Mathematics Lower Secondary Teacher Guide
MathematicsLower SecondaryTeacher GuidePapua New GuineaDepartment of Education
MathematicsIssued free to schools by the Department of EducationPublished in 2006 by the Department of Education, Papua New Guinea Copyright 2006, Department of Education, Papua New GuineaAll rights reserved. No part of this publication may be reproduced, stored in aretrieval system or transmitted by any form or by any means electronic,mechanical, photocopying, recording or otherwise without the prior writtenpermission of the publisher.ISBN 9980–935–68–5AcknowledgmentsThe Lower Secondary Mathematics Teacher Guide was written, edited andformatted by the Curriculum Development Division of the Department ofEducation. The development of the teacher guide was coordinated by BettyJoku Pulpulis.Teachers, inspectors, tertiary educators, community members,representatives from non-government organisations and the MathematicsAdvisory Committee have assisted in the development of this teacher guidethrough meetings, workshops and consultations.This document was developed with the support of the AustralianGovernment through the Curriculum Reform Implementation Project.ii
Lower Secondary Teacher GuideContentsSecretary’s message. ivIntroduction . 5Teaching and learning. 6Assessing Mathematics . 16Planning and programming units . 23Grade 9 units. 27Grade 10 units. 69Recording and reporting. 97Resources . 101References. 105Glossaries . 107Appendix . 111iii
MathematicsSecretary’s messageThis teacher guide is to be used by teachers when implementing the LowerSecondary Mathematics Syllabus (Grades 9 and 10) throughout Papua NewGuinea. This teacher guide has lots of practical suggestions for hands onactivities that students can do in order to learn MathematicsThe lower secondary Mathematics syllabus identifies the unit learningoutcomes and assessment requirements. The lower secondary Mathematicsteacher guide gives more information about what to teach and assess anddescribes ways of implementing the syllabus.There are many ideas on different teaching and learning strategies that canbe used to make mathematics lessons more interesting. The teacher mustprovide a classroom environment for Mathematics that encourages creativityand enjoyment. When planning, teachers must relate mathematical activitiesand problems to relevant, real life situations. Teaching using meaningfulcontexts and ensuring students participate in hands-on experiences assistsstudents to understand mathematical concepts.Teachers are encouraged, where appropriate to integrate Mathematics skillsand content across the whole curriculum.I commend and approve this Mathematics teacher guide for use in allschools with Grades 9 and 10 students throughout Papua New Guinea.DR JOSEPH PAGELIOSecretary for Educationiv
Lower Secondary Teacher GuideIntroductionThe purpose of this teacher guide is to help you to implement theMathematics syllabus. It is designed to stimulate you to create exciting andmeaningful teaching programs and lessons by enabling you to chooserelevant and purposeful activities and teaching strategies. It will encourageyou to research and look for new and challenging ways of facilitatingstudents’ learning.The teacher guide and the syllabus must be used side by side. The syllabusstates the learning unit learning outcomes for the subject and each unit, andoutlines the content and skills that students will learn, and the assessmentrequirements.The teacher guide provides direction for you in using the outcomes approachin your classroom using a step by step approach. Although the syllabusprovides the assessment tasks at the end of each unit, the outcomesapproach requires you to consider the assessment requirements early inyour planning. This is reflected in the teacher guide.This teacher guide provides examples of teaching and learning strategies forMathematics, sample programs for each unit, elaboration of suggestedactivities and content, detailed information on how to mark assessment tasksand the resources needed to teach Mathematics. The section on recordingand reporting shows you how to record students’ marks and how to reportagainst the broad learning outcomes.5
MathematicsTeaching and learningHow students learnWhat I hear I forgetWhat I hear and see I remember a littleWhat I hear, see and discuss I begin to understandWhat I hear, see, discuss and do, I acquire knowledge and skillWhat I teach to another, I master.