This dissertation focuses on improving the non-restoring division and square root algorithm. Although the non-restoring division algorithm is the fastest and has less complexity among other radix-2 digit recurrence division algorithms, there are some possibilities to enhance its performance. To improve its performance, two new approaches are proposed here. In addition, the research scope is extended to seek an efficient algorithm for implementing non-restoring square root, which has similar steps to non-restoring division. For the first proposed approach, the non-restoring divider with a modified algorithm is presented. The new algorithm changes the order of the flowchart, which reduces one unit delay of the multiplexer per every iteration. In addition, a new method to find a correct quotient is presented and it removes an error that the quotient is always odd number after a digit conversion from a digit converter from the quotient with digits 1 and -1 to conventional binary number. The second proposed approach is a novel method to find a quotient bit for every iteration, which hides the total delay of the multiplexer with dual path calculation. The proposed method uses a Most Significant Carry (MSC) generator, which determines the sign of each remainder faster than the conventional carry lookahead adder and it eventually reduces the total delay by almost 22% compared to the conventional non-restoring division algorithm. Finally, an improved algorithm for non-restoring square root is proposed. The two concepts already applied to non-restoring division are adopted for improving the performance of a non-restoring square root since it has similar process to that of non-restoring division for finding square root. Additionally, a new method to find intermediate quotients is presented that removes an adder per an iteration to reduce the total area and power consumption. The non-restoring square root with MSC generator reduces total delay, area and power consumption significantly. / text

This is an action research study which focuses on a didactical model founded on base ten decomposition as an algorithm for performing division on naturals. Base ten decomposition is used to enhance the algebraic structure of division on naturals in an attempt to cross the cognitive divide that currently exists between arithmetic long division on naturals and algebraic long division on polynomials. The didactical model that is proposed and implemented comprises three different phases and was implemented over five one hour lessons. Learners’ work and responses which were monitored over a fiveday period is discussed in this report. The structure of the arithmetic long division on naturals formed the conceptual basis from which shorter methods of algebraic long division on polynomials were introduced. These methods were discussed in class and reported on in this study.

Made available in DSpace on 2016-04-27T16:58:57Z (GMT). No. of bitstreams: 1 Leandro Nhoncance.pdf: 1122912 bytes, checksum: 1f1ae715e54b357ac747e3a2fecf691b (MD5) Previous issue date: 2009-10-05 / The theme of this research arose during observations made in the classroom, and also after attending classes in TIC ( Information and Communication Technology) in the course for Professional Master s Degree at the Pontifícia Universidade Católica if São Paulo. The aim of the work is to relate mathematics classes with the technology presente in the daily lives of our students. It proposes a sequence of activities which would help and lead tothe pupils obtaining the natural rest of an inexact division with the calculator. The euclidian division is a procedure that makes it easier for pupils to find the natural rest working with the calculator of their mobile phones. In this research we performed a study with a group of fifteen pupils in the senior high school of a state school. We worked with diagnosis an intervention. The data collected were analyzed in the light of some presppositions of Didactic Engineering, and while the activities were being carried out, we realized that the difficulties of some students in regard to the operation of division, especially and to obtain the natural rest, appeared when they had to work with the calculator with natural numbers. However, these difficulties were decreased with the sequence of activities proposed. This work contributed to the students learning and showed that the calculator is a worthy ally in the educational process in addition to recovering some concepts about division of natural numbers. It taught the pupils to work with the calculator in the universe of natural numbers and to obtain the rest in an operation of inexact division / A proposta desta pesquisa surgiu durante várias observações feitas em sala de aula e; também, após cursar a disciplina de TIC (Tecnologia da Informação e Comunicação) no curso de Mestrado Profissional da Pontifícia Universidade Católica de São Paulo. Este trabalho visou a interagir as aulas de Matemática com a tecnologia presente no cotidiano de nossos alunos. Propôs uma seqüência de atividades que levasse e auxiliasse os alunos a obterem o resto natural em uma divisão inexata com a calculadora. A divisão euclidiana foi um procedimento que facilitou encontrarem o resto natural, trabalhando com a calculadora do celular. Nesta pesquisa, realizou-se um estudo com um grupo de 15 alunos da 3ª série do Ensino Médio de uma escola pública estadual. Trabalhou-se de uma maneira diagnóstica e interventiva. Os dados coletados foram analisados sob alguns pressupostos da Engenharia Didática, e durante a realização das atividades, verificou-se que as dificuldades de alguns alunos em relação à operação da divisão, principalmente na obtenção do resto natural, surgiram quando os alunos tiveram que trabalhar com a calculadora no universo dos números naturais. Porém foram minimizadas com a seqüência de atividades propostas. Este trabalho contribuiu com a aprendizagem dos alunos, mostrou que a calculadora é uma forte aliada no processo educativo, além de ter recuperado alguns conceitos sobre divisão de números naturais. Ensinou aos alunos trabalharem com a calculadora no universo dos números naturais na obtenção do resto, em uma operação de divisão inexata