FUNDAÇÃO EDUCACIONAL UNIFICADA CAMPOGRANDENSE (FEUC) FACULDADES INTEGRADAS CAMPO-GRANDENSES (FIC) COORDENAÇÃO DE MATEMÁTICA Estrada da Caroba, 685, Campo-Grande/RJ - Tel: 3408-8450 Sites: www.feuc.br , www.sites.google.com/site/FEUCmat
APOSTILA
D I D Á T I C A DA MATEMÁTICA
Professor Alzir Fourny Marinhos
1
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TRANSPOSIÇÃO DIDÁTICA A necessidade de se ensinar o conhecimento leva à necessidade de modificá-lo. Essa modificação chamamos de transposição didática. A transposição didática é o estudo das transformações por que passam os conteúdos de matemática. A transposição didática ocorre permanentemente, por exemplo, quando: - o conteúdo é selecionado ou recortado de acordo com que o professor considera relevante para constituir as competências (um conjunto de conhecimentos que envolve o saber e habilidades que envolve o saber fazer) na proposta pedagógica; - alguns aspectos ou temas são mais enfatizados, reforçados ou diminuídos; - o conhecimento é dividido para facilitar a compreensão e depois o professor volta a estabelecer relação entre aquilo que foi f oi dividido; - distribui-se o conteúdo no tempo para organizar uma seqüência, um ordenamento; - determina-se uma forma de organizar e apresentar os conteúdos, como por meio de textos, gráficos, entre outros. Fazer a transposição didática implica no que é preciso desenvolver e isto deverá estar contemplado no plano de educação continuada da escola, da região ou do sistema de ensino: - saber fazer recortes na sua área de especialidade de acordo com um julgamento sobre relevância, pertinência, significância para o desenvolvimento desenvolvimento das competências que vão garantir a inserção do aluno no mundo moderno; - saber selecionar quais aspectos daquele conhecimento são relevantes; - saber relacionar o conhecimento em questão com os de outras áreas; - saber como contextualizar esse conhecimento; - dominar estratégias de ensino eficazes para organizar situações de aprendizagem que efetivamente promovam no aluno as competências que se quer desenvolver. O fenômeno da transposição didática põe em evidência o fato que a disciplina escolar não é o conhecimento científico, mas uma parte dele modificada. O conhecimento surge dentro de um campo científico, porém, para que possa ser ensinável, precisa passar por transformações que o vão ―moldando‖,
tornando-o assimilável por parte dos alunos. Assim, o saber passa por, pelo menos, três tipos distintos de transposição até se constituir em conteúdo alcançado pelos alunos: o saber científico, o saber a ensinar e o saber ensinado.
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O SABER CIENTÍFICO O saber científico é aquele produzido pelos cientistas. O saber científico não pode ser ensinado como é em seu estado nascente, isto é, em linguagem científica, mas tem de ser publicado em linguagem acessível e por pessoas que não são os seus criadores. Nesse momento, ele começa a passar por algumas transformações, a sofrer a transposição didática, para que possa ser elaborado de forma didática e se constituir em saber a ensinar. O saber científico torna-se público através de publicações em revistas especializadas e após, mais tarde, de livros didáticos, que nas escolas os professores utilizam como auxílio a sua prática pedagógica. O saber científico é validado por parâmetros internos da ciência e é apresentado ao meio científico através da publicação em forma de artigos, teses, relatórios ou livros. O SABER A ENSINAR O saber a ensinar é o que é estabelecido para ser ensinado nas Instituições escolares. O saber a ensinar é encontrado na forma de livros didáticos, cartilhas, softwares e outros materiais de apoio ao professor. O SABER ENSINADO O saber ensinado é o resultado do planejamento da aula e do ato pedagógico. O saber ensinado não é só aquilo que foi planejado no momento de preparar o plano de aula; na verdade, é o resultado final, desde a sua elaboração no planejamento da aula, na definição do conteúdo, na metodologia m etodologia aplicada pelo professor para transmitir ou trabalhar esse conteúdo específico de ensino até o resultado final obtido. O saber ensinado está ligado diretamente diretamente ao programa programa de ensino. ensino. Isso leva a pensar na importância de se estar atento ao que se apresenta ao aluno no início como programa de ensino, pois muitos se esquecem e squecem do plano logo em seguida e só lembram quando se encerra o ano letivo. Torna-se interessante também lembrar a importância da interação e do diálogo entre professor e aluno, pois essa troca entre os pares leva ao desenvolvimento, além de levar o professor a refletir sobre o método utilizado e o que os alunos aprenderam.
