GUIA DO ALUNOVolume 1

Matemática3ª série do Ensino Médio

Entre Jovens 3ª série do Ensino Médio: guia do aluno matemática.

– São Paulo: Instituto Unibanco/CAEd, 2013.

170 p.; Vol I

Coordenação do materialJuliana Irani do Amaral

Pesquisa e conteúdoCAEd – Centro de Políticas Públicas e Avaliação da Educação

Consultoria responsávelCAEd – Centro de Políticas Públicas e Avaliação da Educação

Agradecimentos especiaisSecretaria Municipal de Educação do Rio de Janeiro, Secretarias de Estado de Educação do Rio de Janeiro, São Paulo, Espírito Santo e Minas Gerais e escolas parceiras que contribuíram para a testagem e validação da Metodologia Entre Jovens.

RealizaçãoInstituto Unibanco

PresidênciaPedro Moreira Salles

Vice-PresidênciaPedro Sampaio Malan

ConselhoAntonio MatiasCláudio de Moura CastroCláudio Luiz da Silva HaddadMarcos de Barros LisboaRicardo Paes de BarrosTomas Tomislav Antonin ZinnerThomaz Souto Corrêa NettoWanda Engel

Diretoria ExecutivaFernando Marsella Chacon RuizGabriel Amado de MouraJânio GomesJosé Castro Araujo RudgeLeila Cristiane B. B. de MeloLuis Antônio RodriguesMarcelo Luis Orticelli

Superintendência ExecutivaRicardo Henriques

Gerência de Implementação de ProjetosTiago Borba

Gerência de Desenvolvimento e ConteúdoMarta Grosbaum

Gerência de Gestão do ConhecimentoCamila Iwasaki

Gerência de Administração e FinançasFábio Santiago

Gerência de Escritório de ProjetosJosé Carlos R. de Andrade

Assessoria de Assuntos EstratégicosChristina Fontainha

Assessoria de ComunicaçãoMarina Rosenfeld

Assessoria de VoluntariadoFabiana Mussato

SUMÁRIO

APRESENTAÇÃO

OFICINA 1RESOLUÇÃO DE PROBLEMAS

OFICINA 2PROPORCIONALIDADE

OFICINA 3EQUAÇÃO DO 1º GRAU

OFICINA 4PORCENTAGEM

OFICINA 5EQUAÇÃO DO 2º GRAU

OFICINA 6PROGRESSÕES ARITMÉTICAS E PROGRESSÕES GEOMÉTRICAS

OFICINA 7MATEMÁTICA FINANCEIRA

OFICINA 8SEMELHANÇA DE TRIÂNGULOS E O TEOREMA DE TALES

OFICINA 9ALGUMAS CONSEQUÊNCIAS DO AXIOMA DAS PARALELAS

OFICINA 10FUNÇÃO DO 1º GRAU

RESPOSTAS

9

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Prezado aluno, seja bem vindo ao Guia do Aluno do Projeto Entre Jovens – 3º ano.

Este é um programa de tutoria que tem por objetivo consolidar competências básicas em matemática que possam melhorar seu desempenho escolar e, ainda, colocá-lo em condições de prosseguir, seja com os estudos, seja profissionalmente, em algumas das possibilidades que se apresentam aos concluintes do ensino médio.

Este é o primeiro de dois guias e é composto por dez oficinas, cada uma a ser cumprida em uma semana. Tem-se em média dez atividades por oficina, que serão discutidas e trabalhadas pelos tutores com você.

Esperamos de sua parte uma postura ativa no enfretamento das atividades propostas. Você deverá tentar resolver todas as atividades, desenvolvendo-as ao máximo para, em seguida, discuti-las com os colegas e com o tutor. Ainda que não consiga chegar à solução, partilhe com o grupo o seu raciocínio, o seu encaminhamento parcial da atividade.

Matemática não se aprende de forma contemplativa, assistindo alguém resolver exercícios. Aprender matemática exige dedicação, esforço, persistência e iniciativa.

Procure desenvolver seu próprio raciocínio, compreendendo o enunciado de cada atividade, identificando o que está sendo pedido, os dados fornecidos e as relações existentes entre os dados do problema e a incógnita. Em seguida, elabore uma estratégia que permita relacionar os dados à incógnita e execute-a. Ao obter uma resposta, confira-a. Veja se o valor obtido está compatível com o problema proposto, se cumpre todas as condicionantes. Desenvolver essa rotina de procedimentos é útil na resolução de problemas.

As oficinas que compõem esse guia abordarão os seguintes temas:

1. Resolução de problemas.

2. Proporcionalidade.

3. Equação do 1º grau.

4. Porcentagem.

5. Equação do 2º grau.

6. Progressões aritméticas e progressões geométricas.

7. Matemática financeira.

8. Semelhança de triângulos e teorema de Tales.

9. Algumas consequências do axioma das paralelas.

10. Função do 1º grau.

APRESENTAÇÃO

OFICINA 1

Esperamos que aproveite os encontros, que tenha atitudes colaborativas com o grupo de colegas e com o tutor, de forma que todos os participantes do projeto se apropriem de saberes matemáticos essenciais para quem se encontra concluindo essa importante etapa de escolarização.

Bons estudos e ao trabalho!

RESOLUÇÃO DE PROBLEMASOFICINA 1

Metas:– Introduzir técnicas básicas de resolução de problemas.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• identificar dados e incógnitas em um problema de matemática.

• planejar e elaborar estratégias para a resolução de problemas.

• executar estratégias.• verificar e analisar os resultados obtidos.

Pré-requisito:– Temas diversos de matemática pertinentes ao Ensino

Fundamental.

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1. A arte de resolver problemas

A seguir, apresentamos os quatro passos na resolução de um problema, apontados por Polya.

