Volume XXVI , issue 1 ( 2023 ) | back |

Milosav M. Marjanović (1931--2023) | 1$-$4 |

**Abstract**

A trubute to academician Milosav M. Marjanović, the chief editor of this journal from its founding in 1998, who passed away on May 9, 2023.

Graph solution of a system of recurrence equations | 5$-$13 |

**Abstract**

We define a chain of cubes as a special part of the 3-dimensional cube grid, and on it, we consider the shortest walks from a base vertex. To a well-defined zig-zag walk on the cube chain, we associate a sequence described by a system of recurrence relations and using a special directed graph we determine its recurrence property. During our process, we enumerate and collect some directed shortest paths in the directed graph. In addition, we present two other examples of our graphical method to transform a system of recurrence equations of several sequences into a single recurrence sequence.

**Keywords:** Cube chain; recurrence; directed graph; graphical solution of recurrence equation system.

**MSC Subject Classification:** 97K30, 97N70

**MathEduc Subject Classification:** K35, N75

Smoothness of the signed distance function: A simple proof | 14$-$21 |

**Abstract**

The paper is of pedagogical nature and is aimed mainly at students. It presents a detailed proof of the well-known fact that if the boundary of an open set in ${R}^n$ is of class $C^k$, $k\geq 2$, so is the signed distance to the boundary function. This function plays an important role in problems of Analysis and Geometry. The presented proof could give a teacher a good opportunity to discuss important theorems in Calculus.

**Keywords:** Signed distance function; smooth boundary.

**MSC Subject Classification:** 97I40, 58C07, 26B05

**MathEduc Subject Classification:** I45

The relation between Pappus's and Ceva's theorem | 22$-$28 |

**Abstract**

In this paper, we provide a proof of Pappus's theorem following the idea presented in the book ``Geometry Revisited'' by H.S.M. Coxeter and S.L. Greitzer. Our proof is based on Ceva's theorem instead of Menelaus's theorem.

**Keywords:** Ceva's theorem; Pappus's theorem

**MSC Subject Classification:** 97G40

**MathEduc Subject Classification:** G44

P Problem as a means to encourage students' conceptualization of fractions | 29$-$45 |

**Abstract**

Students connect fractions with pie-charts as they are taught in 3rd grade in Lithuania, but they do not have a deeper understanding of the concept. Word problems could be a tool to find misunderstandings and conceptions of fractions students have. In this study, we use word problems where fractions are shown as a numerical magnitude of a level of liquid in a glass (as an analogy to a number line) and check them with students. Each additional drop of liquid gives a new fraction. This way we raise the discussion in class and find out in which phase of the conception of fractions the students are and what misunderstandings and correct ideas they have about fractions. We did design-based research, conducted the teaching experiment, and analyzed qualitative data. We suggest a lesson plan including word problems that teachers can use to raise a discussion.

**Keywords:** Measurement; fractions; word problems; conception of fraction; synthetic model.

**MSC Subject Classification:** 97F40

**MathEduc Subject Classification:** F41

An application of the quadrilateral's geometry in solving competitive planimetric problems | 46$-$53 |

**Abstract**

In the present publication, which can be considered as a continuation of the paper V. Nenkov, St. Stefanov, H. Haimov, An application of quadrilateral's geometry in solving competitive mathematical problems, Synergetics and reflection in mathematics education, Proceedings of the anniversary international scientific conference, Pamporovo, October 16-18, pp. 121--128, 2020, the application of the geometry of quadrilateral to the solution of exams is considered. Three examples given in the magazine ``Mathematics and Informatics'' have been selected, the solutions of which illustrate well the benefit of studying the recently discovered properties of convex quadrilaterals. Two solutions to the tasks are presented for comparison. The first, proposed by participants in the competition, are relatively complex and longer, and the second---based precisely on elements of the geometry of quadrilateral, are significantly simpler and shorter. These solutions are based on properties of quadrilaterals associated with some of their remarkable points.

**Keywords:** Convex quadrilateral; incenter; pseudocenter; inverse isogonality; competitive planimetric problems.

**MSC Subject Classification:** 97G40

**MathEduc Subject Classification:** G43

Yet another elementary proof of Brauer's theorem | 54$-$55 |

**Abstract**

We present another elementary proof of the theorem of Alfred Bauer from 1952 on eigenvalue perturbation.

**Keywords:** Brauer's theorem; perturbation; eigenvalues and eigenvectors.

**MSC Subject Classification:** 97H60

**MathEduc Subject Classification:** H65