mathematics - arc · 2003-03-31 · philosophy continues that of the nsw mathematics k–6 and...

MathematicsYears 9–10

Syllabus

Advanced Course

Stage 5

# 9 6 1 9 9 M a t h A d v I n t r o 1 0 / 9 / 9 6 4 : 0 2 P M P a g e

http://www.boardofstudies.nsw.edu.au

© Board of Studies NSW 1996

Published by the Board of Studies NSWPO Box 460North Sydney NSW 2059Australia

Tel: (02) 9927 8111

ISBN 0 7310 7514 5

August 1996

96199


Contents

Introduction 5

NSW Mathematics Courses K–12 6

Rationale 7

The Three Courses 8

Aim 10

Objectives 10

Equity Principles and Issues 11

Solving Problems 14

Communication — The Role of Language 16

Collecting, Analysing and Organising Information 17

Using Technology 17

Working With Others and in Teams 18

Planning and Organising Activities 18

Teaching Strategies 19

Programming 19

Syllabus Structure 20

Summary of Years 9–10 Advanced Course — Core 23

Summary of Years 9–10 Advanced Course — Options 25

Outcomes 26

Assessment 37

Evaluation of School Programs 42


Advanced Course Content — Core 43

Geometry (G1 – G4) 45

Number (N1 – N6) 72

Measurement and Trigonometry (M1 – M3) 100

Chance and Data (CD1 – CD3) 120

Algebra and Coordinate Geometry (A1 – A5) 144

Mathematical Investigations 173

Advanced Course Content — Options 181

1. Fractals 183

2. Networks 191

3. Mathematics of Small Business 201

4. Practical Applications of Measurement 207

5. Further Geometry 214

6. Curve Sketching and Polynomials 231

7. Functions and Logarithms 241

8. Modelling 249

Mathematics Years 9–10 Syllabus — Advanced Course


Introduction

Mathematics is one of the eight Key Learning Areas that comprise the secondarycurriculum. This syllabus specifies the content of the Mathematics Key LearningArea for students in Stage 5 (usually Years 9 and 10) of their secondary education.

Students who have completed Stage 4 Mathematics are at various stages in thedevelopment of their mathematical knowledge, understanding and skills. Somestudents demonstrate a high degree of conceptual understanding, while otherstudents still need to practise their basic numeracy skills in a variety ofapplications. This syllabus provides the opportunity for students in Years 9 and 10to study one of three courses in Mathematics — Advanced, Intermediate orStandard. These courses show variation in mathematical abstraction, depth oftreatment and practicality. In this way the syllabus caters for a wide range ofstudents with different learning needs.

The curriculum in NSW requires all students to engage in substantial study ofMathematics each year from Kindergarten to Year 10. Mathematics is one of thefour Key Learning Areas in Years 7–10 that must be studied each year. CurriculumRequirements for NSW Schools (1990) states that 400 indicative hours of Mathematicsare to be completed from Year 7 to Year 10. This syllabus has been designed for aminimum of 200 indicative hours. However, it is more usual for schools to have agreater time allocation for Mathematics over Years 9 and 10 and requirements forgovernment schools mandate 500 hours of Mathematics over Years 7–10.

In each course there are two components:

• the core — this section is mandatory and is designed to be taught in a minimumtime of 160 indicative hours

• the options — option topics can be chosen to meet varying student needs andinterests. It is intended that students spend a minimum of 40 indicative hours onthe options.

This syllabus is designed for mathematics teaching and learning within the contextof mathematical problems that are meaningful and challenging to students. Thisphilosophy continues that of the NSW Mathematics K–6 and Mathematics 7–8syllabuses, and reflects the Mathematics Statement of Principles K–12, The Nature ofMathematics Learning and the Aims of Mathematics Education, which are detailed inthe support document accompanying this syllabus.

5

Introduction


NSW Mathematics Courses K–12

The diagram below summarises the Mathematics courses in NSW for Years K–12.

6


Mathematics K–6

MathematicsYears 7–8

Years 9–10Standard

Years 9–10Intermediate

Years 9–10Advanced

Years 11–12Mathematics in

Practice

Years 11–12Mathematics inSociety (2UG)

Years 11–12Mathematics

2 Unit

Years 11–12Mathematics

3 Unit

Year 124 Unit

Stage 6

Stages 1–3

Stage 4

Stage 5


Rationale

Mathematics involves the study of patterns and relationships and provides apowerful, precise and concise means of communication. Mathematics is a creativeactivity. It is more than a body of collected knowledge and skills. It requiresobservation, representation, investigation and comparison of patterns andrelationships in social and physical phenomena. At an everyday level it isconcerned with practical applications in many branches of human activity. At ahigher level it involves abstraction and generalisation. As such, it has been integralto most of the scientific and technological advances made in Australia and world-wide.

Mathematical demands on people have changed considerably over the past fewdecades. All people need to be numerate — that is, to be able to calculate, measureand estimate in a variety of situations. There is an increased dependence ontechnology, and the amount of information that is available has expanded rapidly.It is vital that Australia has a mathematically competent workforce. There is ademand for people to be innovative, to be able to solve mathematical problems,communicate and to make informed decisions after analysing data. Mathematicseducation provides many opportunities for students to develop these skills.

There is general recognition that the process of mathematical problem solving willprepare students more appropriately to function competently in society and that aproblem-solving approach actually aids mathematical learning. Mathematicalactivity in society frequently involves problem solving — whether the activity isrelated to everyday life or is more abstract in nature. By supporting a problem-solving approach, as in this syllabus, the mathematical education community isrecognising its responsibility to ensure that students are prepared to take theirplace as effective members of society who are able to solve the mathematicalproblems that arise.

The Mathematics 9–10 Syllabus aims to develop mathematical skills andconfidence in students appropriate to their level of development. It emphasises theability to investigate and reason logically, to solve non-routine problems, tocommunicate about and through mathematics, to connect ideas withinmathematics and to be motivated to learn more mathematics. It follows theMathematics K–6 (1989) and Mathematics 7–8 (1988) Syllabuses in presentingmathematics as a dynamic and process-oriented subject, as well as one that has animportant body of knowledge and skills.

These ideas are balanced within the syllabus, while the nature and needs of thestudent and the learning processes are taken into account. Problem solving and theapplications of mathematics in the world are key elements, as is student

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Rationale


communication. By talking to each other about mathematics, reflecting and writingabout mathematics, drawing diagrams and listening to the teacher and otherstudents discussing mathematics, the learning of mathematics is enhanced andstudents are motivated to investigate further mathematical problems.

This philosophy underpins the teaching of mathematics throughout Years K–12.Such an approach represents a shift in philosophy and a resulting change inpedagogy for mathematics since the previous Mathematics 9–10 Syllabus, writtenin the early 1980s.

The approach of this Mathematics 9–10 Syllabus reflects that expressed in thedocument A National Statement on Mathematics for Australian Schools (CurriculumCorporation for the AEC 1990). This statement gives guidelines for curriculumdevelopment in Mathematics across all States and recognises the need forimprovement and change in school mathematics:

We need to aim for improvement in both access and success inmathematics for all Australians. All Australians must leave school wellprepared to meet the demands of their future lives and with theknowledge and attitudes needed to become lifelong learners ofmathematics.

Through material in the core and options, this syllabus provides opportunities forthe solutions of relevant, non-routine problems to be integrated into the teachingand learning of mathematics.

The Three Courses

The Standard course combines a thematic and topical approach to encourage thedevelopment of basic mathematical skills. It is designed for students who needmore time to develop these skills for everyday life by practising these skills in avariety of realistic themes and topics. The mathematical content of the coursebuilds on skills and knowledge from the Mathematics 7–8 course and provides theopportunity for students to experience some of the applications of mathematics totheir lives.

The Intermediate course lies between the Advanced course and the Standardcourse and contains elements of both. The number of new concepts and level ofdifficulty is less than in the Advanced course. The Intermediate course is designedfor students who require more time than those doing the Advanced course todevelop their mathematical ideas, and for students who are still developing a moreabstract approach to mathematical thinking.

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The Advanced course is the most abstract of the three courses. It is designed forthose students who have achieved all, or the vast majority of, the outcomes of theprevious Mathematics syllabus (ie Mathematics 7–8). The Advanced course doesnot repeat material from this course since the assumption is that it has beencompleted. In some areas, material from the Mathematics 7–8 Syllabus isreviewed, particularly where it is then covered in greater depth and at a highercognitive level. The Advanced syllabus contains and extends the content of theIntermediate course, requiring students to develop their reasoning abilities to agreater extent than for the Intermediate course. The course emphasises algebraicprocesses, graphical techniques, interpretation, justification of solutions, advancedapplications and reasoning, which arise in more sophisticated problems fromrealistic applications.

There is a degree of commonality between courses, especially between theAdvanced and Intermediate courses and also between the Intermediate andStandard courses. There is flexibility for students to move between courses,especially during Year 9. The options are designed to provide the opportunity forstudents to proceed to different courses in Years 11–12.

The expected pathways through the Years 9 and 10 Mathematics courses to Years11 and 12 Mathematics are as follows:

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The Three Courses

Advanced Course

3/4 Unit

2 Unit

Mathematics in Society(2 Unit General)

Mathematics in Practice

IntermediateCourse

Standard Course


Aim

The Mathematics 9–10 Syllabus aims to promote students’ appreciation ofmathematics and develop their mathematical thinking, understanding, confidenceand competence in solving mathematical problems.

This aim is to be achieved through developing students’ capacities to:

• acquire the mathematical knowledge, operational facility, concepts, logicalreasoning, symbolic representation and terminology appropriate to their stageof mathematical development and in preparation for further study ofMathematics

• interpret, organise and analyse mathematical information and data

• apply mathematical knowledge and skills to creatively and effectively solveproblems in familiar and unfamiliar situations

• communicate mathematical information and data

• justify mathematical results and give proofs where appropriate, makingconnections between important mathematical ideas and concepts

• value mathematics as an important component of their lives.

Objectives

Students will develop:

• appreciation of mathematics as an essential and relevant part of life

• knowledge, understanding and skills in working mathematically

• knowledge, understanding and skills in Geometry

• knowledge, understanding and skills in Number

• knowledge, understanding and skills in Measurement and trigonometry

• knowledge, understanding and skills in Chance and data

• knowledge, understanding and skills in Algebra and coordinate geometry.

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Equity Principles and Issues

This syllabus and its accompanying support materials, assessment guidelines andexamination specifications for the School Certificate reflect the Board of Studies’Statement of Equity Principles, which relates two of the Board’s corporate objectives:

• to develop high quality courses and support materials for primary andsecondary education suited to the needs of the full range of students;

• to assess student achievement and award credentials of internationalstandards to meet the needs of the full range of students.

Statement of Equity Principles (1996), p 1.

The syllabus supports these objectives by recognising educational research, notonly in relation to the identification of groups that are disadvantaged in gainingaccess to the curriculum and participating fully in its aspects, but also in relation toeffective approaches to teaching and learning involving disadvantaged groups.Research suggests:

… the following groups are disadvantaged in gaining access to thecurriculum and participating fully in its aspects:

• students from low socioeconomic backgrounds

• Aboriginal and Torres Strait Islander students

• students learning English as a second language

• students of non-English speaking background

• students who have physical or intellectual disability.

In addition, both girls and boys are disadvantaged by various forms ofsex stereotyping.

Ibid, p 1.

It should be recognised that children from different cultural backgrounds bringdiverse mathematical experiences to the classroom. Aboriginal children, forexample, bring with them complex understandings of patterns, kinship and spatialconcepts. These different experiences and perspectives can contribute to a deeperunderstanding of the nature of mathematics. For example, many students ofmathematics would take the use of base 10 for granted as being almost a naturalway of doing mathematics. Students may not be aware that this is culturallyspecific and that different bases can be used in counting. Such a realisationprovides insight into the nature of mathematics.

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Aboriginal world-views emphasise an intelligent responsiveness to the environmentthat is characterised by cooperation and co-existence with nature. This cooperationextends to human relationships. Many Aboriginal people show a preference tolearn from each other in groups, using oral language. Individual competitiveness islikely to be at odds with their cultural backgrounds. Western notions of quantitiesand measurements, comparisons (more or less), number concepts, time andpositivistic thinking can be irrelevant and contrary to their established thoughtpatterns. Effort must be made to provide basic linkages between their world andthe social meanings of Western mathematical ideas, which are important for themto develop. (Adapted from Dawe, L, Teaching Secondary School Mathematics, p 243.)

Effective approaches to teaching and learning

The Aboriginal child brings to school a way of communicating that reflects thelanguage used at home. In most cases this is Aboriginal English. AboriginalEnglish is a dialect of standard English and the first (or home) language of manyAboriginal children. Aboriginal English differs not only in words and meanings,but grammatically and pragmatically (see Board of Studies document, AboriginalEnglish). Teachers need to be aware that mathematical language is often veryunfamiliar to Aboriginal children, as it is to many children from a variety of othercultural backgrounds.

Relating mathematics to the students’ lives, using materials as well as makingexplicit connections between the concrete and the abstract, will help the studentsto gain a firmer understanding of new words and their meanings, as well as theirassociated concepts. A variety of teaching methods, including group work, workingin pairs, working outdoors and working with materials, helps to create anenvironment conducive to learning.

The language of mathematics is often the same as everyday language. This can addto some children’s confusion. To avoid ambiguities, explicit teacher explanation isneeded if a word has more than one meaning. Consideration of these similaritiesand differences will help teachers to emphasise the acquisition and use ofmathematical terminology.

How the syllabus and support material address equity issues

Research suggests that equity of access for all groups is increased when strategiesnecessary for success are made explicit and students are able to develop anawareness and control of holistic processes, which enable them to effectivelysynthesise their learning.

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The Mathematics 9–10 syllabus and support documents address equity issues byproviding:

• a focus on the articulation of processes essential for success in mathematics

• a focus on the development of thinking skills through problem solving

• suggestions for a range of relevant resources which will complement andfacilitate good teaching practice

• a wide range of applications, suggested activities and sample questions whichemphasise the use of relevant problems in the learning process

• three courses in Stage 5 so that students’ mathematical needs can be moreappropriately met

• a number of option topics that provide additional flexibility for catering forstudent needs

• suggestions for teaching strategies which include group work, discussion andactive participation

• a range of suggested assessment methods and strategies through which toidentify students’ achievement within a range of modes.

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Solving Problems

A major aspect of mathematics is problem solving. Students learn through solvingproblems. A mathematics teacher should provide opportunities for students tosolve meaningful, non-routine and challenging problems as a significant aspect oftheir learning.

Problem solving promotes processes and skills such as communication, criticalreflection, creativity, analysis, organisation, experimentation, synthesis,generalisation and validation. In addition, teaching through problems that arerelevant to the students encourages improved attitudes to mathematics, and anappreciation of its importance to society. Problem solving should encouragestudents to be systematic when recording information and to persevere.

Four important elements of solving problems are detailed below.

1. Understanding the problem

Teachers can help students to understand problems by giving them practice in:

• text editing, including identification of redundant and irrelevant information

• identifying problems where insufficient information has been given

• restating a problem in a student’s own words

• explaining the meaning of a problem to others (peers and the teacher)

• discussing the meaning of the text of a problem and any ambiguities

• trying a problem and returning to the text a number of times to ensureappropriate interpretation.

Sample questions have been included in this syllabus that encourage students todiscuss and explain the meaning of particular problems, decide what furtherinformation may be needed and identify any redundant or irrelevant informationin a question. Often students need time to try to solve the problem and then rereadthe problem a number of times to ensure appropriate interpretation.

2. Planning a solution

Planning a solution involves categorising a problem and then knowing theappropriate procedures for that type of problem. Teachers can help students toplan solutions by:

• facilitating students’ schema acquisition. This can be helped by reducingstudents’ cognitive load through providing goal-free or open-ended questions,

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integrating text with diagrams and encouraging the study and development ofworked examples

• discussing plans for solving problems

• organising group activities in which students sequence plans for solvingproblems.

This syllabus provides a large number of sample questions that are open-ended.Such questions are also useful when assessing student achievement, since theyallow students to respond on a variety of levels.

Note: Students might not automatically be able to give full responses to open-endedquestions — this needs to be developed over time. Students will need to practisethese types of questions so that they can identify them and give the full range ofanswers or a generalised answer as appropriate. This development will be facilitatedby students working in groups, listening to and discussing the responses of others.

3. Implementing the plan to find the solution

Teachers can help students to improve their problem-solving skills and develop theability to work out a solution by:

• offering students experience with a variety of problems that require differentstrategies for solution (eg using a table, drawing a diagram, looking for patterns,working backwards, guessing and checking, simplifying the problem, breakingthe problem up into smaller parts)

• ensuring that students have a well-structured foundation of basic mathematicalideas on which to build their understanding

• facilitating the development of the necessary knowledge and skills to enablestudents to carry out their plan of solution

• encouraging students to recall any necessary formulae and be competent withroutine skills to ensure that they can carry out the solution phase of a problem.

While specific problem-solving strategies such as those above have been includedin the section A1 of this syllabus, it is intended that such strategies would beencouraged throughout this course. The development of real understanding, in-depth knowledge and competence is emphasised throughout this syllabus.

4. Looking back

Reflection on the problem solution is an important aspect of problem solving, andone which is often ignored. The recording of the problem solution is also a vitalstep, and one that students often find difficult. Both of these aspects can be aidedby:

15

Solving Problems


• discussing errors and a range of solution methods, where students need toidentify the most efficient method

• encouraging ‘internal talk’ throughout the problem, where students sit back, askthemselves how they are going and gain a holistic view of the problem

• ensuring that students write up their work carefully, including:– a statement of the problem in their own words– all the necessary working– a statement of what has been discovered– some discussion of the processes used– a list of any helpful ideas– a generalisation of the result if appropriate.

There are many suggestions throughout the syllabus for students to report on theprocess of their solution, discuss their solution and those of others, evaluatesolutions and make judgements by deciding on the ‘best’ or most efficient (orelegant) solution.

Communication — The Role of Language

The Mathematics 9–10 Syllabus makes a significant contribution towards thedevelopment of the Key Competency of Communicating Ideas and Information. Itfacilitates the development of communication by recognising the importance oflanguage in learning and focusing on numerical, algebraic and graphicalpresentation of information. This is particularly evident in the statisticalinvestigation for Chance and data.

Students’ command of language dramatically affects the quality of learning inschool mathematics classrooms. Students need to develop a deep understanding ofthe meaning of mathematical vocabulary, and facility in communicating theirunderstanding to others. This understanding will allow them to use themathematics terms meaningfully, both inside and outside school. Beyondmathematical vocabulary, unravelling semantic structure places significantdemands on students’ problem-solving skills. For example, for the problem: ‘There are twelve times as many sheep (s) as people (p); write this relationship insymbols’, many students will write ‘12s = p’. Students can lose the meaning of thewords because of the sentence structure. They need to focus on semantic structurerather than a key-words approach. This syllabus supports the teaching ofmathematics to link learners’ personal worlds with their formal mathematical skillsand their formal mathematical language.

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Research suggests that learners’ personal worlds are inherently influenced by theircultural, socioeconomic and/or geographic backgrounds. These factors need to beconsidered in determining the most appropriate means of developing mathematicallanguage concepts.

Collecting, Analysing and Organising Information

This syllabus explicitly addresses knowledge and skills that develop, and providestudents with opportunities to demonstrate, the Key Competency of collecting,analysing and organising information numerically and graphically. Students arerequired to formulate and refine key questions prior to investigation, to design andconduct an investigation with an understanding of issues involved in sampling, andto use this understanding to evaluate the information, its sources and the methodsused in the investigation.

Using Technology

Mathematics provides an opportunity for students to use materials and equipmentin a manner that constitutes a process and that reflects the ‘technology’ or ‘know-how’ of mathematics. It is important for students to determine the purpose of atechnology, to apply the technology, and to evaluate the effectiveness of theapplication. This ability depends not only upon the students learning when andhow to use technology, but also on their learning when the use of technology isinappropriate or even counter-productive.

The use of scientific calculators is mandatory — students must have regular accessto scientific calculators during this course. It is very important, however, thatstudents maintain and develop their mental arithmetic skills, rather than relying ontheir calculators for every calculation.

Other tools such as geometrical instruments and templates are also needed atdifferent times throughout the course. The use of graphics calculators andcomputers is optional, but is suggested in the Applications, suggested activities andsample questions sections to enhance the teaching and learning of mathematics.Some schools may not have access to these tools. It is important to recognise thatthis course can be taught successfully without the use of graphics calculators andcomputers, but that the appropriate use of such technology within this course willenhance students’ mathematical learning.

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Using Technology


Technology also has a role in assisting students with special education needs to gainaccess to the mathematics curriculum. A computer may assist students who have aphysical disability. For example, students who are unable to write may be able touse a wand and produce their own work on a computer.

Further advice on the use of technology in Stage 5 Mathematics is provided in thesupport document.

Working with Others and in Teams

Experience of working in groups can facilitate learning. Group work provides theopportunity for students to communicate mathematically with each other, to makeconjectures, to cooperate and to persist in solving problems and in investigations.This strategy can also promote and improve motivation, enjoyment and confidencein mathematics. Group work should be carefully managed — students need to bevery clear about their tasks and each member of a group should be givenresponsibility for an aspect or part of the task.

Experience of working in groups can not only facilitate learning but also providefoundation experience in the Key Competency of Working with Others and inTeams. Students may elect to develop this competency by working in a team onthe in-depth mathematical investigation or on shorter investigations. Such studentsdevelop their awareness that working with others requires them to establish groupgoals and consensus on individual roles and responsibilities. They recognise theimportance of taking responsibility for individual performance and groupperformance, and develop the ability to work within a given time frame. It isimportant that they focus on evaluating not only the product of the investigationbut also the process of group interaction involved in developing it.

Planning and Organising Activities

The mathematical investigation also allows students to develop the capacity to planand organise their own activities. This involves the ability to set goals, establishpriorities, implement a plan, manage resources and time, and monitor one’s ownperformance.

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Teaching Strategies

To allow students to achieve the outcomes of this course, a range of teaching strategiesmust be employed. If students are to improve their mathematical communication, forexample, they must have the opportunity to discuss interpretations, solutions,explanations etc with other students as well as their teacher. They should beencouraged to communicate not only in writing but orally, and to use diagrams aswell as numerical, algebraic and word statements in their explanations.

Students learn in a range of ways. Students can be mainly visual, auditory orkinesthetic learners, or employ a variety of senses when learning. The range oflearning styles is influenced by many factors, each of which needs to be consideredin determining the most appropriate teaching strategies. Research suggests thatcultural and social background have a significant impact on the way students bestlearn mathematics. These differences need to be recognised and a variety ofteaching strategies used so that all students have equal access to the developmentof mathematical knowledge and skills.

Learning can occur within a large group where the class is taught as a whole,within a small group where students interact with other members of the group, orat an individual level where a student interacts with the teacher or another student,or works independently. All arrangements have their place in the mathematicsclassroom.

Programming

There is no need for teachers to teach a whole strand at once. Rather, the contentof each of these courses, being spiral in nature, provides for the strands to berevisited and concepts developed further over time as students mature in theirunderstanding of mathematics. It is not intended that each strand occupy the sameamount of time. Within each strand, many concepts relate to aspects of otherstrands. This interrelatedness is fundamental to mathematics and should beincluded in students’ learning experiences. The Considerations at the beginning ofeach strand discuss these relationships and identify areas where connections shouldbe made. The incorporation of issues raised in these considerations encouragesstudents to view the course as a whole and helps them to appreciate theinterrelatedness of mathematics. The support document that accompanies thissyllabus has further advice on programming, along with some sample formats forprograms and sequences of teaching for each course.

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Programming


Syllabus Structure

The Mathematics Years 9–10 Syllabus — Advanced Course is divided into core andoptions as below:

The core is divided into six strands, each of which must be studied by all students.All the content of each strand is to be studied. These strands are:

GeometryNumberMeasurement and trigonometryChance and dataAlgebra and coordinate geometryMathematical investigations.

