Undervisning med växande geometriska mönster ses som en bro mellan aritmetiskt och algebraiskt tänkande genom att elever möter uppgifter som möjliggör generalisering av aritmetiska uttryck. Eftersom svenska elever i internationella tester visar upp bättre resultat inom aritmetik än algebra är syftet med studien att ta reda på vilka aspekter som är kritiska för elevers utveckling från ett aritmetiskt till ett algebraiskt tänkande. Studien är en learning study där en lektion planerades och genomfördes i tre klasser i årskurs 5 och 6. Lektionerna kompletterades med ett för- och eftertest som tillsammans med lektionerna bidrog till studiens resultat. Vid planering och analys av lektionerna tillämpades variationsteorin som fokuserar på vilket lärande som möjliggörs, vilket lärande som sker och vad som kan förbättra lärande. Resultatet är att det kan vara kritiskt för elever att urskilja regelbundenheten i växande geometriska mönster och att särskilja regelbundenheten från proportionalitet. En ytterligare kritisk aspekt kan vara att urskilja bokstävers betydelse inom matematik. I resultatet framkommer även att två av de variationsmönster som använts kan möjliggöra urskiljning av aspekter som kan vara kritiska för elever. Vid undervisning med växande geometriska mönster finns det flera aspekter som lärare behöver möjliggöra för elever att urskilja. Genom att använda genomtänkta variationsmönster kan aspekterna synliggöras och därmed utveckla elevers förståelse för algebra. / Teaching of growing geometrical patterns should be seen as a bridge between arithmetic and algebraic thinking, that through giving the students tasks that enables generalization of arithmetic expressions. Swedish students’ results show that they perform better arithmetically than algebraically therefore, the aim of this study is to ascertain which aspects that are critical to students’ development from an arithmetic thinking to an algebraic thinking. This study is a learning study where a lesson was planned and performed in three different classes in grade 5 and 6. The lessons included a pre-test and a posttest to further validate the study and the tests, combined with the lessons, contributed to the result of the study. When planning and analyzing the lessons the theory that was applied was variation theory which focuses on what is learned, what learning that takes place and what can be improved to further the learning. The result of this study shows that it can be critical for students to discern regularities in growing geometrical patterns and also to separate regularity from proportionality. Furthermore, the study found another critical aspect which is to discern the meaning of letters within mathematics. The result also reveals two variation patterns which enables discerning of aspects that appear critical for students. When teaching about growing geometrical patterns there are several aspects teachers need to make possible for students to discern. Through utilization of variation patterns that are well prepared and thought through these aspects can be visualized and consequently auxiliary advance students understanding for algebra.

Thesis (M.Ed. (Mathematics Education)) -- University of Limpopo, 2019 / Alignment between content standards, instruction, assessment and learning materials assists in achieving the intended content in the classroom. The purpose of this study was to explore the alignment between Senior Phase Mathematics Content Standards (SPMCS) and numeric and geometric patterns’ workbook activities. The problem was that teachers sometimes use the Department of Basic Education’s workbooks interchangeably with textbooks, while their purpose is to supplement textbooks and provide worksheets for the learners. The alignment status of the Department of Basic Education (DBE) senior phase mathematics’ workbooks could not be found in the literature. Mixed methods research and document analysis were employed to explore the status of alignment between SPMCS and DBE workbook activities on Numeric and Geometric Patterns (NGP). This was aimed at highlighting the status of alignment in terms of the content structure and the alignment indices through the use of alignment model of Webb (1997) and of Porter (2002). The findings of this study revealed that the alignment between SPMCS and DBE workbook activities on NGP in terms of the categorical concurrence, depth of knowledge consistency and range of knowledge correspondence ranges from ‘acceptable’ to ‘full’ level of agreement. However, content beyond the scope of the content standards was found in Grade 7 and Grade 8 DBE workbook activities on NGP. The computed alignment indices for Grade 7, Grade 8 and Grade 9 range from moderate to strong alignment. Besides, weak and strong discrepancies were identified, which need to be addressed to improve the content structure of the DBE workbooks. This study recommends two alignment models to explore the alignment between educational components for comprehensive results and complementation. In addition, studies such as this should be conducted to enhance the quality in developing assessments in future. KEY CONCEPTS Alignment; assessment; content standards; learning materials; workbooks; numeric patterns and geometric patterns.