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ALGUNS ELEMENTOS DA TRANSPOSIÇÃO DIDÁTICA A TEXTUALIZAÇÃO DO SABER A textualização do saber é um processo de preparação prévia por que passa o conteúdo a ser ensinado na escola, e sua realização ocorre sob o controle de certas regras que visam a estruturação de uma forma didática: - divisão da teoria em várias áreas e bem delimitadas. - separação do saber de qualquer contexto pessoal. - estabelecimento de uma programação de aprendizagem progressiva e racional. - a definição explícita do que deverá ser ensinado. Na textualização do saber devemos destacar duas variáveis fundamentais : o tempo didático e o tempo de aprendizagem. O tempo didático é aquele marcado nos programas escolares e nos livros didáticos em cumprimento a uma exigência legal. O tempo de aprendizagem é aquele que está mais vinculado com rupturas e conflitos do conhecimento, exigindo uma permanente reorganização de informações, e que caracteriza toda a complexidade no ato de aprender. È o tempo necessário para o aluno superar os bloqueios e atingir uma nova posição de equilíbrio. Trata-se de um tempo que não é seqüencial e nem pode ser linear na medida em que é sempre necessário retomar uma concepção para poder transformá-la. Cada sujeito tem o seu próprio tempo de aprendizagem. AS NOÇÕES PARAMATEMÁTICAS PARAMATEMÁTICAS As noções paramatemáticas são idéias que se caracterizam como ferramentas auxiliares à atividade matemática. As noções paramatemáticas não são ensinadas de forma explícita e também são excluídas de uma avaliação direta. Elas são concebidas como idéias possíveis de serem aprendidas no transcorrer da própria aprendizagem. São sempre necessárias ao ensino como à aprendizagem matemática. As noções de equações, sistemas, definição, demonstrações, fatoração, variáveis são exemplos de noções paramatemáticas. Sempre se pede ao aluno para resolver uma equação ou demonstrar um teorema, mas quase nunca se pergunta o que é uma equação, uma demonstração ou uma definição matemática.
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AS NOÇÕES PROTOMATEMÁTICAS As noções protomatemáticas formam uma categoria que não se refere diretamente às noções matemáticas em si, mas são exigidas de forma implícita no estudo escolar. São competências que antecedem o próprio conhecimento matemático, tais como habilidade de raciocínio, percepção de modelos, identificação e formulação de questões. EXEMPLOS DE TRANSPOSIÇÃO DIDÁTICA A TRANSPOSIÇÃO DIDÁTICA DO ESTUDO DE FUNÇÕES. É importante destacar a ação do Ministério da Educação do Brasil em desenvolver um trabalho em prol da qualidade da educação escolar. Esse trabalho, que contou com a experiência e estudos de diversos educadores brasileiros, foi analisado e debatido por professores de diferentes graus de ensino, por especialistas da educação e de outras áreas, além de instituições governamentais e não-governamentais, resultando num projeto que visa uma revisão nos currículos que orientam o trabalho realizado pelos professores no Brasil e propôs um novo currículo que tenha vínculo com os diversos contextos de vida dos alunos e não mais apenas um acúmulo de informações. Assim foi entregue ao Sistema Educacional Brasileiro Os Parâmetros Curriculares Nacionais, Nacionais, (PCNs ), para servirem como mais uma ferramenta de apoio ao educador no momento do planejamento de suas aulas. Os PCNs ressaltam a importância do ensino da matemática para que o aluno possa estruturar o pensamento e o raciocínio dedutivo e considera sua compreensão uma ferramenta que serve para a vida cotidiana e para muitas tarefas específicas em quase todas as atividades humanas. Especificamente no estudo de funções, que é o alvo de nossa análise, os PCNs destacam que o seu ensino isolado como vem ocorrendo não permite a exploração do seu caráter integrador. Segundo os PCNs, a compreensão do conceito de funções é de extrema importância para a interpretação e construção de gráficos do comportamento de certos fenômenos, do cotidiano e de áreas específicas do conhecimento, como a Física, Geografia ou Economia, por exemplo. Deve-se, conforme orientam, estar atento também para a sua relação com outros temas da matemática. Os PCNs enumeram uma série de relações entre funções e outros tópicos da matemática, a título de exemplificação, conforme transcrito abaixo: ―...Uma
