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• Examples of analogies and generalizations in synthetic geometry

  19-39

  Ioana E. Agut

  Calin M. Agut

  Views:

  29

  Teaching tools and different methods of generalizations and analogies are often used at different levels of education. Starting with primary grades, the students can be guided through simple aspects of collateral development of their studies. In middle school, high school and especially in entry-level courses in higher education, the extension of logical tools are possible and indicated.
  In this article, the authors present an example of generalization and then of building the analogy in 3-D space for a given synthetic geometric problem in 2-D.
  The idea can be followed, extended and developed further by teachers and students as well.

• Analyse von Lösungswegen und Erweiterungsmöglichkeiten eines Problems für die Klassen 7–11

  231-249

  Gabriella Ambrus

  Views:

  30

  Making several solutions for a problem i.e. the generalization, or the extension of a problem is common in the Hungarian mathematics education.
  But the analysis of a problem is unusual where the connection between the mathematical content of the task and of its different formulations is examined, solutions from different fields of mathematics are presented regarding the knowledge of different age groups, the problem is generalized in different directions, and several tools (traditional and electronic) for solutions and generalizations are presented.
  This kind of problem analysis makes it viable that during the solution/elaboration several kinds of mathematical knowledge and activities are recalled and connected, facilitating their use inside and outside of mathematics.
  However, an analysis like this is not unfamiliar to the traditions of the Hungarian problem solving education – because it also aims at elaborating a problem – but from several points of view.
  In this study, a geometric task is analysed in such a way.

• Some Pythagorean type equations concerning arithmetic functions

  157-179

  Ildikó Kézér

  Views:

  59

  We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x² + y² = z², we study equation f(x²) + f(y²) = f(z²) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.

  Subject Classification: 97F60, 11A25

• A geometric application to the third-order recurrence relations for sequences

  287-302

  Calin M. Agut

  Ioana E. Agut

  Views:

  31

  Using a third-order linear homogeneous recurrence relation with constant coefficients, it is found a limit-point of a sequence of affixes in plane. Starting from a classic geometric problem, an application is so created and few more nice properties are found and described.

• Illustrated analysis of Rule of Four using Maple

  383-404

  György Maróti

  Views:

  37

  Rule of Four, as a basic didactic principle, was formulated among the NCTM 2000 standards (see [14]) and since then it is quoted by numerous books and publications (see [4], [9], [12]). Practically we can say it is accepted by the community of didactic experts. The usage of the Rule of Four, however, has been realized mainly in the field of calculus, in fact certain authors restrict the wording of the principle to the calculus itself (e.g. [3]).
  Calculus is a pleasant field, indeed. A sequence of values of a function provides us with example for numeric representation, while the formula and the graph of the function illustrate symbolic and graphical representations, respectively. In the end by wording the basic features of the function on natural language we gain textual representation.
  This idyllic scene, however, becomes more complex when we leave the frame of calculus. In this paper we investigate the consequences of the usage of Rule of Four outside calculus. We discuss the different types of representations and show several examples which make the multiple features of representation evident. The examples are from different fields of mathematics and are created by the computer algebra system Maple, which turns out to be an excellent tool for illustration and visualization of the maim features of mathematical objects.
  Next we introduce the concept of basic representation and rational representation, which is considered as the mathematical notion of "didactic usable" or "didactic rational" representation. In the end we generalize the notion of numeric representation, which leads us a more widely usable didactic principle which can be considered as a generalization of Rule of Four.