As well as the core, eight options are provided. Teachers must ensure that at least40 hours (indicative) are spent in the study of the option topics included here.

Each core strand and option is introduced by Considerations, which teachers shouldread before teaching sections of the strand. The considerations raise issues relatedto the teaching and learning of the concepts and skills in the strand or option. Theissues relate to:

• the syllabus aim and objectives

• the syllabus outcomes

• possibilities for integration with other strands and subjects

• language development

• assumed knowledge and skills from Stage 4

• other specific aspects of the syllabus.

Content

The content statements on the left-hand pages (core and options) describe in detailwhat students should know, understand and be able to do as a result of appropriateand relevant learning experiences facilitated by the teacher. These statements guideteachers on the extent and depth of treatment expected. The content provides thebasis for the achievement of syllabus outcomes and includes the skills that students

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Core (160 hours minimum) Options(40 hours minimum)


acquire as they undertake the learning experiences described. They are groupedunder subsections and have been arranged somewhat sequentially. This does notimply that teachers must follow this particular sequence when teaching eachsection.

Applications, suggested activities and sample questions

The applications, suggested activities and sample questions on the right-hand pagesare optional. They are suggestions for learning experiences and relevant problemsthat will aid students in their achievement of syllabus outcomes. They reflectcurrent research on the teaching and learning of mathematics. They give a range ofproblem types and investigations to aid the teaching and learning process. Theactivities included highlight the relevance of mathematics. Their use within theteaching program facilitates a problem-solving approach to student learningexperiences. They are also intended to provide teachers with a guide to the level ofdifficulty intended by the syllabus. The list of suggestions provided is not intendedto be exhaustive, nor is it intended that students must experience every one of theactivities and questions listed. Teachers should choose those activities and questionsthat are appropriate for their students and will need to use additional applications,activities and questions to ensure that the students have broad experiences inmathematics.

Some suggested activities and sample questions are introduced by the symbol (E).This indicates that these suggestions represent a level of difficulty that goes beyondthe general intention of the syllabus, but which some students and teachers maywish to use as extension or enrichment activities.

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Syllabus Structure

Geometry

Geometrical facts, properties andrelationshipsCongruenceSimilarity

Further reasoning in geometry

Considerations

Much of the material in this strand will have beenintroduced in Years 7–8, though without theemphasis on deductive reasoning and precisedefinitions. It may be necessary to review theYears 7–8 geometry work, especially the use ofgeometrical instruments, language and angleproperties of parallel and perpendicular lines, ifstudents have not developed competence in theseaspects. The assumption is that students arecompetent with the following constructions fromYears 7–8 (section G2.4) using compasses and astraight edge, ie:

• triangles, quadrilaterals and circles givendimensions

• angles equal to a given angle

• bisector of an interval, and an angle …

Considerationsfor the strand


The core

The core of essential learning has been designed for a minimum of 160 indicativehours for Stage 5. All students should undertake the appropriate mathematicalexperiences so that they have ample opportunity to achieve the outcomes of thiscourse.

The strands Geometry, Number and Algebra and coordinate geometry each include asection on further reasoning. While reasoning is integrated throughout the contentof the course (as evidenced by the wording of many content statements, eg solve,justify, explain, write proofs etc), the content in the further reasoning sectionsrepresents a higher level of reasoning and highlights the need for students tosynthesise their understanding of earlier concepts.

Students must undertake at least one mathematical investigation that would takearound five hours as part of the core of this course. This could be done when thecore has been completed or could be integrated into the teaching of the core.Teachers should refer to the considerations for Mathematical investigations on page173 for further information.

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G1: Geometrical Facts, Properties andRelationships

Content

Facts and relationships about angles at a point andangles associated with transversals (from theMathematics 7–8 course) are taken as assumedknowledge for this part of the course.

i) Drawing geometrical figures

Learning experiences should providestudents with the opportunity to:• construct figures satisfying given

conditions using a variety of geometrictechniques and tools

• draw a sketch from a given verbaldescription, and describe a given sketchin sufficient detail for it to be drawn

• describe a diagram concisely usingappropriate language

• explain that there may be constraintson the drawing of figures (for exampletwo sides of a triangle must together belonger than the third).

G1: Geometrical Facts, Propertiesand Relations

Applications, suggested activitiesand sample questions


Students could:

◊ construct a rhombus ABCD on a giveninterval AB, in which angle A = 60°

◊ draw a figure accurately given specifications,eg ‘an equilateral triangle ABC has BCproduced to D and D joined to A. The pointE is the midpoint of AD and is joined to C’

◊ discuss and agree upon a set of conditionsfor the lengths of the sides of a triangle sothat it can always be drawn

◊ write a set of geometric instructions thatwould enable another person to construct adiagram with the same geometric featuressuch as the one below …

Contentof thesectionsof thestrand

Applications,suggestedactivities

and samplequestions forthe sections

of the strand


The options

The remainder of the time spent on Mathematics in Years 9–10 is to be taken up bythe options component. Students should study the options for at least 40 hours. It isnot the case that a particular number of options needs to be taught — parts ofoptions and/or whole options can be chosen from this syllabus. Option topics orparts of option topics should be chosen that best meet the needs and interests of thestudents. The options will give students experience in applications of mathematicsthat are relevant to them, and also provide further preparation for their chosencourse of Mathematics for Years 11 and 12.

Students completing the Advanced course who intend to continue their study ofMathematics at 3 Unit level in Years 11 and 12 should study the following optionsas preparation for this course:

Option 5: Further geometry

Option 6: Curve sketching and polynomials

Option 7: Functions and logarithms.

A summary of the core and options for this course follows.

Summary of Years 9–10 Advanced Course — Core

(continued next page)

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Syllabus Structure

Geometry

G1: Geometricalfacts, propertiesand relationships• Drawing

geometricalfigures

• Triangles• Quadrilaterals• Polygons

Number

N1: Number andcomputation skills• Calculation

and numbersense

• Approximation

Measurementand

trigonometry

M1: Techniquesand tools formeasuring• Measuring• Estimation

Chance anddata

CD1: Collectingand organisingdata• Defining the

question• Designing the

investigation• Collecting data• Organising and

displaying data

Algebra andcoordinategeometry

A1: Generalisationin a problem-solving context

Mathematical Investigations


Summary of Years 9–10 Advanced Course — Core

Geometry

G2: Congruence• Congruence of

general figures• Congruence of

triangles

G3: Similarity• Similarity of

general figures• Similarity of

triangles

G4: Furtherreasoning ingeometry

Number

N2: Consumerarithmetic• Earning• Interest• Sales• Consumer

problems

N3: The realnumber system• Rational

numbers• Real numbers

N4: Surds andindices• Surds• Indices• Exponential

relationships• Scientific notation

N5: Rates andvariation• Rates• Variation

N6: Furtherreasoning in number

Measurementand

trigonometry

M2: Perimeter, areaand volume• Perimeter• Area• Surface area• Volume• Problems in

perimeter, areaand volume

M3: Trigonometry• Trigonometric

ratios• Right-angled

triangles andtrigonometry

• Non-right-angledtriangles andtrigonometry

Chance anddata

CD2: Summarisingand interpreting data• Measures of

location• Measures of

spread• Interpreting

displays of data• Evaluating

results

CD3: Chance• Informal concept

of chance• Simple

experiments• Probability• Probability

problems withcompoundevents

• Furtherprobability

Algebra andcoordinategeometry

A2: Linearexpressions andrelationships• Expressions• Relationships• Equations• Inequalities• Simultaneous

equations

A3: Coordinategeometry• Distance,

gradient andmidpoint

• Equation of astraight line

• Parallel andperpendicularlines

• Circles• Coordinate

exercises

A4: Quadraticrelationships• Quadratic and

relatedexpressions

• Quadraticrelationships

• Quadraticequations

• Graphs ofparabolas

A5: Furtherreasoning in algebra• Relating algebra

to physicalphenomena

• Graphs• Solving literal

equations• Understanding

variables

24




Summary of Years 9–10 Advanced Course —Options

25

Summary — Options

Option 1: Fractalsi) Iterationii) Fractals in two dimensionsiii) Fractals in three dimensions

Option 2: Networksi) Paths and circuits in networksii) Using matrices in networks

Option 3: Mathematics of small businessi) Paying wagesii) Paying taxesiii) Investmentiv) Running costs of small businesses

Option 4: Practical applications ofmeasurementi) Surveyingii) Navigationiii) Navigation on land

Option 5: Further geometryi) Further constructionsii) Proofs of geometrical relationships

involving triangles andquadrilaterals

iii) Right-angled triangles andPythagoras’ theorem

iv) Circlesv) Chord properties of circlesvi) Angle properties of circlesvii) Tangents and secantsviii) Proofs using circle theorems

Option 6: Curve sketching and polynomialsi) Curve sketchingii) Polynomialsiii) Sketching polynomials

Option 7: Functions and logarithmsi) Functionsii) Logarithms

Option 8: Modelling


Outcomes

Outcomes for Mathematics K–10 are written for each of the five stages. Theoutcomes for the Mathematics Stage 5 Advanced course are derived from thecontent of this syllabus and express the specific intended results of the teaching ofthis syllabus. They provide clear statements of the knowledge and understanding,skills, values and attitudes expected to be gained by most students as a result of theeffective teaching and learning of this course. The objectives of the syllabus act asorganisers for the outcomes.

Outcomes can help teachers to:

• understand the intent of this syllabus

• set clear expectations and focus on what is to be achieved

• indicate to students and parents what has been achieved and what is to beachieved

• focus on student growth and progress, and make informed judgements aboutstudent achievement

• determine student needs, whether it be for consolidation, extension activities,remediation, or progress to another stage

• clarify the type of student achievement to be assessed by indicating appropriateknowledge and understandings, skills, and values and attitudes for students ineach stage

• encourage student self-assessment and independent learning

• plan the learning environment, program appropriate learning activities andselect teaching resources

• focus upon the product as well as the process of teaching, thereby taking greaterresponsibility for the result of their efforts

• evaluate the effectiveness of their teaching programs.

The Advanced, Intermediate and Standard Mathematics courses for Stage 5 (Years9–10) each have their own set of outcomes derived from the content of thesyllabus. As there is some commonality between the courses, there is also someoverlap between the outcomes for each course. In particular, there is overlapbetween the outcomes for the Advanced and Intermediate courses. Because it isintended that students who undertake the Advanced course will have achieved theoutcomes of the Stage 4 (Years 7–8) Mathematics syllabus, there is little overlapbetween the outcomes of Mathematics 7–8 and those of the Stage 5 Advancedcourse.

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The outcomes for the core of the Advanced course are organised in the categoriesValues and attitudes, Working mathematically, Geometry, Number, Measurement andtrigonometry, Chance and data and Algebra and coordinate geometry. The outcomes foreach strand of the syllabus have been linked to the content by the identification ofthe sections of the strand. The outcomes for the Geometry strand are included as anexample below.

The use of syllabus references to relate outcomes to sections of the syllabus doesnot necessarily imply that all aspects of the syllabus will be represented by theoutcomes. The set of outcomes reflects the main components of the knowledge,understandings and skills relevant to the Mathematics Stage 5 Advanced course.Students may also achieve unintended outcomes through the teaching and learningprocess.

In some strands, outcomes may relate to more than one section. Completion of asection might provide the opportunity for students to achieve part of a singleoutcome or set of outcomes, or in some cases the complete outcome(s). For

Knowledge, Understanding and Skills

Objectives Outcomes Syll Ref

27

Outcomes

A student:• uses geometric techniques and tools to construct

diagrams• interprets and describes diagrams using appropriate

and concise language• identifies and uses geometrical facts, properties and

relationships to solve geometrical problems relatingto angles, triangles, quadrilaterals and polygons andjustifies the results

• recognises congruent figures as those that can besuperposed through a series of transformations

• applies the four congruence tests to prove thattriangles are congruent and solves problemsinvolving congruence

• recognises similar figures as those that can besuperposed through a series of transformations

• applies the four tests for similarity to prove thattriangles are similar and solves problems involvingsimilarity

• recognises and implements suitable strategies to solvegeometrical problems, including those that requirecongruence and/or similarity

• constructs arguments which prove general geometricresults.

Students will developknowledge,understanding andskills in:• Geometry.

G1

G2

G3

G4


example, the first two outcomes for Geometry link fairly closely to the section G1(Geometrical Facts, Properties and Relationships) of the syllabus, but the last outcome‘constructs arguments which prove general geometric results’, whilst relating to G4(Further reasoning in geometry), could also relate to other sections of the Geometrystrand.

The outcomes for the options of the syllabus have been organised within eachtopic. Students will work towards the achievement of the relevant outcomes fromthe option topics which they study.

It is intended that most students undertaking the Advanced course should achievemost of the course outcomes by the end of Stage 5. However, a small proportion ofthe outcomes represent a high degree of reasoning and application ofunderstanding, knowledge and skills, and it is perceived that only a smallpercentage of students might achieve these outcomes by the end of Stage 5.

The outcomes for Working mathematically relate to the important and overarchingskills that are expected to be achieved by students while undertaking the learningexperiences in each of the six strands of this course.

Outcomes statements for the Stage 5 (Years 9–10) Mathematics Advanced courseare included in the following pages.

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Objectives and OutcomesAdvanced Course

29

Outcomes

Values and Attitudes

Objectives Outcomes

A student:

• appreciates that mathematics involves observing, generalising andrepresenting patterns and relationships

• demonstrates a positive response to the use of mathematics as a toolin practical situations

• shows an interest in and enjoyment of the pursuit of mathematicalknowledge

• demonstrates the confidence to apply mathematics and to seek andgain knowledge about the mathematics they need from a variety ofsources

• shows a willingness to work cooperatively with others and to valuethe contributions of others

• appreciates the importance of visualisation when solving problems• shows a willingness to take risks when working mathematically• shows a willingness to persist when solving problems and to try

different methods• uses mathematics creatively in expressing new ideas and discoveries• recognises the economy and power of mathematical notation,

terminology and convention in helping to develop andcommunicate mathematical ideas

• appreciates that conventions, rules about initial assumptions,precision and accuracy enable information to be communicatedeffectively

• appreciates that a mathematical model is a simplified image ofsome aspect of the social or physical environment

• realises that justification of intuitive insights is important• appreciates how mathematics is used in a range of aspects of society• appreciates the contribution of mathematics to our society• recognises that mathematics has its origins in many cultures and is

developed by people in response to human needs• appreciates aspects of the historical development of mathematics• appreciates the impact of mathematical information on daily life.

Students willdevelop:

• appreciation ofmathematics as anessential andrelevant part of life.


Objectives and OutcomesAdvanced Course — Core

30



Objectives Outcomes

A student:• estimates the results of calculations and checks the reasonableness of

results using approximations and exact values as appropriate• uses appropriate technology effectively to assist in the solution of

problems• carries out algebraic and arithmetic calculations efficiently and

accurately• selects and uses appropriate mathematical techniques effectively• interprets diagrams, symbols and texts, moving between them as

appropriate• uses a range of problem-solving strategies, including those involving

generalising from one problem to another, and compares the relativemerits of different methods of solution

• interprets the results of problem solutions in different contexts,considering the range of possible solutions and any conditions andconstraints

• plans, carries out and reports on an extended mathematicalinvestigation with persistence, autonomy and flexibility

• communicates mathematical knowledge and understanding clearly andlogically, presenting arguments concisely.

Outcomes

A student:• uses geometric techniques and tools to construct diagrams• interprets and describes diagrams using appropriate and

concise language• identifies and uses geometrical facts, properties and

relationships to solve geometrical problems relating toangles, triangles, quadrilaterals and polygons and justifiesthe results

• recognises congruent figures as those that can besuperposed through a series of transformations

• applies the four congruence tests to prove that trianglesare congruent and solves problems involving congruence

• recognises similar figures as those that can be superposedthrough a series of transformations

• applies the four tests for similarity to prove that trianglesare similar and solves problems involving similarity

• recognises and implements suitable strategies to solvegeometrical problems, including those that requirecongruence and/or similarity

• constructs arguments which prove general geometricresults.

Students willdevelopknowledge, under-standing and skillsin: • Working

mathematically.

Syll. Ref

G1

G2

G3

G4

Objectives

Students willdevelopknowledge, under-standing and skillsin:

• Geometry.



31

Outcomes


Objectives Outcomes Syll. Ref

A student:• selects and uses an appropriate mental, written or

calculator technique to perform a variety of operationsinvolving fractions, decimals, percentages, integers andratios

• interprets and uses written and graphical information tosolve problems related to consumer arithmetic

• demonstrates an understanding of real numbers as pointson the number line and as decimals, and distinguishesbetween rational and irrational numbers

• performs operations and solves problems using surds andindices, including scientific notation

• moves between representations of numbers as appropriate• recognises and represents exponential and reciprocal

relationships in tables, graphs and symbols• interprets and uses rates to solve problems presented in

written and graphical form and solves problems involvingdirect and inverse variation

• presents clear arguments for some general properties ofnumbers.

A student:• estimates measurements appropriately in various contexts• understands and uses formulae to find lengths, perimeters

and areas of triangles, quadrilaterals, circles and compositefigures

• understands and uses formulae to find the surface area andvolume of simple and composite solids

• recognises and uses the length, area and volumerelationships between similar figures

• understands and uses the trigonometric ratios sine, cosineand tangent for angles between 0° and 180°

• draws sine and cosine curves for angles between 0° and180°

• uses trigonometry to solve practical problems involvingright-angled triangles

• uses the sine, cosine and area rules to solve problemsinvolving non-right-angled triangles.

Students willdevelopknowledge, under-standing and skillsin:

• Number.

Students willdevelopknowledge,understanding andskills in:

• Measurementandtrigonometry.

N1

N2

N3

N4

N5

N6

M1M2

M3



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A student:• investigates a problem by determining its focus and using

appropriate statistical processes and techniques• organises and displays data in a variety of ways and

interprets the displays• finds and interprets measures of location and spread from

sets of scores, including standard deviation andinterquartile range

• compares sets of data using graphical displays andmeasures of location and spread

• evaluates statements about data and draws informalconclusions

• draws conclusions considering the factors that influencethe reliability of results from investigations

• designs and performs simple chance experiments and usesthese experiments to estimate probabilities

• solves simple probability problems• constructs organised lists, tables and/or tree diagrams to

help assign probabilities to compound events.

Students willdevelopknowledge,understanding andskills in:• Chance and

data.

CD1

CD2

CD3


Objectives and OutcomesAdvanced Course — Options



33

Outcomes

A student:• uses symbolic language to generalise patterns and the

solutions to problems• understands, simplifies and manipulates algebraic

expressions, including those involving indices• understands and uses relationships expressed in tables of

values, symbols and graphs• uses a variety of techniques to solve linear equations,

inequalities and linear simultaneous equations, and solvesrelated problems

• finds the length, gradient and midpoint of an interval inthe coordinate plane

• understands and uses various standard forms of theequation of a straight line and the equation of a circle

• graphs straight lines, parabolas, circles and linearinequalities

• applies the techniques of coordinate geometry to solveproblems

• applies various techniques of factorisation• solves quadratic equations using a variety of techniques• understands and uses graphs to represent physical

phenomena• compares and contrasts the graphs of straight lines,

parabolas, circles, hyperbolas and exponential curves andtheir algebraic equations, recognising the relationshipbetween the equation and its graphical representation

• changes the subject of formulae, considering anyrestrictions on the values of the variables

• solves algebraic problems that require the application ofvarious techniques.

Students willdevelopknowledge,understandingand skills in:• Algebra and

coordinategeometry.

A1

A2

A3

A4

A5




Objectives Options Outcomes

34


A student:• uses a variety of methods to produce fractals• recognises, describes and generalises patterns

arising from 2D and 3D fractals• develops rules to describe the perimeter, area

and volume of fractals at the nth stage anddescribes these measures as n .

A student:• draws and interprets networks appropriately, in

both theoretical and practical situations• identifies paths and circuits within networks• operates arithmetically with matrices and

interprets information from practical situationsand networks in terms of matrices and theirpowers.

A student:• makes calculations related to wages, taxes and

investments• interprets tables of information related to

running costs and draws appropriate conclusions• draws conclusions related to the running costs of

small businesses after making appropriatecalculations.

A student:• selects and uses the most appropriate method of

surveying areas including the traverse, radialand triangulation methods

• constructs scale drawings from sketches andpractical exercises and calculates areas andperimeters

• uses navigational terminology and techniques• demonstrates understanding of the conventions

used in navigation on sea and land and solvesproblems involving bearings

• uses appropriate techniques to navigate a simplecourse.

→ ∞

Students willdevelopknowledge,understanding andskills in:• Working

mathematically• Geometry• Number• Measurement

andtrigonometry

• Chance and data• Algebra and

coordinategeometry.

Fractals

Networks

Smallbusiness

Practicalapplications

ofmeasurement





35

Outcomes

A student:• uses a variety of straight edge and compass

constructions to produce geometrical figures,and explains why they are valid

• proves and applies theorems related to trianglesand quadrilaterals

• proves and applies theorems relating to chord,angle and tangent properties of circles

• constructs arguments using appropriatetheorems and techniques to prove generalgeometric results related to circles.

A student:• sketches the graphs of a variety of simple

polynomial functions• recognises, interprets and graphs the equations

of circles represented in different forms• finds the points of intersection of a line with a

variety of curves, using both graphical andalgebraic methods

• identifies polynomial expressions and uses theassociated terminology

• performs operations with polynomials andunderstands and applies the remainder andfactor theorems

• factorises polynomials completely and solvesrelated equations

• given the graph of a polynomial, usestransformations to sketch the graphs of otherpolynomials.

Students willdevelopknowledge,understanding andskills in:• Working


andtrigonometry

• Chance anddata

• Algebra andcoordinategeometry

Furthergeometry

Curvesketching

andpolynomials





36


A student:• sketches and interprets the graphs of data arising

from practical situations and describes therelationship between the graphed variables

• defines a function, uses f (x ) notation, andexplains any restrictions on the x and y valuesfor a variety of functions

• finds the inverse of a linear function• relates inverse functions to reflection through

y = x, and understands the conditions necessaryfor the existence of an inverse function

• applies transformations to sketch functions• understands and uses logarithmic notation, and

uses the laws of logarithms, relating them to theindex laws

• graphs exponential and logarithmic functions,and recognises that these are mutual inverses

• solves simple index equations using appropriatemethods.

A student:• constructs, solves, interprets and reports on

mathematical models of real situations• applies some standard modelling techniques• recognises the provisional nature of any model,

is aware that assumptions are used, andsubsequently refines or changes the model asappropriate.

Students willdevelopknowledge,understandingand skills in:• Working


andtrigonometry

• Chance anddata

• Algebra andcoordinategeometry.

Functionsand

logarithms

Modelling


Assessment

Assessment is the process of gathering, judging and interpreting information aboutstudent achievement in order to inform different decisions about education,including decisions about students, curriculum, and educational policy. Assessmentforms an integral and continuous part of any teaching program. The purposes ofassessment include:

• providing reliable information that can be used to inform teaching and learning

• providing feedback to students about progress

• generating information to be used in reporting processes.

Assessment can be diagnostic, formative and/or summative.

diagnostic: the identification of students’ needs, strengths and weaknesses (usedto determine the nature of students’ misconceptions or lack ofunderstanding)

formative: the measurement of students’ achievement (used to find out whatstudents know and can do so that the next steps in learning can beplanned)

summative: the measurement of the result of teaching and learning (used torecord information that shows overall achievement of a student atthe end of a unit or course).