Thesis (M. Ed. (Mathematics Education)) -- University of Limpopo, 2019 / Alignment between content standards, instruction, assessment and learning materials assists in achieving the intended content in the classroom. The purpose of this study was to explore the alignment between Senior Phase Mathematics Content Standards (SPMCS) and numeric and geometric patterns’ workbook activities. The problem was that teachers sometimes use the Department of Basic Education’s workbooks interchangeably with textbooks, while their purpose is to supplement textbooks and provide worksheets for the learners. The alignment status of the Department of Basic Education (DBE) senior phase mathematics’ workbooks could not be found in the literature. Mixed methods research and document analysis were employed to explore the status of alignment between SPMCS and DBE workbook activities on Numeric and Geometric Patterns (NGP). This was aimed at highlighting the status of alignment in terms of the content structure and the alignment indices through the use of alignment model of Webb (1997) and of Porter (2002). The findings of this study revealed that the alignment between SPMCS and DBE workbook activities on NGP in terms of the categorical concurrence, depth of knowledge consistency and range of knowledge correspondence ranges from ‘acceptable’ to ‘full’ level of agreement. However, content beyond the scope of the content standards was found in Grade 7 and Grade 8 DBE workbook activities on NGP. The computed alignment indices for Grade 7, Grade 8 and Grade 9 range from moderate to strong alignment. Besides, weak and strong discrepancies were identified, which need to be addressed to improve the content structure of the DBE workbooks. This study recommends two alignment models to explore the alignment between educational components for comprehensive results and complementation. In addition, studies such as this should be conducted to enhance the quality in developing assessments in future. KEY CONCEPTS Alignment; assessment; content standards; learning materials; workbooks; numeric patterns and geometric patterns.

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Este trabalho propÃe uma forma de abordagem para as TÃcnicas de IntegraÃÃo â que serve como um complemento para aquelas que sÃo trabalhadas por parte dos autores dos livros de CÃlculo Diferencial e Integral (C.D.I). Seu objetivo geral à estruturar e propor situaÃÃes de ensino apoiadas na Tecnologia Digital, mais precisamente no software Geogebra, relativa Ãs TÃcnicas de IntegraÃÃo, onde sejam explorados os padrÃes grÃfico-geomÃtricos relacionados com as funÃÃes integrandas e suas primitivas. A organizaÃÃo da pesquisa seguiu as duas fases iniciais da Engenharia DidÃtica (E.D) â AnÃlises Preliminares e AnÃlise a Priori. A estruturaÃÃo das sessÃes didÃticas, envolvendo situaÃÃes-problema diferenciadas, respeitou as fases da SequÃncia Fedathi â Tomada de posiÃÃo, MaturaÃÃo, SoluÃÃo e Prova. Iniciamos o trabalho com o levantamento da problemÃtica â identificamos que o conteÃdo, TÃcnicas de IntegraÃÃo, Ã, nos livros didÃticos da disciplina de C.D.I, citados no Programa de Unidade DidÃtica (PUD) do Curso Licenciatura em MatemÃtica do IFCE â Juazeiro do Norte, trabalhado unicamente por meio do carÃter algÃbrico. Com a intenÃÃo de registrarmos essa observaÃÃo, fizemos comentÃrios sobre as formas de abordagens dos autores Stewart (2010), Guidorizzi (2011) e Leithold (1994), em que pudemos deixar registrado que, de fato, hà uma limitaÃÃo sobre a exploraÃÃo dos padrÃes grÃfico-geomÃtricos relativos Ãs TÃcnicas: SubstituiÃÃo de VariÃveis, Por Partes, FraÃÃes Parciais e SubstituiÃÃo TrigonomÃtrica. Como produto educacional, foi desenvolvido um âsiteâ em que disponibilizamos as videoaulas e as respectivas sessÃes didÃticas.

L’intégration du langage informatique dans l’architecture et l’ingénierie a commencé dès les années 1960, mais a connu un réel essor dans les années 1990. Les outils de modélisation ont alors progressivement fait émerger une nouvelle architecture. Tout d’abord baptisée « architecture blob », on lui a ensuite attribué le nom de « blobisme » et ses dérivés sont aujourd’hui appelés « architecture non standard ». Au cours de la dernière décennie, certains chercheurs comme Mario Carpo ou praticiens comme Patrick Schumacher ont introduit les termes de « paramétrisme », « customisation de masse », et « architecture non standard » pour définir ces nouvelles constructions complexes. Elles ne sont pas nécessairement savantes, mais ont été conçues grâce à des outils numériques et réalisées avec des méthodes de fabrication digitales. Cette thèse prend appui sur le postulat qu’une approche non standard dans un projet est loin de se résumer à une démarche purement plastique ou fonctionnelle : elle est destinée à produire des objets constructibles. Le domaine d’étude est celui des murs et enveloppes construites en bois et plus particulièrement les parois de type cellulaire. Ce travail inclut donc un outil paramétrique d’aide à la conception (ACPT, Aided-Conception Parametric Tool) et à la fabrication qui peut accompagner les concepteurs dans l’exploration de solutions non-standards pour des problèmes architecturaux spécifiques Cet outil paramétrique et intégré s’appuie sur une modélisation géométrique et technique des parois murales et des différents dispositifs cellulaires qui les composent. L’approche paramétrique autorise le développement de nombreuses variantes morphologiques. Sa dimension intégrée permet la production et l’optimisation des données tant topologiques que constructives. La production grandeur nature d’une paroi réalisée par un robot de coupe a servi de cadre expérimental pour démontrer les potentialités de notre approche mais aussi en identifier les difficultés. Les améliorations effectuées ont conduit à produire une série de clusters (groupes de fonctions) pour Rhinoceros-Grasshopper (RGH) qui ont permis de mettre en oeuvre la première version opérationnelle de cet outil, baptisée Fab-Cell / The integration of computing language into architecture and engineering has been present since the 1960s but it only proved practical by the 1990s when modeling tools started to participate in an architectural shift that has conceptually mutated through the years. By then, the trend was called “blob architecture” and thirteen years ago, trends succeeding “blobism” were named as non-standard architecture. In the last ten years, academicians like Mario Carpo and practitioners such as Patrick Schumacher nested the terms parametricism, mass customization, and nonstandard architecture to define those complex –not necessarily complicated architectures created by using of digital tools and aided-manufacturing methods. This thesis is underpinned on the fact that using the non-standard approach in any architectural project needs more than just a plastic or functional intention but the means to translate that intention into actual buildable objects. The aims of this study are therefore oriented towards architectural elements using cellular-like patterns as morphologic resource. This work brings up an Aided-Conception Parametric Tool (ACPT) that actually helps designers to explore non-standard solutions to specific architectural problems regarding timber-built walls and envelopes. This ACPT is meant then to succeed architectural intentions in which geometric patterns –as morphologic modifiers- are used to provide walls and envelopes with a particular language (a cellular structure) that might require morphologic form searching (Carpo, 2015a) and topologic optimization by means of parametric generative modeling. The previously mentioned aims were validated by means of a full-scale prototyping exercise in which the first version of the ACPT is tested. Furthermore A series of modeling improvements regarding pattern generation, jointing calculation and fabrication simulation, helped fixing the difficulties found during the first validation stage in order to produce a set of Rhinoceros-Grasshopper (RGH) functional clusters that embody the early operational state of this ACPT called Fab-Cell