1º. É preciso compreender o problema

O primeiro passo para resolver um problema é compreendê-lo bem. Deve-se saber exatamente o que é dado e o que é pedido.

• Leia o problema com muita atenção. Se necessário leia-o em voz alta e tente explicá-lo aos seus colegas.

• Tome nota das quantidades e das condições que são dadas – os chamados dados do problema.

• Identifique as incógnitas. Identifique as condicionantes. O que é que se pretende exatamente: calcular ou provar?

• Desenhe uma figura ou um esquema que possa lhe ajudar a organizar a informação e a visualizar o problema.

• Uma possível ajuda é reformular o problema de formas diferentes, ou pensá-lo numa situação concreta que te seja mais familiar. Tente isto!

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2º. É preciso planejar uma estratégia para resolver o problema.

Agora que já compreendeu bem o problema, deve imaginar um plano para resolvê-lo. Esse é o passo mais difícil porque requer algumas capacidades que tem que aprender a desenvolver – criatividade, espírito de organização e experiência. É claro que só irá consegui-lo com esforço e trabalho. Não há outra hipótese! Mas verá que vale a pena.

• Tente pensar num problema semelhante que, eventualmente, já tenha resolvido antes.

• Tente fazer um diagrama que explique a estratégia que tenha imaginado para a resolução.

• Identifique as ferramentas necessárias para a resolução - ferramentas analíticas, geométricas, combinatórias, etc...

• Se não conseguir resolver o problema proposto, procure antes resolver algum problema do mesmo tipo. É possível imaginar um problema parecido mais acessível? Um problema mais genérico? Um problema mais específico? Um problema análogo? É possível resolver uma parte do problema?

• Fixe apenas uma parte das condicionantes, deixe outras de lado; até que ponto fica assim determinada a incógnita? Como pode ela variar? É possível obter dos dados alguma coisa de útil? É possível pensar em outros dados apropriados para determinar a incógnita? É possível variar a incógnita ou os dados, ou todos eles, se necessário, de tal maneira que fiquem mais próximos entre si?

• Utilizou todos os dados? Utilizou todas as condicionantes?

3º. Execute a estratégia que planejou. Reveja tudo, desde o início se necessário!

Agora já tem um plano para resolver o problema. Tem que o executar. Sem receios, é claro! Seja auto-confiante!

• Guarde sempre as notas do teu trabalho, já que podem ser úteis se precisar revê-lo.

• Verifique de novo cada passo da resolução e confirme se não há erros que afetem a solução final.

• Teste sempre a estratégia que imaginou no passo 2. Se encontrar erros, contradições ou falta de informação para prosseguir, é natural que tenha que repensar o teu plano. Tenha paciência, volte atrás! A paciência é também fundamental na resolução de problemas!

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4º. Verifique e interprete os resultados a que chegou.

Agora já tem uma solução. Mas, estará correta? É muito importante validar os resultados obtidos.

• Confirme que os resultados que obtive fazem sentido. Verifique que, por exemplo, eles têm as unidades esperadas e têm valores numéricos que não sejam absurdos em relação ao contexto apresentado pelo problema.

• Verifique de novo os seus cálculos, ou faça-os de outra forma diferente.

• Teste a consistência dos teus resultados considerando casos particulares ou situações limites.

• Escreva a solução final de forma clara e concisa, usando uma linguagem simples sem qualquer margem para ambiguidade.

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Resolva as atividades propostas abaixo.

Atividade 1: Um gato está sobre um muro de 4m de altura quando avista um rato a uma distância de 8m da base do muro. Quando o rato dirige-se à sua casa (em linha reta até o muro) é comido pelo gato, que pula diagonalmente, andando o mesmo comprimento que o rato tinha andado até então. Qual a distância que cada um percorreu?

Atividade 2: Nicolau gastou tudo o que tinha no bolso em cinco lojas. Em cada loja gastou R$ 1,00 a mais do que a metade do que tinha ao entrar. Quando tinha Nicolau no bolso antes das compras?

Atividade 3: Um número palíndromo é aquele que é “o mesmo” lido da esquerda para a direita e vice-versa. Exemplos: 343; 1.001; 245.542, etc.

Um motorista dirige em uma rodovia cuja velocidade máxima permitida é de 100 km/h. O motorista obedece a esse limite e viaja a uma velocidade constante. Então observa que o marcador de quilometragem indica 15.951 km, e diz pra si mesmo:

“Um palíndromo - e isso aconteceu há um bom tempo”.

Mas exatamente duas horas depois o marcador apresenta um novo número palíndromo. A que velocidade viaja o motorista?

Atividade 4: Entre os papéis do vovô foi encontrado um recibo:

72 frangos = $ _67,9_

O primeiro e o último algarismos do número, que evidentemente representava o preço total das aves, aparecem aqui substituídos por espaços em branco porque se apagaram e estão ilegíveis. Quais serão os algarismos apagados e qual era o preço de um frango?

Atividade 5: Para numerar as páginas de um grosso volume de um livro, o tipógrafo utilizou 2.989 algarismos, começando a numerar a 1ª página pelo número 1. Quantas páginas tem esse livro?

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Atividade 6: André irá decorar azulejos quadrados, de 8 cm de lado, pintando uma figura geométrica decorativa, conforme ilustrado na figura abaixo.

A quantidade de tinta que ele dispõe é suficiente para pintar somente 0,5 m² de superfície. Quantos azulejos André conseguirá pintar completamente?