parte importante da Trigonometria diz respeito às funções trigonométricas e seus gráficos. As seqüências, em especial progressões aritméticas e progressões geométricas, nada mais são que particulares funções. As propriedades de retas e parábolas estudadas em Geometria
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Tais conexões entre os temas devem ser observadas e exploradas pelo professor no momento do planejamento para que o ensino se torne mais dinâmico e significativo. É também importante que não seja esquecido o aspecto da contextualização, que possibilitará ao aluno uma visão ampla e científica da realidade. Logo, de acordo com os PCNs, o ensino de Matemática deve propiciar ao aluno um nível de compreensão que o permita trabalhar com o conceito de função na resoluções de situações –problemas tanto em Matemática como em outras disciplinas. Sendo a Transposição Didática um conjunto de ações que visam favorecer o desenvolvimento de uma atividade didática que contribua efetivamente para um aprendizado significativo, a elaboração dos PCNs pode ser considerada uma dessas ações de valiosa contribuição. Em edições recentes de livros, encontramos aplicações ilustradas, dando um visual mais moderno, estimulando o aluno ao conhecimento e trabalhando alguns conceitos matemáticos, como preliminares ao estudo de funções. Desses, alguns se preocupam em apresentar resolução de problemas envolvendo situações do cotidiano, problemas relacionando outras disciplinas ou mesmo fatos de interesse científico onde possa ser demonstrado a utilização desse tópico matemático. O estudo de funções deve iniciar atendendo às recomendações dos PCNs, utilizando as idéias intuitivas como: lei, associação, regra, correspondência, para desenvolver a noção de função, para depois se retomar o estudo de funções de maneira mais formal, com a finalidade de se demonstrar como se estrutura a argumentação nessa Ciência, através de linguagem precisa e dedução lógica. O conceito de função, apesar de ser um dos do mais importantes na área da Matemática, como pode ser notado, é relativamente simples e formaliza o que ao longo da história foi sendo observado pelo homem, nas relações existentes na natureza e, principalmente, nas relações de grandezas existentes em suas atividades científicas. A medida em que o homem evolui, suas necessidades e atividades vão se tornando cada vez mais complexas. O homem avança em busca de explicações e soluções e com isso o conhecimento da humanidade evolui. Com a somatória de conhecimentos, muitos desses, agregados ao saber científico, ou apenas em informações que vão enriquecendo o conhecimento na esfera individual, o homem foi tendo condições de criar, perceber relações já existentes e formalizá-las. As diversas relações ou o que, desde Leibniz, conhece-se como funções, também foram, no decorrer dos estudos matemáticos, classificadas e nomeadas. Exemplificando, pode-se citar função constante, função linear,
