Title: The Collection of Exercises About Qaudrangles Author: Bc. Pavel Nohál Department: Department of Mathematics Education Supervisor: doc. RNDr. Jarmila Robová, CSc., Department of Mathematics Education Abstract: The master thesis is focused on exercises regarding qaudrangles. This gives a chance to pupils of primary schools and lower secondary schools to repeat and practice math curriculum with a view of further study, but also develop pupils' common sense and logical thinking. The quadrangles are being encountered by us every day and therefore the exercises in this area are largely connected with the practical life. As not many current Collections on the market are dealing exclusively with quadrangles, author would like to fill this vacancy by this thesis. Keywords: Quadrilaterals and their division Parallelograms and trapezoids Circumference and area Tops, sides and interior angles

Made available in DSpace on 2016-04-27T16:58:34Z (GMT). No. of bitstreams: 1 Valdenir Francisco Duarte.pdf: 2275326 bytes, checksum: 64718a9487076854ec47a0150adb082d (MD5) Previous issue date: 2007-12-10 / This work, carried out as part of the research project AprovaME developed at the Pontifical Catholic University of São Paulo, has as its aim to verify the advances and the difficulties presented by students in the elaboration of proofs related to the properties of parallelograms. The research procedures adopted during the study drew from the theories of de Parzysz (2001) concerning formal and empirical proofs; the four dimensions involved in the construction of geometrical thinking perception, representations, construction and conception presented by Machado (1995); the representation of information from the point of view of Duval (1995); and the considerations related to logical sequences in Duval e Egret (1989). Using the methodology Didactical Engineering, a sequence of activities was designed and carried out with two groups of students. One group was composed of 8th grade Middle School students and the second of students from the first year of High School. The activity sequence was planned to involve students in, first, the construction of hypotheses and theses on the basis of empirical explorations and, in term, organise, in a deductive form, their propositions in order to elaborate proofs of various properties of parallelograms. The analysis of the students productions illustrates difficulties experienced in the process of argumentation and proof that can be grouped into three categories: difficulties related to the elaboration of proofs, difficulties associated with the acceptance of empirical arguments and difficulties linked to problems in interpreting the problems proposed. The analysis also suggested that as the activity sequence progressed, certain advances in relation to these difficulties occurred. Some students began to carry out calculations without needing to consider particular cases, others presented complete formal proofs and even those who produced incomplete proofs made use of logical reasoning in attempts to express valid arguments. In addition, a positive factor related to the activity sequence was the engagement of students in the analysis of proofs constructed by others / O presente trabalho, vinculado ao projeto AprovaME, desenvolvido na Pontifícia Universidade Católica de São Paulo, tem por objetivo verificar os avanços e as dificuldades apresentadas pelos alunos na elaboração de provas sobre as propriedades dos paralelogramos. Os procedimentos dessa pesquisa foram fundamentados nas teorias de Parzysz (2001) sobre provas formais e empíricas; Machado (1995) sobre as quatro dimensões da construção do pensamento geométrico: percepção, representação, construção e concepção; Duval (1995) sobre formas de representação de informações; Duval e Egret (1989) sobre seqüências lógicas. Usando a metodologia da Engenharia Didática, foram concebidas e aplicadas a alunos da oitava série do Ensino Fundamental e do primeiro ano do Ensino Médio, uma seqüência de atividades que visaram levá-los, de forma empírica, a construir o conceito de hipótese/tese e, de forma dedutiva, a ordenar proposições de modo a elaborar provas das propriedades dos paralelogramos. A análise das produções dos alunos mostra dificuldades no processo de argumentação e prova que podem ser agrupadas em três categorias: dificuldades ligadas à elaboração de uma prova, dificuldades oriundas da aceitação de provas empíricas e dificuldades ligadas à leitura e interpretação de enunciados. A análise também apontou que houve certos avanços nesse processo. Alunos realizaram cálculos sem o apoio empírico, outros apresentaram algumas provas formais completas, e mesmo aqueles que produziram provas incompletas mostraram um raciocínio lógico até o ponto onde elas foram feitas. Além disso, verificou-se que o fato de dois alunos poderem analisar a prova feita pela outra dupla foi um fator muito positivo na seqüência apresentada