(Active Learning Credo statement by Silberman, 1996)In support of this are the findings that we remember: 20% of what we hear40% of what we see90% of what we see, hear, say and do or what we discover for ourselves.A student-centred approach to learningDifferent students learn in different ways. Some students learn best bywriting, others by talking and discussing, others by reading and others bylistening. Most students learn by using a combination of these. All studentslearn skills through practicing and repetition. You need to use a variety ofteaching strategies to cater for the different ways your students learn.Teaching and learning strategiesTo assist and encourage students to learn, you perform certain tasks. Theseare referred to as teaching strategies. You need to engage students directlyin learning but there are times when you have to take charge of the learningin the class and teach particular concepts or ideas.To have common understanding of the different teaching and learningstrategies, there should be collaboration among teachers. This is to ensurethat teachers are using the strategies in a similar way. This is collaborativeplanning.Teaching strategies include: 6group workrole playskills practiceresearch/inquiryclass discussionsproblem-solving activitiesteacher talk, instructions, explanations, lectures or reading alouddirected question and answer sessions
Lower Secondary Teacher Guide audio-visual presentationstext books or worksheetsdirected assignmentsdemonstration and modellingguest speakersfield workclassroom displays.Using groups as a teaching and learning strategyUsing groups is an important strategy in Mathematics as students learn fromeach other, not just from the teacher. Group work encourages students toparticipate in achieving a shared goal and collaborative learning.In deciding whether to use groups or not, you need to consider: your intended outcomesthe extent to which the unit learning outcomes can be achieved by agroupthe lesson contentthe time allocated for the completion of the taskthe classroom settingavailable materials and resourcesthe structure of the group based on gender, ability, cultural backgroundand student preferences.Groups work well when: the group decides upon their goal, timelines and tasksstudents realise that success depends on the achievement of the wholegroup, not individualsthe task is broken into subtasks which must be finished to successfullycomplete the overall taskeveryone has a role to play, e.g. group projects or investigationsmembership of small groups is changed regularly to provide a variety oflearning experiences for all students.Strategies for organising and managing groups: mixed-ability groups the more able learners in the group can help theothers to master the work so that you need not teach some parts same-ability groups you can leave the groups of faster learners to get onwith the work on their own. She/he can give extra help to individuallearners in the slower groups using group leaders you can appoint faster, more able learners as groupleaders who can help slower learners.7
MathematicsDeveloping skillsPrinciples and proceduresStudents need to develop skills to help them learn. Skills developmentshould happen as a part of a student’s learning experience and the learningand practicing of skills needs to occur in the context of the units beingtaught.Skills learning tends to be most effective when: students go from the known to the unknownstudents understand why it is necessary to gain mastery of specific skillsskills are developed sequentially at increasing levels of difficultystudents identify the components of the skillthe whole skill and the components of the skill are demonstratedthere are frequent opportunities for practice and immediate feedbackthe skills being taught are varied in terms of amount and type, accordingto the needs of studentsthe skill is used in a range of contexts.To teach skills effectively you need to include learning activities that spanthe range from teacher-directed to student-centred learning, use groups ofdifferent sizes ranging from the whole class to small groups and use a rangeof teaching strategies which use higher order skills as your students’progress.Bloom’s taxonomy of skillsHigh levelBloom's Taxonomy is a way to classifyskills, activities or questions as theyprogress in difficulty. The lower levelsrequire less in the way of thinking skills.As you move up the hierarchy, theactivities require higher level ionlevelLow luatejudge
Lower Secondary Teacher GuideMathematical skillsApplying Mathematics usually involves one or more of the following skills: Gathering, recording, reading and interpreting dataUsing calculation tools (calculators, computers etc)Using rules and mation and problem-solving are probably the hardest skills to learn.Students’ estimation skills will improve with practice. When estimating followthese steps: Make a first guess or estimateDiscuss the estimate and improve it if possibleCalculate the actual and compare it with the estimateUse these comparisons to help with future estimates.Language skills for MathematicsStudents need to learn how to speak and listen, read and write, view andobserve. Students learn language skills through, for example: discussionsoral and written reportsinterviewing opportunities.Activities can help with regard to language skills and you should provideopportunities for your students to practice whenever you can. Studentsshould be encouraged to use new terms or concepts learnt in different units.Providing opportunities for students to listen is very important. Guestspeakers, oral presentations, group discussions, CD’s, tapes, radio andtelevision