1 Abstract: The aim of this thesis is to create websites focused on teaching linear equations and functions and verify their utility from the perspective of pupils and selected teachers. Websites are one of the form called E-learning. Therefore, the thesis deals not only with websites but also with the e-learning. The work is divided into theoretical and practical part. The theoretical part is mainly based on selected literature search and analysis of textbooks from the point of view of linear equations and functions. The practical part is about creating own website and about qualitative research implemented at the basic schools in Obříství and Neratovice. The research was conducted through questionnaires, which were enriched by interviews with selected pupils and teachers. It was found that pupils and teachers consider these created websites beneficial. The List of the strengths and weaknesses of the websites, as well as the possibilities for its improvement are one part of the thesis results. Keywords: websites, e-learning, linear equations, functions, models of implementation linear equations

Recently there has been a keen interest in the area of mathematics and finding the best methods of instruction. For instance, the No Child Left Behind Act (NCLB) has placed new levels of accountability on educators for the success of students with their education, especially in mathematics. Certain areas of mathematics, such as Algebra, have been known to challenge students to think abstractly. This has become a difficult task for educators to accomplish. The challenge of teaching algebra becomes apparent when students do not comprehend the abstract reasoning of algebra. Many students need help with the transition from numerical calculation to the abstract reasoning required for algebra. This dissertation focuses on the best approaches to helping students with this transition. This dissertation investigates a mathematics program called Hands-On Equations (HOE), which is designed to help students learn abstract concepts taught in algebra with less difficulty. The program concentrates on the transition from numerical calculation to abstract reasoning by utilizing manipulatives. The objective of the study is to investigate the effectiveness of HOE in 9 th and 10 th grade. The research uses three pretests, three posttests, a three week retention test, a six week retention test, and benchmark tests to evaluate the academic growth of students in two set groups. The collected data is then quantitatively analyzed by applying simple t-tests and an ANOVA. Analysis of the data endorses HOE as being effective with solving linear equations at Level 1, Level 2, Level 3, and with the three-week retention tests, which indicates HOE may be a positive factor in achieving success with linear equations. However, analysis of the data revealed HOE is not as effective with the six-week retention test or the benchmark test which indicates after six weeks and beyond, students do not retain the information to be successful on end of the year exams such as benchmarks. The findings from this study may be useful to educators who are contemplating implementing HOE or other such programs at their schools.

Este trabalho tem como objetivo salientar a importância do estudo de sistemas de equações lineares no ensino médio através de aplicações que estão inseridas no dia-a-dia, e, tornar significativo o saber matemático, visto que, os alunos demonstram ter uma grande dificuldade em relacionar os conteúdos aprendidos ao longo de suas vidas com situações problemas cotidianas. Para tanto, esse trabalho foi organizado em três capítulos. O capítulo I apresenta uma introdução ao estudo de sistemas de equações lineares sobre números reais, parte da matemática fundamental para a compreensão dos métodos de resolução. O capítulo II apresenta algumas aplicações de sistemas de equações lineares frequentes no dia-a-dia. O capítulo III apresenta uma proposta de sequência didática de uma situação problema contextualizada para o Ensino Médio / The aim in this work is to highlight the importance of studying systems of linear equations for secondary school students through applications that are embedded in day-today, and make significant mathematical knowledge, since students have demonstrated great difficulty relate the content learned throughout their lives with situations everyday problems. For this purpose, this work was organized in three chapters. Chapter I provides an introduction to the study of systems of linear equations over the real numbers, the mathematics fundamental to understanding the methods of resolution. Chapter II presents some applications of linear systems of equations frequently in day-to-day. Chapter III proposes a didactic sequence of a problem situation in context to Secondary School