Assessing requires measuring student achievement of syllabus outcomes. Within anassessment program it is important to consider the selection of assessmentstrategies in relation to the outcomes being assessed. The most appropriate methodor procedure for gathering assessment information is best decided by consideringthe purpose for which the information will be used, and the kind of performancethat will provide the information. For example, the assessment of achievement ofoutcomes for Chance and data involves consideration of the students’ statisticalinvestigation, while assessment of achievement of outcomes for Measurementwould require different assessment strategies and often practical tasks.

Assessment throughout Stage 5 would usually be diagnostic as well as formativeand, at times, summative. Assessment during the Stage 5 Mathematics course helpsidentify students’ needs and measures students’ achievement so that the next stepsin learning can be planned.

37

Assessment


Assessment of student achievement in relation to the objectives and outcomes ofthe syllabus should incorporate measures of students’:

• ability to work mathematically

• knowledge, understanding and skills related to Geometry, Number, Measurementand trigonometry, Chance and data, and Algebra and coordinate geometry.

While achievement of Values and attitudes outcomes need not be reported upon,schools may choose to do so. Assessment of these outcomes could occur informallythrough student feedback, observation etc.

Students indicate their level of understanding and skill development in what theydo, what they say, and what they write and draw. Consequently there is a varietyof ways to gather information in mathematics for assessment purposes. No one wayalone is adequate, but each makes a valuable contribution to the overall assessmentprocess. Each assessment instrument should be appropriate for the outcomes it isdesigned to measure.

The following are points to consider when developing effective assessment tasks tomeasure student achievement of syllabus outcomes. (‘Tasks’ refers to anythingstudents are given to do from which assessment information will be gathered, egprojects, investigations, oral reports or explanations, tests, practical assignments etc.)

• Which syllabus objectives are to be assessed?

• What are the associated syllabus outcomes?

• What type of task will be used?

• What should be considered when designing the task? – the requirements of the task need to be clear to students– the task needs to allow students to demonstrate achievement of the

appropriate outcomes– the language used needs to be clear to students– any stimulus material or practical materials need to be appropriate to the

task– students need to have the appropriate tools to complete the task– the task needs to be accessible to students.

• Does the task measure what is intended? – it should assess the appropriate balance of knowledge, understanding and

skills– it should allow for valid judgements to be made of the students’

achievements– will the task be commented upon, graded, and/or marked?

38



• How will the task be designed to produce consistent results?– it should be challenging and promote interest– it should be of sufficient length and level of difficulty– it should facilitate the achievement of the relevant outcomes regardless of

gender or cultural background– it should not disadvantage students who have a particular physical disability– the method of drawing information from the task should be consistent for all

students.

Teachers have the opportunity to observe and record aspects of learning. Whenstudents are working in groups, teachers are well placed to determine the extent ofstudent interaction and participation — aspects that can enhance the learningexperience for many students. By listening to what students say, including theirresponses to questions or other input, teachers are able to collect many clues aboutstudents’ existing understandings and attitudes. Through interviews (which mayonly be a few minutes in duration), teachers can collect specific information aboutthe ways in which students think in certain situations. The students’ responses toquestions and comments will often reveal levels of understanding, interests andattitudes. Records of such observations form valuable additions to informationgained using other assessment strategies and enhance teachers’ judgement of theirstudents’ achievement of outcomes. Consideration of students’ journals or theircomments on the process of gaining a solution to a problem can also be veryenlightening for teachers and provide valuable insight to the degree of students’mathematical thinking.

Possible sources of information for assessment purposes include the following:

• student responses to questions, including open-ended questions

• student explanation and demonstration to others

• questions posed by students

• samples of students’ work

• student-produced overviews or summaries of topics

• practical tasks such as measurement activities

• investigations and/or projects

• students’ oral and written reports

• short quizzes

• pen and paper tests involving multiple choice, short-answer questions andquestions requiring longer responses, including interdependent questions (whereone part depends on the answer obtained in the preceding part)

39

Assessment


• open-book tests

• comprehension and interpretation exercises

• student-produced worked examples

• teacher/student discussion or interviews

• observation of students during learning activities, including listening to students’use of language

• observation of students’ participation in a group activity

• consideration of students’ portfolios

• students’ plans for and records of their solutions of problems

• students’ journals and comments on the process of their solutions.

Teachers may wish to use some of the suggested activities and sample questionsfrom the syllabus when assessing students.

The Board’s document Assessing Students with Special Education Needs: Guidelines forthe Provision of Alternative Assessment Tasks for Students with Severe Physical Disabilitiesin Stage 5 and Stage 6 (1996) provides advice on the adjustment of assessmentstrategies for special needs students. This document will be very useful for teacherswho have students with physical disabilities in their class.

Assessment for the School Certificate

One aspect of assessment during Stage 5 is assessment for the School Certificate.Such assessment is summative in nature and is for the specific purpose ofmeasuring student achievement in relation to all other students in the stage whohave studied the same Mathematics course. For the purpose of the SchoolCertificate, schools need to produce a rank order of their students in each of theMathematics courses based on their achievement in mathematics.

The assessment process for this purpose involves the design of assessment tasks that willallow decisions to be made on students’ achievement of their Mathematics course inrelation to other students in their school who are studying the same course. The tasksneed to validly discriminate between students. They must be based on the relevantcourse of the Mathematics 9–10 syllabus and could employ a variety of strategies.

Where tasks are scheduled throughout a course, greater weight would usually begiven to those tasks held towards the end of the course. Generally, it will benecessary to use a number of different assessment tasks in order to ensure thatstudent achievement in all the knowledge and skills objectives is assessed. For thepurpose of grading for the award of the School Certificate, values and attitudesshould not be included in assessment.

40



Achievement at the School Certificate in all courses is reported as a grade, basedon each school’s assessment of their students’ learning. In each Mathematicscourse, the pattern of grades awarded is:

Top 10% ANext 20% BNext 40% CNext 20% DNext 10% E

In order to ensure a common standard statewide, the grades in Mathematics aremoderated by the performance of the students on a Reference Test in each course.These tests are prepared by the Board of Studies and are based on the content ofthe core of each course.

Schools are advised how many of each grade they can award, determined by thenumber of their students who were in the percentile band for each grade on theReference Test. Schools then award grades to their students in accordance with theresults achieved on the school’s assessment program.

41

Assessment


Evaluation of School Programs

A regular evaluation of class and school programs should be implemented byteachers within each school. The purpose of the evaluation of teaching programs isto improve the teaching and learning of mathematics. Evaluation is concerned withreviews and judgements concerning the effectiveness, quality and need formodification of all aspects of the Mathematics curriculum.

The evaluation should include the following aspects:

• the extent to which the aims, objectives and outcomes of the syllabus are met

• the appropriateness of the assessment procedures adopted

• the adequacy of the teaching program for the development of knowledge,understanding, skills, values and attitudes specified in the syllabus

• the adequacy of the resource material available for the course

• the extent to which the syllabus supports teachers to facilitate studentmotivation and involvement in mathematics.

Schools should conduct effective ongoing evaluation that addresses questions such as:

• Have the overall aims of the syllabus been achieved?

• Were the objectives as stated in the syllabus implemented?

• Have the syllabus outcomes been achieved through the teaching program set bythe school?

• What types of assessment procedures were used? How effective were they?

• What teaching strategies were used? How effective were they?

• How did the students respond to the course as presented?

• Has the course been relevant to the students?

• What revisions to the teaching program have been worthwhile?

• What resources were used? Were others available? Were they effective?

This will involve qualitative as well as quantitative measures.

Informal evaluation is a continuous process in which teachers monitor and react tothe needs of their students in the teaching/learning environment.

Informal evaluations should be complemented by formal monitoring to enableschools to coordinate and plan more effective mathematics programs. The Boardof Studies will conduct a formal evaluation of the syllabus document at least oncethrough its period of implementation.

42



Advanced Course

Content — Core

Geometry

Number

Measurement and trigonometry

Chance and data

Algebra and coordinate geometry

Mathematical investigations

# 9 6 1 9 9 M a t h A d v C o r e 1 0 / 9 / 9 6 4 : 0 7 P M P a g e

Geometry

Geometrical facts, properties and relationshipsCongruenceSimilarity

Further reasoning in geometry

Considerations

Much of the material in this strand will have been introduced in Years 7–8, thoughwithout the emphasis on deductive reasoning and precise definitions. It may benecessary to review the Years 7–8 geometry work, especially the use of geometricalinstruments, language and angle properties of parallel and perpendicular lines, ifstudents have not developed competence in these aspects. The assumption is thatstudents are competent with the following constructions from Years 7–8 (sectionG2.4), using compasses and a straight edge, ie:

• triangles, quadrilaterals and circles given dimensions

• angles equal to a given angle

• bisector of an interval, and an angle

• angles of 60°, 120°, 30° and 90°

• perpendicular to a line through a point on the line and not on the line

• perpendicular bisector of an interval.

By studying this strand, students should begin to develop the ability to reasondeductively, appreciate the importance of geometrical language and understandthe need for clarity and concision in their arguments. Successive steps in reasoningcan often be clearly understood on a diagram. Diagram use in geometryencourages alternative modes of understanding verbal statements and arguments,and provides a visual means of exploring and supporting steps in thinking.Students in this course should use clear diagrams to help them explore proofdevelopment. They could begin by justifying their results and working informallytowards a likely argument, then move towards setting out the written argument toexplicate the reasoning. The argument should ultimately be logically developedand succinct, with correct use of geometric language and symbols. Creative, correctarguments should be encouraged and discussed. The end result should be anunderstanding of some principles of deductive reasoning and the use of theseprinciples in the context of formal geometry.

45

Advanced Course — Core


This form of reasoning is often difficult and students benefit in the developmentalprocess from experimentation and empirical investigation of particular cases.

Sketching, accurate construction, measurement by ruler and protractor, paperfolding and cutting and the use of templates, graphics calculators and geometricalsoftware are all recommended. Dynamic geometry software tools are a powerfulmeans of encouraging investigation and building understanding.

The structure of this strand rests on assumptions and theorems that can beestablished intuitively or formally. Within the classroom, some students may be atthe level that allows them to formally prove such theorems. If the theorems ontriangles and quadrilaterals are developed in this way, the students are effectivelycovering some of the material in Option 5 — Further geometry. The examples given inthe applications are intended to indicate the depth of treatment as well as therigour required in justifications and proofs of exercises. Further examples of proofsare provided in the support document accompanying this syllabus.

The term ‘corresponding’ is often used to refer to angles or sides in the sameposition, but it also describes corresponding angles within parallel lines. Thissyllabus has used the word ‘matching’ to describe angles and sides in the sameposition; however, the use of the word ‘corresponding’ is not incorrect. The word‘superposed’ is used to describe the placement of one figure upon another in such away that the parts of one coincide with the parts of the other.

References to ‘establish and apply’ theorems and properties in the content specificoutcomes are intended to provide teachers with the opportunity for a flexibleapproach. ‘Establish’ in this sense could mean informally, through investigationinvolving symmetry, for example, or formally through written proof, which ofteninvolves congruence. Students would not be expected to reproduce a proof forthese properties in an external examination.

When proving a result, students should be able to quote any reasonable fact ortheorem that they have met previously. Rote learning of proofs is to bediscouraged. In an external examination, students will not be asked to provetheorems such as ‘opposite sides of a parallelogram are equal’, but rather will begiven problems involving unfamiliar situations in which they will need to applythese theorems. Students should develop from initially solving arithmetic andalgebraic problems in geometry to later reasoning deductively, using moretheoretical arguments. Diagrams would normally be given to students, with theimportant information labelled on the diagram to aid reasoning. Students wouldsometimes need to produce a careful diagram from a set of instructions.

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There are many relationships between Geometry and other parts of this course. Forexample:

• the work on areas and volumes depends on triangles, circles and specialquadrilaterals

• coordinate geometry makes constant reference to geometry and providesalternative ways of proving many theorems

• trigonometry is based on similarity of triangles and the cosine rule generalisesPythagoras’ theorem

• areas and volumes of similar figures are also considered in the strandMeasurement

• curve sketching of families of curves often relies upon translations andreflections of graphs.

The order of sections of this strand does not prescribe the order of teaching.Congruence and similarity could be taught initially, especially if students areproving each theorem in relation to special triangles and quadrilaterals. Thesection Reasoning in geometry is not necessarily intended to be done after all theother sections, and could be integrated throughout the strand if students are at theappropriate level of deductive reasoning. Throughout the strand, the examplesgiven in the applications indicate the level of difficulty intended.

47



G1: Geometrical Facts, Properties and Relationships

Content

Facts and relationships about angles at a point and angles associated with transversals (fromthe Mathematics Years 7–8 course) are assumed knowledge for this part of the course.


Learning experiences should provide students with the opportunity to:

• construct figures satisfying given conditions using a variety of geometrictechniques and tools

• draw a sketch from a given verbal description, and describe a given sketch insufficient detail for it to be drawn

• describe a diagram concisely using appropriate language

• explain that there may be constraints on the drawing of figures (for example,two sides of a triangle must together be longer than the third).

48






Students could:

◊ construct a rhombus ABCD on a given interval AB, in which angle A = 60°

◊ draw a figure accurately given specifications, eg ‘an equilateral triangle ABC hasBC produced to D and D joined to A. The point E is the midpoint of AD and isjoined to C’

◊ discuss and agree upon a set of conditions for the lengths of the sides of atriangle so that it can always be drawn

◊ write a set of geometric instructions that would enable another person toconstruct a diagram with the same geometric features as the one below.

49


A B

D C

E



Content

ii) Triangles


• prove and apply properties of triangles:

The sum of the interior angles of a triangle is 180°.

An exterior angle of a triangle is equal to the sum of the opposite interior angles.

• recognise and apply the following definitions:

A scalene triangle is a triangle in which no two sides are equal in length.

An isosceles triangle is a triangle in which there are two sides equal in length.

An equilateral triangle is a triangle in which all sides are equal in length.

• establish and apply the following theorems about triangles:

If two sides of a triangle are equal, then the angles opposite those sides are equal.

Conversely, if two angles of a triangle are equal, the sides opposite those angles areequal.

Each angle of an equilateral triangle is 60°.

• apply theorems and properties of triangles to solve problems, justifying thesolutions using appropriate language.

50





ii) Triangles

Students could:◊ comment on multiple classifications of triangles, eg could a triangle be scalene

and obtuse at the same time, or isosceles and obtuse, or right-angled and obtuse,or equilateral and obtuse?

◊ answer questions like:a) an isosceles triangle has one side 10 cm and an angle 25°. What might the

triangle look like?b) in a triangle where one angle is twice the size of one of the other angles,

write algebraic expressions for the sizes of all the angles c) deduce that a triangle with one axis of symmetry has two sides equald) construct the three possible triangles in which one side is 5 cm and two

angles are 90° and 60°◊ write down as much information as possible about this triangle

◊ discuss whether an equilateral triangle is isosceles◊ explain why an equiangular triangle must be equilateral◊ construct all four isosceles triangles in which one side is 5 cm and one angle is 30°◊ show that the longest side of a triangle is opposite the largest angle◊ answer questions like:

a) find the value of x and y in the diagram below and give reasons. Whatconclusions can you draw? (An example of a possible solution is given)

Solution: x = 66° (vertically opposite angles)y = 180° – 114° (straight line)y = 66°∴ AB = AC (opposite equal angles)∴ ∆ ABC is isosceles (two sides equal)

b) in the diagram, AB is a diameter and PQ achord of a circle centre O. PQ and ABmeet at X, and QX = OQ. Prove that∠ AOP = 3∠ QOB

◊ (E) investigate what happens to the size of the internal angles of a triangledrawn on the surface of a sphere (taking great circles as the sides).

51


72° 36°

x y 114°B

E

A

D66° C

A XO B

PQ



Content

iii) Quadrilaterals


• establish and apply the following theorem:

The sum of the interior angles of a quadrilateral is 360°.

• recognise and apply the following definitions:

A trapezium is a quadrilateral in which one pair of opposite sides is parallel.

A parallelogram is a quadrilateral in which both pairs of opposite sides areparallel.

A rhombus is a parallelogram in which a pair of adjacent sides is equal .

A rectangle is a parallelogram in which one angle is a right angle.

A square is a quadrilateral which is both a rhombus and a rectangle.

• apply the following properties of quadrilaterals:

The opposite angles of a parallelogram are equal.

The opposite sides of a parallelogram are equal.

The diagonals of a parallelogram bisect each other.

The diagonals of a rhombus bisect each other at right angles.

• apply theorems and properties of quadrilaterals to solve problems

• justify the solutions to geometrical problems involving quadrilaterals usingappropriate language.

52





iii) Quadrilaterals

Students could:

◊ establish that the sum of the interior angles of a quadrilateral is 360° by drawingin a diagonal and recognising that the quadrilateral is made up of two trianglesand that the sum of the angles of the quadrilateral is equal to the sum of theangles of the two triangles, eg:

◊ use a diagram or flowchart to show the relationships between differentquadrilaterals

◊ generate and classify shapes that satisfy a given condition (eg adjacent sides equal)

◊ decide, from the definitions of special quadrilaterals, whether the followingstatements are true:a) a parallelogram is a trapeziumb) all sides of a rhombus are equalc) all angles in a rectangle are equal

◊ discuss whether the definition for rectangles implies that any result proven forparallelograms would also hold for rectangles. Students could also considerwhether the reverse holds

◊ establish and apply the properties of quadrilaterals: the diagonals of a rhombusbisect the vertex angles through which they pass, and the diagonals of arectangle are equal

◊ determine the common properties for shapes with equal diagonals which bisecteach other and consider what happens if a further constraint is added, eg thediagonals meet at right angles

◊ explore the types of quadrilaterals that are formed if: a) two intersecting diagonals bisect each otherb) only one diagonal is bisectedc) the diagonals do not bisect each other

(continued)

53




Content

54





iii) Quadrilaterals (continued)

◊ use the fact that the diagonals of a rhombus bisect each other at right angles toapply Pythagoras’ theorem to problems, and to establish the area formula for arhombus

◊ prove that of the four triangles into which a trapezium is dissected by itsdiagonals, two have equal area

◊ relate properties of special quadrilaterals to the geometrical constructions fordrawing the:– perpendicular from a point to a line– bisector of an interval– bisector of an angle– perpendicular to a line from a point within a line

◊ use the properties of the diagonals of a rectangle to explain why the midpoint ofthe hypotenuse of a right-angled triangle is equidistant from all three vertices

◊ use the properties of a rhombus or kite to construct the bisector of an angle

◊ answer questions like:a) for the parallelogram below, the line AE

is perpendicular to a diagonal. What twoangles could be given so that the sizes ofall angles in the diagram can be found?Explain your reasoning, giving severalanswers.

55


A

E

123° x°

b) decide what additionalinformation might beneeded to find the size ofthe marked angle in thediagram below.



Content

iv) Polygons


• establish, and apply the following theorems:

The interior angle sum of a polygon with n sides is 180(n – 2)°.

The sum of the exterior angles of a polygon is 360°.

• find the interior and exterior angles of a given regular polygon

• apply the above relationships to solve problems involving polygons, justifyingthe solutions using appropriate language.

56





iv) Polygons

Students could:

◊ construct some regular polygons in a given circle

◊ show, by tearing off the corners and arranging them in a spiral, that the anglesof a heptagon add to 2.5 revolutions

◊ draw various polygons (triangle, quadrilateral, pentagon, hexagon, octagon …)and, by dissection into triangles, find the angle sum of each. They coulddevelop a rule that describes the relationship between the number of sides of apolygon and its angle sum (computer software such as LOGO or othercommercially available geometrical software can enhance this activity)

◊ investigate the sum of the exterior angles for regular quadrilaterals, pentagons,hexagons and octagons, and predict the rule for this sum

◊ interpret the sum of the exterior angles of a polygon as the amount of turningduring a circuit of the boundary and generalise to circles and any closed curves

◊ compare perimeters of inscribed and circumscribed polygons to approximatethe length of the circumference of a circle (this is how Archimedes developedan approximation for π)

◊ use their knowledge of regular polygons to explain why regular hexagonstessellate but some other regular polygons may not. They could determinewhich regular polygons tessellate

◊ find the size of the interior angles in a regular pentagon

◊ find the number of sides of a regular polygon if the exterior angles are each 45°

◊ determine whether there is a regular polygon whose exterior angles are all 15°.

57



G2: Congruence

Content

i) Congruence of general figures


• demonstrate understanding that two figures are congruent when one figure canbe superposed on the other so that they coincide exactly

• demonstrate understanding that such superpositions can be achieved by asequence of translations, rotations and/or reflections

• demonstrate understanding that when two figures are congruent, areas, volumesand matching lengths and angles are equal.

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G2: Congruence


i) Congruence of general figures

Students could:

◊ investigate congruence in a variety of patterns used by other cultures (eg tapacloths, Aboriginal designs, Indonesian ikat designs, Islamic designs, designsused in ancient Egypt and Persia, window lattice, woven mats and baskets)

◊ given two congruent figures, find several sequences of rotations, translationsand/or reflections that will superpose one figure onto the other

◊ decide whether figures are congruent and specifywhich transformations might have beenperformed to superpose the figures, eg for thefigures shown

◊ copy a given triangle or quadrilateral using straight edge and compasses, eitherby copying lengths and angles, or by copying only lengths.

59


1 2 21


G2: Congruence

Content

ii) Congruence of triangles


• determine what information is needed to show that two triangles are congruent,ie:

If the three sides of one triangle are respectively equal to the three sides of anothertriangle, then the two triangles are congruent (SSS).

If two sides and the included angle of one triangle are respectively equal to twosides and the included angle of another triangle, then the two triangles arecongruent (SAS).

If two angles and one side of one triangle are respectively equal to two angles andthe matching side of another triangle, then the two triangles are congruent (AAS).

If the hypotenuse and a second side of one right-angled triangle are respectivelyequal to the hypotenuse and a second side of another right-angled triangle, then thetwo triangles are congruent (RHS).

• demonstrate these four tests by constructions using ruler, compasses, set squareand protractor in particular cases where appropriate sides and/or angles of atriangle are given

• apply the four congruence tests above to solve numerical and algebraicexercises

• justify that two given triangles are congruent using congruence tests for triangles

• write proofs of congruence of triangles, preserving matching order of vertices.

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G2: Congruence


ii) Congruence of triangles

Students could:

◊ decide how much information is needed to show that two triangles arecongruent

◊ experiment to suggest sets of minimum conditions under which triangles will becongruent

◊ consider a question where information is given that two sides of one triangle areequal to two sides of another, and decide what further information is needed inorder to say that the two triangles are congruent

◊ show that there is more than one triangle which can be drawn with sides 5 cm,7 cm and angle 40°, but only one if the angle is included

◊ investigate whether a triangle can be copied with the following techniques:using compasses and drawing the three sides; using a protractor only andtransferring the three angles; using two sides and any angle; using two anglesand any one side

◊ given dimensions of a triangle (eg side lengths of 5 cm, 7 cm and 9 cm, or twosides of 4 cm and 6 cm and included angle 35°), show by construction andsuperposition that all triangles with these dimensions are congruent

◊ use the congruence tests to pick pairs of congruent triangles from a given listlike:

◊ explain the geometrical constructions using congruence, ie perpendicular from apoint to a line, bisector of an interval, bisector of an angle, perpendicular to aline from a point within a line

(continued)

61


A60° 40°

7cm C

B

E60° 80°

7cm 7cm

F

G

H60° 80°

7cm I

J


G2: Congruence

Content

62



G2: Congruence


ii) Congruence of triangles (continued)

◊ answer questions like:a) in the diagram below, AB || QR and

ACR is a straight line, with AC = CR.Prove that ∆ABC ≡ ∆RQC, and findx, y and z

◊ prove by congruence that the angles opposite equal sides in an isosceles triangleare equal

◊ prove that if an angle bisector of a triangle is perpendicular to the opposite side,then the triangle is isosceles

◊ (E) develop a set of congruence tests for quadrilaterals.

