Volume XIII , issue 1 ( 2010 ) | back |

A contribution to the development of functional thinking related to convexity | 1$-$16 |

**Abstract**

When a liquid (water) flows into a vessel at the constant inflow rate, then the height filling function is convex or concave depending on the way how the level of the liquid changes. When the level changes accelerating or slowing down, the function is convex or concave, respectively. This vivid interpretation holds in general, namely we prove that given a strictly increasing convex (concave) continuous function, then there exists a vessel such that its height filling function is equal to the given function (a fact that seems to be new). We also hope that our paper could exemplify the case of a research project to be assigned to excellent students.

**Keywords:** Height filling function; convex and concave functions.

A Socratic methodological proposal for the study of the equality $0.999\ldots = 1$ | 17$-$34 |

**Abstract**

Our objective is to produce a learning experience in the form of an interview which is structured in an introduction and three phases with the purpose of making the equality $0.999\ldots = 1$ acceptable providing the tail of dots ``$\dots$'' with a precise meaning. Essentially the interviewee will have to understand that a symbol is not what its aspect may suggest, but what we want it to be in a precise way. What we want will be the result of an evolutionary process of change of meaning, dictated by the context in which we move in each conceptual phase in which the experience is structured. In more detail, the Introduction will serve him/her to reflect on what a symbol is and to appreciate the usefulness of the positional system of numerical symbols being aware of the hidden character of the involved algebraic operations. The experience will run assigning different meanings to the symbol ``$\dots$'', each meaning reconciled with the previous one. Thus Phase 1 will extend the positional system of symbols to the rational numbers with the appearance of a new algebraic operation, the division. Phase 2 will state that the habitual algebraic operations are not sufficient to equip the symbol $0.999\dots$ with a numerical meaning, which will force us to the introduction of a new algebraic operation in Phase 3: interpreting the tail of dots either as a dynamic process (movement) or as its stabilization in an end product (rest), we will choose the last one and deal with the non trivial problem of how to formulate algebraically what means that a dynamic process becomes stabilized. To attain this objective, logical quantifiers need to appear in scene and the use of a suitable mathematical assistant will encourage their understanding through visualization.

**Keywords:** Meanings of symbols; meaning of ``$\dots$''; equality\hfil\br $0.999\dots = 1$; visualization.

Investigation and proof of a property of interior angle bisector in a triangle | 35$-$49 |

**Abstract**

The paper is focused on educational practices which may enable students to discover the property of interior angle bisector in a triangle with the help of exploration of dynamic constructions. Interactive geometry software is used for exploring relationships between geometrical objects, for transition from conjecture to verification of a statement, and for development of discovered relationships. Second part of the paper presents various methods to prove the formulated theorem based on the similarity of triangles, trigonometric law of sines, and analytic method.

**Keywords:** Investigation; triangle; angle bisector; dynamic construction; proof; Apollonius circle.

The ancient problem of duplication of a cube in high school teaching | 51$-$61 |

**Abstract**

The paper is devoted to exposition of constructions with straightedge and compass, constructible numbers and their position with respect to all algebraic numbers. Although the large number of constructions may be accomplished with straightedge and compass, one of the known problems of this kind dating from Greek era is duplication of a cube. The given proof in this paper is elementary and self-contained. It is suitable for teachers, as well as for high school students.

**Keywords:** Duplication of a cube; construction with compass and straightedge; constructible number; algebraic number.