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Linear Diophantine equations and congruences
Kaňáková, Natálie ; Beran, Filip (advisor) ; Jančařík, Antonín (referee)
This bachelor's thesis summarizes and systematizes knowledge about congruences and linear Diophantine equations. This work is divided into two parts. The first part is dedicated to congruences. At first, it shows where we can find congruences in real life, congruence as a relation, its properties, and applications in calculating the last ciphers of large numbers, proofs of divisibility rules, or calculating the date of Easter. Afterward, we look into congruences containing unknowns - linear congruence equations. It looks into methods of solving linear congruences and illustrates them in exercises. The last topic of the first part is oriented on systems of linear congruences and the Chinese remainder theorem, both for non-coprime and coprime moduli, the algebraic version, applications in various types of problems, and modular representation of numbers. The second part of this thesis is dedicated to linear Diophantine equations - equations with integer solutions. It shows various methods of solving linear Diophantine equations with two, three, or more unknowns - the extended Euclidean algorithm, reduction method, substitution method, and others. This part also describes the relationship between linear congruences and linear Diophantine equations and the use of this relationship in solving both linear...
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Divisibility in 6th and 7th grade - textbooks and knowledge of pupils
Mašatová, Zora ; Pilous, Derek (advisor) ; Vondrová, Naďa (referee)
My theses called Divisibility at second grade - textbooks and knowledge of the pupils is devoted to different aspects of schooling of this topic at elementary schools. This work is divided into three chapters. Whilst the first chapter is dedicated to the theory, the subject of two other chapters are two different researches conducted under the divisibility theme. Theoretical resources common to both parts of the research can be found in the first chapter. Among the topics mentioned in this chapter are for example the differences between formal and informal knowledge, constructivist and transmissive way of teaching or the process of solving word problems. The essence of my first research are the textbooks for elementary school concerned with the divisibility theme. Except the part devoted to a textbooks search (all the textbooks approved by the Ministry of Education, Youth and Sports which deal with divisibility), this chapter also contains the propaedeutics of divisibility and interviews with teachers, in which I have tried to cover the most important moments in the schooling of divisibility. Second part of my research is focused on checking of the pupils' knowledge of divisibility with a special emphasis on formal and informal knowledge. I have used the test to this purpose, which was...
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Divisibility in mathematics textbooks
Šmídová, Lucie ; Vondrová, Naďa (advisor) ; Pilous, Derek (referee)
The bachelor thesis is focused on the concepts of divisibility in Czech mathematics textbooks for lower secondary schools. Namely, it describes which divisibility rules they include, if pupils are led to participating on their derivation, how are the rules mathematically justified, which models are used in their explanation and if there are real word problems included in the textbook. The thesis is divided into two parts. The first part is theoretical and it introduces main concepts that are essential for the chosen topic. Its big part is dedicated to the divisibility, its characteristics and divisibility rules. The second part consists of the analysis of the selected textbooks focused on the divisibility rules. It was found that the selected textbooks differs not only in which rules they contain but also in the approach to studying the rules of divisibility. Diversity of approaches shows significantly in the number of models used to reasoning the divisibility rules. KEY WORDS Divisibility, divisibility rules, mathematics textbook, primary school
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Divisibility at the lower secondary level
Strnádková, Ivana ; Novotná, Jarmila (advisor) ; Jančařík, Antonín (referee)
This diploma thesis deals with the topic of divisibility of natural numbers in mathematics in the 6th year of primary schools. The first, theoretical part of the diploma thesis introduces the concept of didactic situations from the Theory of Didactic Situations in Mathematics by G. Brousseau, the theory of multiple intelligences according to H. Gardner and the theory of cognitive development according to J. Piaget. All chapters are projected into the experimental part. The diploma thesis is focused on the topic of divisibility, therefore a chapter of the same title is included in this part. From the history point of view, the sieve of Eratosthenes and Mersenne prime numbers are described here in two chapters. The practical part of the thesis includes a chapter containing an analysis of various didactic methods used in six sets of mathematics textbooks. Attention is paid to similarities and differences in the sets of tasks. In each textbook the topic of divisibility is compared in regard to the introduction of the divisibility rules. The following two chapters are devoted to an experiment designed to help pupils understand the divisibility rules for numbers 2, 5, 10, 8 and 4. The experiment took place over several days in two parallel classes, each of which was pedagogically led for a long time by a...
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Divisibility in 6th and 7th grade - textbooks and knowledge of pupils
Mašatová, Zora ; Pilous, Derek (advisor) ; Vondrová, Naďa (referee)
My theses called Divisibility at second grade - textbooks and knowledge of the pupils is devoted to different aspects of schooling of this topic at elementary schools. This work is divided into three chapters. Whilst the first chapter is dedicated to the theory, the subject of two other chapters are two different researches conducted under the divisibility theme. Theoretical resources common to both parts of the research can be found in the first chapter. Among the topics mentioned in this chapter are for example the differences between formal and informal knowledge, constructivist and transmissive way of teaching or the process of solving word problems. The essence of my first research are the textbooks for elementary school concerned with the divisibility theme. Except the part devoted to a textbooks search (all the textbooks approved by the Ministry of Education, Youth and Sports which deal with divisibility), this chapter also contains the propaedeutics of divisibility and interviews with teachers, in which I have tried to cover the most important moments in the schooling of divisibility. Second part of my research is focused on checking of the pupils' knowledge of divisibility with a special emphasis on formal and informal knowledge. I have used the test to this purpose, which was...
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