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The design of inquiry-based tasks and problem situations for daily mathematics teaching is still a challenge. In this article, we study the implementation of two tasks as part of didactic scenarios for inquiry-based mathematics teaching, examining teachers’ classroom orchestration supported by these scenarios. The context of the study is the Erasmus+ project MERIA – Mathematics Education: Relevant, Interesting and Applicable, which aims to encourage learning activities that are meaningful and inspiring for students by promoting the reinvention of target mathematical concepts. As innovative teaching materials for mathematics education in secondary schools, MERIA scenarios cover specific curriculum topics and were created based on two well-founded theories in mathematics education: realistic mathematics education and the theory of didactical situations. With the common name Conflict Lines (Conflict Lines – Introduction and Conflict Set – Parabola), the scenarios aim to support students’ inquiry about sets in the plane that are equidistant from given geometrical figures: a perpendicular bisector as a line equidistant from two points, and a parabola as a curve equidistant from a point and a line. We examine the results from field trials in the classroom regarding students’ formulation and validation of the new knowledge, and we describe the rich situations teachers may face that encourage them to proceed by building on students’ work. This is a crucial and creative moment for the teacher, creating opportunities and moving between students’ discoveries and the intended target knowledge. (DIPF/Orig.)

Osnovna tema diplomskog rada je rješavanje problema u matematici. Za učinkovito rješavanje problema primjenjuju se različite strategije. Diplomski rad posebno je usredotočen na strategije usko vezane za rješavanje odredenih problemskih situacija i njihovo poučavanje u nastavi matematike. Cilj ovog diplomskog rada je istražiti i analizirati strategije rješavanja problema u nastavnom predmetu Matematika. Svrha ovog rada je prezentirati potencijale u učenju i poučavanju strategija za rješavanje problema u nastavi matematike. Time je osobito važno ukazati na značaj pristupa nastavnika tematici rješavanja problema. U radu su predstavljeni praktični primjeri koje nastavnici mogu koristi pri poučavanju strategija za rješavanje problema. Ovim radom želi se potaknuti dublje razumijevanje problematike u području poučavanja strategija rješavanja problema. The basic topic is problem solving in mathematics. Different strategies are used to effectively solve the problem. This document is particularly focused on strategies closely related to solving certain problem situations and teaching them in mathematics classes. The aim of this thesis is to investigate and analyze strategies for problem solving in the teaching subject Mathematics. The purpose of this paper is to present potentials in learning and teaching strategies for problem solving in mathematics teaching. With this, it is particularly important to point out the importance of the teacher’s approach to problem solving. The paper presents practical examples that teachers can use when teaching problem solving strategies. This work aims to encourage a deeper understanding of the problem in the area of teaching problem solving strategies.

U ovom diplomskom radu obrađena je racionalna funkcija u nastavi matematike. To je elementarna funkcija koja je dio kurikula za nastavni predmet matematika u srednjoj školi, ali pomalo zapostavljena u nastavi. Obrađen je pojam racionalne funkcije te njena svojstva počevši od obrnute proporcionalnosti do funkcije s kvadratnim izrazom u brojniku i nazivniku. Posebno je istražen utjecaj koeficijenata na graf pripadne funkcije te su izrađeni metodički materijali pomoću kojih učenici mogu samostalno, ali i uz pomoć programa dinamične geometrije, istraživati svojstva racionalnih funkcija. Na kraju, dan je projektni zadatak koji objedinjuje sve spoznaje o svojstvima racionalne funkcije, ali i prikazuje primjenu iste u medicini. In this master’s thesis, the rational function in the teaching of mathematics is treated. It is the elementary function that is part of the curriculum for the subject of mathematics in high school, however it is somewhat neglected in teaching. The notion of a rational function and its properties are processed, starting from inverse proportionality to a function with a square expression in a denominator. The influence of coefficients on the graph of the corresponding function was especially investigated, and methodological materials were developed with which students can independently, but also with the help of dynamic geometry programs, investigate the properties of a rational function. Finally, a project task is given that brings together all the knowledge about the properties of a rational function, but also shows the application of the same in medicine.

U ovom diplomskom radu spominju se geometrijska preslikavanja s kojima se učenici susreću u osnovnoj i srednjoj školi, ali i pojam matrice koji se uvodi u gimnazijski program. Iako se matrice unutar matematike najčešće povezuju sa sustavima linearnih jednadžbi, u ovom se radu spominje i veza s geometrijskim preslikavanjima. Iz tog su razloga u ovom radu opisane aktivnosti kroz koje učenici, osim istraživanja svojstva geometrijskih preslikavanja, otkrivaju i njihov matrični prikaz. Rad je podijeljen u četiri glavna poglavlja: Geometrijska preslikavanja, Matrice, Veza geometrijskih preslikavanja i matrica te Aktivnosti. The focus of this master’s thesis are geometric transformations which students learn both in primary and secondary school. Additionally, the notion of matrices, which are being introduced to the curriculum of the gymnasium, is another focus of the thesis. The matrices are mostly associated with systems of linear equations, however in this thesis the connection with geometric transformations is more emphasized. For this reason, the described activities enable students to research the properties of geometric transformations and their matrix representation. The thesis is divided into four chapters: Geometric transformations, Matrices, The link between geometric transformations and matrices and Students’ activities.