are listening resources. When students have listening experiencesas a regular part of classroom activities, their ability to listen and theircomprehension will improve.Place of vernacular in lower secondaryMaintenance of the student’s language is something that continues at lowersecondary as stated in the Department of Education’s Language policy in allschools. At times it will be appropriate to use vernacular, Motu or Tok Pisinto explain concepts or ideas. Vernacular can be used to describe andillustrate those things that do not have English translations. There will alsobe opportunities to use exercises which target vernacular development suchas from English to the vernacular Motu or Tok Pisin. For example, it wouldbe appropriate to use the vernacular when using surveys or findinginformation from the community.9
MathematicsWriting skillsStudents must be able to choose the right word to get the message acrossand be able to put words together in a way that makes sense to the reader.The ability to write well and use appropriate Mathematical terms andvocabulary takes a lot of practice and writing skills and techniques should beemphasised in Mathematics.Thinking and questioning skillsMathematics assists students to analyse and think critically aboutinformation they come across. By processing information rather than rotelearning, students are more likely to understand and retain what they havelearnt. Students must be involved in the process of thinking instead of simplyaccepting the end products of someone else’s thoughts. The ability to thinkcritically can be taught effectivelyBy asking the types of questions listed below: what do you notice/see/find?what difference do you .?what similarities do you .?which ones belong together? why?why don’t these belong to this group .?what could have happened if .?what would . be like if .?how would you .?what explanation would you give for .?is this always so?does evidence of . change the original explanation?how can this be tested/checked?suppose .what would happen?what makes you think this would happen?what would be needed for that to happen?is there a different explanation?if happened, what would happen next?Measuring skillsMeasuring is often done using special instruments or maps, but it can alsobe done with simple items such as a ruler, tape measure, trundle wheel,cubes, and measuring cylinders. Students can measure length, height,weight, speed, volume, capacity. Estimation of measurement is also a skillthat must be developed in the students. Lots of opportunities should beprovided for students to practice estimation and hands on experiences usingmeasurement.10
Lower Secondary Teacher GuideTeaching and learning strategies for MathematicsHere are teaching and learning strategies which can be used to makelearning more meaningful and interesting in Mathematics. You should varyyour lessons by using different teaching strategies, making sure that theones you use for the lessons are suitable for your lesson outcomes. Many ofthese strategies work together, for example developing a relationship chartbetween the different quadrilaterals helps students to understand visually thesimilarities and differences.BrainstormingThis is a technique in which a class or group meets in order to record all theinformation already known on a topic, to develop new ideas or to stimulatecreative thinking. Participants let the ideas come into their heads, write themdown, sort them and decide which require further research. Brainstorming isa useful way of determining prior knowledge of a topic. It would beappropriate to use when introducing a unit or topic or when summarising theunit or topic.Classroom displaysA classroom display provides a way of focusing on the current unit. Itstimulates learning, provides a record of learning as well as encouragingstudents to interact and to respond to learning.ChartsHelping students use charts, that is, to organise information in variousgroupings under different headings, is valuable. Charts (for example,properties of different shapes and solids) are a powerful teaching aid and ofconsiderable help in getting students to think about data.DiagramsDiagrams are employed by teachers in a variety of situations. They may beused to illustrate properties of lines and shapes. They can show somethingcomplex like a scale drawing. The best diagrams are clear, with all thenecessary details, and labels to identify features and explain processes.Flow chartsA flow chart is a diagram showing a series of step-by-step operations whichmake up a particular process. The main elements of the process are shownin picture form and are linked by arrows to indicate how one operation leadsto the next. A flow chart can, for example be used to show stages in the flowof money or probability.11