Atividade 7: Um velho e justo mercador de Bagdá deixa seus bens para serem divididos igualmente entre seus três filhos. Entre os bens existiam 21 vasilhames: 7 cheios de mel; 7 com mel pela metade e 7 vasilhames vazios. Como fazer a divisão equitativa, de forma que cada um dos filhos receba o mesmo número de vasilhames e a mesma quantidade de mel, sem que haja nenhuma transposição de qualquer quantidade de mel de um vasilhame para outro?

Atividade 8: Em ambas margens de um rio existem dois mastros, um em frente ao outro. A altura de um é 9 metros e a do outro é de 4 metros. A distância entre eles é de 25 metros. Sobre cada mastro está um pássaro. Subitamente, os dois pássaros descobrem um peixe que aparece na superfície da água, no alinhamento dos dois mastros. Os pássaros, que voam a uma mesma velocidade, lançam-se sobre o peixe e alcançam-no ao mesmo tempo. A que distância do mastro maior apareceu o peixe?

Atividade 9: Um cavalo e uma mula caminhavam juntos levando no lombo sacos pesados. Lamentava-se o cavalo de sua pesada carga quando a mula lhe disse: “De que se queixa? Se eu levasse um dos seus sacos, minha carga seria o dobro da sua. E se eu lhe desse um saco, sua carga seria igual à minha”. Quantos sacos levava o cavalo e quantos levava a mula?

Atividade 10: O professor de matemática resolveu dividir os alunos de sua turma em grupos. Ao dividi-los em grupos de 6, sobraram 4 alunos. Ao dividi-los em grupos de 9, também sobraram 4 alunos. Qual é o menor número possível de alunos desta turma?

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OFICINA 2

PROPORCIONALIDADEOFICINA 2

Meta:– Apresentar as noções de:

• grandezas proporcionais.• grandezas inversamente proporcionais.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• reconhecer grandezas direta e inversamente proporcionais.

• resolver problemas envolvendo grandezas proporcionais, direta ou inversamente.

• resolver problemas onde uma grandeza é proporcional à várias grandezas.

Pré-requisito:– Operações com números inteiros e frações.

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Resolva as atividades propostas abaixo.

Atividade 1: Uma empresa de engenharia consegue asfaltar 60 km de estrada em 20 dias. Quantos dias seriam necessários para a mesma empresa asfaltar uma estrada de 84km?

Atividade 2: Uma lata de leite em pó, pesando 400g, custa R$ 5,20. O mesmo leite, na embalagem de 900g, custa R$ 11,20. Qual das duas opções é economicamente mais vantajosa?

Atividade 3: Trabalhando 8 horas por dia, 3 trabalhadores constroem um muro de 40 m de altura em 12 dias. Se o número de horas de trabalho diário for reduzido para 6 e o número de trabalhadores aumentado para 5, qual o comprimento de um muro de mesma altura que eles construirão em 15 dias?

Atividade 4: Se 5 torneiras (com mesma vazão) enchem um tanque em 2 horas (estando o tanque inicialmente vazio), quanto tempo demorará para encher esse tanque (estando inicialmente vazio) quando somente 3 dessas 5 torneiras estiverem abertas?

Atividade 5: Um fazendeiro possui ração suficiente para alimentar suas 16 vacas durante 62 dias. Após 14 dias, ele vende 4 vacas. Passados mais 15 dias, ele compra 9 vacas. Quantos dias, no total, durou sua reserva de ração?

Atividade 6: Uma torneira A enche um tanque em 5 horas. Uma torneira B enche esse mesmo tanque em 15 horas. As duas torneiras juntas encherão esse tanque em quantas horas?

Atividade 7: Um barco com 7 pessoas, à deriva no mar, tem suprimento de água suficiente para 28 dias. Após 4 dias, o barco recolhe mais 1 náufrago. Se o consumo diário de água por pessoa se mantiver o mesmo, em quantos dias mais durará a reserva de água?

Atividade 8: Uma escola lançou uma campanha para seus alunos arrecadarem, durante 30 dias, alimentos não perecíveis para doar a uma comunidade carente da região. Vinte alunos aceitaram a tarefa e, nos primeiros 10 dias, trabalharam 3 horas diárias, arrecadando 12 kg de alimentos por dia. Animados com os resultados, 30 novos alunos somaram-se ao grupo e todos passaram a trabalhar 4 horas por dia nos dias seguintes até o término da campanha. Admitindo-se que o ritmo de coleta tenha se mantido constante, qual a quantidade de alimentos arrecadados ao final dos 30 dias de campanha?

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Atividade 9: Uma caixa d’água de 1000 litros tem um furo no fundo, por onde escoa água a uma vazão constante. Às 6h da manhã de certo dia ela foi cheia e, ao meio dia desse mesmo dia, só tinha 850 litros. Quando o volume d’água restante na caixa alcançará a metade da capacidade da caixa?

Atividade 10: Uma caravana com 7 pessoas deve atravessar o Sahara em 42 dias. Seu suprimento de água permite que cada pessoa disponha de 3,5 litros por dia. Após 12 dias, a caravana encontra 3 beduínos sedentos, vítimas de uma tempestade de areia, e os acolhe. Pergunta-se:

a) Quantos litros de água por dia caberão a cada pessoa se a caravana prosseguir sua rota como planejado?

b) Se os membros da caravana (beduínos inclusive) continuarem consumindo água como antes, em quantos dias, no máximo, será necessário encontrar um oásis?

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OFICINA 3

EQUAÇÃO DO 1º GRAUOFICINA 3

Meta:– Apresentar equação do 1º grau como ferramenta de resolução

de problemas.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• resolver equações do 1º grau.• resolver problemas que envolvam equações do 1º grau.

Pré-requisito:– Operações entre números inteiros e racionais.

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Resolva as atividades propostas abaixo.