63


C

Q R

A B4cm

6cm6cm

x

z

y

A D

EB C

b) in the diagram below, ∠ ACB = ∠ DBC and AE = DE.Prove that ∆ABC ≡ ∆BDC


G3: Similarity

Content

i) Similarity of general figures


• demonstrate understanding that two figures are similar when one figure can beenlarged and superposed on the other so that they coincide exactly

• demonstrate understanding that such superpositions can be achieved by asequence of enlargements, translations, rotations and/or reflections

• demonstrate understanding that two figures are similar when they have thesame shape but different sizes

• identify the elements preserved in similar figures, namely shape, angle size andthe ratio of corresponding sides

• find the scale factor or similarity factor that relates two similar figures

• recognise that similarity is demonstrated in scale drawings, models,photographs, plans etc

• demonstrate understanding that when two figures are similar with scale factor k : 1,then matching angles are equal, matching lengths are in the ratio k : 1, matchingareas are in the ratio k 2 : 1, and matching volumes are in the ratio k 3 : 1.

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G3: Similarity


i) Similarity of general figures

Students could:

◊ find the scale factor of enlargement obtained by the overhead projector, orenlargements of photos

◊ find the scale factors of reduction for maps and plans to draw consequencesabout lengths

◊ given two similar figures, find several sequences of enlargements, rotations,translations and/or reflections that will superpose one figure onto the other

◊ investigate the similarity of triangles in the construction of fractals such as theSierpinski triangle

◊ given a number of triangles, investigate sets of triangles that are congruent,similar, or neither

◊ construct a threefold enlargement of a given triangle or quadrilateral usingstraight edge and compasses, either by copying lengths and angles, or bycopying only lengths

◊ find classes of figures (eg circles) such that any two members of that class aresimilar

◊ answer questions like:a) a square has side length 5 cm. If it is enlarged by a scale factor of 2.5, what is

the ratio of the areas of the two squares?b) two dice in the shape of cubes have side length 1 cm and 5 cm. What is the

ratio of their volumes?

65



G3: Similarity

Content

ii) Similarity of triangles


• determine what information is needed to establish that two triangles are similar,ie:

If the three sides of one triangle are proportional to the three sides of anothertriangle, then the two triangles are similar.

If two sides of one triangle are proportional to two sides of another triangle, andthe included angles are equal, then the two triangles are similar.

If two angles of one triangle are respectively equal to two angles of anothertriangle, then the two triangles are similar.

If the hypotenuse and a second side of one right-angled triangle are proportional tothe hypotenuse and a second side of another right-angled triangle, then the twotriangles are similar.

• demonstrate these four tests by constructions using ruler, compasses, set squareand protractor in particular cases where appropriate sides and/or angles of atriangle are given

• apply the four similarity tests above to solve numerical and algebraic exercises

• justify that two given triangles are similar using similarity tests for triangles

• write proofs of similarity of triangles, preserving matching order of verticesthroughout the proof.

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G3: Similarity


ii) Similarity of triangles

Students could:

◊ construct triangles with given lengths and angles to justify the similarity tests (egconstruct two triangles, one with sides 3 cm, 5 cm and 7 cm and another withsides 6 cm, 10 cm and 14 cm, and check that the angles of both triangles areequal)

◊ draw sets of triangles with equal angles and measure the sides, then check thatthe sides are in proportion

◊ use similarity tests to choose pairs of similar triangles from a given list like:

◊ use practical measurement and similar triangles to find lengths which cannot bemeasured, eg find the height of a tree by measuring the lengths of the shadowscast by it and by a vertical metre ruler

◊ prove that there is a pair of similar triangles and find the length of the unknownside in the diagram below

◊ find the area of the triangle ABC above, if the area of the triangle ADE is 3 cm2.

67


A60° 40°

C

B

E60° 80°

F

G

H40° 100°

I

J

CB

D4cm 3cm

2cmE

A

y


G4: Further Reasoning in Geometry

Content


• prove, using isosceles triangles, that the angle in a semi-circle is a right angle

• prove Pythagoras’ theorem using any method

• prove properties of triangles and quadrilaterals, such as:


• prove and apply a test for quadrilaterals, eg:

If all sides of a quadrilateral are equal, then it is a rhombus.

• use geometrical relationships and properties of triangles, quadrilaterals andpolygons to prove statements about geometrical figures

• prove results in geometric figures by first establishing a preliminary result usingcongruency or similarity of triangles

• identify and use suitable strategies for solution of geometrical problems,including when additional constructions and congruency or similarity oftriangles are required.

68



69




Students could:

◊ find out who first proved that angles in semi-circles are right angles and/orPythagoras’ theorem

◊ describe how to test whether a figure is a rhombus

◊ prove tests based on diagonals for parallelograms (diagonals bisect each other),for rhombuses (diagonals bisect each other at right angles) and for rectangles(diagonals are equal and bisect each other)

◊ establish whether or not a quadrilateral in which each diagonal bisects thevertex angle through which it passes is necessarily a rhombus

◊ work out what extra information is needed to answer a geometrical question, oridentify redundant information

◊ solve problems like: in the diagram, G and I aremidpoints of the sides ED and EF respectively. EH isan altitude of the triangle EFD. What kind ofquadrilateral is GEIH? (Give reasons for your answer)

◊ solve problems like: take a rectangular sheet of paper and fold it once diagonally.Then fold the two single flaps over the sections labelled AB and AD, as shown.

Open out the sheet, trace andlabel the creases as shown.Prove that ABCD is a rhombus

◊ solve problems like: a) ABCD is a rhombus. DEA

is an equilateral triangleand EAB is a straight line.Find the value of x, givingreasons

(continued)

D F

E

H

G I

B B

A A

B

D

C

b) ABC is an isosceles triangle with AB = AC.BX ⊥ AC and CY ⊥ AB. BX and CYintersect at O. Prove that ∆AOY ≡ ∆AOX,and that AO bisects ∠ BAC

AE

D C

Bx A O

BY

CX



Content

70



71




(continued)

◊ using the diagram provided below, prove that the triangle AXY is isosceles. (Anexample of such a proof is included)

In ∆s ACX and ABY, AC = AB and CX = BY (given)∠ ACX = ∠ ABY (opposite equal sides in an isosceles triangle)∴ ∆ACX ≡ ∆ABY (SAS)∴ AX = AY (corresponding sides of congruent triangles)∴ ∆ AXY is isosceles by definition

◊ solve a problem like: given that ABCD is aparallelogram with diagonals produced so thatBF = DH and EA = CG, prove in at least twoways that EFGH is a parallelogram

◊ prove that the quadrilateral formed by the lines joining the midpoints of thesides of any quadrilateral will result in a parallelogram

◊ solve problems like the following: ∆ABC is a righttriangle with ∠ A = 90° and AD ⊥ BC. Whatrelationships are there between the two smallertriangles ADB and ACD and the triangle ABC?Prove that BD × DC = AD2

◊ (E) read how Euclid or later texts based on Euclid deduce the assumptionsabout angles associated with transversals from more primitive assumptions

◊ (E) find out why Gauss took sightings of the summits of three mountains tomeasure the sum of the angles of a large triangle.

B

AC

X

Y

AE

H

F

G

BD C

C BD

A


Number

Number and computation skillsConsumer arithmetic

The real number systemSurds and indices

Rates and variationFurther reasoning in number

Considerations

The skills developed in this strand have strong links with other strands. Manycould be taught by integration across strands. Students will practise and refinethese numerical skills throughout the course. The skill of approximation, inparticular, should be practised throughout, with students continually encouraged tolook at their answers and ensure that they make sense in the context of thequestions. The examples given in the applications indicate the intended level ofdifficulty.

Algebra is introduced as generalised number. A firm basis of understanding innumerical concepts is essential for students to help develop clear understandingand confident application of algebraic concepts. Some areas of the Number strand,especially Indices, Exponential relationships and Variation, include specific reference tosymbolic application of number concepts. Students may need to have completedsome of the early parts of the Algebra strand to ensure that they have the necessaryalgebraic skills for these topics.

The section on Calculation and number sense is intended as a review of the numberwork from Years 7–8. Time spent on calculation with fractions, integers, decimals,ratios and percentages at this stage should be reasonably brief and would dependon the competence of the class and individuals within the class. It is important thatstudents are able to perform operations with fractions (using written techniques),since they will need this skill when dealing with algebraic fractions. This skill couldbe practised at the time of need, ie when working with algebraic fractions, so thatstudents see the reason for the skill.

Some aspects of Consumer arithmetic have already been introduced in Years 7–8, butit is important that students are competent in applying their number skills to realsituations related to their lives. Examples that reflect current wage and salaryconditions and commercial practice should be used. Students will need practice inusing weekly, fortnightly and monthly PAYE tax tables. It is not intended thatstudents calculate superannuation using a ‘series’ approach, but rather understand

72



that it is a deduction from gross income. Consumer arithmetic provides anopportunity for students to appreciate the usefulness of spreadsheets. The use ofprepared spreadsheets provides a tool that will give quick results when items arevaried (eg interest rate, time period etc).

Students need to be aware that different representations of numbers are needed inparticular contexts (eg scientific notation for very large or very small numbers).They should be confident in using different representations of numbers andappreciate the size of such numbers.

Students should recognise that different kinds of numbers can be distinguishedwithin the real number system, for example integers, rationals and irrationalnumbers, and that each point on the number line can be represented by a realnumber. Students should understand the distinction between a number in exactform and an approximation to it. They should be able to use numbers in exactform or as approximations, depending on the context and the requirements of thequestion.

Students should already be competent in using scientific calculators from theMathematics 7–8 course. In this syllabus, calculators are not treated as a separatetopic, and students should be introduced to new keys on their scientific calculatoras they need them. However, it is most important that students maintain anddevelop their mental arithmetic skills. They should be encouraged not to rely ontheir calculators for every calculation, but should choose the most efficient meansfor answering questions. Students should recognise that different scientificcalculators may require different sequences of keys and should be confident inusing their own calculators.

Rates and variation and Exponential relationships should be considered in real-lifecontexts as far as possible and linked to other areas such as measurement,consumer arithmetic and graphs.

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N1: Number and Computation Skills

Content

i) Calculation and number sense


• perform calculations involving fractions, integers, decimals, ratio andpercentages using: – mental techniques– calculator techniques– written techniques as appropriate

• interpret and apply commonly used and efficient numerical techniques, eg anincrease by 15% is the same as 115% of the number

• restate word problems symbolically in terms of the operation(s) needed

• use appropriate representations of numbers for particular contexts

• check whether the answer makes sense in context

• solve problems that involve the above skills.

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i) Calculation and number sense

Students could:

◊ answer questions like: a) give two fractions with different denominators that add/subtract to give b) two partners invest in a business in the ratio 4:9. The smaller investment was

$12 000. What was the value of the other investment?c) what number, when divided by 0.8, gives 16?d) mentally square 1.2, 0.12, 0.012 …e) you were given a 20% discount followed by a 5% discount. What single

discount is equivalent to this combination, and what other combinationswould give the same result?

f) find the wholesale price of an item which sells at $650 if the selling price is130% of the wholesale price

g) a tap fills a tank in six minutes while a hose will fill the tank in eight minutes.How long will it take to fill the tank using both the tap and the hose?

h) if A is 80% of B, express B as a % of A

◊ perform operations on fractions without using calculators

◊ calculate the ratio of the volume of two cones where one has radius 2 cm andheight 6 cm, and the other has radius 4 cm and height 4 cm

◊ use mental arithmetic to square integers, eg all the integers from 0 up to 20

◊ compare the result of squaring numbers between 0 and 1 with squaringnumbers greater than 1 and sketch a graph of the result so that a generalisationcan be made

◊ state what ‘increased by 200%’ means and give an example where it might beused appropriately

◊ identify results that are obviously wrong (eg an answer of 9.1 kg for the averageweight of newborn babies, or investing $1000 for 5 years and receiving$25 000).

511

75




Content

ii) Approximation


• estimate the results of calculations to check the reasonableness of calculations

• decide on an appropriate level of accuracy for results of calculations

• use the language of estimation appropriately, including: – rounding– approximate– estimate– exact– level of accuracy

• round numbers sensibly, including expressing numbers correct to a specifiednumber of: – decimal places– significant figures

• provide exact answers rather than decimal approximations as appropriate

• determine the effect of truncating or rounding during calculations on theaccuracy of the results.

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ii) Approximation

Students could: ◊ estimate the size of the result of calculations (eg 4.8 × 58.2 is about 5 × 60 or 300)◊ recognise that the result of 8.6 × 84.4 is between 8 × 80 and 9 × 90 (ie between

640 and 810)◊ consider large numbers such as Australia’s Gross Domestic Product (GDP) or

crowd sizes etc, and discuss the level of accuracy which is used in differentcontexts

◊ discuss why knowledge of the number of significant figures in a measurement isnecessary, eg a measurement of 1.70 m is the same as 1.7 m but gives anindication of the level of accuracy of the measurement, ie that the measurementis to the nearest centimetre

◊ answer questions like: a) write 457.31 to two significant figures, 0.01357 to one significant figure etcb) a result is 3 000 000 correct to one significant figure. What might have been

measured?c) a number is rounded to 2.15. What could the number have been? What is

the smallest the number could have been? Is there a largest number whichcan be rounded to 2.15? What interval on the number line could representnumbers which round to 2.15?

d) seven people have a restaurant meal. They decide to share the bill of $187.45equally. Discuss the level of accuracy needed for the result and explain howto use rounding off to find the amount each person pays

◊ recognise that calculators show approximations to repeating decimals, eg = 0.66666667

◊ consider the effects of rounding inappropriately (eg rounding the area of a room,measured as 2.73 m × 4.14 m, to 3 m × 4 m if working out the area for tiling)

◊ answer the following two questions and compare the results:a) the circumference of a circle is 11 cm.

Find the radius, giving your answer toone decimal place. Find the area of thecircle, giving the answer to one decimalplace.

23

77


b) the circumference of a circleis 11 cm. Find the area of thecircle, giving the answer toone decimal place.


N2: Consumer Arithmetic

Content

i) Earning


• interpret and use the language of earning (eg wage, salary, commission,piecework, overtime, net income)

• calculate wages, salary, piecework, commission and overtime for various timeperiods

• calculate net earnings, considering deductions such as taxation andsuperannuation.

ii) Interest


• calculate simple interest on investments and loans using the formula I = PRTwhere R =

• calculate compound interest on investments and loans using repetition of thesimple interest formula

• explain how the formula A = P(1 + R)n (where R = ) arises and use it tocalculate compound interest.

r100

r100

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i) Earning

Students could:

◊ using class data, investigate and use rates of pay from students’ part-time jobs,or jobs advertised in the newspaper, to calculate for wages or salaries, includingovertime, superannuation and taxation

◊ investigate and calculate other payments such as bonuses and holiday loadings.

ii) Interest

Students could:

◊ find the interest payable by reading a prepared table

◊ use a calculator or spreadsheet to calculate interest and costs when using acredit card

◊ establish the compound interest formula by considering examples like: $1000 isinvested for 5 years compounded annually at 6% per annum

◊ use a spreadsheet to compare simple and compound interest, eg the interestearned on $10 000 invested at 8% simple interest per annum and 0.75%compounded monthly

◊ investigate what happens if an amount is invested at a certain interest rate andthe interest is compounded over an increasingly small time interval (eg interestpaid six-monthly, three-monthly, monthly, fortnightly, daily)

◊ answer questions like:a) if the simple interest paid on $50 000 is $8000, what would the interest rate

be for different time periods?b) Juan borrowed $1200 at 1.52% simple interest per month. Which is the best

estimate of the interest charges for six months: $20, $100, $200 or $1800?

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Content

iii) Sales


• calculate profit and loss on purchases

• calculate percentage discount on items.

iv) Consumer problems


• identify best buys

• compare the cost of loans using flat and reducible interest for a small number ofrepayment periods

• find the value of an item after a certain time period of depreciation orappreciation

• calculate and compare the cost of purchasing using different methods ofpayment including:– cash– lay-by– buying on terms– loan

• interpret and use step graphs and conversion graphs related to consumerarithmetic.

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iii) Sales

Students could:◊ investigate advertisements relating to sales that may be vague or misleading (eg

100% off)◊ consider whether the purchase price of an item will be the same when:

a) a fixed percentage mark-up is added to the cost price and then a 10%discount is given

b) the 10% discount is taken from the cost price and then the fixed percentagemark-up is added

◊ calculate the amount of mark-up for different items given a fixed profit margin◊ answer questions like: a shop owner marks everything up by 30%, and an item

has been marked up by $45. What is the cost price and selling price of the item?

iv) Consumer problems

Students could:◊ devise and compare strategies to determine best buys in a realistic context◊ compare the cost of the same item in different sizes: does the ratio of cost to size

remain constant as the size of the item increases?◊ use a spreadsheet and graph to investigate the effect of different repayment

schedules on the cost of a housing loan◊ investigate credit cards and terms as methods of payment and compare the total

amounts paid◊ interpret step graphs for situations such as: phone rates, taxi fares, parking rates◊ interpret and draw conversion graphs for situations such as: comparison of

Australian currency to an overseas currency, hours of work to weekly pay ◊ investigate the costs involved in running a car, going on an overseas trip,

providing food for the family for a week ◊ plan a disco to raise $100, estimating costs and a reasonable ticket price. They could

graph costs ($) versus number of people, and on the same axes graph income ($)versus number of people. They could then use the graphs to determine how manytickets need to be sold to break even, and the number of tickets for a profit of $100

◊ use prepared data from banks to draw and interpret a conversion graph for repay-ments on a loan of $100 000 for a fixed rate of interest over different time periods.

81



N3: The Real Number System

Content

i) Rational numbers

A rational number is the ratio of two integers where b ≠ 0.


• define a rational number and explain the definition

• explain why all integers are rational numbers

• use the notation for recurring decimals (eg = 0.3333…, = 0.241241241…)

• express rational numbers as terminating or recurring decimals

• express simple recurring decimals (eg ) as rationals.

ii) Real numbers

Real numbers are represented by points on the number line.

Irrational numbers are real numbers that are not rational.

Some students might benefit by seeing the proof that is irrational in the development oftheir understanding of irrationals.


• demonstrate understanding that real numbers are represented as points on thenumber line

• demonstrate understanding that real numbers can be expressed in decimal form

• demonstrate understanding that not all real numbers are rational

• use compasses and a straight edge to construct simple rationals and surds on thenumber line.

2

0.3̇

0.2̇41̇0.3̇

ab

82



N3: The Real Number System


i) Rational numbers

Students could:

◊ investigate families of rationals such as ninths or elevenths, writing the rationalsas decimals and using patterns to predict decimal equivalences for fractions suchas , or

◊ write fractions such as , and as recurring decimals

◊ write , , – as rationals

◊ explain why = 1

◊ check that arithmetic operations involving rational numbers always result inrational numbers.

ii) Real numbers

Students could:

◊ investigate how many real numbers are represented by points on the numberline between any two points

◊ write a variety of real numbers (fractions, percentages, ratios, integers, surds, π)as decimals or decimal approximations

◊ place rational and irrational numbers on the number line by estimation

◊ investigate the history of the calculation of π◊ choose integers, rationals and irrationals from a list of reals, eg 5, –2.7, ,

, 3 × 104, 26%, π, 2 : 3, , ,

◊ construct and other simple surds on the number line

◊ place π on a number line by rolling a coin along the number line that has unitsequal to the diameter of the coin

◊ check that arithmetic operations involving irrational numbers do not alwaysresult in irrational numbers (eg )

◊ discuss how to construct and on the number line.3 22 2

3 × 12

2 , 3 , 6 , – 2

83 , 93 , – 100

2,

5 502

97252

4.5̇

0.9̇

42.5̇3.1̇0. 4̇

16

57

13

1110

1900

190

83



N4: Surds and Indices

Content

i) Surds


• use the following results for x, y > 0:

= x = , = . or × ,

• use the four operations of addition, subtraction, multiplication and division tosimplify surds

• rationalise the denominator of surds of the form

• demonstrate understanding that is undefined or x < 0, = 0 when x = 0,

and is the positive square root when x > 0.x

xx

a b

c d

x

y= x

yyxyxxyx2x( )2

84





i) Surds

Students could:

◊ estimate surds such as , and check by calculator

◊ investigate , , …, and predict what will happen after a large numberof iterations

◊ decide whether statements such as the following are true or false: = 2.2361,

= 2.24 (to 3 significant figures), = 2.236067977, the exact value of

is 2, 2 = + , ( )2 = 5, 2 = ,

◊ find pairs of surds that satisfy conditions such as:– their product is rational – their product is irrational– their quotient is rational – their quotient is irrational

◊ investigate the relationship between the diagonals of different standard sizes ofpaper

◊ simplify expressions such as: , , , ,

,

◊ rationalise the denominators for the following:

◊ graph the relationship y =

◊ expand expressions involving surds of the form ( + )2 or (2 – )(2 + )(this may be considered in the algebra section on binomial expansion)

◊ (E) rationalise surds with binomial denominators.

3353

x

23

, 42

, 3

4 5

2 ± 8

2

2( )2– 2 2

4

1502

94

8 + 2 18 – 2 5018 − 5 2

50 = 5 210555555

55

5

88

8

85




Content

ii) Indices


• use appropriate language to describe numbers written in index form (eg base,power, exponent, index)

• use patterns of indices or other approaches to demonstrate the reasonableness of

the definitions: x0 = 1, and x–m = where x > 0, m > 0

• translate numbers to index form (integral indices) and vice versa

• verify the index laws (integral indices)

• use the following results for positive and negative indices:

xa × xb = xa+b, xa ÷ xb = xa–b, (xa)m = xam

• use the index laws to demonstrate the reasonableness of the definitions for

fractional indices (since, for x positive, , etc, ,

)

• translate expressions in surd form to expressions in index form and vice versa

• verify the index laws (fractional indices)

• solve numerical and algebraic problems involving indices.

xm

n = xmn

x1

n = xnx1

2 = xx1

2 × x1

2 = x

1xm

86





ii) Indices

Students could:

◊ consider patterns like 54, 53, 52, 51, 50, 5–1, 5–2 to illustrate the use of the zeroand negative index, and to write terms with negative indices as fractions

◊ evaluate 2 × 3–1, 4 × 32 etc and pay attention to the number to which the indexrelates

◊ consider a series of statements about indices, deciding which are true or false, eg 1024 = 210, 8 = 24, 3–2 = , 4–1 = , 2(3)–1 = , –28 = (–2)8, 2–1 > 1, 1 < 3–1

◊ write 4a–2 as an expression with a positive index

◊ investigate the effect of raising a fraction to the power –1, leading to ( )–1 =

◊ find pairs of terms, expressed in index form, which can be multiplied to give 27

◊ find pairs of terms involving indices which can be divided to give 3x2

◊ simplify expressions like , , 20a–3 ÷ 60a 4

◊ explain the difference between 7x0 and (7x)0

◊ what pairs of expressions could be multiplied together to obtain 30xy 2?

◊ write , … as expressions with fractional indices anddiscuss the pattern

◊ find values of a and b, given that (x a )3b = x

◊ explain why =

◊ find some values that x, p and q could take if = 2

◊ solve problems like: a sheet of paper is folded in half, forming two regions. Thisfolding is repeated as many times as possible. Investigate how the number ofregions increases for each new fold, and generalise the number of regions for nfolds.

xp

q

2 28 = 23

2 = 2( )3= 23

x2, x

3, x

4, x

5x

(ab)12

ab6104 × 10–2

(103)2

ba

ab

16

12

19

87




Content

iii) Exponential relationships


• generate exponential relationships arising from practical problems

• use tables and graphs to describe relationships of the form y = ax, y = bax and y = ba–x (where a is a positive integer)

• describe the features of the graphs of y = bax and y = ba–x (where a is a positiveinteger), including discussion of intercepts and asymptotes.