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A análise e interpretação de gráficos é um conceito muito importante nos diversos ramos das Ciências, portanto, é de grande valor que o professor atente para o fato de que o ensino desse tema não deve restringir-se à mera resolução de uma série de exercícios repetitivos que favorecem a resolução mais pelo adestramento do que pela motivação da resolução de problemas reais, relacionados com o cotidiano do aluno ou com outros temas matemáticos e científicos. Encontrar aplicações significativas para esse tópico da matemática deve ser uma preocupação constante. Além disso, é importante que o aluno ao estudar funções se familiarize com suas aplicações em outras áreas científicas, para sentir-se motivado a reconhecer o valor desse conceito. O ensino de funções deve propiciar ao aluno a possibilidade de uma compreensão ampla que possa ser utilizada em benefício de outras disciplinas. Acreditamos ser propício que a noção intuitiva de função seja desenvolvida desde as primeiras séries do ensino fundamental, através de jogos, construção e interpretação de gráficos e tabelas simples, com o intuito de que o aluno comece a visualizar as operações em questão e perceber relações a partir de problemas concretos e interessantes, o mais próximos possível de seu universo. EXERCÍCIOS ATIVIDADE 1 AAAA Com as contas de luz de sua casa em mãos, analise os gastos dos últimos cinco meses que constam na fatura. Para entender melhor, suponha que os últimos 5 meses sejam: janeiro, fevereiro, março, abril e maio. Complete o quadro a seguir: MÊS JANEIRO
CONSUMO(KWH)
VALOR MENSAL (R$)
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c) Compare a sua conta de luz com a de seus colegas. Existe diferença entre a conta de sua casa e a de seus colegas? d) Escreva a expressão matemática que a companhia de luz utiliza para calcular o valor mensal, em reais, de cada residência em função do consumo (kwh). ATIVIDADE 2 Suponhamos que, por uma torneira, passem 10 litros de água por minuto. Baseado nessa informação, complete o quadro, considerando que o tempo 0 (zero) equivale ao momento de abertura da torneira. Tempo 0 minuto Volume litro
1
2
3
4
5
6
10
15
20
30
Com base nas informações do quadro , responda: a) O que ocorre com o volume volume de água que passa passa pela torneira a medida que o tempo aumenta? As duas grandezas envolvidas, volume e tempo, são proporcionais? Direta ou inversamente? Por quê? b) Nesta situação, a relação entre as grandezas volume e tempo definem uma função? Se definir função, que função é essa? A TRANSPOSIÇÃO DIDÁTICA NA RESOLUÇÃO DE EQUAÇÕES DO PRIMEIRO GRAU A resolução de equações era o tema principal da obra Hisab al- jabr w’ al muqabalah de al-Khowarizmi (825 dC), que representa o marco inicial de nossa álgebra. Como as equações desempenham um papel tão importante na Matemática, se faz necessário um grande empenho no seu aprendizado.
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A definição de raiz indica o conhecimento básico de que um aluno necessita para dar início a um estudo significativo da resolução de equações. Como achar a raiz de uma equação? Seqüência evolutiva: a) Método de gerar e avaliar. Dada uma equação básica, nossos alunos seriam orientados a pensar em seu conceito de números e a gerar diferentes valores para serem testados. Gerar números mentalmente fazendo os testes na equação e, por tentativa-e- erro ir operando até encontrar a raiz. b) O método de esconder. O método de esconder se insere num esquema mais amplo e vem depois do método de gerar e avaliar. Consiste em achar a incógnita pensando sobre o que a equação diz. Por exemplo: 16 – (14 – x) = 8 16 menos que número dá 8? Busca-se a resposta 8. Então (14 – x) vale 8. 14 menos que número dá 8? Busca-se a resposta 6. Então a raiz da equação é 6. c) O método de desfazer. O método de desfazer baseia-se nas noções de inversos operacionais. Pode ser substituído pelo método de esconder, à medida que a complexidade da equação aumenta. Como exemplo: 7(2 x
3) 10
5
3
Ao dividirmos um número por 10 estamos perguntando quantas vezes o número 10 está ―dentro‖ desse número? A resposta dada na equação é 3. Logo temos 3 vezes 10 sendo o número. Sabemos que o número é 30. Então temos 7(2x-3) – 5 = 30. Se a b = c, pela operação inversa, temos a como c acrescido de b.
