Pagrindinėje mokykloje tiesinių nelygybių sprendimo mokoma aštuntoje klasėje. Tačiau nėra pakankamai vaizdžių priemonių, kurios padėtų mokiniams geriau įsisavinti ir suvokti nelygybių sprendimą. Jas dažniausiai mokoma aiškinant nelygybių sprendimo algoritmą. Mokiniai dažniausiai daro klaidas pritaikydami nelygybių savybes dauginant ar dalijant nelygybės abi puses iš neigiamo skaičiaus. Taip pat sunkiai sekasi pavaizduoti nelygybės sprendinius skaičių tiesėje ar užrašyti intervalu, nes intervalai iki begalybės nėra realiai matomi. Didžiausia problema – tekstinių (probleminių) uždavinių sprendimas. Mokiniai, mokėdami spręsti tiesines nelygybes, nesugeba žinių pritaikyti praktikoje. Dažnai daromos klaidos sudarant nelygybes bei užrašant sprendinius. Labai dažnai pamirštama, kad sprendžiama nelygybė yra tik būdas surasti tekstinio uždavinio sprendiniams. Šios problemos sprendimas – interaktyvių priemonių kūrimas ir panaudojimas ugdymo procese. Integruotos matematikos ir informacinių technologijų pamokos mokiniams ���domios, skatina loginį mąstymą bei motyvaciją. Todėl atsižvelgiant į mokinių daromų klaidų analizę bei ieškant efektyvaus sprendimo būdo, sukurta interaktyvi mokymo priemonė, kuri padės mokiniams aiškiai suprasti tiesinių nelygybių sprendimo algoritmą. / Solutions for linear inequalities are taught in the eight form in basic school. But there are still lack of examples that would greatly contribute to pupils’ knowledge concerning solving tasks of linear inequalities. Usually they are being taught by explaining algorithm of solution of inequalities. The most common mistakes pupils face proceed by multiplicating or divisioning both sides of inequality with the negative count. Also they face with difficulties expressing inequalities solutions in the linear counts or noting that according interval rules, because intervals are without end and the rest of counts can not be seen by them. One of the biggest problem – text’s (problemic) tasks solutions. Even having ability to solve tasks of linear inequalities they are not able to adopt it practically. There are often maddening such mistakes in composing inequalities and writing the solutions. Also they often forget that method of inequality is appointed just to find solution for the text’s tasks. Solution for this problem – creation of interactive measures and applying them in the process of education. Integrated lessons of mathematics and IT are very interesting for the pupils, prompt logical thinking and promote their motivation. Regarding to this problem I created an interactive methodical material, that would greatly contribute to pupils’ abilities to understand tasks connected to solving of linear inequalities and its algorithm solutions.

Orientador: Cristiano Torezzan / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T15:05:42Z (GMT). No. of bitstreams: 1 Monticeli_AndreRodrigues_M.pdf: 7043231 bytes, checksum: 683696a5c1b284a08a9d19c54647edaa (MD5) Previous issue date: 2010 / Resumo: Neste trabalho abordamos o problema de descrever o conjunto solução de um sistema de inequações lineares. Este problema está fortemente relacionado com o problema clássico da enumeração de vértices de um poliedro. Descrevemos o método de Fourier-Motzkin que pode ser utilizado para eliminar variáveis de um sistema de inequações lineares e projetar a região de solução num espaço de dimensão menor. Mostramos como o problema da enumeração de vértices pode ser convertido em um problema de encontrar o fecho convexo do conjunto de pontos dual ao sistema de inequações lineares, uma vez encontrado um ponto interior factível. Alguns algoritmos para o fecho convexo de um conjunto finito de pontos e também para encontrar um ponto interior factível são estudados. Nosso interesse, além de listar os vértices e as faces é também visualizar a região de solução utilizando um programa computacional. Para tanto propomos um método que constrói a lista dos vértices e faces do poliedro definido por um dado sistema de inequações lineares e grava o resultado num arquivo de texto puro com extensão obj, que é compatível com os principais softwares de visualização gráfica 3D. O método foi implementado no Octave e diversos testes foram feitos, analisando o custo computacional e possíveis dificuldades que podem surgir devido a erros numéricos ou falta de memória / Abstract: In this work we approach the problem of describing the solution of a system of linear inequalities. This problem is closely related to the classical problem known as vertex enumeration. We describe the method of Fourier-Motzkin, that can be used to eliminate variables in a system of linear inequalities, projecting its solution in a lower dimensional space. We show how the vertex enumeration problem can be converted into an equivalent problem of finding the convex hull of a set of dual points, once found a feasible interior point. Some algorithms for convex hull and also for finding a feasible interior point are studied. Our interest is not only to store the vertices and faces but also visualize the correspondent polyhedron using a computer graphics software. In this way we propose a method that stores the polyhedron's vertices and faces and output the results into a plain text _le with extension obj, which is a geometric definition file format that can be opened with all major 3D graphics software. The method was implemented in Octave and several tests were made, analyzing the computational cost and possible difficulties that may arise due to numerical errors or memory requirements / Mestrado / Matematica / Mestre em Matemática