MathematicsCultural activitiesThrough participation in cultural mathematical activities, students areexposed to a variety of activities such as patterns in weaving, measurementin their traditional society, their traditional counting systems that give theminsight into their own culture or that of others.Decision-makingDecision-making is the process of choosing from two or more alternatives.Part of the process is the analysis and evaluation of the possible outcomesof the decision. Decision making is important when choosing whichoperations or formula to use to solve problems.EvaluationEvaluation involves weighing options, consequences and evidence indecision making contexts in order to make decisions and take action ineffective ways. The evaluation process often requires us to make decisionsbetween values which are in apparent or real conflict.Guest speaker or visitorA guest speaker or visitor is a person who is invited to share his/herknowledge and skills with the students. This may be a teacher from anotherclass, a parent, a member of the local community or a representative from agroup, organisation or institution. It would be appropriate to invite a guestspeaker from a small business in the community to speak to students aboutmoney matters in Unit 10.1.InterviewsAn interview involves asking questions in order to find out more informationabout a subject. In this way, students can learn about things and peoples’opinions first hand. There are usually many people with special knowledgeabout a topic. Students can invite them to the classroom. To conduct aninterview successfully students need to: 12prepare their questions beforehandmake sure questions are simple, relevant, to the point and that theyrequire more than a single word answermake sure they tell the interviewee their purpose and thank them at theendlisten carefully to answerstake notes if possible.
Lower Secondary Teacher GuideLearning gamesLearning games are activities that involve students in simulated experiencesto develop concepts and understandings, record information or demonstrateknowledge and understandings. Learning games can be made by teachersor students. For example, in chance students can create a game of chanceand explain it to their friends.MappingIt is important for students to use mathematical techniques to interpret maps.Finding directions, computing distances, locating places, relative location;use of scale and symbols, exposure to different kinds of projections,comparisons between different kinds of maps and drawing inferences frommaps are all important.Mind maps/concept mapsA mind or concept map is a way of recording information. It allows studentsto organise their ideas either as a class, small group or individually. A mindmap is often associated with brainstorming and is useful for drawingconnections between ideas and concepts, assisting in the further research ofa topic such as probability.ModelsModels provide demonstration of a concept in concrete form. Models caninclude items made from cardboard which are useful in the topics on solids.PresentationsPresentations are used to share information obtained through individual andgroup research and study. Presentations can be spoken, written ormultimedia. They give students experience in organising, planning andpresenting information and material to a particular audience and aretherefore valuable experiences for both the presenter and the audience.Problem solvingA particularly relevant teaching and learning strategy for mathematical topicsis problem solving. Students can be involved in identifying and workingtowards solutions to problems. The classroom, school grounds, communityand home all contain problems which are appropriate starting points forinvestigation by students.The purpose of learning through the application of problem solving skills is tolink conceptual understandings with practical experiences. It is important thatstudents be given opportunities to apply problem solving techniques to arange of issues.13
MathematicsThe teacher’s role is to: assist students identify problems that are relevant and solvableorganise learning that develops skills in problem solvingchoose learning activities which encourage responsible actions.ResearchOne of the best ways to learn about Mathematics is to think of the questionsyou want answered or what you want to know and inquire about the thingswhich interest you. This means doing your own research to find the answers.The same applies to your students.There are a number of steps involved in doing research and the best resultsare achieved if students do things in the right order and ask the followingquestions.Defining what do I want to find out?what is my purpose?what are the key words and ideas of this task?what do I need to do it?Locating where can I find the information I needwhat do I already know?what do I still need to find out?Selecting what information do I really need?what can I leave out?how relevant is the information I have found?how reliable is the information I have found?how will I record all the information?Organising how can I best use this information?do I need to use all the information?how can I best combine information from different sources?Presenting 14how can I present this information?with whom will I share this information?how does the audience affect my presentation?