88





iii) Exponential relationships

Students could:

◊ investigate the number of creases formed by repeatedly folding a piece of paperin half, set up a table of values, sketch the relationship and find a rule thatrelates the number of times it is folded to the number of creases. They could usethe rule to predict the number of creases after 15 folds.

◊ consider historical puzzles such as the grain of rice problem. (One grain of riceis placed on the first square of the chess board, two on the second, four on thethird etc. How many grains will be on the last square? How many grainsaltogether?)

◊ (E) play the game ‘Tower of Hanoi’ and investigate the minimum number ofmoves needed to win the game, using one, two, three and four disks. Theycould set up a table to show the number of moves for each number of discs andgraph the relationship. They could then describe the relationship in symbols,using the investigation to predict the minimum number of moves for more discsand, given an appropriate number of moves, predict the number of disks.

◊ graph y = 3x, y = 2 × 3x and y = × 3x , describing the graphs with considerationof intercepts, asymptotes, and whether the curves are always increasing ordecreasing. They could discuss what happens in the general case y = b × ax.

◊ graph y = 2–x, y = 3–x and describe the effect on the graphs of replacing the x by –x

◊ determine the dimensions of standard paper sizes (A0 to A6), set up a table ofseries number, length and width, sketch the relationship between series numberand length or width, and use the graph to determine the dimensions of an A7sheet of paper.

12

89




Content

iv) Scientific notation


• express numbers in scientific notation

• enter and read scientific notation on the calculator

• use index laws to make order of magnitude checks for numbers in scientificnotation

• convert numbers expressed in scientific notation to ordinary form

• order numbers expressed in scientific notation

• solve problems involving numbers in scientific notation.

90





iv) Scientific notation

Students could:

◊ explain the difference between 2 × 104 and 24, particularly with reference tocalculator displays

◊ express the distances from Earth to different stars in scientific notation andorder the numbers from smallest to largest

◊ investigate nanotechnology (the technology of very small machines where partsmay measure only a few micrometres), making comparisons between the size ofcomponents

◊ make a reasonable estimate for the thickness of paper, converting a decimalestimate to scientific notation

◊ answer questions like:a) have you lived for a million seconds?b) how long ago was a million minutes?c) order the following numbers from smallest to largest: 3.24 × 103, 5.6 × 10–2,

6, 9.8 × 10–5, 1.2 × 104, 2.043, 0.0034, 5.499 × 102

◊ use the distance between the sun and Earth to work out the time it takessunlight to reach Earth, then compare this value with that for other stars.

91



N5: Rates and Variation

Content

i) Rates


• calculate rates in a variety of contexts, eg measurement, consumer arithmeticetc

• convert from one rate to another, eg from km/h to m/s

• use rates to solve problems

• draw and interpret travel graphs.

92





i) Rates

Students could:

◊ count the pulse and the number of breaths taken in one minute and calculatethe number of heartbeats/breath

◊ investigate the rate of population growth in different countries

◊ work out routine questions on price, speed, scale etc

◊ convert speeds in metres per second to kilometres per hour, nautical miles perminute to knots

◊ answer questions like:a) a surfboard is purchased in Hawaii for US $430. Use exchange rates from a

newspaper or teletext to calculate how much is paid in A$b) find the speed in m/s and km/h for the Olympic record for the men’s or

women’s 100 m, and compare to the speed for the 200 m, 400 m, 800 m,3000 m and Marathon

c) the reaction time for a driver is 1.2 seconds. A pedestrian steps onto the roadahead. How far will the car travel before the brakes are applied, if its speedis 60 km/h?

◊ interpret travel graphs such as the following:

93


time (hours)

km



Content

ii) Variation


• decide when a situation involves direct variation

• solve problems involving direct variation

• generate relationships that involve reciprocal relationships from problems

• describe relationships of the form y = using tables and graphs

• describe the features of hyperbolas of the form y = , including asymptotes

• solve problems involving inverse variation.

kx

kx

94





ii) Variation

Students could:

◊ calculate the scale factor for enlargements of different photos

◊ draw conclusions about what will happen to one variable as the other changes(eg doubles, increases by 10%) in cases of direct and inverse variation

◊ investigate the relationship: distance = speed × time (D = S × T), and considerthe relationship between any two of these variables when the third is heldconstant

◊ answer questions like:a) the volume of a sphere varies as the cube of the radius. If the radius is

doubled, what effect will it have on the volume?b) the number of people who can attend a concert varies inversely according to

the amount of space allowed per person. If each person is allowed 3600 cm2,the ground can hold 3600 people. How many people could beaccommodated if only 2500 cm2 were allowed per person?

◊ investigate the relationship between the size of the external angle of a regularpolygon and the number of sides of the polygon, draw up and complete a tableto relate the number of sides to the size of the external angle, graph therelationship and use the graph to predict angle size for further polygons. Theycould also discuss what happens to angle size as n (the number of sides) getsvery large

◊ investigate the possible dimensions of a backyard given that the area is 200 m2

(draw up a table of values for length v width, graph the relationship, describethe relationship in symbols and words, discuss questions such as: if the length istwice the width, what are the dimensions?)

◊ graph y = , y = , y = and y = and generalise to describe the graph of y =

◊ investigate what happens to the value of y as x gets larger and also as x getssmaller for the relationship y = .k

x

kx

–2x

2x

–1x

1x

95



N6: Further Reasoning in Number

Content


• solve problems involving skills with rational and irrational numbers

• prove some general properties of numbers, such as:

The sum of two odd integers is even.

The product of an odd and even integer is even.

The sum of 3 consecutive integers is divisible by 3.

• compare and contrast different methods of solution for problems in number,deciding on the best method as appropriate.

96





Students could:

◊ determine the relationship between the lengths of the diagonals and sides (andarea) of any square

◊ use Pythagoras’ theorem to investigate the dimensions of rectangles that havediagonals whose lengths are irrational or rational

◊ investigate the areas of squares formed by joining the midpoints of the sides ofsquares as below:

◊ find the value of the last digit of 720

◊ explain why the sum of three odd integers is always odd, and why the sum ofthree consecutive even integers is divisible by 6

◊ solve problems like:a) the sum of five two-digit numbers is greater than 100. For each of these

statements, decide whether it is true or false. Explain your reasoning.i) each number must be more than 20ii) if four numbers are less than 20, one is greater than 20iii) one is less than 20 and four are greater than 20iv) one number is less than 20v) if all five numbers are different, their sum must be less than or equal to

490b) Given that p, q and r are non-zero integers, and p > q > r, what can be said

about the relative sizes of following fractions?

(continued)

rp

pr

rq

qr

qp

pq

97




Content

98





(continued)

Students could:

◊ explain why, when dividing by a fraction, we multiply by the reciprocal

◊ find different ways of finding the sum of all the positive integers up to 100

◊ (E) find the sum of all positive fractions that are less than 1 and havedenominators less than or equal to 100, ie

+ + + + + + … +

◊ find the smallest number n, such that each or all of the following apply:a) n divided by 10 leaves a remainder of 9, n divided by 9 leaves a remainder of 8b) n divided by 8 leaves a remainder of 7, n divided by 7 leaves a remainder of 6c) n divided by 6 leaves a remainder of 5, n divided by 5 leaves a remainder of 4d) n divided by 4 leaves a remainder of 3, n divided by 3 leaves a remainder of 2e) n divided by 2 leaves a remainder of 1

◊ decide whether the order of taking successive discounts (eg 10% then 5% or 5%then 10%) affects the result and generalise the result for any two successivediscounts

◊ (E) investigate other forms of numbers, eg absolute values n , factorials n!,integer value [n], and use these forms in defining integers using the digits of thecurrent year (eg 1997: 19 – 9! ÷ 7! = 53, 1996: [19 ÷ 9] + 6 = 8).

99100

34

24

14

23

13

12

99



Measurement and Trigonometry

Techniques and tools for measuringPerimeter, area and volume

Trigonometry

Considerations

Students should develop an idea of the levels of accuracy that are appropriate to aparticular situation (eg a bridge may be measured in metres to estimate thequantity of paint required but needs to be measured far more accurately forengineering work). For students to investigate and model their world, they need tobe able to measure accurately, and be aware of the importance of accuratemeasurements in this technological age (eg electronic timing for competition meansthat the measurements of competition swimming pools and running tracks need tobe very accurate).

It is assumed that, from Years 7–8, students are familiar with the size of the mostcommonly used units — metres, millimetres, centimetres, kilometres, litres,millilitres, kilolitres, cubic centimetres, cubic metres, grams, kilograms, tonnes,seconds, minutes, hours, days, weeks, months, years, square centimetres, squaremetres and hectares — and can convert between measures of length, mass, time,area, volume and capacity. Students should already be able to measure using avariety of standard measuring equipment (linear and non-linear scales, eg dials). Ifstudents are not competent with these skills from Years 7–8, revision of this contentwill be required.

Students should develop familiarity with less common units of measurement suchas millennium, nautical mile, micrometre, nanometre, megametre, megabyte,gigabyte and light year, but would not be expected to convert such units frommemory. There are a number of situations where imperial units may still be used(eg altitude for planes). While students are not expected to remember theconversions, it might be useful in some special situations for students to convertfrom imperial to metric measurements when given the conversion factor, so thatthey might estimate the size of the imperial measures used.

Estimation is a very important skill for students. They should develop the ability tomake reasonable estimates for quantities using metric units.

Students have already covered perimeter, area and volume of basic shapes in Years7–8. This strand extends their knowledge and skills to more complex figures andassociated problems. While it is expected that students will be able to developformulae for the perimeter, area and volume of common shapes, some formulae

100



cannot be easily derived — these formulae will be included in a list in any externalexamination and are included with the School Certificate ExaminationSpecifications for Mathematics from 1998. In this situation, students will need tochoose the appropriate formulae and apply them correctly.

Trigonometry for angles greater than 90° can be introduced through circles orsimilar triangles. The choice of the appropriate approach is left to the individualteacher. If the circle method is chosen, students should be familiar with theequation of a circle before going on to define the trigonometric ratios for obtuseangles. Alternatively, they could develop facility for working with trigonometricratios of obtuse angles by investigating ratios using their calculators. The intentionof this syllabus is for students to concentrate on angles only up to 180°. Studentswould be given the exact ratio triangles so that they can use the exacttrigonometric ratios for 30°, 45° and 60° in solving problems.

Trigonometry reinforces ratio, similarity, Pythagoras’ theorem and scale drawing,and has wide applications to problems in a number of areas. It is important toemphasise the real-life applications of trigonometry in building construction,surveying and navigation etc. Students must have access to a scientific calculatorand be aware of the approximate level of accuracy required. They should haveexperience in using clinometers (for finding angles of elevation and depression)and magnetic compasses (for bearings). Students should be encouraged to set outtheir solutions carefully and use correct mathematical language, working andsuitable diagrams. The proofs of the sine, cosine and area rules are not examinablebut should be done with the students if appropriate. These rules will be providedin any external examination. Students should be competent in choosing and usingthe appropriate rule.

101



M1: Techniques and Tools for Measuring

Content

i) Measuring


• demonstrate familiarity with less common units of measurement

• use and interpret practical scales, including those that involve indices, egdecibel, Richter and pH

• use calculators for time calculations by using the degrees, minutes, secondsbutton.

ii) Estimation


• relate significant figures to the level of accuracy of measurements

• decide when it is more appropriate to estimate than to measure

• make reasonable estimates involving length, area, volume, angle, capacity,temperature, time

• recognise that all measurements are approximate.

102



M1: Techniques and Tools for Measuring


i) Measuring

Students could:◊ discuss the use of less common units such as millennium, nautical mile,

micrometre, nanometre, megametre, megabyte, gigabyte, light year, anddescribe the contexts in which they are used

◊ convert less common units such as those above to more common units◊ use measures of loudness of sounds, intensity of earthquakes and acidity levels

to demonstrate the usefulness of logarithmic scales◊ answer questions on logarithmic scales like: the scale for measuring the loudness

of sounds is the decibel scale. If ordinary conversation has a loudness of 60decibels (relative intensity = 106 or 1 000 000) and lawn mowers a loudness ofabout 120 decibels (relative intensity = 1012 or 1 000 000 000 000), by how manytimes is the relative intensity of the mower greater than that of conversation?

◊ represent a logarithmic scale on an axis and discuss the difference between areading of 5 and 6, for example, on the Richter scale

◊ explain the difference between 8.25 hours and 8 hours 25 minutes◊ work out the difference between any two given times, eg between 1:32 am and

10:19 pm on the same day◊ answer questions like: if a car travels 25 km at an average speed of 55 km/h,

how long (to the nearest minute) will the journey take?

ii) Estimation

Students could:◊ use practical measurement to consider level of accuracy and significant figures,

eg a measurement of 2.5 m indicates that a length has been measured to thenearest tenth of a metre, but a measurement written as 2.50 m indicatesmeasurement to the nearest centimetre

◊ estimate lengths and improve estimations by measuring◊ identify ways to improve estimations of quantities not easily measured◊ estimate a variety of quantities, eg the amount of material needed to make a

skirt for a size 14, the material needed to make a cylinder to hold a half-kilo ofcoffee, the dimensions of the classroom, the relative area of a state or countryfrom a map of the country, walking/running speed in m/s and km/h, a car’s fuelconsumption, crowd size etc

◊ discuss the accuracy of measurements and whether measurements can ever betruly accurate.

103



M2: Perimeter, Area and Volume

Content

i) Perimeter


• find the perimeter of polygons and circles

Circumference of a circle = πd (π × diameter).

• find the perimeter of composite figures involving triangles, quadrilaterals andcircles

• apply Pythagoras’ theorem and other perimeter results to solve problemsinvolving perimeter.

ii) Area


• explain the relationship between units of length (eg cm) and units of area (cm2)

• establish and justify formulae for finding the area of triangles, quadrilaterals(including area of a square, rectangle, parallelogram, rhombus, trapezium, kite)and circles

Area of a circle = πr2.

• find the areas of composite figures involving triangles, quadrilaterals and circlesby dissections of composite shapes into several simpler shapes

• establish and use relationships between side length and the area of similar two-dimensional figures (ie areas of similar figures are proportional to the squares onmatching sides).

104





i) Perimeter

Students could:

◊ investigate ways of finding the length of an arc and hence the perimeter of asector

◊ undertake an historical investigation of attempts to calculate π and experimentto produce estimates of π

◊ find the perimeter of the figure below (the curved shape is a semi-circle)

◊ calculate the minimum length of a straw for a drink packed in a container in theshape of a rectangular prism

◊ given the diagonals of a rhombus, find its perimeter.

ii) Area

Students could:◊ explain the need for ‘units2’ in the measurement of area◊ demonstrate appreciation of the size of a square metre and hectare, by

measuring if necessary ◊ use dissections and/or transformations to establish the formula for the area of a

parallelogram, trapezium, rhombus or kite, and justify their results◊ establish the formula for the area of a trapezium by dividing it into triangles◊ verify the formula for the area of a circle by dividing a circle into small sectors◊ divide a length of string into two pieces and use them to form two different

shapes with the same area◊ when calculating the areas of the figures below, identify any unnecessary

information and discuss

(continued)

105


5cm

8cm

4cm

33 10 67

4 8

510



Content

iii) Surface area


• devise and use methods for finding the surface area of right prisms, cylindersand pyramids

• use formulae to find the surface area of cones and spheres:Surface area of a sphere = 4πr2.Curved surface area of a cone = πrl (where l = slant height).

• establish and use relationships between the surface area of similar solids (ie thesurface area of similar figures is proportional to the squares on matching edges).

106





ii) Area (continued)

Students could:◊ describe which measurements would be needed to find the areas for a variety of

quadrilaterals◊ find the area of composite figures such as the one in the Perimeter section◊ undertake an investigation relating to area, eg design a carpark to hold x cars ◊ answer questions like the following:

a) do rectangles with the same perimeter have the same area?b) do parallelograms with the same perimeter have the same area?c) the dimensions of a room are 3500 mm by 4200 mm and the height of the

room is 2200 mm. If the combined area of the door and windows is 5.5square metres, calculate the amount of paint needed to paint the room if 1 litre of paint will cover 12 square metres of wall

◊ compare the surface areas of countries and, by drawing the areas of countries asrectangles, represent the relative areas of a group of countries, eg Australia,Indonesia, Malaysia, Japan, China

◊ estimate the relative area of each of the States from a map of Australia (or frommemory)

◊ draw similar rectangles and investigate the relationship between the ratio of thesides and the areas of the rectangles

◊ compare the sizes of photographs or postcards and posters, eg a postcard is 8 cmwide and has an area of 80 cm2. Find the width if the postcard is enlarged to anarea of 3920 cm2.

iii) Surface area

Students could:◊ investigate a variety of solid shapes and, using their knowledge of area, decide

which solids they can find the surface area of, and work out the surface area ofthese solids

◊ given a surface area of 3200 cm2, design the best container and describe why itis the ‘best’ container

◊ investigate why some Kit Kats are wrapped ‘on an angle’◊ given a sector of a circle to be formed into a cone, calculate the radius of the

base for different perpendicular heights(continued)

107




Content

iv) Volume


• explain the relationship between units of length (eg cm) and units of volume (cm3)

• use the relationship between litres and other measures of volume (cm3 and m3)

• apply Pythagoras’ theorem to solve problems involving volume of solid shapes

• use formulae to find the volume of right prisms and cylinders:

Volume of right prisms = Ah.

• use formulae to find the volume of right pyramids, cones and spheres:

Volume of a pyramid or a cone = Ah.

Volume of a sphere = πr3.

• dissect composite shapes into several simpler shapes so that volume can becalculated

• establish and use relationships between the dimensions of similar solids (ie thevolume of similar three-dimensional figures is proportional to the cubes onmatching edges).

43

13

108





iii) Surface area (continued)

Students could:◊ find the amount of paper needed to cover a cone of base radius 2 cm and

vertical height 5 cm if it requires a maximum of cm overlap at the join◊ investigate, by building if necessary, different sized cubes and find the surface

area of each in order to establish the relationships between the surface areas ofsimilar solids.

iv) Volume

Students could:◊ demonstrate appreciation of the size of a cubic metre, by building if necessary◊ discuss the need for ‘units3’ when measuring volume◊ investigate the possible dimensions for a container to hold a litre of milk or a kL

of water◊ measure the volume of solids, both regular and irregular, by displacement of water◊ calculate the volume of a variety of prisms, cylinders, cones, spheres and

composite solids◊ answer questions like:

a) if the volume of a prism is 5400 cm3, what might the dimensions be?b) how many cubic metres of concrete would be needed to lay a slab 3000 mm

by 4000 mm and 60 mm thick? c) a tap was left dripping overnight. Find the amount of water wasted and

investigate the drip rate and volume of the drops

◊ discuss how to find the volume of air inside a building that is a hexagonal prismwith a hexagonal pyramid on top

◊ find the volume of composite figures such as:

(continued)

12

109


3cm10cm

5cm 3cm

1cm



Content

v) Problems in perimeter, area and volume


• use relationships between measurements in one, two and three dimensions tosolve problems involving length, area and volume.

110





iv) Volume (continued)

Students could:

◊ investigate, by building if necessary, different sized cubes and find the volume ofeach in order to establish the relationships between the volumes of similar solids

◊ answer questions like: the ratio of the volume of two cubes is 8:1. What couldthe side lengths of each cube be?

◊ consider applications in biology (eg larger animals have to develop differentstrategies for temperature regulation than smaller creatures; the size of birds’wings increases greatly with the size of the bird)

◊ consider the effect of enlarging a model aeroplane, eg a glider may fly but, whenit is enlarged, the increase in mass (proportional to the cube of dimensions) is farmore than the corresponding increase in wing-surface area (proportional to thesquare of the dimensions).

v) Problems in perimeter, area and volume

Students could:

◊ consider problems like:a) what happens to area, volume and surface area if the lengths are doubled in

figures? b) if the side length of a cube is tripled, by what factor will the surface area

increase?c) if the lengths of a rectangular prism are doubled, the heights tripled and the

widths halved, how does the volume of the new prism relate to the volumeof the original prism?

◊ establish the ratio between the volumes of a sphere, cone and cylinder that havethe same diameter and where the height of the cone and cylinder also equaltheir diameters

◊ discuss the question: which is the better fit, a round peg in a square hole or asquare peg in a round hole, where the radius of the circle is r?

111



M3: Trigonometry

Content

i) Trigonometric ratios


• demonstrate understanding that the ratio of matching sides in right-angledtriangles (such as opposite to adjacent) is constant for equal angles

• define the sine, cosine and tangent ratios for angles in right-angled triangles

• express the tangent ratio in terms of sine and cosine

• use calculators to find trigonometric ratios of angles

• use calculators to find angles given trigonometric ratios (angles to be measuredin degrees and minutes and in decimal degrees)

• establish and use the relationship between the sine and cosine ratios ofcomplementary angles in right-angled triangles.

112



M3: Trigonometry


i) Trigonometric ratios

Students could:

◊ investigate the ratios of the sides of similar right-angled triangles

◊ find the value of the sine, cosine and tangent ratios of angles in right-angledtriangles, eg:

◊ use calculators to find cos 25°, tan72.57°, sin63° 5' etc

◊ relate the tangent ratio to slope, eg for a water ski jump where the horizontaldistance is 8 m and the vertical rise is 3 m

◊ determine the range of values for the sine, cosine and tangent ratios

◊ verify that the sine ratio divided by the cosine ratio gives the tangent ratio

◊ consider questions like: sin40° = cos x°; find the value of x°

◊ discuss why cos 25° = sin 65°.

113


8

6x°

12

7z°

10

4x°

5

2a°


M3: Trigonometry

Content

ii) Right-angled triangles and trigonometry


• use sine, cosine and tangent ratios to find the unknown sides in right-angledtriangles

• use sine, cosine and tangent ratios to find unknown angles in right-angledtriangles (angles to be measured in degrees and minutes or decimal degrees)

• determine exact sine, cosine and tangent ratios for angles 30°, 60° and 45°,given the appropriate exact value triangles (shown below), and solve problemsusing these exact values

• solve problems involving three-figure bearings

• solve problems involving angles of elevation and depression.

114


60°1

2 2√3

11

1 √245°


M3: Trigonometry


ii) Right-angled triangles and trigonometry

Students could:

◊ find x in three different ways in the diagram below

◊ find out everything they can about a right-angled triangle like the one below

◊ find the exact value for x in each of the diagrams below and hence find theareas of each shape

The exact value triangles:

◊ by rationalising the denominators, order the exact ratios for sin 30°, tan 30° andcos 30°

◊ use a directional compass to obtain three-figure bearings for objects from a setpoint in their playground

◊ work with bearings, interpreting directions like 035°, 158°, 315°, drawingdiagrams from word problems and finding lengths of unknown sides in theright-angled triangles which result

◊ answer questions like: an aircraft leaves Sydney and flies 400 km on a bearingof 135°. How far south of Sydney is the plane at this time?

◊ use a clinometer to read angles of elevation and depression and find the heightsof buildings, trees etc in their school environment

◊ solve problems from surveying and navigation involving right-angled triangletrigonometry.

115


4 3

5x

43°

35cm

60°1

2 2√3

11

1 √245°

x

5

4√360° 45°

x30°

12x

6

6 6


M3: Trigonometry

Content

iii) Non-right-angled triangles and trigonometry

The proofs of the sine, cosine and area rules are not examinable but should be done with thestudents if appropriate.


• find the trigonometric ratios of obtuse angles either from redefinition of thetrigonometric relationships using the circle (usually the unit circle) or by use of acalculator

• find the possible acute and/or obtuse angles, given a trigonometric ratio

• establish and use the following relationships for obtuse angles (0° ≤ A ≤ 90°):

sin (180° – A) = sin A

cos (180° – A) = – cos A

tan (180° – A) = – tan A

• draw graphs of the sine and cosine curves for 0° ≤ A ≤ 180°

• solve angle and length problems using the sine rule in acute- and obtuse-angledtriangles

Sine rule: = =

• solve angle and length problems using the cosine rule in acute- and obtuse-angled triangles

Cosine rule: c2 = a2 + b2 – 2ab cos C, cos C =

• use the area formula to find the area of a triangle

Area = ab sin C

• select and use appropriate trigonometric ratios and formulae to solve problemsinvolving trigonometry that require the use of more than one triangle (twodimensions), where the diagram is provided.