本研究的主要目的是探討國一學生在一元一次不等式的錯誤類型，並分析錯誤原因。 本研究的設計採用調查研究法，共分成兩個階段進行，第一階段為準備階段，主要工作為文獻探討、分析國中數學教材、自編「一元一次不等式錯誤類型分析研究」試卷，進行試卷的施測，預試樣本共56名國三學生，抽樣方式非隨機取樣，採方便取樣進行，再由預試結果經修改編製正式施測之試卷。第二階段為正式施測與分析階段，正式施測樣本共30名國一學生，男生12名、女生18名，根據施測結果，依成績分成高分組、中分組、低分組三組，再隨機抽取男女生各2名進行半結構的晤談，以瞭解學生答題的想法，分析學生錯誤的原因。 本研究的主要結果如下： 一、不等式答題表現在文字問題的錯誤比率、空白比率最高。 二、在一元一次不等式的錯誤類型為： （一）同義詞的轉換：1.不等號的同義詞概念；2.不等號的符號認知。 （二）範圍解與圖示：1.數的運算；2.無法判斷範圍解；3.數線的認知；4.不等號的圖示認知；5.不等號的方向。 （三）解不等式：1.不等號改變方向；2.去括號；3.數的運算；4.遺漏或增加符號；5.移項的錯誤；6.等量公理的誤用；7.未化簡；8.不等號概念的錯誤9.多項錯誤；10.抄錯題目或答案；11.胡亂猜測答案。 （四）文字問題：1.無法理解題意；2.列式錯誤；3.三角形面積公式錯誤；4.忽略題目的已知條件；5.答案遺漏或錯誤；6.不等號的概念；7.數的運算；8.多項錯誤9.不等號的同義詞概念。 三、一元一次不等式的錯誤原因：1.先備知識的不足；2.資料使用錯誤；3.新舊學習經驗的互相干擾；4.錯誤的使用運算規則；5.由題目所給數字直接產生答案；6.不清楚題目設計或文字敘述而產生錯誤；7.忽略題目所給條件或答案不夠完備而產生錯誤；8.沒有從離散量概念延伸至連續量概念。 / The main purpose of this study is to investigate “seventh grade students' misconceptions and error types of linear inequalities in one unknown”and analyze the causes of the errors. This study adopts survey research and includes two phases. The first stage is the preparation phase, including literature review, analysis of mathematics textbooks, self-compiled test papers on “misconceptions and error types of linear inequalities in one unknown,” and the pretest. The pretest adopts convenience sampling- totally 56 students from ninth grade. The results were later revised to compile the formal test papers. The second stage is the official survey and analysis phase. The 30 samples are seventh grade students, 12 boys and 18 girls. According to the results of the test, these sample students are divided into three groups-high, medium and low performances. Out of each group, two boys and two girls are randomly sampled and conduct semi-structured interviews to analyze the causes of the errors. The findings of this study are as follows: 1.Most of the errors are due to text problems. 2.Error types of linear inequalities in one unknown are: (1)The conversion of synonyms: a. the concept of synonyms; b. symbolic cognitive. (2)The range of solution: a. calculation; b. to determine the range of solution; c. number line; d. notation of inequality sign; e. direction of inequality sign. (3)Problem-solving in inequality: a. to reverse symbol; b. to remove bracket; c. calculation; d. to omit or add symbols; e. transposition errors; f. isometric axiom errors; g. lack of simplification; h. the misconception of inequality sign; i. multinomial errors; j. to copy wrong questions or answers; k. to speculate answers. (4)Text problem: a. not to understand the questions; b. mistakes in formulating expressions; c. triangle area formula errors; d. to ignore provided conditions; e. omission or wrong answers; f. notation of inequality sign; g. calculation; h. multinomial errors; I. the concept of synonyms. 3.Cause of errors: (1) lack of prior knowledge; (2) to use wrong data; (3) the mutual interference of old and new learning experiences; (4) to use wrong calculating rules; (5) to generate answers from given numbers of the questions; (6) not to understand description of the questions; (7) to ignore the provided conditions or the answers are not complete; (8) not extend to continuous volume.

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