Este trabalho tem como objetivo salientar a importância do estudo de sistemas de equações lineares no ensino médio através de aplicações que estão inseridas no dia-a-dia, e, tornar significativo o saber matemático, visto que, os alunos demonstram ter uma grande dificuldade em relacionar os conteúdos aprendidos ao longo de suas vidas com situações problemas cotidianas. Para tanto, esse trabalho foi organizado em três capítulos. O capítulo I apresenta uma introdução ao estudo de sistemas de equações lineares sobre números reais, parte da matemática fundamental para a compreensão dos métodos de resolução. O capítulo II apresenta algumas aplicações de sistemas de equações lineares frequentes no dia-a-dia. O capítulo III apresenta uma proposta de sequência didática de uma situação problema contextualizada para o Ensino Médio / The aim in this work is to highlight the importance of studying systems of linear equations for secondary school students through applications that are embedded in day-today, and make significant mathematical knowledge, since students have demonstrated great difficulty relate the content learned throughout their lives with situations everyday problems. For this purpose, this work was organized in three chapters. Chapter I provides an introduction to the study of systems of linear equations over the real numbers, the mathematics fundamental to understanding the methods of resolution. Chapter II presents some applications of linear systems of equations frequently in day-to-day. Chapter III proposes a didactic sequence of a problem situation in context to Secondary School

This work is focused on overdetermined systems of interval linear equati- ons. First part consists of introduction to interval arithmetics and interval linear algebra and basic theory of interval linear systems. In the second part various methods for solving overdetermined interval linear systems are de- scribed. By solution of overdetermined interval system we mean union of all solutions of all subsystems. Known and our variants of algorithms are discussed. We introduce our subsquare method. All mentioned methods are implemented in one toolbox for Matlab. Methods are tested on solvable and unsolvable overdetermined systems. For solvable systems we test solution enclosure, time and special features of methods. For unsolvable systems we test detection of unsolvability. At the end of this work we provide basic in- troduction to Intlab. 1

Kritická místa matematiky na základní škole (Analýza didaktických praktik učitelů - lineární rovnice) Critical Areas of Mathematics at the Primary School (Analysis of Teachers' Didactic Practices - Linear Equations) Abstract The goal of the diploma thesis is to describe didactic practices used by primary teachers when teaching linear equations, seen as one of the critical moments of primary school mathematics. First, semantic and structural models of linear equations and ways of teaching linear equations as suggested by selected textbooks for primary schools are described. Next, on the basis of literature review, pupil' problems and types of mistakes they make in this area are given. My research consisted of semistructured interviews with five experienced mathematics teachers and observations of their lessons. The interviews and observations were transcribed and the transcriptions were analysed by techniques of grounded theory. It was found out that the teachers use common didactic practices as well as their own which they develop. The practices, models which the teachers use when teaching linear equations, approaches to introducing equivalent operations, as well as a list of critical places within linear equations are the main results of the work. Keywords: Linear equations, models, pupils' problems,...

Os problemas de engenharia atualmente têm aumentado a sua ordem de grandeza, por conta de diversos fatores. A modelagem em ambiente computacional dos mesmos esbarra em limitações, como grandes quantidades de tempo de processamento gastos com diversas simulações da modelagem e a pouca quantidade de memória disponível para alocar propriamente os problemas. A resolução de grandes sistemas de equações lineares, comumente abordado nos problemas atuais de engenharia, necessita da exploração das duas situações mencionadas anteriormente. A subárea computacional que permite explorar a redução do tempo e a possibilidade de alocação na memória de tais problemas é chamada de computação de alto desempenho. O objetivo deste trabalho é ilustrar o uso de softwares de resolução de sistemas de equações lineares, chamados de solvers, projetados para os ambientes computacionais de alto desempenho, testando-os e avaliando-os em um conjunto de matrizes conhecido, bem como abordar os detalhes computacionais envolvidos em tais procedimentos. / Engineering problems today have increased their order of magnitude, due to several factors. Modeling these problems with computers brings up certain limitations, as the amount of processing time needed for several simulations and the lack of available memory to properly allocate them. The resolution of large systems of linear equations, commonly discussed in current engineering problems, needs the exploration of the two situations mentioned above. The subarea that allows exploring the computational time reduction and the possibility of allocating memory in such problems is called high performance computing. The aim of this paper is to illustrate the use of software to solve systems of linear equations, called solvers, designed for high performance computing environments, to test and evaluate them for a set of matrices as well as to address the computational details involved in such procedures.