12

a2 + b2 – c2

2ab

csin C

bsin B

asin A

116



M3: Trigonometry


iii) Non-right-angled triangles and trigonometry

Students could:

◊ given two sides and an angle (not included) of a triangle, draw the triangle intwo ways (eg sides 5 and 7 units and angle 42° opposite the side 5). They couldfind pairs of values for each of the other angles

◊ relate the above example to the need for an included angle in the SAScongruence test

◊ use a calculator to compare sin 10° with sin 170°, sin 20° with sin 160°, sin 30°with sin 150° etc and use the results to graph sin x for 0° ≤ x ≤ 180°. They couldpredict from the graph the general result for sin (180° – x) where 0° ≤ x ≤ 90°(similarly for cos x )

◊ define cos A and sin A as in the diagram below and relate the sine and cosineratios of 180° – A to the coordinates of the corresponding point on the circle

◊ show that cos 150° = – cos 30°, sin150° = sin 30° and hence tan150° = – tan 30°

◊ investigate the graphs of the sine and cosine curves between 0° and 180° (thiscould be done using a graphics calculator or computer graphing package ifavailable)

◊ use the sine and cosine rule to approximate sides and angles in non-right-angledtriangles, eg find the size of angle θ in the triangle below

◊ find the area of any triangle (eg in the above example)

◊ find the area of a regular hexagon with sides 5 cm

(continued)

117


O 1A

x

y

(cos A, sin A)

82

71°63

θ


M3: Trigonometry

Content

118



M3: Trigonometry


iii) Non-right-angled triangles and trigonometry (continued)

Students could:

◊ develop an expression to find the area of an equilateral triangle

◊ measure triangles and find their areas, discussing the different methods possible

◊ explain the relationship between the cosine rule and Pythagoras’ theorem

◊ explain what happens to the area rule when one of the angles is 90°

◊ explain what happens to the sine rule when one of the angles is 90°

◊ solve problems involving non-right-angled triangle trigonometry from surveyingand navigation

◊ report on what happens to the third side of a triangle where two sides stayconstant in length but the angle between them changes

◊ answer a question like: an observer, 200 m from a wall, notes that the angles ofelevation to the bottom and the top of a flagpole on top of the wall are 35° and38° respectively. Find the height of the flagpole

◊ (E) use the graph of y = sin x to find all possible values of x, 0° ≤ x ≤ 180°, forwhich sin x = 0.3.

119


35°38°

200 m


Chance and Data

Collecting and organising dataSummarising and interpreting data

Chance

Considerations

Within this strand, students should carry out an investigation (individually or in agroup) in which they experience the main aspects and methods of planning,organising, analysing and evaluating data. There are many suggestions forstatistical investigations included in the accompanying support document. Whendifferent groups in a class work on different investigations, the varied questions anddata sets that arise will promote consideration of a range of different strategies foranalysing the data and drawing conclusions — strategies that are appropriate for theparticular investigations and data sets.

Many aspects of Collecting and organising data could be discussed as part of theprocess of students investigation. However, there are specific aspects of displayingand summarising data, eg stem-and-leaf plots, which would need to be taughtbefore students begin their investigations. While grouping of data would be neededfor their investigation if it involves continuous data, it is not intended that studentsspend a lot of time looking at the effect of different groupings on the shape of thedata display.

Students should be aware of the extensive uses of statistics in society. Newspapers andmagazines are very useful sources. Students should be able to interpret and criticise theway data are used. They should have experience in using tools such as spreadsheets orstatistical software packages to organise, display and analyse their data.

It is assumed from the Mathematics K–6 and 7–8 courses that students are able touse and interpret histograms, frequency polygons, sector graphs, picture graphs,bar graphs and column graphs. The drawing of these graphs is not repeated here;however, students may need to use and interpret these graphs as appropriate totheir investigation.

Students could be shown the method of obtaining a standard deviation by finding itfor a small set of scores, which have an integral mean, by hand. It is not necessaryfor students to find the standard deviation using a formula; however, students shouldunderstand that standard deviation is a measure of the spread of scores. Theintention of this course is that students could use their scientific calculator or statisticssoftware to find the standard deviation of a set of scores and be able to makecomparisons between the standard deviations of different groups of scores.

σn( )

120



The scope of this course does not extend to inferential statistics. References tosamples and to drawing conclusions from sample data are included so that theseaspects will be discussed informally.

The support document accompanying this syllabus includes further information onthis strand, particularly the statistical investigation and methods of displaying datasuch as stem-and-leaf plots and box-and-whisker graphs. It will be helpful to referto this document in conjunction with the syllabus.

Students will have varied experience with chance before beginning this course. It isimportant to consider such differences when deciding upon the starting point forchance.

Experiments in probability provide students with the opportunity to gain aninformal notion of probability and the language of probability through experience,before moving on to theoretical probability. The use of experiments also links tothe data analysis in this strand.

By exploring games and activities involving chance, and using computers andcalculators for counting, random number generation and simulation whereappropriate, students should develop the idea of outcomes of experiments and thenotion of equally likely outcomes. For experiments having a finite number n ofequally likely, mutually exclusive outcomes, the probability P(A) of a single eventA is given by P(A) = . The notion of independentevents should be discussed so that students understand that for two or more eventswhich are independent, the occurrence or non-occurrence of one in no way affectsthe occurrence or non-occurrence of the others.

number of outcomes that produce An

121



CD 1: Collecting and Organising Data

Content

i) Defining the question

Learning experiences should provide students with the opportunity to:• formulate key questions for a problem of general interest in order to investigate

issues that can be answered by the collection, organisation, display and analysisof data

• refine the key questions (if necessary).

ii) Designing the investigation


• distinguish between a sample and the population, recognising the differencebetween sampling and taking a census

• design a simple questionnaire to answer key questions, trial and improve it tohelp collect appropriate data; or present a plan for an experiment that involvesmeasurements, including making comparative trials

• consider the suitability of sampling as a procedure for their investigation

• plan how data will be recorded.

iii) Collecting data


• collect data from a variety of sources

• ensure data are collected as consistently and fairly as possible

• recognise that collecting data from an atypical group may result in data that donot represent the general population.

122





i) Defining the question

Students could:

◊ talk about statistical situations they have experienced

◊ discuss what questions need to be asked in order to investigate the problem, andrefine the statement of the problem if necessary.

ii) Designing the investigation

Students could:◊ discuss the difference between a census and a sample, and illustrate with

examples of where each is used◊ consider different ways of presenting questions, eg open questions, yes/no

questions, tick a box, responding on a scale of (say) 1 to 5◊ trial a few questions to test whether they are understood and serve the intended

purpose◊ investigate whether the wording of the questions encourages appropriate

responses (eg Do they have more than one meaning? Can they be answered viaa single response? Do they encourage a particular response?)

◊ identify the target population to be investigated◊ discuss bias, misrepresentation and other issues that may affect the

interpretation of the results ◊ decide whether data will be recorded using an organised table or list, or using

some other method.

iii) Collecting data

Students could:◊ perform data collection for their investigation◊ discuss factors that may affect the consistency of the data, including whether the

group chosen is representative◊ explore some of the various methods of data collection and recording or sources

of data, both historical and contemporary (eg census, tally sticks, weatherbureau, computer networks, church records, Guiness Book of Records, ATSIC,Australian Bureau of Statistics).

123




Content

iv) Organising and displaying data


• check raw data for obvious and gross errors

• select and use an appropriate method to organise data, using grouped intervalswhere necessary

• organise data into frequency tables, using a format suitable for analysis and theavailable tools

• organise data into cumulative frequency tables

• display data in:– dot plots– stem-and-leaf plots

• make sensible statements on the distinctive features of displays (outliers,clusters, general shape of data displays)

• choose appropriate techniques to display and summarise data.

124





iv) Organising and displaying data

Students could:

◊ consider raw data, identify any scores that cannot be valid and remove themfrom the data set, eg a score of 12 beats/minute as someone’s pulse rate

◊ decide upon the most appropriate grouping for the organisation of a data set

◊ collaborate in planning how to organise measurement data to answer specificquestions

◊ having gathered data, use a scientific calculator, graphics calculator and/orspreadsheets or databases or other appropriate software to organise the data anddraw graphs

◊ from the organised data, identify any scores that are very different from themain data set and explain their inclusion

◊ make quickly produced dot plots in order to explore data initially

◊ display data in a stem-and-leaf plot after considering the appropriate stem forthe data

◊ recognise that outliers are scores that are a long way from the main group ofscores, and identify outliers in sets of scores

◊ decide on the most appropriate type(s) of display from a histogram, stem-and-leaf plot, polygon, or other types of graphs

◊ change the type of display to see if a different impression is given by the shapeof the data display

◊ (E) discuss the stems needed for data sets that include negative scores.

125



CD 2: Summarising and Interpreting Data

Content

i) Measures of location


• find the mean , mode and median of a set of scores

• describe the advantages of using different methods of summarising information(mean, mode and median)

• use stem-and-leaf plots on the same scale to compare two sets of data

• use stem-and-leaf plots to find the median, upper and lower quartiles andinterquartile range

• draw simplified box-and-whisker plots from stem-and-leaf plots

• compare sets of data using simplified box-and-whisker plots.

x( )_

126





i) Measures of location

Students could:

◊ order data and find the middle score (median) for a small set of scores; find themedian from a table; and estimate the median graphically where data has beengrouped

◊ compare the mean, mode and median of various sets of data and makedecisions about the most appropriate measure(s) to summarise the information

◊ discuss the circumstances in which the mean is less than the median

◊ consider questions like: which is the lower, the mean or median house price inSydney?

◊ use stem-and-leaf plots to find the critical points of a box-and-whisker plot(median, upper and lower quartiles, extremities)

◊ having gathered data, use a scientific calculator, graphics calculator and/orspreadsheets or other appropriate software to analyse the data and calculateappropriate summary statistics

◊ use scientific calculators to help summarise data, along with databases,spreadsheets, statistics packages and graphing software as appropriate. Studentscould use prepared sets of data or spreadsheets and learn to adapt them fortheir needs and work towards designing and setting up their own fields

◊ consider two sets of data, such as heights of girls and boys in a particular year,and draw stem-and-leaf plots, histograms or double-column graphs and bargraphs to compare the data.

127




Content

ii) Measures of spread


• find the following measures of spread:– range– interquartile range – standard deviation (using a scientific calculator )

• interpret the above measures of spread and relate to plots of data to makesensible statements about the spread of the data

• compare different sets of data using the above measures of location and spread.

σn( )

128





ii) Measures of spread

Students could:

◊ given eight scores, 5, 7, 3, 4, 2, 5, 3, 3, find the deviations from the mean andthe squared deviations from the mean. They could calculate the standarddeviation by dividing the sum of the squared deviations by the number ofscores and taking the square root

◊ decide how many scores are more than two standard deviations from the meanin a particular data set

◊ discuss the following question: a set of 15 scores has a mean of 10 and standarddeviation of 1, while a different set of 15 scores has a mean of 10 and a standarddeviation of 4. Which set of scores is clustered more closely?

◊ consider displays of data such as histograms and make sensible statements aboutthe mean and standard deviation from the displays, eg compare the range,mean and standard deviation for the following histograms

◊ compare two sets of data, considering their means and standard deviations

◊ (E) investigate what happens to the range, mean and standard deviation of a setof scores if a constant is added to each score or if the values are multiplied by aconstant.

129


f

5 6 7 8 9Scores

f

5 6 7 8 9Scores

f

5 6 7 8 9Scores



Content

iii) Interpreting displays of data


• interpret and report on a range of visual displays of data (eg dot plots,histograms, stem-and-leaf plots, simplified box-and-whisker plots, tables, graphsand diagrams), including using summary statistics (measures of location andspread)

• pay attention to the scales on the axes when interpreting graphs

• compare different representations of the same data and comment on theappropriateness and effectiveness of various displays

• use appropriate language to discuss features of the shape of distributions,including the informal notion of ‘skewness’, after consideration of the displaysand summary statistics for the data

• identify and describe graphs of data that are misleading.

130





iii) Interpreting displays of data

Students could:

◊ interpret a range of displays of data as found in the media

◊ interpret information from lists of figures (eg about the comparativeachievement in mathematics by males and females, the length of fibres from thefleeces of two breeds of sheep etc)

◊ find displays of data where irregular scales are used on the axes and discuss theeffect of this on their impression of the data

◊ display data in at least two different ways and decide on the most appropriatedisplay

◊ use the same data to support different sides in a debate

◊ make comparisons and judgements about the information (eg 20% of students inYear 9 are taller than 160 cm but only 10% of students in Year 8 are taller than160 cm)

◊ from a stem-and-leaf plot, decide whether thescores are clustered together, whether theshape of the display indicates possible skewingof scores or any particular tendency in the data,eg for the following stem-and-leaf plot showingthe heights of 30 Year 9 students

◊ interpret simplified box-and-whisker plots such as the one below for the abovedata for the heights of 30 Year 9 students, and compare with a histogramshowing the same data

131


13 614 5 7 915 1 2 2 4 5 6 6 8 9 916 0 0 1 2 3 5 5 7 8 8 917 0 2 2 3 9

130 140 150 160 170 180

f

123456

137 142 147 152 157 162 167 172 177

Class centres

(continued)



Content

iv) Evaluating results


• find, interpret and criticise published statistical information from differentsources (eg media)

• evaluate statements that have been made about situations represented by a datadisplay

• interpret displays that show two sets of data and make comparisons

• draw informal conclusions based on sample data, appreciating that summarystatistics may vary from sample to sample

• consider informally the reliability of conclusions from investigations; this couldinclude consideration of:– the factors which may have masked the results– the accuracy of the measurements taken– whether the results can be generalised to other situations

• report conclusions from investigations.

132





iii) Interpreting displays of data (continued)

◊ identify the distinctive features of data as evident from graphs and summarystatistics such as outliers, clusters of scores and the shape of the distribution, egfor the histograms below:

◊ find examples of displays of data where the data has been misrepresented anddecide how to display the same data to give a more reasonable impression ofthe data.

iv) Evaluating results

Students could:

◊ investigate the role of statistics in shaping and describing aspects of public life(eg describing and influencing consumer tastes, developing public policy oncontroversial issues)

◊ critically review surveys, polls and reports and use published information toassist in the development of informed opinions and arguments (eg about loggingforests)

◊ interpret and evaluate data from their own surveys, drawing conclusions thatcan be justified

◊ consider other statistical investigations that have the same focus (if available),and discuss the effect of the sample on the results

◊ report orally and in writing on an investigation, discussing what wasinvestigated, how the investigation was planned and the data collected, thedisplay and analysis of the data and the conclusions that can be drawn

◊ write a letter to a newspaper summarising the results of an investigation,suggesting the implications, explaining and justifying conclusions.

133


f f f f

score score score score


CD 3: Chance

Content

i) Informal concept of chance


• order simple events from least likely to most likely

• use language associated with chance events appropriately

• place informal expressions of chance on a scale of 0 to 1

• explain the meaning of a probability of 0, and 1 when there are finitely manypossible outcomes.

ii) Simple experiments


• define an experiment to investigate a situation involving chance where there ismore than one possible outcome

• list all possible outcomes

• repeat the experiment a number of times and record the outcomes

• estimate probabilities from experimental data using relative frequencies

• describe the effect on probability estimates as the number of trials increases

• assign probabilities to simple events by reasoning about equally likelyoutcomes.

12

134



CD 3: Chance


i) Informal concept of chance

Students could:

◊ draw up a list of events (eg rain, snow, trip overseas, holiday on 25 December,living over the age of 92, counting to a million in a minute) and define thechance of each event happening from a list of chance words such as certain,probable, impossible, unlikely etc

◊ investigate the use of chance language in the printed media by collecting wordsof chance and organising them from most likely to least likely, then assigningthe words an associated probability between 0 and 1

◊ comment on statements of chance from newspapers or magazines

◊ consider the texts of different authors, investigating the frequency of commonwords used, eg ‘a’ and ‘that’. They could consider whether this method serves todistinguish between different authors

◊ use the language of chance in statistical reports (eg it is more likely for a certainevent to occur than another event)

◊ describe events that have a probability of 0.5, 0 and 1.

ii) Simple experiments

Students could:

◊ investigate chance situations, eg throwing a die, tossing a coin, drawing a cardfrom a pack, spinning a wheel, tossing thumbtacks, drawing discs from a bag,matching names and pictures

◊ prepare an organised list of the sample space for an experiment

◊ estimate the relative frequency of an event by performing a series of trials andrecording the number of times the event occurs. They could use the result to predictthe relative frequency of the event in the future, eg tossing a coin a large number oftimes and then using the observed proportion to predict heads in the future

◊ graph the results of a probability experiment, eg toss a coin 100 times and graphthe proportion of heads obtained in 10, 20, … 100 tosses of the coin, discussingany tendencies in the graph

◊ consider whether it is reasonable to compare results from different-sized datasets, eg compare results in sets of 10 with results in sets of 20

(continued)

135



CD 3: Chance

Content

136



CD 3: Chance


ii) Simple experiments (continued)

Students could:

◊ discuss the idea of randomness and decide whether all results are equally likelyfor an experiment, ie that each result has an equal chance of happening

◊ use random generators (coins, dice, cards) to develop the notion of equallylikely events and simulate probability situations

◊ make a spinner that has two colours, eg red and blue, where the result will beblue 3 out of 5 times on average

◊ design a four-coloured circular spinner that would give one colour twice thechance of being chosen as any one of the other colours

◊ design a probability device to produce a specified relative frequency

◊ use technology to generate random numbers and simulate probabilityexperiments

◊ for randomly chosen telephone numbers from their local area, decide therelative frequency of numbers ending in 9

◊ estimate the probability of an event by considering the relative frequency ofevents

◊ estimate, by sampling, the probability of drawing a red counter from a bagcontaining an unknown number of red and green counters

◊ discuss and decide whether outcomes of a list of experiments are equally likely,eg spinning a tennis racket to determine ‘rough’ or ‘smooth’ in order to start agame, selecting the winner of a lottery, choosing a job applicant for anemployment position from six people interviewed

◊ (E) investigate how code breakers use relative frequencies to decipher codes.

137



CD 3: Chance

Content

iii) Probability


• express probabilities using fractions, decimals and percentages

• use published data to assign probabilities to events

• solve simple probability problems

• describe complementary events

• solve simple probability problems by reasoning about complementary events

• describe the difference between relative frequency and theoretical probability ina simple experiment, eg rolling a die.

138



CD 3: Chance


iii) Probability

Students could:

◊ find the probability of events such as drawing a black card from a deck ofplaying cards

◊ make up some probability questions for a particular situation, eg choosing aletter at random from the word MATHEMATICS. Students could swapquestions with another person in the group and discuss the solutions once thequestions are answered.

◊ comment critically on statements involving probability, eg ‘Since there are 26letters in the English language, the probability that a person’s name starts withX is 1 in 26’, ‘Since traffic lights can be red, amber or green, the probability thata light is red is ’

◊ suggest complementary events for given events, eg what is the complementaryevent for getting an even number after rolling a die, or for drawing a red cardfrom a deck of playing cards?

◊ answer questions like: a) what is the chance of winning a prize in the $2 lottery?b) what is the chance of not winning any prize in the $2 lottery?c) what is the probability of obtaining at least a value of 5 when rolling a die?

◊ consider the assumptions made in comments like: 30% of the population isunemployed so the probability of being unemployed is 0.3

◊ discuss the fairness of a chosen game of chance, considering the chances ofwinning or losing in the long term

◊ consider a game involving chance and answer questions on probability such asthe following:a) in a game where a six must be rolled on a die before starting, what is the

probability of starting the game on the first roll of the die?b) in Scrabble, find the chances of drawing a ‘Z’ from the tiles in the first draw.

13

139



CD 3: Chance

Content

iv) Probability problems with compound events


• distinguish informally between dependent and independent events

• describe all possible outcomes for two-stage events by constructing organisedlists, tables and/or tree diagrams

• simulate two-stage probability experiments that include sampling with andwithout replacement

• assign probabilities to compound events based on information from a table,diagram or graph

• solve probability problems involving two independent events.

140



CD 3: Chance


iv) Probability problems with compound events

Students could:

◊ discuss efficient ways of describing all possible outcomes, eg use a list, table, ortree diagram to decide the total number of ways of matching three photographsof babies to three adults

◊ consider an experiment involving two stages, eg draw two counters from a bagcontaining 3 blue, 4 red and 1 white counter, write down the sample space andfind the probability of an event such as obtaining two blue counters

◊ discuss the implications of probability in games of chance involving more thanone stage

◊ make up a game of chance that is unfair, explain why it is unfair, and thenmodify it to ensure fairness in the long term

◊ identify dependent and independent events from each pair of listed events, eg– a person wearing glasses and a person having big feet– a person who works hard and a person who succeeds in their career– a pair of fours rolled with two dice and an ace of hearts drawn from a pack

of cards

◊ give examples of independent events and dependent events

◊ discuss questions like: a) the four children in a family are all boys. What is the probability that the

next child will be a boy?b) a fair die is tossed six times and a 5 comes up each time. Will the probability

of gaining a 5 on the next toss be greater than ? Less than ? Explain youranswer

◊ discuss misunderstandings in probability, eg ‘If you get four tails in a row whenflipping a coin, there is a greater chance that the next one is a head’

◊ answer questions like:a) a die is rolled twice and the numbers added. What events have a probability

of ?b) a set of six cards are numbered one to six. Find the probability of choosing

two odd numbers from the cards in two drawsi) with replacementii) without replacement (continued)

19

16

16

141



CD 3: Chance

Content

v) Further probability


• undertake a probability investigation that can be simulated and:– predict the outcome– conduct the simulation– draw conclusions– report on the investigation– discuss alternative ways to simulate the situation.

142



CD 3: Chance


iv) Probability problems with compound events (continued)

Students could:

◊ use technology to generate random numbers and hence simulate two-stageprobability experiments

◊ use a two-way table to assignprobabilities, eg the table shownrepresents data collected on 300athletes and compares height withweight. They could use the table tofind the probability of choosing a light,short athlete from this population,given that all people have the samechance of being chosen.

v) Further probability

Students could:

◊ consider the use of probability by governments and companies (egdemography, insurance, planning for roads, calling elections)

◊ investigate chance situations; for example, the three-door quiz show problemwhere the following situation occurs. The prize for the quiz show is behind onedoor and the contestant selects one of the quiz doors but does not open it. Thequiz show host, knowing where the prize is, then opens one of the other doorsto show the contestant that the prize is not behind that door. The contestant isthen asked if they want to change their mind. Should the contestant changetheir mind? Explain your answer. (This situation could be simulated by usingthree cups and a coin)

◊ investigate chance situations, such as the probability that for a group of 5 peoplethere are at least 2 people who have their birthdays in the same month; or theprobability that of 25 people there are at least 2 people who have the samebirthday

◊ simulate a situation that involves medical testing (% of false negatives gained).

143


WeightHeavy Light

Height

Tall 70 20 90

Short 50 160 210

120 180 300


Algebra and Coordinate Geometry

Generalisation in a problem-solving contextLinear expressions and relationships

Coordinate geometryQuadratic relationships

Further reasoning in algebra

Considerations

Algebra is introduced and developed through problem solving. The first section,A1, extends the general problems from Years 7–8 which the students should beable to investigate using a range of strategies, and which can be generalisedsymbolically. In this way, students see the need for algebra and move intoinvestigating linear and then quadratic relationships in depth. These sections areagain developed through carefully chosen problems.