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expressão em x e o segundo membro por uma expressão apenas numérica. Isso vai facilitar o processo da formação das equações equivalentes, até encontrarmos a equação equivalente x = raiz. Como exemplo: 3( x + 1) – 8 = x + 1 Ao somarmos 8 aos dois membros temos a equação equivalente 3(x+1) – 8 + 8 = x + 1 + 8 ou 3 ( x+1) = x + 9. Ao subtrairmos x nos dois membros teremos a equação equivalente 3 ( x + 1) – x = x + 9 – x ou 3 ( x + 1 ) – x = 9. Podemos desenvolver: 3x + 3 – x = 9 e 2x + 3 = 9. Ao subtrairmos 3 aos dois membros temos a equação equivalente 2x + 3 - 3 = 9 – 3 ou 2x = 6. Ao dividirmos os dois membros por 2 teremos a equação equivalente 2x/2 = 6/2 ou x = 3 ( a raiz da equação). equação). EXERCÍCIOS: 1- Fazer o método de gerar e avaliar, como uma transposição didática, nas resoluções das equações: a) 28 – 2x = 6. b) 3x + 4 = -12 c)
2 x
3
4
6
2- Fazer o método de esconder, como uma transposição didática, nas resoluções das equações: a) 15 + ( 10 – x ) = 22 b)
3( x
4)
5
6
c) 15 + ( 10 – x ) = 22 d) 15 – (10 – x) = 8 3- Fazer o método de desfazer, como uma transposição didática, nas resoluções das equações: a)
4( 2 x
4) 3
2
2
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e) Desenvolver estratégias de ensino eficazes eficazes para organizar situações situações de aprendizagem que efetivamente promovam no aluno as competências que se quer desenvolver. 4.1- Desenvolver a transposição didática no estudo de funções no Ensino Fundamental: definição de função; definição de domínio; definição de contradomínio; definição de imagem; definição de conjunto imagem. 4.2- Desenvolver a transposição didática no estudo de frações no Ensino Fundamental: definição de fração; frações equivalentes; fração própria; fração imprópria. 4.3 Desenvolver a transposição didática no estudo do máximo divisor comum, no Ensino Fundamental. 4.4- Desenvolver a transposição didática no estudo do mínimo múltiplo comum, no Ensino Fundamental. 4.5- Desenvolver a transposição didática no estudo da adição com frações, no Ensino Fundamental. 4.6- Desenvolver a transposição didática no estudo da multiplicação e divisão com frações, no Ensino Fundamental, 4.7- Desenvolver a transposição didática no estudo das regras de sinais nas operações aritméticas, no Ensino Fundamental. 4.8- Desenvolver a transposição didática no estudo do conjunto dos números irracionais, no Ensino Fundamental. 4.9- Desenvolver a transposição didática no estudo do sen, cos, tg, cotg, sec, cossec dos arcos em graus: 0 0, 900, 1800, 2700 e 3600, no Ensino Médio. .