Atividade 1: Num país chamado Aposentadolândia, um trabalhador tem direito à aposentadoria quando a soma do número de anos trabalhados com a idade do trabalhador é igual a 100. Com que idade poderia aposentar-se uma pessoa que tenha trabalhado ininterruptamente a partir dos 22 anos?

Atividade 2: Resolva as equações abaixo:

a) 2 3 4 2x x+ = −

b) 3 1 1

1,52 2

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c) 4 3

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Atividade 3: Em um teste de 25 questões, por cada acerto ganha-se 4 pontos e por cada erro perde-se 1 ponto. Pedro respondeu a todas as questões e marcou 70 pontos. Quantas questões Pedro acertou?

Atividade 4: Azul e Branca são duas locadoras de automóvel. A locadora Azul cobra R$ 1,00 por quilômetro rodado mais uma taxa de R$ 100,00 fixa. A locadora Branca cobra R$ 0,80 por quilômetro rodado mais uma taxa fixa de R$ 200,00. Discuta em que situação é mais vantajoso se alugar um carro em uma ou outra locadora.

Atividade 5: Um grupo de 50 pessoas fez um orçamento inicial para organizar uma festa, que seria dividido entre elas em cotas iguais. Verificou-se ao final que, para arcar com todas as despesas, faltavam R$ 510,00, e que 5 novas pessoas haviam ingressado no grupo. No acerto foi decidido que a despesa total será dividida em partes iguais pelas 55 pessoas. Quem não havia ainda contribuído pagaria a sua parte e cada uma das 50 pessoas do grupo inicial deveria contribuir com mais R$ 7,00. De acordo com essas informações, qual foi o valor da cota calculada no acerto final para cada uma das 55 pessoas?

Atividade 6: Resolva o sistema: 2 1

3 11

x y

x y

− = + =

Atividade 7: Eu tenho o dobro da idade que tu tinhas quando eu tinha a idade que tu tens. Quando tiveres a idade que eu tenho, juntos teremos 135 anos. Qual é a minha idade?

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Atividade 8: Para promover um baile, um clube fez o seguinte levantamento de gastos:

Banda R$ 3.000,00

Decoração R$ 2.400,00

Iluminação R$ 400,00

Além dos gastos acima, o buffet cobrará R$ 35,00 por pessoa. O preço do convite individual é R$ 70,00. Qual o número mínimo de convites que o clube deve vender para que o baile não dê prejuízo?

Atividade 9: A herança do Antônio foi dividida, entre 10 pessoas, seus 2 irmãos e seus 8 filhos, em partes iguais. Caso a partilha da herança fosse feita somente entre seus 8 filhos, cada um deles teria recebido R$ 5.000,00 a mais do que receberam na partilha por 10. Qual o valor da herança partilhada?

Atividade 10: Foram pesados 3 cubos idênticos e 2 esferas idênticas, todos juntos e, nessa pesagem, obteve-se 11,5 kg. Em seguida, pesou-se 2 desses cubos juntos com 3 esferas idênticas às da primeira pesagem e obteve-se 11kg. Qual o peso de cada cubo e de cada esfera?

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OFICINA 4

PORCENTAGEMOFICINA 4

Meta:– Introduzir a noção de porcentagem.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• reconhecer o conceito de porcentagem.• calcular porcentagem.• resolver problemas envolvendo porcentagens.

Pré-requisito:– Operações com números inteiros e frações e equação do 1º grau.

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Resolva as atividades propostas abaixo.

Atividade 1: O salário mensal de um trabalhador é R$ 980,00. Ao receber um aumento salarial de 3,6%, qual passou a ser seu novo salário?

Atividade 2: O preço da entrada do cinema foi reajustado em 25% e, com isso, passou a valer R$ 11,25. Qual era o valor da entrada antes do reajuste?

Atividade 3: Numa empresa há 620 funcionários. Desse total, 341 são homens. Qual a porcentagem de funcionárias nessa empresa?

Atividade 4: Em uma liquidação, um lojista diminuiu em 20% o preço de todas as suas mercadorias. Terminado o período da liquidação, o lojista resolveu reajustar todos os preços de forma a restaurá-los aos preços praticados antes da liquidação. Qual deverá ser o percentual de reajuste?

Atividade 5: No 1º semestre de 2009, Carlos teve 5% de aumento salarial. No 2º semestre desse mesmo ano Carlos teve um novo aumento salarial, agora de 10%.

a) Qual o aumento salarial de Carlos, após esses dois semestres?

b) Teria sido melhor ou pior mudar a ordem (mas não os valores) dos aumentos?

Atividade 6: Num salão com 200 pessoas, 99% são mulheres. Mantido os homens no salão, quantas mulheres devem sair para que 96% das pessoas que fiquem no salão sejam mulheres.

Atividade 7: O tanque contém 60 litros de combustível, dos quais 30% são de álcool e o restante de gasolina. Quantos litros de gasolina devem ser acrescentados a esse tanque de modo que o combustível resultante tenha apenas 20% de álcool?

Atividade 8: A água do mar contém 2,5% de seu peso em sal. Quantos quilogramas de água do mar são necessários para obter 500g de sal?

Atividade 9: Numa promoção, uma bermuda sofreu um desconto de 15% e, no último dia da liquidação, sofreu um novo desconto de 10%. Qual foi o desconto percentual total no preço da bermuda no último dia da liquidação, em relação ao preço anterior à liquidação?

Atividade 10: Pedro realizou um trabalho pelo qual recebeu R$ 2.349,00 líquidos, após o desconto de 10% de INSS.

a) Qual foi a remuneração (bruta) paga a Pedro por esse trabalho?

b) Quanto Pedro deveria ter cobrado por esse trabalho de forma a receber (líquidos) R$ 2610,00, após a dedução do INSS?