Algebra and coordinate geometry has strong links with all the other strands in thissyllabus, particularly when situations are to be generalised symbolically as in thesections on indices, exponential relationships, inverse variation and reasoning.Students need to be encouraged to use the most efficient and appropriatetechnique from a range of strategies and integrate their knowledge and skills fromother strands to solve problems. Students should move from the concept of apronumeral as standing for a particular number towards understanding the conceptof a variable as standing for a range of numbers.

Students have already been introduced to algebra in Years 7–8 and have a clearidea that the pronumerals or ‘letters’ stand for numbers rather than items (eg acould stand for the number of apples but not the apples themselves). Each student’sknowledge of order of operations and basic number laws should be used todevelop skills and confidence in algebraic manipulation. It is assumed for thiscourse that students are competent with the algebra included in the Mathematics7–8 Syllabus.

Graphing is a powerful tool that enables algebra to be visualised. The emphasisshould be on developing students’ intuitive knowledge of the shape of the graphsof different relationships (eg linear, parabolic, simple hyperbolic, exponential andcircles) and their understanding of the effects on the graphs of making a change tothe relationship (eg adding or subtracting a constant, multiplying by a constant).Students can use mathematical templates, computer graphing software or graphicalcalculators as tools to help in graphing and comparing graphs of relationships.Spreadsheets are useful tools for teaching algebra skills since formulae can easilybe entered, values for variables changed and results given immediately.

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When dealing with expressions in the section on Linear expressions and relationships,it should be noted that an amount of general manipulation has been included andthat some non-linear expressions such as 3ab are included here. The same appliesto expressions under Quadratic relationships, where expressions like 3a2b and 2xy – 3y + 2x – 3 have been included. When working with quadratic expressions,students should develop operational facility with expansions and factorisations ofcommon expressions (binomial squares, difference of squares) so that they giveready responses.

Formal treatment of equations should be introduced through practical problemsolving. Students should be able to write an equation to show a simple relationshipbetween pairs of numbers and choose from a number of methods to solveequations, such as guess, check and improve, analytic, ‘backtracking’ (reverse flowcharts) and ‘do the same thing to both sides’. Solutions should be set out clearly,logically and carefully, and students should be able to explain their solutions.Students should be able to use the ‘=’ sign appropriately, ie usually only one equalssign per sentence when solving equations. They should check their solutions as amatter of course, and consider whether a solution makes sense in the context of thequestion. Inequalities should be restricted to the linear type.

Some students assume that any expression (eg 3 – 2(a + 5) or x2 – x – 6) must equalzero and get into the habit of solving this ‘equation’. The difference between anexpression and an equation should be emphasised. Students should realise that ifab = 0, possible solutions are either a = 0 and/or b = 0, and relate this idea to thesolution of quadratics. Students will not be expected to prove the quadraticformula but it should be established through consideration of completing thesquare. It is important that they can use this formula to find the zeros of aquadratic relationship and the roots of a quadratic equation.

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A1: Generalisation in a Problem-Solving Context

Content


• try a number of strategies to solve unfamiliar problems, such as:– using a table– drawing a diagram– looking for patterns– working backwards– guess and check– simplify a problem– trial and error

• solve non-routine problems by generalising the solution symbolically

• report on the process of obtaining a solution (oral and written), including detailsabout interpreting the problem, the plan, how the generalisation was developedand the result checked

• compare and contrast different methods of solution.

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A1: Generalisation in a Problem-Solving Context


Students could:

◊ look for different patterns in tables of numbers to decide whether a sequencecould be linear, quadratic or exponential; for example, the general terms for:

3, 6, 9, … can be expressed as 3n, for n = 1, 2, …

5, 8, 11, … can be expressed as 3n + 2, for n = 1, 2, …

1, 4, 9, … can be expressed as n2 for n = 1, 2, . . .

1, 2, 4, 8, 16, … can be expressed as 2n , for n = 0, 1, 2, …

81, 27, 9, 3, … can be expressed as for n = 0, 1, 2, …

◊ look for patterns in rows and diagonals of Pascal’s triangle and generalise theresults

◊ generalise solutions to problems like:– the number of handshakes needed so that everybody shakes hands with

everybody else– the number of cubes that have 0, 1, 2 or 3 faces painted when a cube formed

by a set of smaller cubes is painted and then pulled apart– the number of squares on a chessboard– the angle sum of a polygon– the number of digits used for the pages of a book– the one thousandth term of the sequence of 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5 …

81

3n

147



A2: Linear Expressions and Relationships

Content

i) Expressions


• translate simple word sentences into algebraic expressions and vice versa

• generate expressions from problems

• evaluate algebraic expressions by substitution

• identify equivalent algebraic expressions

• simplify algebraic expressions, including those that contain fractions

• identify common factors and hence factorise expressions.

ii) Relationships


• generate linear relationships from problems and describe them using tables,graphs and symbols

• from the table of a linear relationship, describe the relationship in symbols

• set up a table of values for the relationship y = mx + b and hence graphequations of the form y = mx + b.

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i) Expressions

Students could:

◊ write the following as algebraic expressions: 5 more than twice n, 7 less thanhalf a, 15 less than negative m, one third of the sum of x and 24

◊ translate expressions like [(3 × m ) + 4] into words

◊ answer questions like: a rectangular garden has one side 3 metres longer thanthe other. Write an algebraic expression for the perimeter

◊ make substitutions into algebraic expressions, eg if x = 3 and y = –7, find thevalue of 2x – y, 4xy, 3 – (x – y), , –3(4x + 3y)

◊ draw a diagram using rectangles and an array of dots to show equivalences suchas 3(n + 2) = 3n + 6

◊ distinguish between 3(a + b) and 3a + b and explain the difference

◊ give algebraic expressions which would describe odd and even numbers

◊ simplify expressions like: –11x + 2y + 7x – 8y + 5, 4(3x + 2) – (x – 1), 1.5x + 4.8y – 2.5x, 4a – 3ab + 5b – , – .

ii) Relationships

Students could:

◊ consider the pattern and represent therelationship between the number ofcircles and the number of intersectionpoints in a table, and in symbols. Theycould then graph the relationship

◊ express in symbols a linear relationshipbetween a and b which represents thenumbers in the table

◊ construct tables of values and use the coordinates to graph straight lines, eg y = x,y = 2x – 1, y = 3 – x, y = + 2, y = , y = 5, x = – 4, 2x – 3y = 6

◊ recognise that not all graphs are linear, eg compare the graphs of y = x and y = x2.

3x – 52

2x3

x +15

x3

7a2

xy6

149


a –4 –2 0 2 4b –13 –7 –1 5 11



Content

iii) Equations


• generate and solve linear equations which arise from the linear relationships

• develop algebraic expressions to represent situations described in words

• decide whether a suggested value is a solution to a linear equation bysubstitution

• use analytical and graphical methods to solve a range of linear equations,including equations that involve brackets and fractions

• check solutions, ensuring that the result makes sense in context

• solve problems involving linear equations

• solve linear equations resulting from substitution into formulae.

150





iii) Equations

Students could:

◊ answer questions like: for the relationship c = 3a + 5, find a when c = 10. If y = 5 – 2(x + 3), find x when y = 11

◊ solve linear equations of the type: 3(2a – 6) = 5 – (a + 2), = , + =

◊ write equations for word problems such as: ‘Seven more than the number is onemore than double the number’ as n + 7 = 2n + 1; ‘The rectangle is twice as longas it is wide’ as l = 2w)

◊ solve word problems that result in equations like: 2x – 3 = x + 7, 3(x + 4) = 5(x – 3)

◊ compare different ways of answering questions like: the solution to 3(x – 2) = 4 – (2x + 1) is x = . Change one term or sign in the initial equationso that the answer will be x = 3

◊ answer questions like:a) a bicycle wheel travels 12 metres in 6.5 revolutions. What is the diameter of

the wheel?b) a student earns $5 for the first hour of babysitting and $4 for each hour after

that. Write an equation to represent the information. If he earned $29 for ababysitting job, how many hours did he work?

c) a farmer is restocking after a drought. She can buy as many sheep as shenow owns and 50 additional lambs from one neighbour. Another neighbouroffers twice as many sheep as she now owns, less 100. Either way, she wouldget the same number of animals. How many sheep does she now own?

95

12

2x + 33

x – 14

x + 75

2x – 53

151




Content

iv) Inequalities


• use <, >, ≥, ≤, ≠ to generate linear inequalities from problems

• find a range of values that satisfy inequalities using strategies such as guess andcheck, and graphing

• solve linear inequalities analytically, including changing the direction of theinequality when multiplying or dividing by a negative number

• graph the solution to linear inequalities on a number line

• solve problems involving inequalities (in one variable).

v) Simultaneous equations


• generate linear simultaneous equations from problems

• find values that satisfy pairs of simultaneous equations using non-algebraicmethods such as guess and check, setting up tables, and looking for patterns

• solve simultaneous equations by finding the point of intersection of their graphs

• solve simultaneous equations algebraically using the analytical methods ofsubstitution and elimination

• check the solutions

• solve problems involving simultaneous equations.

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iv) Inequalities

Students could:

◊ write in symbols statements like: 3 times a number is always smaller than 8; 4 lessthan twice a number is greater than 9

◊ by guess and check methods, find solutions to 8 + 3x ≤ 4 and discuss thenumber of possible solutions

◊ solve inequalities such as 2x + 3 > 4, 2a – 5 < a + 3

◊ beginning with inequalities with numbers (eg 3 < 5), establish the need toreverse the direction of the inequality when multiplying by a negative number

◊ solve inequalities like the following and graph their solution on the number line:5 – x > 2x + 1, – 4y ≤ 6, 5 – 7c > 3.

v) Simultaneous equations

Students could:

◊ use guess and check or other methods to solve problems like: a) at the school disco there were 52 more girls than boys. Total attendance was

420. How many boys and girls attended?b) a zoo enclosure contains wombats and emus. If there are 50 eyes and 80

legs, find the number of emus and wombats

◊ give examples of pairs of linear equations that would have 0, 1 or an infinitenumber of points of intersection

◊ answer questions like: a) write a story connecting the equations 2x + y = 8 and x + 2y = 7. Draw the

graphs of the two lines and find the values of x and of y that satisfy bothequations at the same time

b) draw graphs of y = 2x – 3 and y = 3x – 4, and use the graph to solve 3x – 4 = 2x –3

◊ solve linear simultaneous equations and check the solutions, eg:a) x + y = 3 and x – y = 1b) 3x – 4y = 2 and 2x – y = 3c) y = 2x + 5 and y = x – 3.

153



A3: Coordinate Geometry

Content

i) Distance, gradient and midpoint


• interpret and use subscript notation

• find the distance between two coordinate points using Pythagoras’ theorem andthe distance formula

• find the midpoint of an interval

• define gradient as

• find the gradient of an interval using and by the formula

• apply the ratio of the sides of similar triangles to explain why the ratio rise:run isconstant for a straight line

• identify m and b in the equation of the line y = mx + b as the gradient and y-intercept

• use the graph of a straight line to find its gradient

• from the graph of a straight line, determine its equation in the form y = mx + b.

y2 – y1x2 – x1

riserun

riserun

154





i) Distance, gradient and midpoint

Students could:

◊ use Pythagoras’ theorem to find the distance between two points on the numberplane

◊ find pairs of points on the number plane which are units apart

◊ discover the midpoint formula by plotting pairs of points that form horizontal,vertical and oblique lines

◊ relate b to the y-intercept for a variety of lines in the form y = mx + b

◊ find the gradient of a variety of lines from the graph of the line

◊ explain why the gradient of a straight line is given by m in the equation y = mx + b

◊ draw a line by using just the y-intercept and gradient

◊ explain the effect on a line of changing the gradient or y-intercept

◊ find the gradient of lines where the scales on the axes are different

◊ answer questions like:a) if x + 2y = 9, what are the possible values of x and y?b) draw the line through (1,3) and (–2,4) and find its equation. Does the point

(7,13) lie on the line? What is the y value when x = 2?

◊ estimate the gradient of a line from a graph

◊ find the gradient given two points on the line

◊ describe what ‘gradient’ means, and describe when a line will have a negativegradient

◊ answer questions like:a) if the midpoint of an interval is (1,4), what could the endpoints of the interval

be?b) the equation of a line is 3x + 2y + 6 = 0. Write down everything you can

about this line.

2

155




Content

ii) Equation of a straight line


• describe the equation of a line as a relationship between the x and y coordinatesof any point on the line

• find the equation of a line passing through a point with a given gradient using:

y = mx + b

y – y1 = m(x – x1)

• find the equation of a line passing through two points

• recognise and find the equation of a line in general form ax + by + c = 0

• given the equation of a line, sketch its graph by finding the x- and y-intercepts.

iii) Parallel and perpendicular lines


• explain that two lines are parallel if their gradients are equal

• describe two lines as perpendicular if the product of their gradients is –1

• find the equation of a line that is parallel or perpendicular to a given line.

156





ii) Equation of a straight line

Students could:

◊ recognise equations that result in the graph of a straight line from a list ofequations

◊ describe the equation of a line given the gradient and a point on the line, eggradient of – and x-intercept –3

◊ find the equation of the line joining pairs of points, eg (3,–5) and (–1, 4)

◊ draw three lines through the point (1,3) and write down the equations

◊ draw four lines with a gradient of and write down the equations

◊ rearrange equations of the form ax + by + c = 0 to the form y = mx + b andhence graph the line

◊ rearrange equations like y = + 5 in the form ax + by + c = 0.

iii) Parallel and perpendicular lines

Students could:

◊ investigate conditions for lines to be parallel or perpendicular

◊ explain why the product of the gradients of perpendicular lines cannot be equalto 1

◊ show that if two lines passing through the origin are perpendicular then theproduct of their gradients is –1

◊ answer questions like:a) find the equations of the lines that are parallel/perpendicular to y = 5x + 3

and passing through the point (2,5)b) the line through (1,2) and (4,6) is met by another line that is perpendicular to

it. Give the general form for such a line

◊ discuss the equations of graphs that can be superposed by a translation orreflection through the y-axis, eg consider the graphs y = 2x, y = – 2x and y = 2x + 1and describe the transformation that would superpose one graph onto the other.

x2

13

12

157




Content

iv) Circles


• use Pythagoras’ theorem to establish the equation of a circle, centre the origin,radius r and graph equations of the form x2 + y2 = r2

• recognise and describe the algebraic equations that represent circles with centrethe origin and radius r.

v) Coordinate exercises


• graph a region represented by a linear inequality

• check whether a particular point lies in a given region specified by a linearinequality

• determine whether a particular point is inside, on, or outside a circle

• prove the formula for the distance between two points

• use the techniques of coordinate geometry for simple exercises.

158





iv) Circles

Students could:◊ show that the coordinates of all points on the circle centre the origin and radius

2 satisfy the equation x2 + y2 = 4 ◊ graph circles with equations like x2 + y2 = 9◊ recognise the equation for circles with centre the origin from a list of equations◊ from the graph of a circle with centre the origin, determine the equation.

v) Coordinate exercises

Students could:◊ graph inequalities like y ≤ x on the number plane by considering the position of

the boundary of the region as the limiting case◊ check whether the following pairs of coordinates — (1,5), (5,3), (–1,4), (7,–2),

(0,5), (9,5) — would lie in the shaded region for the graph of x + y < 7. Find otherpairs of coordinates that will lie in this region and hence satisfy the inequality

◊ graph the regions formed by inequalities such as x + y < 7, y < 2(x – 3), 2x – y ≥ 5◊ show that two intervals with equal gradients and a common point form a

straight line◊ find the area of the triangle enclosed by the lines y = 0, y = 2x and x + y = 6◊ use coordinate geometry to investigate and describe the properties of triangles

and quadrilaterals◊ find the length of an altitude and the area of the triangle, given the vertices of

an isosceles triangle◊ investigate the intersection of the perpendicular bisectors of the sides of acute-

angled and right-angled triangles by using cut-out triangles and paper, or bycoordinate geometry

◊ draw any quadrilateral on the number plane, join the midpoints of each side andfind out everything they can about the new quadrilateral that has been formed

◊ show that four specified points form the vertices of a parallelogram◊ use the vertices of the rectangle (0,0), (5,0), (5,3) and (0,3) to investigate the

properties of the diagonals of rectangles◊ give the set of possible coordinates of the vertices of a rhombus◊ answer questions like: if a parallelogram has vertices (0,0), (4,0) and (1,3), find a

possible fourth vertex.

159



A4: Quadratic Relationships

Content

i) Quadratic and related expressions


• generate quadratic expressions from problems

• evaluate quadratic expressions by substitution

• expand and simplify the product of expressions that result in a quadraticexpression

• recognise and apply the special products:

(a + b)(a – b)= a2– b2

(a ± b)2 = a2 ± 2ab + b2

• recognise expressions that are perfect squares and be able to complete thesquare

• factorise expressions, including the use of:– common factors– difference of two squares– perfect squares– trinomials – grouping in pairs for four-term expressions

• simplify expressions involving algebraic fractions.

160





i) Quadratic and related expressions

Students could:

◊ answer questions like:a) a rectangle has its length 3 cm longer than its width. What is an expression

for its area?b) What is an expression for the area of the rectangle with sides (x + 4) and (x + 1)?

c) if x = 2, y = –3, find the value of 3x2 – x + 5, (x + 3)(2x – 5), 4y2 + 2y – 7,2x2y, –

◊ become familiar with the expansion of binomial products through finding thearea of rectangles as below

or by expanding binomials algebraically, eg:(x + 2)(x + 3) = x (x + 3) + 2(x + 3)

= x2 + 3x + 2x + 6= x2 + 5x + 6

◊ discover and readily recall expansions of special products such as (x ± 3)2 and (x – 5)(x + 5)

◊ complete the following in many different ways:

(x … )(x … ) = x2 … x … 15

(x … )(x … ) = x2 … 5x …

(5x …)(x … ) = 5x2 … x … 2

◊ answer questions like the following:a) evaluate 1022, 982, 1002 – 972, using knowledge of factoringb) two terms have a common factor of 3x2. What could the terms be?c) what two expressions could have been multiplied to give 4a2 + 8a?d) what should be added to x2 – 6x to complete the square?

(continued)

4y2

32x2

5

161


x 2

x x2 2x

3 3x 6



Content

ii) Quadratic relationships


• generate quadratic relationships from problems and describe them using tables,graphs and symbols

• informally relate the shape of the graphs of quadratic relationships to parabolicshapes in the environment.

162





i) Quadratic and related expressions (continued)

Students could:

◊ factorise:

9x 3– 3xy x 2– 9 4x 2 – 25

x 2 – 5x + 6 2x 2 – 12x + 18 4x 2 – 20x + 25

6x 2 + 13x – 5 2xy – 3y + 2x – 3 2a2b – 6ab – 3a + 9

◊ simplify expressions like:

–

× –

ii) Quadratic relationships

Students could:

◊ answer questions like the following: the perimeter of a rectangle is 40 cm andone side is 6 cm longer than the other. Find an expression for the area of therectangle

◊ represent triangular numbers pictorially, draw up a table that relates the numberof dots for each triangular number to the position of the triangular number anddevelop an expression for the nth triangular number

◊ find a relationship that describes the number of diagonals in a polygon with n sides

◊ look for parabolic shapes in the environment, eg the path taken by a ball whenthrown, satellite dishes, the reflector of a torch, car headlights.

x2 + 5x + 4x2 – 2x – 8

x2 – 16x + 2

8mm2 – 2m

3m – 64

3x2 – 1

4x2 + x

x2 + 3x + 2x + 2

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Content

iii) Quadratic equations


• generate quadratic equations that arise from quadratic relationships

• solve equations of the form ax2 = c

• solve quadratic equations using:

– quadratic factors

– completing the square

– the quadratic formula x =

• identify whether a given quadratic equation has zero, one or two solutions

• check the solutions of quadratic equations

• compare and contrast different methods of solving quadratic equations

• solve problems involving quadratic equations

• solve quadratic equations resulting from substitution into formulae.

–b ± b2 – 4ac

2a

164





iii) Quadratic equations

Students could:

◊ find an equation that describes y in terms of x for the table of values below

◊ solve quadratic equations by a variety of methods including guess, check andimprove

◊ solve equations like:

3x2 = 4 x2 – 5x = 6 x2 – 8x – 4 = 0

2x (x + 4) = 0 x (x – 3) = 4 (y – 2)2 = 9

◊ answer questions like:a) if 5a × 6b = 0, what values must a or b (or both) take?b) a circle has an area of 3.5 cm2. What is its radius and diameter?c) solve x2 – 4x + 3 = 0 in as many different ways as possible

◊ solve word problems that result in equations like 2x2 – 5x – 3 = 0

◊ after solving a number of quadratic equations, discuss the possible number ofroots for any quadratic equation

◊ (E) prove the quadratic formula.

165


x 0 1 2 3 4 5

y 1 2 5 10 17 26



Content

iv) Graphs of parabolas


• graph equations of the form y = ax2 and describe the effect on the graph ofdifferent values of a

• graph equations of the form y = ax2 + c and describe the effect on the graph ofdifferent values of the constant c

• find the x- and y-intercepts for the graph of y = ax2 + bx + c, given a, b and c

• graph y = ax2 + bx + c for various values of a, b and c, using the availabletechnology

• find the coordinates of the vertex of a parabola by:– finding the midpoint of the x-intercepts and substituting– completing the square– using the formula

• identify and use features of parabolas and their equations to assist in sketchingquadratic relationships, eg x- and y-intercept, vertex, axis of symmetry andconcavity

• determine a possible equation, given a graph that shows some of these features.

166





iv) Graphs of parabolas

Students could:

◊ draw the graph of a quadratic relationship by plotting points, factorising andfinding the x- and y-intercepts, using the graphics calculator and/or computergraphing packages

◊ graph y = x2, using a graphics calculator or otherwise, and write down as manythings as they can about this graph. Using the calculator or otherwise, theycould then graph y = x2 + 1, describing this graph and predicting the equationof the graph which looks the same but has its vertex at (2,0)

◊ graph y = (x – 2)2 and y = (x – 2)2 + 3, describing what effect adding 3 has hadon the original graph

◊ investigate the graphs of parabolas of the form y = x2, y = x2 + 1, y = ax2, y = (x – a)2 and y = (x – a)2 + k

◊ answer questions like:a) if a parabola cuts the x axis at 0 and 2, what might the equation be?b) if the perimeter of a rectangle remains fixed at 18 cm, draw up a table for

length v area and graph the results. What type of relationship is indicated bythe graph? Estimate the dimensions that give the maximum area

◊ find the axis of symmetry for the parabola y = x2 – 6x + 8 and use it to find thevertex, then draw the graph

◊ find the vertex and x-intercepts for y = x2 + x + 1 and hence sketch its graph

◊ sketch the graphs of y = x2 + 6x – 7 and y = 2x2 + 12x – 14, comparing the twographs and describing the differences between them

◊ from a graph of a parabola, with the main features clearly marked, discuss andpredict the equation of a curve, using computer graphing software if available

◊ find the equation of the graph of a quadratic relationship that has its vertex at(1,12) and crosses the x axis at (3,0).

167



A5: Further Reasoning in Algebra

Content

i) Relating algebra to physical phenomena


• decide whether a particular graph is a suitable representation of a given physicalphenomenon

• sketch graphs to show important features of phenomena

• describe the key features of a graph (including rates of change) that represents agiven physical phenomenon.

ii) Graphs


• identify graphs of lines, parabolas, exponential curves, hyperbolas and circleswith centre the origin

• describe graphs of lines, parabolas, exponential curves, hyperbolas and circleswith centre the origin, using the appropriate equations.