CONTRATO DIDÁTICO A relação professor – aluno está subordinada a muitas regras e convenções que funcionam como se fossem cláusulas de um contrato. Essas
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A prática pedagógica mais comum em matemática parece ser aquela em que o professor: - cumpre seu contrato dando aulas expositivas e passando exercícios para os alunos; - em suas aulas seleciona partes do conteúdo que o aluno possa aprender e propõe problemas cujos enunciados contenham os dados necessários e tão-somente esses, cuja combinação racional, aliadas aos elementos da aula, permite encontrar a solução do problema. O aluno, por seu lado cumpre o contrato se ele bem ou mal compreende a aula dada e consegue resolver, corretamente ou não, os exercícios. Se isso não acontecer, o professor deverá ajudá-lo, dirigindo o seu trabalho através de indicações que esclareçam suas dúvidas ou de pequenas questões elementares que conduzam o resultado. r esultado. Há casos extremos em que o professor se refugia na segurança dos algoritmos prontos, fraciona a atividade matemática em etapas pelas quais passa mecanicamente, esvaziando o seu significado. Sua atuação se resume em apresentar uma definição, dar alguns exemplos e solicitar exercícios idênticos aos dos exemplos dados. Aos alunos cabe memorizar as regras para repeti-las nas provas repletas de questões rotineiras que permitem a reprodução do professor. Na estratégia de ensino em que o aluno trabalha individualmente ou em duplas, seguindo as orientações contidas em seqüências didáticas organizadas pelo professor, e a institucionalização do saber se dá através de sessões coletivas, o contrato didático é totalmente diferente. O professor se apóia nas
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EXERCÍCIOS 1 - No ensino da representação de reta a partir de uma equação dada na forma geral, no sistema cartesiano ortogonal, fazer um contrato didático. 2- No ensino dos algoritmos da adição e subtração usando o ábaco, fazer um contrato didático. 3. No ensino da resolução de sistema com duas equações e duas incógnitas, fazer um contrato didático. Sugestão: - O trabalho deve ser realizado individualmente; - É permitida a consulta de todo material (anotações, livros, calculadoras, etc); - Não haverá consulta ao professor; - Após a realização do exercício de forma individual serão formadas duplas; - A dupla deve deve discutir sobre os seus resultados. Estão iguais? Estão diferentes? Se diferentes, houve erros? Se houve erros, detectá-los. Se houver dúvidas, saná-las com os seus materiais; - Anotar todas as conclusões ou dúvidas surgidas da discussão;
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A matemática trabalha com objetos abstratos, ou seja, os objetos não são diretamente acessíveis acessíveis à percepção, necessitando necessitando para a sua apreensão apreensão o uso de uma representação. Neste caso as representações através de símbolos, códigos, tabelas, gráficos, algoritmos, desenhos são bastante significativas, pois permitem a comunicação entre os sujeitos e as atividades cognitivas do pensamento, permitindo registros de representação diferentes de um mesmo objeto matemático. Por exemplo, a função pode ser representada através da expressão algébrica, tabelas e/ou gráficos que são diferentes registros de representação. A ÁLGEBRA E OS REGISTROS DE REPRESENTAÇÃO Consideramos que a Álgebra está presente no ensino desde as séries iniciais, porém é importante que os alunos tenham um aprendizado significativo dos conceitos algébricos. Para o aluno construir um significado concreto para as expressões algébricas é necessário necessário haver uma fundamentação teórica rica em significados desde o início do aprendizado. Ao tratarmos do ensino de álgebra, precisamos considerar os diferentes registros de representação, no sentido de contribuir com o aprendizado significativo, produzindo a construção de conceitos algébricos pelos alunos. REGISTROS DE REPRESENTAÇÕES REPRESENTAÇÕES ALGÉBRICAS: - Registro Funcional – expressa relações e variáveis; Como exemplo, uma função função afim y = 2x + 2. - Registro de Equações – as letras entendidas como incógnitas; i ncógnitas; Como exemplo, as soluções da equação y = 2x + 2.
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EXERCÍCIOS 1 - Fazer o registro de representação figural para: a) o estudo do sinal da da função f(x) = x – 2. b) o estudo do sinal da da função f(x) = x2 – 3x + 2 c) a resolução da inequação inequação x - 2 > 0 d) a resolução da inequação inequação ( x - 1) x ( - x + 2 ) > 0. 2 - Desenvolver uma atividade atividade em que haja os registros de representações: representações: -Registro da língua materna. -Registro figural. -Registro de equações.