The purpose of this study was to investigate the effects of teaching linear equations with Dynamic Mathematics Software (GeoGebra) on seventh grade students&rsquo / achievement compared to the regular instruction. Randomized posttest-only control group design was utilized in the study. 60 seventh grade students (32 girls and 28 boys) of a public school in Yenimahalle district in Ankara participated in the study. The study was conducted in 2011-2012 fall semester, lasting 9 class hours in three weeks. The data was collected by three Mathematics Achievement Tests: Cartesian coordinate system achievement test (MAT1), linear relation achievement test (MAT2) and graph of linear equation achievement test (MAT3). The quantitative analysis was conducted by using analysis of covariance (ANCOVA). The results revealed that teaching Cartesian coordinate system and linear relation by using Dynamic Mathematics Software had no significant effect on seventh grade students&rsquo / achievement compared to the regular instruction. On the other hand, the results also indicated that teaching graph of linear equations by using Dynamic Mathematics Software had a significant effect on seventh grade students&rsquo / achievement positively.

This thesis is a short survey for the comparison theorems and oscillation theorems for the second order half-linear equation [c(x)u'^{(p-1)}]'+a(x)u^{(p-1)}=0, where u^{(p-1)}=|u|^{p-2}u. Some examples are also given. The above equation is said to be oscillatory ( O ) if there exists a nontrivial solution having an infinite number of zeros in (0,¡Û); and non-oscillatory ( NO ) if otherwise. Oscillation theorems help to determine whether an equation is ( O ) or ( NO ). These comparison theorem and oscillation theorems give information for the number and position of zeros in (0,¡Û) for a nontrivial solution of the above equation. Materials in this thesis originate from the papers of Li-Yeh and the monograph of Dosly and Rehak. But Reid type comparison theorem is new.

This study used the multiple methods of classroom observation, interview with teachers and students, and paper-and-pencil test to investigate the ability development of seventh-grade students in learning two-dimensional linear systems of equations and the corresponding error analysis. Hopefully, the results of this study can be as a reference for the middle school math teachers to plan the suitable teaching strategies when they teach two-dimensional linear systems of equations to their students. At the beginning, the researcher entered two seventh-grade classrooms of one middle school in Kaohsiung to make the preliminary observations and let students (also the teachers) to get used to the appearance of the researcher in the classroom during the period that one-variable linear equations were taught. Subsequently, the formal observations were carried out for 40 class periods that two-dimensional linear systems of equations were taught. All the observations made about how teachers taught and how students learned were recorded and content analyzed. Two paper-and-pencil tests were administered during the period of preliminary observations. And three paper-and pencil tests were given during the period of the formal observations. All the test results were collected and analyzed in numerous ways. Based on the literature survey and the interviews with six middle school math teachers, all relevant abilities of mastering two-dimensional linear systems of equations were classified into three categories: Character Symbols (10 sub-abilities), Operational Principals (five sub-abilities), and Other Abilities (16 sub-abilities). Based on the results of the content analyses of classroom observations and the error analyses of five paper-and-pencils tests for each sub-abilities of mastering the subject, it was observed that during the period of developing the abilities on solving two-dimensional linear systems of equations, most students showed some signs of obstacles and puzzles. Even by the end of the course on two-dimensional linear systems of equations, most students still did not master the subject well. Based on the results of this study, it is proposed that the length of teaching period needs to be increased and more efficient learning strategies need to be introduced to the students when two-dimensional linear systems of equations are taught.