M. Tech. (Department of Mechanical Engineering, Faculty of Engineering and Technology), Vaal University of Technology. / The purpose of this research was to develop optimal numerical solutions that can be employed during the design of cross bored thick-walled compound cylinders. The geometric design parameters of a cross bored compound cylinder that were optimized include shrinkage pressure, cross bore size, shape, location, and obliquity. Finite Element Analysis (FEA) modeling software called Abaqus version 2019 was used to generate numerical solutions. A total of 48 different part models were created and analyzed in this work. The generated FEA results from these models were validated using analytical solutions developed from Lame’s theory. The effects of shrinkage pressure on hoop stresses and Stress Concentration Factor (SCF) were studied to determine the optimal conditions. The optimum shrinkage pressure obtained was henceforth used for further analysis in this work. In addition, using one factor at time optimization technique, an optimization process was carried out to determine the optimal combination of the cross bore configuration geometry that gives minimum SCF. These parameters of cross bore configuration geometry include different sizes of either circular or elliptical-shaped cross bore, positioned at radial, offset, and/or inclined. The analyses of the effects of shrinkage pressure ranging from 4.4733 to 223.662 MPa on 11 different part models, established that the shrinkage pressure of 89.464 MPa generated the minimum SCF magnitude of 3.02. After analyzing 8 different circular cross bore size ratios ranging from 0.1 to 0.8, at the radial position, it was established that the hoop stress increases with an increase in a cross bore size. The smallest cross bore size ratio of 0.1 gave the lowest hoop stress and minimum SCF of 3.02. Whereas the highest stress was developed at the cross-size ratio of 0.8 with an SCF magnitude of 6.75. The minimum magnitude of SCF translates to a reduction of the pressure carrying capacity of the compound cylinder by 67% than a similar plain compound cylinder. Generally, offsetting of the circularly shaped cross bore from the radial position, led to a reduction of the magnitude of SCFs. For instance, from the 8 offset positions analyzed, the minimum SCF occurred at the offset position of 0.006 m with a magnitude of 2.50. This SCF magnitude indicated a reduction of pressure carrying capacity of 60% in comparison to a similar plain compound cylinder. Evaluation of 12 different diameter ratios of elliptical-shaped cross bore ranging from 0.5 to 10, at the radial position, established the lowest SCF magnitude of 1.33 that occurred at a diameter ratio of 5. Henceforth, this optimum diameter ratio was used for further analysis. This aforesaid SCF magnitude translated to a reduction of the pressure carrying capacity of the compound cylinder by 24.81% when compared to a similar plain compound cylinder. Besides, offsetting of elliptically shaped cross bore generally decreased the magnitudes of SCFs. Therefore, for elliptically shaped cross bore, the lowest SCF occurred at radial position with magnitude of 1.33. A general comparison between the effects of circular and elliptical cross bore, established that the elliptical-shaped cross bores generated both lower hoop stresses and SCFs than those of circularly shaped cross bores. On the other hand, oblique elliptical offset cross bores along the Z-axis of the compound cylinder led to an increase in SCFs. As the oblique angle increased from 0 0 to 75 0, the SCFs also increased progressively, however, there was a significant increase in SCF when the inclination angle increased from 60 0 to 75 0. The lowest and highest SCF magnitude was 1.52 and 1.92 at 15 0 and 6.19 at 75 0, respectively. Overall, the optimum geometric configuration of a cross bore in a thick compound cylinder was found to be elliptically shaped, offset at radial position which is an obliquity angle of 0 0 having a diameter ratio a/b of 5.