Lower Secondary Teacher GuideAssessing what did I learn from all this?did I achieve what I set out to achieve?how did I go with each step of the information process?how did I go with presenting my information?where do I go from here?Task cardsTask cards are teacher-defined activities or pieces of research work,presented in a written form and assigned to individual students or groups.They are a method of directing student learning. You can devise task cardsto direct activities on an aspect of a topic.Using the internetThe Internet encompasses a number of facilities including the World WideWeb and electronic mail (e-mail. It is both a useful source of information onmany topics and a means of communicating with people in other places.Specific skills are required to access information on the Internet and moreimportantly to critically evaluate and validate such information.15
MathematicsAssessing MathematicsAssessment is an important part of teaching and learning. It is used to: evaluate and improve teaching and learningreport achievementprovide feedback to students on their progress.Assessment in Mathematics measures students’ achievements of the unitlearning outcomes described in the syllabus. It is an ongoing process ofidentifying, gathering and interpreting information about students’achievement and can be integrated into the students’ normal learningactivities.Assessment for learningAssessment for learning is often called formative assessment and isassessment that gathers data and evidence about student learning duringthe learning process. It enables you to see where students are havingproblems and to give immediate feedback which will help your students learnbetter. It also helps you plan your program to make student learning, andyour teaching more effective. Often it is informal and students can mark theirown work or that of their friends. An example is a quick class quiz to see ifstudents remember the important points of the previous lesson.Assessment of learningAssessment of learning is often called summative assessment. It is used toobtain evidence and data that shows how much learning has occurred,usually at the end of the term or unit. End of year examinations areexamples of summative assessment. It is usually done for formal recordingand reporting purposes.Assessing Mathematics unitsIn the Mathematics syllabus, the unit learning outcomes, which link to thebroad learning outcomes, are assessed through specified assessment tasksusing a range of assessment methods. Assessment criteria for each unitoutcome provide clear indications of how, and to what extent, theachievement of the unit learning outcomes may be demonstrated.Performance standards, marking guides and assessment criterion helpteachers with the marking process and ensure that assessment is consistentacross schools.Students must complete the assessment tasks for the unit. You will expandeach task and provide clear guidelines to students for how the task will becompleted and how the criteria will be applied.When you set a task make sure that: 16the requirements of the task are made as clear as possible to the student
Lower Secondary Teacher Guide the assessment criteria and performance standards are provided to thestudent so that they know what it is that they have to doany sources or stimulus material used are clear and appropriate to thetaskachievement is measured in terms of more than one outcomeinstructions are clear and concisethe language level is appropriate for the gradeit does not contain gender, cultural or any other biasmaterials and equipment needed are available to studentsadequate time is allowed for completion of the task.FeedbackWhen you assess the task, remember that feedback will help the studentunderstand why he/she received the result and how to do better next time.Feedback should be: constructive so that students feel encouraged and motivated to improvetimely so that students can use it for subsequent learningprompt so that students can remember what they did and thought at thetime focused on achievement, not effort the work should be assessed, notthe studentspecific to the unit learning outcomes so that assessment is clearlylinked to learning. Feedback can be: informal or indirect - such as verbal feedback in the classroom to thewhole class, or person to personformal or direct– in writing, such as checklists or written commentary toindividual student either in written or verbal formformative – given during the topic with the purpose of helping the studentknow how to improvesummative – given at the end of the unit with the purpose of letting thestudents know what they have achieved.TestsA test is a formal and structured assessment of student achievement andprogress which you administer to the class.Tests are an important aspect of the teaching and learning process if theyare integrated into the regular class routine and not treated merely as asummative strategy. They allow students to monitor their progress andprovide valuable information for you in planning further teaching and learningactivities.Tests assist student learning if they are clearly linked to the unit learningoutcomes. Evidence has shown that several short tests are more effective17