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OFICINA 5

EQUAÇÃO DO 2º GRAUOFICINA 5

Meta:– Apresentar técnicas resolutivas de equações do 2º grau.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• resolver equação do 2º grau.• resolver problemas envolvendo equação do 2º grau.

Pré-requisito:– Módulo de um número real, equação do 1º grau.

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Resolva as atividades propostas abaixo.

Atividade 1: Um terreno, que tem a forma de um quadrado, foi reduzido da maneira indicada na figura abaixo, para dar lugar a uma calçada com 2m de largura. Ao final, sua área passou a ser de 484m². Qual era a medida do lado do terreno original?

Atividade 2: Resolva as equações abaixo:

a) 2 4x =

b) ()22 3 25x − =

c) 2 2 15 0x x− − =

d) 22 3 1 0x x+ + =

Atividade 3: Antônio, com 20m de cerca, construiu um cercado retangular de 32m² de área, utilizando seu muro como um dos lados. Quais as medidas dos lados desse cercado retangular?

Atividade 4: Para presentear os colegas no natal, Ana comprou alguns exemplares de um livro por R$ 540,00. Por ter obtido um desconto de R$ 15,00 no preço de cada exemplar do livro, ela conseguiu comprar 3 exemplares a mais do que previra originalmente. Com o desconto concedido, quantos exemplares desse livro Ana comprou?

Atividade 5: Quero fazer um cercado retangular com 22 m² de área. O material que possuo dá para erguer 18 m de cerca. Que medidas devem ter os lados do meu cercado retangular?

Atividade 6: A equação 2 9 0x x c− + = tem raízes inteiras. Qual é o maior valor possível para c?

Atividade 7: Para quais valores de m e n as equações 2 0x mx m n− + − = e ()23 6 1 0x m x n− + + − = têm as mesmas raízes?

Atividade 8: Duas torneiras juntas enchem um tanque em 12 horas. Uma delas sozinha levaria 10 horas mais do que a outra para enchê-lo. Quantas horas leva cada uma das torneiras para encher esse tanque?

Atividade 9: Dois empresários formam uma sociedade cujo capital é de 100 mil reais. Um deles trabalha na empresa três dias por semana e o outro 2. Após um certo tempo, vendem o negócio e cada um recebe 99 mil reais. Qual foi a contribuição de cada um para formar a sociedade?

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Atividade 10: Três homens, A, B e C, trabalhando juntos, realizam uma tarefa em x horas. Se trabalhassem sozinhos, A executaria a tarefa em x + 1 horas; B, em x + 6 horas; C, em 2x horas. Calcule x.

Atividade 11: Resolva a equação 4 215 16 0x x+ − = .

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OFICINA 6

PROGRESSÕES ARITMÉTICAS EOFICINA 6

Meta:– Introduzir as noções de Progressões Aritméticas e Geométricas.

Objetivos: Ao término dessa oficina o aluno deverá ser capaz de resolver problemas envolvendo:

• o termo geral e/ou a soma dos n primeiros termos de uma PA ou de uma PG.

• a soma de infinitos termos em PG.

Pré-requisito:– Operações com números inteiros e frações.

PROGRESSÕES GEOMÉTRICAS

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Resolva as atividades propostas abaixo.

Atividade 1: Em uma fila, a trigésima segunda pessoa ocupa o lugar central. Qual o número de pessoas nessa fila?

Atividade 2: Verifique se as sequências dadas abaixo são progressões aritméticas ou geométricas, em caso positivo, identifique sua razão.

a) ( )2,5,8,11,15

b) 1 1 1 11, , , ,

2 4 8 16 − − −

c) ( )3,3,3,3,3

d) ( ), , 3a b a b a b− + +

Atividade 3: Uma fábrica de bijuterias, para aumentar suas vendas de pulseiras, propôs a seguinte meta mensal a seus vendedores: 1º dia útil: vender 1 pulseira, 2º dia útil: vender 3 pulseiras, 3º dia útil: vender 5 pulseiras e assim por diante, vendendo a cada dia útil duas pulseiras a mais do que havia vendido no dia útil anterior, até o 20º dia útil. O preço de um pulseira é R$ 15,00. No final do mês, os vendedores que cumprissem essa meta receberiam 20% de comissão sobre o total em reais de suas vendas.

a) Quantas pulseiras devem ser vendidas no 20º dia do mês?

b) O vendedor que cumprir a meta terá vendido quantas pulseiras no total?

c) Quanto ele receberá de comissão?

Atividade 4: O valor do faturamento anual de uma pequena empresa em 1997 foi de R$ 100.000,00, em 1998 foi de R$ 110.000,00, em 1999 foi de R$ 121.000,00 e, nos anos subsequentes, manteve um crescimento anual de 10% em seu faturamento, em relação ao faturamento do ano anterior.

a) Mantido esse crescimento, quanto essa empresa teria faturado em 1992?

b) Qual seria o faturamento total acumulado de 1997 a 1992?

Atividade 5: Calcule a geratriz da dízima periódica 3,177777...

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Atividade 6: Uma partícula parte do ponto ()0 0,0P , desloca-se até o ponto

()1 0,1P , segue para o ponto 2

1,1

2P

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1 3,

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sucessivamente vai descrevendo uma linha poligonal, no sentido horário, onde, o comprimento de cada lado dessa poligonal é igual à metade do lado anterior. Qual é o ponto de destino da partícula?

Atividade 7: Uma pessoa obesa, pesando em um certo momento 156 kg, recolhe-se a um spa onde se anunciam perdas de peso de até 2,5 kg por semana. Suponha que isso de fato ocorra.

a) Encontre uma fórmula que expresse o peso mínimo, P, que essa pessoa poderá atingir após n semanas.

b) Calcule o número mínimo de semanas completas que a pessoa deverá permanecer no SPA para sair de lá como menos de 120 kg de peso.