SITUAÇÕES DIDÁTICAS A teoria das situações didáticas analisa as diferentes formas de apresentação do conteúdo matemático ao aluno. O significado do saber matemático escolar para o aluno é fortemente influenciado pela forma didática com que o conteúdo lhe é apresentado. O envolvimento do aluno dependerá da estruturação das diferentes atividades de aprendizagem através de uma situação didática. Através das situações situações didáticas realizam-se as transposições transposições didáticas. As situações didáticas devem efetuar não a simples comunicação de um conhecimento, mas a devolução de um problema. A devolução é a
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Poderão ser utilizados recursos didáticos variados, por exemplo: problematização matemática a partir da exploração de material concreto de manipulação ou de situações -problema contextualizadas; desafios matemáticos com o uso de programas computacionais; desenvolvimento de atividades baseadas em seqüências didáticas previamente elaboradas pelo professor; enfim há um grande número de problemas e opções que possibilita um aumento considerável de situações a-didáticas. a -didáticas. O que impulsiona o processo de ensino –aprendizagem matemática são as atividades envolvendo a resolução de problema. O trabalho pedagógico tem início exatamente com a escolha com a escolha de um problema que deve ser compatível com o nível de conhecimento do aluno. CATEGORIAS DE SITUAÇÕES 1- Situações de ações. Um determinado contexto de aprendizagem é uma situação de ação quando o aluno, que se encontra ativamente empenhado na busca de solução de um problema, realiza determinadas ações mais imediatas, que resultam de uma operação mais operacional. Numa situação de ação há sempre o predomínio quase que exclusivo do aspecto experimental do conhecimento. Este é o caso, por exemplo, quando na solução de um problema de construção geométrica o aluno se contenta exclusivamente através da realização de um desenho utilizando régua e compasso. Ele realiza uma ação mais experimental sem no entanto se preocupar com a explicação de um resultado que esclareça ou justifique a
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As demonstrações são determinados tipos de provas aceitas pela comunidade matemática. matemática. A estrutura da uma demonstração demonstração se constitui constitui numa seqüência de deduções lógicas formais através de regras bem definidas, apoiando-se em proposições verdadeiras, que permitem concluir a verdade de uma dada proposição. Na origem dessas deduções estão algumas proposições cuja validade é admitida como evidente evidente por si mesma. È necessário destacar que a atividade de validação é indissociável i ndissociável da de formulação.
EXEMPLOS DE SITUAÇÕES DIDÁTICAS 1- SITUAÇÃO DIDÁTICA DIDÁTICA NO ESTUDO ESTUDO DA TRIGONOMETRIA. TRIGONOMETRIA. A) CONSTRUIR UMA TÁBUA TRIGONOMÉTRICA COM OS VALORES VALORES DO SENO, COSSENO E TANGENTE DOS ÂNGULOS ABAIXO ( COM DUAS DECIMAIS). APRESENTAR TODO O PROCESSO DA CONSTRUÇÃO. ÂNGULOS: 250; 300; 450; 600; 800
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Você pode concluir que o ângulo com a horizontal é 30º porque os ângulos agudos em um triângulo retângulo são complementares. C) COM O USO DO TEODOLITO CONSTRUÍDO, DAS RELAÇÕES TRIGONOMÉTRICAS NUM TRIÂNGULO RETÂNGULO E DE UMA TÁBUA TRIGONOMÉTRICA, CALCULE A ALTURA APROXIMADA DO PRÉDIO FRONTAL DA FEUC. 2- SITUAÇÕES DIDÁTICAS ENVOLVENDO A CONTEXTUALIZAÇÃO. O crescimento exponencial é característico de certos fenômenos
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corporal, que ocasiona prejuízos à saúde do indivíduo. A obesidade coincide com um aumento de peso mas nem todo aumento de peso está relacionado à obesidade, a exemplo de muitos atletas, que são ―pesados‖ devido à massa
muscular e não pela gordura corporal.
