zhi
School of Mathematics and Statistics, Yunnan University, Kunming, PR China


In this paper we prove that every nonnegative strictly concave function on the unbounded closed interval $[0,+ınfty)$ is strictly increasing, provided it vanishes at the origin. With the help of this result, we then show that the strict monotonicity condition of the theorem concerning the metric transforms is redundant. We also provide a companion version of this result for merely concave nonnegative function which vanishes only at the origin.

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Keywords: Strict concavity; strict monotonicity; metric transform

MSC Subject Classification: 97I20

MathEduc Subject Classification: I25

A power series approach to an inequality and its generalizations76$-$83
Xu Yan-Hui
College of Mathematics and Information Science, Wenzhou University, Zhejiang, 325035, P.R. China


In this paper, we present and prove an inequality and its several generalizations by using power series and Muirhead inequality.

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Keywords: Inequalities; Muirhead inequality; power series.

MSC Subject Classification: 97I30

MathEduc Subject Classification: I35

Exploring quadratic equations with digital tools in mathematics teacher education84$-$100
Sergei Abramovich
SUNY Potsdam, USA


The paper offers teaching ideas for a technology-enhanced mathematics teacher education course related to an advanced inquiry into one-variable quadratic equations with parameters in place of coefficients. Mathematical activities behind the ideas have been interpreted using the notions of the technology immune/technology enabled (TITE) problem, digital fabrication, collateral learning, and hidden mathematics curriculum. The activities, leading to explorations that can be considered as rudiments of real problems arising in science and engineering, are in support of recommendations by the Conference Board of the Mathematical Sciences [The Mathematical Education of Teachers II, Washington, DC: Mathematical Association of America, 2012] for the preparation of secondary teacher candidates.

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Keywords: Quadratic equation; localization of roots; symbolic computation; digital fabrication; hidden inequalities; Sturm's Theorem; teacher education.

MSC Subject Classification: 97D40, 97H30

MathEduc Subject Classification: D45, H35