Atividade 8: A Festa Junina promovida por uma escola foi enfeitada com bandeirinhas coloridas. As bandeirinhas foram colocadas em linha reta na seguinte ordem: 1 bandeirinha azul, 1 vermelha, 1 verde, 2 azuis, 2 vermelhas, 2 verdes, 3 azuis, 3 vermelhas, 3 verdes, e assim por diante. Depois de colocadas 108 bandeirinhas, quantas bandeirinhas de cor vermelha foram colocadas?

Atividade 9: Uma bobina de papel tem raio interno medindo 5 cm, raio externo medindo 10 cm e a espessura do papel é 0,01 cm. Qual é o comprimento da bobina desenrolada?

Atividade 10: Os inteiros de 1 a 1000 são escritos ordenadamente em torno de um círculo. Partindo de 1, riscamos os números de 15 em 15, isto é, riscamos 1, 16, 31, .... O processo continua até se atingir um número já previamente riscado. Quantos números sobram sem serem riscados?

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OFICINA 7

MATEMÁTICA FINANCEIRAOFICINA 7

Meta:– Introduzir os temas de Matemática Financeira.

Objetivos: Ao término dessa oficina o aluno deverá ser capaz de: • reconhecer o que são juros, capital e montante.• resolver problemas envolvendo taxa de juros.• avaliar diversas opções de financiamento.

Pré-requisito:– Potenciação e PG.

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Resolva as atividades propostas abaixo.

Atividade 1: Flávia aplicou R$ 500,00 a uma taxa de juros de 10% ao mês. Qual será o montante de Flávia três meses depois?

Atividade 2: André tomou um empréstimo de R$ 600,00 a juros mensais de 5%. Dois meses após, André pagou R$ 300,00 e, um mês depois desse pagamento, liquidou seu débito. Qual o valor desse último pagamento?

Atividade 3: Silvio tem duas opções de pagamento na compra de uma guitarra: três prestações mensais iguais, cada uma no valor de R$ 160,00, ou sete prestações mensais de R$ 70,00 cada. Em ambos os casos, a primeira prestação é paga no ato da compra. Se o dinheiro vale 2% ao mês para Silvio, qual das duas opções é economicamente a mais vantajosa?

Atividade 4: Pedro comprou um rádio, pagando R$ 180,00, um mês após a compra e R$ 200,00, dois meses após a compra. Se os juros são de 25% sobre o saldo devedor, qual é o preço do rádio à vista?

Atividade 5: Ângela tomou um empréstimo de R$ 400,00, por dez meses. Os juros foram de 3% ao mês durante os quatro primeiros meses, de 5% ao mês durante os cinco meses seguintes e de 9% ao mês no último mês. Qual o montante pago?

Atividade 6: Uma lanterna de Gol, original, custa R$ 280,00 e tem vida útil de 5 anos. Uma lanterna alternativa custa R$ 70,00 e tem vida útil de 1 ano. Gilmar precisa trocar a lanterna de seu Gol. Considerando que o dinheiro vale 12% ao ano, do ponto de vista econômico, qual lanterna ele deve preferir?

Atividade 7: Uma geladeira custa R$ 1.000,00 à vista e pode ser paga em três prestações iguais. Se são cobrados juros de 6% ao mês sobre o saldo devedor, determine o valor da prestação, supondo que a primeira prestação é paga:

a) no ato da compra;

b) um mês após a compra;

c) dois meses após a compra.

Atividade 8: Uma conta de R$ 700,00 vencia no dia 25 de outubro de 1996 e foi paga em 5 de novembro de 1996. Quanto se pagou de juros, se os juros de mora são de 12% ao mês?

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Atividade 9: As cadernetas de poupança renderam 100% em 10 anos e, a inflação nesse período foi de 72%. Qual foi a rentabilidade real da cardeneta de poupança nesse período?

Atividade 10: Um equipamento pode ser alugado por R$ 75,00 mensais ou comprado por R$ 2.000,00. A vida útil do equipamento é de 30 meses e o valor residual ao fim desse período é de R$ 300,00. Se o equipamento for comprado, há um custo mensal de R$ 5,00 de manutenção. Considerando o valor do dinheiro de 1% ao mês, qual deve ser a decisão: comprar ou alugar?

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OFICINA 8

SEMELHANÇA DE TRIÂNGULOS EOFICINA 8

Meta:– Introduzir a noção de semelhança de triângulos e o Teorema de Tales.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de resolver

problemas envolvendo: • o Teorema de Tales.• a semelhança de triângulos.

Pré-requisito:– Grandezas proporcionais.

O TEOREMA DE TALES

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Resolva as atividades propostas abaixo.

Atividade 1: Este mapa mostra quatro estradas paralelas que são cortadas por três vias transversais. Algumas das distâncias entre os cruzamentos dessas vias e estradas estão indicadas no mapa (em km), mas as outras precisam ser calculadas. Calcule as distâncias x, y e z.

Atividade 2: Consideremos um triângulo ABC de lados 10BC = cm. Seja um segmento CD interno ao triângulo tal que D seja um ponto do lado AB. Sabendo que 4BD = cm e os ângulos BACS e BCDS são congruentes, determine a medida de AD.

Atividade 3: Um terreno, em forma de triângulo, está repartido em dois lotes, por meio de um muro paralelo a um dos lados do terreno, conforme indicado na figura abaixo. Qual a extensão desse muro?