Mathematicsfor student progress than one long test. It is extremely important that testsare marked and that students are given feedback on their performance.There are many different types of tests. Tests should be designed to find outabout student knowledge of content and about the development of thinkingprocesses and skills. Open questions provide more detailed informationabout student skills than a question to which there is only one answer.Principles of designing classroom testsTests allow a wide variety of ways for students to demonstrate what theyknow and can do. Therefore: students need to understand the purpose and value of the testthe test must assess intended unit learning outcomesclear directions must be given for each section of the testthe questions should vary from simple to complexmarks should be awarded for each sectionthe question types (true/false, fill-in-the-blank, multiple choice, extendedresponse, short answer, matching) should be varied.Tests should: be easy to read (and have space between questions to facilitate readingand writing)reflect an appropriate reading levelinvolve a variety of tasksmake allowance for students with special needsgive students some choice in the questions they selectvary the levels of questions to include gathering, processing andapplying informationprovide sufficient time for all students to finish.Who assesses?Teacher assessmentAssessment is a continuous process. You should: 18always ask questions that are relevant to the outcomes and contentcheck understanding of the previous lesson at the beginning of the nextlesson through questions or a short quizconstantly mark/check the students’ written exercises, class tests,homework activitiesuse appropriate assessment methods to assess the tasks.
Lower Secondary Teacher GuideFrequency of assessmentYou should schedule the specified assessment tasks to fit in with theteaching of the content of the unit that is being assessed. Some assessmenttasks might be programmed to be undertaken in the first few weeks of theunit, others at the end of the term. You should take care not to overloadclasses with assessment tasks at the end of the term. You can also checkwith other subjects so students can do a task where certain parts can beassessed for the different subjects the student studies.Judging student performanceStudent achievement is recorded and reported against standards. You mustuse the performance standards provided in each unit of this teacher guidewhen making a decision about the achievement of your students in relationto the unit learning outcomes. The performance standards describe the levelat which the student has to be working to achieve a particular standard ormark.Students should always have access to a copy of the assessment criteriaand the performance standards so that they know what it is they have toknow and be able to do to get a good mark in a particular task. Theperformance standards will help you in your marking and will help thestudent improve their performance in the future. They are useful whenproviding feedback to students as they explain what it is the student needsto do to improve.ModerationTo ensure that you are interpreting the performance standards correctlywhen ass
Secondary Mathematics Syllabus (Grades 9 and 10) throughout Papua New Guinea. This teacher guide has lots of practical suggestions for hands on activities that students can do in order to learn Mathematics The lower secondary Mathematics syllabus identifies the unit learning outcomes and
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especially in lower secondary school (year 7 to year 9) and upper secondary school (year 10 to year 12). Secondary school dropout rate was 19.60 % in lower secondary school and 11.80% in upper secondary school in 2011 (Ministry of Education Youth and Sport, MoEYS,
(1-6), Lower secondary level (7-9) and upper secondary level (9-12). In primary level, students seek how to count and how to use numbers. In lower secondary level and upper secondary level, students study different parts of mathematics, like geometry, triangonomtry, algebra, function, integral and etc.
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7 Lower Secondary Science Textbook 1B (E/NA) New Marshall C 3.40 8 Lower Secondary Science Activity Book 1A (E/NA) New Marshall C 3.35 9 Lower Secondary Science Activity Book 1B (E/NA) New Marshall C 3.25 Total 13.65 10 Singapore A Journey Through Time 1299-1970s Secondary 1 New Star Pub 5.00
Asam folat dapat diperoleh dari daging, sayuran berwarna hijau, dan susu. Gizi buruk (malnutrisi) merupakan penyebab utamanya. Anemia jenis ini jugaberkaitan dengan pengerutan hati (sirosis). Sirosis hati menyebabkan cadangan asam folat di dalamnya menjadi sedikit sekali. Kekurangan asam folat juga dapat menyebabkan gangguan kepribadian dan hilangnya daya ingat. Gejala-gejalanya hampir sama .