Existem diversas maneiras de se classificar e diagnosticar a obesidade, no entanto, uma das mais utilizadas atualmente baseia-se na gravidade do excesso de peso, o que se faz através do cálculo do Índice de Massa Corporal (IMC), utilizando-se a seguinte fórmula: IMC = Peso atual (kg) / altura2 (m2 ) O uso do IMC resulta ser prático e simples. A aplicação do IMC não é indicada para crianças em função das rápidas alterações corporais decorrentes do crescimento. A classificação a seguir, demonstra os diferentes graus de obesidade em indivíduos com idade igual ou superior a 18 anos: Grau de obesidade - Valor do IMC (kg/m2) – 0 Menor ou igual a 24.9
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como é o caso do Japão. A Organização Mundial de Saúde passou a considerar a obesidade como um problema de saúde pública tão preocupante quanto a desnutrição. No Brasil, estima-se que a obesidade infantil atinja 20% das crianças e cerca de 32% da população adulta apresenta algum grau de excesso de peso. A obesidade é um problema sério em todas as regiões do país, mas a situação é ainda mais crítica no Sul. Nos Estados Unidos, 34% dos homens e 55% das mulheres adultos são obesos. Causas São quatro os fatores que ajudam o indivíduo a engordar: ou ele come muito ou tem gasto calórico diminuído ou acumula gorduras mais facilmente ou têm mais dificuldade de queimá-las. Todas essas condições ocorrem não apenas por mecanismos orgânicos mas também devido a fatores genéticos. O gasto calórico significa a queima de energia que uma pessoa
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Não
ter horários fixos para comer, ou seja, ―beliscar‖ a toda hora, a pessoa
perde o controle da quantidade que comeu e come muito sem nem perceber; Exagerar no consumo de alimentos gordurosos como frituras, manteiga, óleos, doces cremosos, chocolates, etc.; Fazer ―dietas da moda‖, responsáveis pelo efeito ioiô, isto é, o ―emagrece -eengorda‖ dos que fazem esses tipos de dieta;
Ficar longos períodos em jejum, o que aumenta a fome e o apetite, favorecendo o consumo de maior quantidade de alimentos; Fazer poucas refeições durante o dia e em grandes volumes, podendo aumentar o volume do estômago e também a quantidade de alimentos que a pessoa consegue ingerir.
Outros fatores relacionados à obesidade Dados de uma pesquisa sobre a obesidade na população adulta brasileira realizada em 1989 associaram alguns possíveis fatores que explicariam os altos índices de obesidade no país. São eles: Dieta desequilibrada onde predominam alimentos muito calóricos e de fácil
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existentes são alarmantes: estima-se que mais de 80 mil mortes ocorridas no país poderiam ter sido evitadas se tais indivíduos não fossem obesos. Tratamento O objetivo atual do tratamento t ratamento da obesidade consiste em se alcançar um peso saudável e não mais o peso ideal. Mas o que seria o peso saudável? Aqueles das modelos e bailarinas extremamente magras ou dos artistas de televisão? Com certeza não. O peso saudável é aquele que se está no ―tamanho‖ adequado para desempenhar as atividades (internas e externas) do
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Seja realista: faça pequenas mudanças no modo como você se alimenta e no seu nível de atividade física. Não comece com grandes alterações, vá passo a passo. Após o primeiro pequeno sucesso estabeleça um novo objetivo e prossiga; Seja aventureiro: saia da mesmice! Experimente alimentos e preparações que você não conhece, especialmente as que são à base de frutas e verduras. Aquele alimento que você nunca experimentou: tente, ele pode ser gostoso e irá contribuir para melhorar o seu consumo de nutrientes; Seja flexível: não fique pensando se você cumpriu ou não os seus objetivos em apenas uma refeição. O ideal é que se tenha um plano diário, mas caso
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4- SITUAÇÃO DIDÁTICA NAS OPERAÇÕES ARITMÉTICAS 1) a) b) c)
Mostre, usando o ábaco, ábaco, como fazer fazer a adição de: 124 + 232 675 + 137 208 + 202
2) Mostre, usando o ábaco, ábaco, como fazer a subtração: a) 4 3 5 – 2 1 2 b) 3 4 2 - 1 6 5
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COMO DIVISÃO DO TODO EM PARTES
DIVIDIR POR DUAS PESSOAS. QUANTO TEM CADA PESSOA?
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6 4
REPRESENTAM A MESMA PORÇÃO? SÃO FRAÇÕES EQUIVALENTES?
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MULTIPLICAÇÃO DE FRAÇÕES.
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ser feito através do atalho no teclado: Ctrl + i. Para acessar as opções de
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