Atividade 4: Para a realização de uma experiência, uma rampa reta e plana de 2m de comprimento, de plástico transparente, foi colocada sobre um terreno plano e horizontal. Quando os raios de sol eram perpendiculares ao terreno, fez-se rolar uma bola desde o ponto mais alto da rampa até o solo, observando-se que a sombra da bola sobre o terreno percorreu uma distância de 1,6 m. Que distância percorreu essa sombra, quando a bola se deslocou 50 cm sobre a rampa?

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Atividade 5: Para medir a altura de uma árvore, foi usada uma vassoura de 1,5 m, verificando-se que, no momento em que ambas estavam em posição vertical em relação ao terreno, a vassoura projetava uma sombra de 2 m e a árvore, de 16 m. Qual a altura da árvore?

Atividade 6: Os lados de um triângulo medem 8 cm, 10 cm e 12 cm. De quanto precisamos prolongar o menor lado para que ele encontre a bissetriz do ângulo externo oposto a este lado?

Atividade 7: No triângulo ABC da figura abaixo, AS é bissetriz interna do ângulo A e AP é bissetriz externa. Calcule a medida do segmento SP (todas as medidas estão dadas em cm).

Atividade 8: Na figura abaixo, o quadrado está inscrito no triângulo ABC. Qual a medida do lado desse quadrado?

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Atividade 9: Determine o valor de x no triângulo abaixo.

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OFICINA 9

ALGUMAS CONSEQUÊNCIAS OFICINA 9

Meta:– Introduzir alguns resultados decorrentes do axioma das paralelas.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de resolver

problemas envolvendo: • a soma dos ângulos internos de um triângulo.• o Teorema de Pitágoras.• relações métricas num triângulo retângulo.

Pré-requisito:– semelhança de triângulos

DO AXIOMA DAS PARALELAS

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Resolva as atividades propostas abaixo.

Atividade 1: O triângulo retângulo ABC é retângulo em B. Qual a medida h da altura relativa à hipotenusa?

Atividade 2: No trapézio dado na figura abaixo, qual é a medida da base maior?

Atividade 3: Três lotes quadrados delimitam um lago em forma de um triângulo retângulo, conforme indicado na figura abaixo.

Sabe-se que as medidas das áreas desses três lotes somam 800 m². Qual a medida do maior lado do lago?

Atividade 4: Um motorista, partindo de um ponto A, corre 10 km em direção ao sul. Depois, 6 km em direção a leste e finalmente, 2 km em direção ao norte, parando em um ponto B. Em cada etapa, ele sempre corre em linha reta. Qual a distância do ponto B ao ponto A?

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Atividade 5: O ponto P é interior ao retângulo ABCD e tal que 3PA = cm, 4PB = cm e 5PC = cm. Qual a medida de PD?

Atividade 6: Dado o triângulo ABC, constrói-se a poligonal 1 1 2 2 3 3L BCBC B C B C=

Qual o comprimento da poligonal L?

n

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Atividade 7: De um ponto A exterior a uma circunferência de raio 6 cm conduz-se uma tangente AT à mesma, medindo 8 cm. Calcule a distância de A ao centro da circunferência.

Atividade 8: Na figura abaixo as quatro circunferências tangenciam-se mutuamente e O é o centro da maior, cujo raio mede 30 cm. Calcule o raio da circunferência menor.

Atividade 9: Uma escada de 5,50 m de comprimento está apoiada em uma parede, sendo que seu pé está distante 1,50 m dela. Um pintor quer que a extremidade superior da escada alcance 30 cm mais alto. Que distância ele precisa deslocar o pé da escada em direção à parede?

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Atividade 10: Um helicóptero sai de um ponto P do solo e faz os seguintes movimentos sucessivos: 500m verticalmente para cima, 900m horizontalmente na direção norte, 200m verticalmente para cima, 700m horizontalmente na direção oeste e 100m verticalmente para baixo, pousando no ponto M de uma montanha próxima. Determine um valor aproximado para a distância entre os pontos P e M.

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OFICINA 10

FUNÇÃO DO 1º GRAUOFICINA 10

Meta:– Introduzir função do 1º grau.

Objetivos:– Ao término dessa oficina o aluno deverá ser capaz de:

• reconhecer uma função do 1º grau, seja por sua expressão analítica, seja por seu gráfico.

• resolver problemas que envolvam inequação do 1º grau.

• resolver problemas que envolvam função do 1º grau.

Pré-requisito:– Equação do 1º grau.

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Resolva as atividades propostas abaixo.

Atividade 1: Resolva a inequação 2

11

xx

≥+

.

Atividade 2: Determine a expressão analítica de cada uma das funções cujos gráficos encontram-se representados no plano cartesiano abaixo.

Atividade 3: O custo C, em reais, para José produzir picolés é dado por ( ) 300 1,2C x x= + , onde x representa o número de unidades produzidas. A receita,

em reais, que ele obtém com a venda desses picolés, em função do número x de unidades vendidas, é dada por ( ) 2R x x= . Considere que José vende tudo que produz. Quantos picolés devem ser vendidos para se obter um lucro de R$ 1300,00?

Atividade 4: Resolva a inequação 1 1

2 1 1x x<

+ −.

Atividade 5: Na cidade de João e Maria, haverá shows em uma boate. Pensando em todos, a boate propôs pacotes para que os fregueses escolhessem o que seria melhor para si.

Pacote 1: taxa de R$ 40,00 por show.Pacote 2: taxa de R$ 80,00 mais R$ 10,00 por show.Pacote 3: taxa de R$ 60,00 para 4 shows e R$ 15,00 por cada show a mais.

João assistirá a 7 shows e Maria, a 4. Quais são as melhores opções para João e Maria?

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Atividade 6: Um experimento consiste em colocar certa quantidade de bolas de vidro idênticas em um copo com água e medir o nível da água nesse copo. O quadro a seguir mostra alguns resultados do experimento realizado.

Número x de bolas Nível y da água

5 6,35 cm

10 6,70 cm

15 7,05 cm

Qual é a expressão da função que fornece a grandeza y em função da grandeza x?

Atividade 7: O cálculo do Imposto de Renda na Fonte (IRF) é feito de acordo com a tabela seguinte (fevereiro/2008)

Base de Cálculo Alíquota (%) Parcela a deduzir

Até R$ 1.372,81 Isento ----

De R$ 1.372,81 até R$ 2.743,25 15% R$ 205,92

Acima de R$ 2.743,25 27,5% R$ 548,82

Baseado na tabela acima, construa o gráfico do imposto a pagar em função do rendimento e dê a expressão analítica da função imposto a pagar.

Atividade 8: Um veículo de transporte de passageiros tem seu valor comercial depreciado linearmente, isto é, seu valor comercial sobre desvalorização constante por ano. Veja o gráfico abaixo.

Esse veículo foi vendido pelo seu primeiro dono, após 5 anos de uso, por R$ 24.000,00. Sabendo-se que o valor comercial do veículo atinge seu valor mínimo após 20 anos de uso e que esse valor mínimo corresponde a 20% do valor que tinha quando era novo, então qual é esse valor mínimo?

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Atividade 9: Os gráficos das funções de 1º grau cujas expressões analíticas são dadas por 5y x= − + e 3y x= − se interceptam no ponto A. Os pontos onde essas retas interceptam o eixo das abscissas são B e C. Qual a medida da área do triângulo ABC?

Atividade 10: Um supermercado está fazendo uma promoção na venda de alcatra: um desconto de 10% é dado nos quilos que excederem a 3. Sabendo que o preço do quilo de alcatra é de R$ 10,00, pede-se:

a) o gráfico do total pago em função da quantidade comprada.

b) o gráfico do preço médio por quilo em função da quantidade comprada.

c) a determinação de quantos quilos foram comprados por um consumidor que pagou R$ 45,00 na compra de alcatra.

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RESPOSTAS

RESPOSTAS

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Oficina 1: Resolução de ProblemasAtividade Resposta

01 5 m

02 R$ 30,00

03 55km/h

04 3 e 2. R$ 5,11

05 1024

06 312

07Filhos I e II: 3 cheios + 1 metade + 3 vazios

Filho III: 1 cheio + 5 metade + 1 vazio

08 11,2 m

09 Cavalo: 5 Mula: 7

10 22

Oficina 2: ProporcionalidadeAtividade Resposta

01 28 dias

02 A de 900g

03 62,5 m

04 2h30min

05 57 dias

06 3h e 45 min

07 21 dias

08 1720 kg

09 2h do dia seguinte

10 a) 2,45 litros b) em 21 dias ou menos

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Oficina 3: Equação do 1º grauAtividade Resposta

01 61 anos

02 a) 5/2 b) IR c) ∅

03 19

04A abaixo de 500 km

B acima de 500 km

05 R$ 32,00

06 2x = e 3y =

07 30 anos

08 166 convites

09 R$ 200.000,00

10 2,5 kg e 2 kg

Oficina 4: PorcentagemAtividade Resposta

01 R$ 1.015,28

02 R$ 9,00

03 45%

04 25%

05 15,5% indiferente

06 150 mulheres

07 30 litros

08 20 kg

09 23,5%

10 a) R$ 2.610,00 b) R$ 2.900

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Oficina 5: Equação do 2º grauAtividade Resposta

01 26 m

02 a) {}2,2− b) {}1,4− c) {}3,5− d) {}1, 1 2− −

03 2m ×16m ou 8m ×4m

04 12 exemplares

05 Não existe tal retângulo

06 20

07 4m = e 3n =

08 20h e 30h

09 55 mil e 45 mil

1023

11 1x = − e 1x =

Oficina 6: Progressões Aritméticas e Progressões GeométricasAtividade Resposta

01 63 pessoas

02a) não é PA, PG, 1,6q = b) não é PA, PG, 0,5q = −

c) PA, 0r = e PG 1q = d) PA, 2r b= , não é PG

03 a) 39 b) 400 c)R$1200,00

04 a) R$ 161.051,00 b) R$ 771.561,00

0528690

062 4

,5 5

07 a) ()2,5 158,5P n n= − + b) 16 semanas

08 36

09 Aproximadamente 236m

10 800

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Oficina 7: Matemática FinanceiraAtividade Resposta

01 R$ 665,50

02 R$ 379,58

03 3× R$ 160,00

04 R$ 272,00

05 R$ 626,30

06 A original

07 a) R$ 352,93 b) R$ 374,11 c) R$ 396,56

08 R$ 30,80

09 16,28%

10 Comprar

Oficina 8: Semelhança de triângulos e Teorema de TalesAtividade Resposta

01 10x = km, 30y = km, 22,5z = km

02 21 cm

03 15 m

04 40 cm

05 12 m

06 40 cm

07 40 cm

08 4 cm

09 14

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Oficina 9: Algumas consequências do Axioma da ParalelaAtividade Resposta

01 2,4 cm

02 36 cm

03 20 m

04 10 km

05 3 2

06 2p

07 10 cm

08 10 cm

09 Não é possível

10 1300 m

Oficina 10: Função do 1º grauAtividade Resposta

01 1x < − ou 1x ≥

02

: 3

3:

51

23

u y

r y x

s x

= −

=

= − +

03 R$ 2000.00

04 1 2x < − ou 0 1x≤ <

05 Pacote 3

06 0,07 6y x= +

07

08 R$ 6.000,00

09 1 unidade de área

10 4,666kg