168





i) Relating algebra to physical phenomena

Students could:

◊ investigate the relationship between the depth of water in containers at differenttimes where the rate of filling is constant but the shapes differ, and draw graphsto represent the relationship

◊ match graphs of physical phenomena with the physical situations, eg graphs thatshow velocity or distance in different sports against time

◊ interpret a graph such as the one shown,making sensible statements about the rateof increase or decrease, the initial and finalpoints, constant relationships as denoted bystraight lines etc.

ii) Graphs

Students could:

◊ match a series of equations with the appropriate graphs

◊ from graphs, describe appropriate equations for the curves, eg in the graphsbelow.

169


depth

time

y

x3

y

x

(2,2)

(-2,-4)

y

x (3,8)y

x

(1,2)y

x 1



Content

iii) Solving literal equations


• change the subject of a formula, using examples from a range of subject areas

• consider the restrictions on the values of the variables implicit in the originalformula and after rearrangement of the formula.

iv) Understanding variables


• replace a variable in an equation by another variable or expression

• use variable substitution to simplify expressions and equations so that specificcases can be seen to belong to general categories

• interpret algebraic expressions and equations given additional conditions

• interpret algebraic expressions or equations in which conditions are implied.

170





iii) Solving literal equations

Students could:

◊ answer questions like: make r the subject of = + , make b the subject of

x = , discussing any restrictions on the variables

◊ consider the restrictions that apply to formulae, eg consider what restrictionsthere would be on the variables in the equation Z = 3x2 and what additionalrestrictions are assumed if the equation is rearranged to x = .

iv) Understanding variables

Students could:

◊ find an expression for x2 + 4 if x = 2at

◊ factorise expressions like: 3a4b – 12a2b3, x3 + x2 + x + 1, a2 – b2 – a + b, x 4 – y 4,x 4 – 32x 2 + 225

◊ factorise x 2 – 4x + 4, and hence factorise x 4 – x 2 + 4x – 4

◊ simplify expressions like: + , (x + 3)2 – (x – 2)2, {(x + 3)2 – 4 }{(x + 3)2 + 4}

◊ replace x by in the expression x 2 + – 2

◊ replace x by x + 1 in the expression x 2 – 5x + 6 and simplify, then investigatethe zeros of the resulting expression, relating them to the zeros of the originalexpression

◊ factorise x 4 – 13x 2 + 36 and hence solve the equation x 4 – 13x 2 + 36 = 0

◊ consider problems like:a) if p is a real number, decide when 2p is larger than p + 2 and explain whyb) a and b are real numbers. If a + b = 10 and a > b, give some possible values

of a and bc) (y + 2)3 + y = 220 when y = 4; find a value of y that makes (2y + 2)3 + 2y = 220

true.

1x2

1x

1x2

1x3

Z

3

b2 – 4ac

1s

1r

1x

171



172




Considerations

Investigation and problem solving are integral to this course. In addition to theinvestigation that students should undertake in the Chance and data strand, it isintended that students will undertake at least one longer investigation which mighttake up to five hours. In doing such an investigation, students would use all theprocesses of Working mathematically, ie investigating, conjecturing, solving problems,applying and verifying, communicating and working in context. While students usethese processes throughout this course, they are identified here specifically, sincethe choice of investigation precludes the identification of specific content from thestrands.

Students should present a written and/or verbal report on their investigation,outlining processes and stages. The use of technology as a tool to aid theinvestigation or report is to be encouraged. Such an investigation could beundertaken during the time when the core content is being taught, ie inconjunction with a topic or strand, or when the core content is complete. Thiswould depend on the choice of investigation — some investigations rely onknowledge, skills and understanding of a particular strand, while others requirestudents to synthesise their knowledge, skills and understanding from a number ofstrands.

The investigations listed here and in the support document suggest topics and arenot exhaustive. The list is intended to indicate the types of investigations studentsmight undertake, but students might choose other investigations. Otherinvestigations mentioned throughout the Applications, suggested activities and samplequestions of the strands could be developed into in-depth investigations. Studentscould also undertake an investigation that relates to a section of the option topics,especially for Fractals, Networks and Modelling.

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Content


• undertake a mathematical investigation

• report on the investigation, including the following:– a description of the problem– a description of any constraints and assumptions – an explanation of how any relevant technology was used– any relevant printouts or records– a description of any problems encountered– a description of any conclusions reached– a description of any possible extensions for further investigation.

174





Students could:

◊ perform one of the following investigations:a) a soft-drink can will have a volume of 375 mL. Find the dimensions that

would require the least amount of sheet metal to construct it. Comment onits suitability for use

b) you pay $1000 a year into an annuity and receive 8% interest on average.Estimate the value of the annuity if you work for 35 years. Would investing$800 a year but averaging 9% interest be more profitable over this time?

c) the LOGO procedure to polystar :N :Prepeat :P [FD 80 RT :N * 360 / :P]end

can be used to draw regular star polygons. Investigate the stars produced forvarious values of ‘N’ and ‘P’. How would you produce a star (exclude regularpolygons) with any number of points? Test your hypothesis for a star with 53points. Can a star with six points (exclude a hexagon) be drawn using thismethod? Explain

d) write a set of procedures using a programming language such as LOGO toinvestigate spirograms. You will need a number of inputs for distances andone for the angle. Can you determine a pattern that can be used to predictwhen a design will be closed (returning to the starting point) or open (doesnot return to the starting point)? Can you determine a pattern involving therepetition factor and the order (number of distances) of the spirograms?

e) a cereal manufacturer includes a phonecard in each box of a certain brand.A young child would like the complete set of six. Use a random process tosimulate collecting the complete set a number of times. What is the expectednumber of boxes of cereal you will have to buy before you obtain acomplete set?

f) write LOGO procedures or use a spreadsheet to simulate the tossing of acoin or a die any number of times. You should be able to type TOSSCOIN1000, for example, to obtain the number of heads and tails from 1000 tossesof a coin. Investigate what happens for various numbers of tosses

g) write a computer program to generate prime numbers up to 1 000 000 andinvestigate aspects of these prime numbers, eg count the prime pairs

(continued)

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Content

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(continued)

Students could:

◊ perform one of the following investigations:

h) the midpoints of adjacent sides of a polygon can be joined to form anotherpolygon. Investigate the ratio of the area of the new polygon to the area ofthe original polygon for various polygons

i) investigate the significance of a, b and c in the graph of the equation y = ax2 + bx + c, and a, b, c and d in the graph of the cubic equation y = ax3 + bx2 + cx + d

j) you are presented with two empty bags and 16 marbles, eight white andeight red. You are to distribute the marbles in the bags in any way youchoose. The only requirement is that every marble must be placed in a bag.Next you are blindfolded so that you can no longer see the bags or themarbles in them. You are to choose a bag at random and then, if possible,choose one marble at random from the bag you have chosen. If you select awhite marble you receive $100. Otherwise you win nothing. How would youdistribute the marbles in the bags?

k) player 1 has eight white marbles to distribute in two empty bags. Player 2has eight red marbles to distribute after seeing how player 1 has distributedthe white marbles. Player 1 is blindfolded so that she can no longer see thebags or the marbles in them. Player 1 is to choose a bag at random and then,if possible, choose one marble at random from the bag chosen. If player 1selects a white marble, she receives $100. Otherwise player 2 wins $100.Does player 1 have an advantage?

l) a tray is to be made by cutting squares from the corners of a rectangular pieceof metal and folding up the side pieces. For various-sized rectangles, what sizesquares should be cut from the corners to give a maximum volume for the tray?

m) investigate population growth over a number of years for a population whosebreeding pattern is governed by the rule that at the end of the year there aretwice as many as at the beginning of the year. What if the populationincreased by 10% a year? What happens for other growth factors?

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Content

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(continued)

Students could:◊ perform one of the following investigations:

n) in the diagram, ABCD and PQRS are squares withAB = 12 cm and AP = x cm. Find an expressionfor the area of PQRS in terms of x, and hence findthe minimum possible area of PQRS

o) investigate the stability of a population for various values of b and x,assuming the growth model y = bx (1 – x), where x = 0 means extinction andx = 1 means maximum capacity

p) think of a number. If it is odd, triple it and add 1. If it is even, halve it. Nowstart over again with the resulting number. Keep repeating this process. Whatdid you find? Investigate other numbers

q) investigate the chance of two people at a party having the same birthday.How many people would need to be at a party before you would be veryconfident that two people had the same birthday?

r) investigate the position of the centroid, circumcentre, orthocentre andincentre for triangles of different shapes. List any interesting properties ofthese points

s) investigate Pythagorean triads by first generating them using a formula.Decide whether it is always true that one of the values is divisible by 5.Investigate the Pythagorean triads for other possible properties

t) investigate properties of the Fibonacci sequenceu) investigate the values of a, b and c for which equations of the form ax + by = c

have integral solutions for x and y. What about positive integral solutions?v) investigate ratios of sides in a pentagramw) a number and its reciprocal differ by one. Can you find the number and any

other interesting facts about it?x) investigate various monthly repayments for paying off a $10 000 loan on a

car if interest is charged at 9% pa. What would be your preferredrepayment? Why?

y) read about Cardano’s solution to cubic equations and write a report, or readsome of Euclid and explain some of his procedures.

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A

P

B

R

D CS

Q


180



Advanced Course

Content — Options

1. Fractals

2. Networks

3. Mathematics of small business

4. Practical applications of measurement

5. Further geometry

6. Curve sketching and polynomials

7. Functions and logarithms

8. Modelling

# 9 6 1 9 9 M a t h A d v O p t i o n s 1 0 / 9 / 9 6 4 : 1 3 P M

Option 1: Fractals

IterationFractals in two dimensions

Fractals in three dimensions

Considerations

Fractals provide students with an opportunity to experience an area of recentlydeveloped mathematics. Through the consideration of fractals, students will furtherdevelop their facility in a number of other areas of mathematics, including scale,measurement, place value and computational accuracy. By investigating theperimeter and area of fractals, students can gain an intuitive understanding ofsequences and limits. Teachers may wish to investigate fractals other than thosementioned here.

Students should gain an informal understanding of an iterative process by drawingsome simple fractals to about the fourth or fifth stage (isometric grid paper would behelpful). Computer programs and calculators could also be used to generate fractals.

The Chaos game could be used to generate fractals. If enough iterations are used,the result will be a Sierpinski triangle. Self-similarity occurs when parts of a figureare small replicas of the whole figure. The Sierpinski triangle could also be used toinvestigate the idea of self-similarity.

In considering dimensions, students could investigate what happens when objects aresliced into self-similar shapes (eg a line can be cut into two intervals of equal length,a square into four equal squares and a cube into eight equal cubes with sides half thelength of the original). When a two-dimensional shape is reduced by a factor of a ,four of the reduced shapes will fit into the original shape. Similarly, eight of thereduced shapes will fit into the original three-dimensional shape. If a three-dimensional shape is reduced by a factor of , then you can fit S 3 of them into theoriginal shape. In general, the number of pieces (N ) that can fit into the originalshape of dimension (D ) when the reduction factor is is given by N = SD.

Some fractals can be shown to have a dimension that is neither 1, 2 or 3, but adecimal. For example, the Von Koch snowflake has a reduction factor of , and N = 4 (since each side of the original has been replaced by four pieces ofequivalent length). This gives 4 = 3D, and by using logarithms or by trial and error,the result is a dimension of approximately 1.26.

13

1S

1S

12

183

Advanced Course — Options


Option 1: Fractals

Content

i) Iteration


• use construction methods to produce fractals based on simple shapes such astriangles and squares

• recognise that fractals are produced using an iterative process, and develop arule to describe the iteration process for fractals.

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Option 1: Fractals


i) Iteration

Students could:

◊ explore the idea of iteration by using a process such as squaring modulo 100and investigate the cycle for numbers between 0 and 99, eg 722 = 5184, 5184 mod 100 = 84, 842 = 7056, 7056 mod 100 = 56. Continuing this process of squaring modulo 100 results in the following cycle:

(Starting with other numbers will producedifferent cycles. In total, there are six differentcycles produced using the squaring modulo 100iteration)

◊ draw fractals for several stages, for example the Sierpinski triangle, a fractal tree,the Von Koch snowflake or a fractal carpet illustrated below

Sierpinski triangle Fractal tree Von Koch Fractal carpetsnowflake

Stage 2 Stage 3 Stage 1 Stage 1 Stage 2

◊ use a diagram of the first three stages of a fractal to determine the rule used togenerate the fractal

◊ vary the construction algorithm of a fractal to produce another fractal, egproduce the Sierpinski triangle using another reduction factor, or vary thefractal carpet by shading only the four corner squares

◊ use the software LOGO to generate fractals

◊ find other examples of fractals or shapes exhibiting self-similarity, eg thecoastline, fern leaves etc.

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36

16

96568472


Option 1: Fractals

Content

ii) Fractals in two dimensions


• recognise number patterns arising from fractals by considering the number ofshapes produced at different stages of the fractal construction, and generaterules to describe these patterns

• make sensible statements about the change in area and perimeter of fractals assuccessive iterations are taken

• develop rules to describe the perimeter and area of a fractal at the nth stage ofiteration

• predict the area and perimeter of fractals at the nth stage of iteration as n → ∞

• recognise self-similarity in fractals

• develop the relationship between the scale factor, the number of self-similarshapes in a fractal and the dimension of the fractal.

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Option 1: Fractals


ii) Fractals in two dimensions

Students could:

◊ use a fractal that they have constructed (eg Sierpinski triangle, Von Koch snowflake,fractal carpet, fractal tree or other) to investigate:– number patterns that can be found in the fractal, eg the number of shapes

produced at different stages of the fractal construction– the change in the perimeter and area of the fractal as successive iterations are

taken– the dimension of the fractal– the patterns formed by changes in perimeter and area as n → ∞, eg for the

Sierpinski triangle these patterns appear:

Stage 0 Stage 1 Stage 2 Stage 3

Number of shaded triangles 1 3 9 27

Area of shaded triangles A

Perimeter of shaded triangles P

◊ investigate the pattern produced in Pascal’s triangle by shading odd numbers (thiswill produce the Sierpinski triangle)

◊ play the Chaos game to produce fractals as follows: start with a large equilateraltriangle. Label the vertices X, Y, Z. Mark a point inside the triangle. Roll a die andmove according to the following rules: if the die shows 1 or 2, move halfwaytowards the vertex X; for 3 or 4 move halfway towards the vertex Y; and for 5 or 6move halfway towards the vertex Z. Mark the new point. Start from this new point,roll the die and again move halfway to the appropriate vertex. Continue theprocess. If this process is continued a large number of times the Sierpinski trianglewill eventually result (this can also be simulated on a computer)

(continued)

27P16

9P4

3P2

27A64

9A16

3A4

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Option 1: Fractals

Content

iii) Fractals in three dimensions

Learning experiences should provide students with the opportunity to:• generate fractals in three dimensions• make sensible statements about the change in volume of fractals in three

dimensions• find a rule to describe the volume of a fractal at the nth stage of iteration• predict the volume of fractals at the nth stage of iteration as n→∞.

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Option 1 : Fractals


ii) Fractals in two dimensions (continued)

Students could:

◊ construct the Cantor set, which is produced by removing the middle third of aunit interval and leaving the end points, then removing the middle third of eachresulting interval and so on, as in the diagram. The Cantor set is the set ofpoints left at the end of this process, ie:

They could determine how many sub-intervals there are at the nth stage. Avariation on this could be to produce a tree with branches formed at 45° anglesand with the lengths of each successive branch one third of the length of theprevious branch. Students could investigate what happens to the height andwidth of the tree as the number of iterations increases

◊ investigate the effect that a change to the construction algorithm has on fractals,eg what effect will a scaling factor of have on the area of the Sierpinskitriangle, or if the original shape was a square or another polygon?

◊ investigate other fractals such as the one below, considering the relationshipbetween the fractal, Pythagoras’ rule, and the area of each successive square.

iii) Fractals in three dimensions

Students could:

◊ use a tetrahedron and the construction algorithm for the Sierpinski triangle toproduce a fractal

◊ use the fractal carpet to create a three-dimensional sponge and investigate thevolume of the sponge and the dimension of the fractal (the dimension of thefractal could be found using trial and error or logarithms). They could describethe rule for the volume at the nth stage and predict the volume as n → ∞.

13

189


Stage 0

Stage 1

Stage 2


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Option 2: Networks

Paths and circuits in networksUsing matrices in networks

Considerations

Networks are useful for simplifying a situation where there are a number of paths,roads or routes that can be taken. There are many practical applications ofnetworks (eg mail delivery, garbage pick-up, communications, couriers’ routes).While the intention is an introduction to networks, students should appreciate theirusefulness, and gain an understanding of the requirements for networks to beconnected. Some of the history of networks could be discussed, eg Euler’sconsideration of the bridges of Königsberg.

Students should be able to decide whether a vertex is even or odd. An even vertexis one where an even number of edges leads to it or from it. A vertex is odd if anodd number of edges leads to it or from it. An investigative approach should beused to introduce connectedness. Eulerian paths are those which pass along everyedge of the network exactly once.

Students should appreciate that conventional maps indicate comparative distanceswhile network diagrams show only the relationships between locations. Othertypes of networks could be included here, eg family trees.

Networks provide a useful context in which to introduce the concept of a matrix. Itis intended that students have some experience of matrix operations in a problem-solving context.

Matrices are a convenient and efficient way to store information. They could beintroduced as rectangular or square arrays used to hold and organise informationrelating to networks, eg the number of paths between destinations. Appropriatelychosen matrices could then lead to addition of matrices and multiplication of amatrix by a scalar. Matrices could also be used to help solve simultaneousequations. This section is intended as an introduction to matrices and could beextended. Examples of matrix applications which relate to real-world situationsencourage students to appreciate the relevance of matrices.

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Option 2: Networks

Content

i) Paths and circuits in networks

A network is a set of vertices in which any two vertices either are or are not connected by anedge.

A path is any sequence of edges connected end to end (a path may visit a vertex or pass alongan edge more than once).

A circuit is a path that begins and ends at the same vertex.

A network is called connected or traversable if every pair of vertices is connected by a pathbeginning at one and ending at the other.

The degree of a vertex is the number of edges connected to it.

An even vertex is one where an even number of edges leads to it or from it.

A vertex is odd if an odd number of edges leads to it or from it.

An Eulerian path is a path that passes along every edge of the network exactly once. AnEulerian circuit is a path that begins and ends at the same vertex and passes along every edgeof the network exactly once.

A Hamiltonian path is a path that visits every vertex exactly once (and which thereforecannot be a circuit). A Hamiltonian circuit is a path that begins and ends at the same vertexand visits every other vertex exactly once.


• use the language of networks appropriately

• draw and interpret network diagrams arising from various practical situations

• identify the degree of a vertex, and demonstrate understanding that a networkmust have an even number of vertices of odd degree

• identify Eulerian and Hamiltonian circuits and paths

• demonstrate understanding of the conditions for a network to have an Euleriancircuit or to have an Eulerian path that is not a circuit, ie:

A connected or traversable network has an Eulerian circuit if and only if every vertex haseven degree.

A connected or traversable network has an Eulerian path that is not a circuit if and onlyif exactly two vertices have odd degree.

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Option 2: Networks


i) Paths and circuits in networks

Students could:◊ investigate the historical Königsberg bridges problem whose solution by Euler

was the origin of network theory◊ identify the physical features to be represented in a network (eg rooms in maths

block as vertices and doors or corridors as edges)◊ plan bus routes to connect a new housing estate to other areas ◊ investigate connectedness by drawing doodles that

(a) start and finish at the same point(Eulerian circuits)

◊ interpret network diagrams used within their community such as for railways,airlines, couriers

◊ distinguish between situations where it is useful to find circuits visiting eachvertex only once, and those in which it is useful to find circuits visiting eachedge only once

◊ draw networks for suburban trains, bus routes, overnight parcel deliveries◊ identify most efficient paths and locations (start and finish at the bus stop

without retracing one’s steps, locate a recycling depot, design a tour of countrytowns by a rock group)

◊ investigate some polyhedra, regular or otherwise, to see whether or not theyhave Eulerian circuits or paths through their vertices and edges

◊ find Hamiltonian circuits through the vertices and edges of a dodecahedron(this was Hamilton’s original game), and generalise to other polyhedra

◊ take a small section of a street map for the local area and draw a network for it.They could investigate the network for the degree of the vertices, number ofedges and whether it is connected, indicating the number of paths one couldtake to get from A to B, considering whether the paths are Eulerian

(continued)

193


(b) start and finish at different points(Eulerian paths that are not circuits)


Option 2: Networks

Content

194



Option 2: Networks


i) Paths and circuits in networks (continued)

Students could:◊ answer questions like:

a) does the following network have an Eulerian path or circuit? If not, whatmodifications are needed to create i) a path ii) a circuit?

b) a courier has to deliver parcels to six places represented by the verticesshown. The edges represent roads. Is it possible for her to start and finish thetrip at vertex 2 (the depot), and visit each place once?

c) a network is formed whose vertices are the squares of a chessboard andwhose edges are the moves of a knight. Show that this network has noEulerian path or circuit, but find one of the many Hamiltonian paths

◊ use network diagrams to analyse games, eg ‘Sprouts’. (This game is played bytwo or more people and begins with a number of dots or vertices, usuallybetween three and ten. Each player takes turns to draw edges between twovertices, placing another vertex on the new edge. A maximum of three edges isallowed for each vertex. The last player able to draw an edge wins.)

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1 2

4

6 5

3

1

5 73

46

2


Option 2: Networks

Content

ii) Using matrices in networks

Learning experiences should provide the opportunity for students to:

• use a matrix to store information from a practical situation, eg networks

• interpret the information presented in a matrix format

• find the order of pairs of matrices and decide whether they can be added orsubtracted

• add and subtract matrices

• multiply a matrix by a scalar quantity

• recognise the zero and identity matrix

• multiply two matrices and recognise that in general AB ≠ BA when A and B arematrices

• interpret powers of a single matrix in terms of networks.

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Option 2: Networks


ii) Using matrices in networks

Students could:

◊ consider matrices for practical situations, as inthe following: a company makes four differenttypes of chairs. The matrix shows the numberof hours needed to assemble, sand and paintthe chairs:

a) what does the row stand for?b) what does the column under B represent?c) add the numbers across each row to form a column vectord) add down each column to produce a row vector and describe the result

e) the cost of labour per hour for assembling, sanding and painting is given bythe matrix . Use this to find the cost of labour to complete atype B chair

f) find the labour cost to make each of the different chairs and represent this ina matrix

◊ use a ruler and compass to draw a map that showsthe distances correctly between 3 cities, X, Y, andZ, as given in the following distance table

◊ for matrices such as , and , investigate

whether the following statements are true: A + B = B + A, A – B = B – A, AB = BA, AC = CA

◊ for the above matrices A, B and C, find 3A, –2B, 4C + A

◊ design a problem that could be represented numerically by a 2 × 2 matrix

(continued)

C =1 0

0 1

B =1 3

4 5

A =2 3

1 4

14 10 9[ ]

5 2 3 4[ ]

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fromX Y Z

toX

Y

Z

0 30 50

30 0 60

50 60 0

ChairsA B C D

5 2 3 4

2 2 1 0

0 1 2 0

assemble

sand

paint


Option 2: Networks

Contents

198



Option 2: Networks


ii) Using matrices in networks (continued)

Students could:

◊ construct a route matrix to describe anetwork. The matrix shows thenumber of paths of one edge betweeneach pair of destinations

◊ square the matrix above. Whatinformation does the new matrixcontain?

◊ examine a diagram such as the one below,showing the line of management in a smallfirm, ie Q is in charge of R and S, and R isin charge of T. An entry of 1 indicates thatthere is a direct line of management, 0 thatthere is no line of management. Studentscould multiply this matrix by itself andconsider what information the elements ofthe new matrix represent.

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A B C D0 1 1 0

1 0 1 2

1 1 0 1

0 2 1 0

A

B

C

D

A D

C

B

Q R S TQ

R

S

T

0 1 1 0

0 0 0 1

0 0 0 0

0 0 0 0

Q

T

R S


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Option 3: Mathematics of Small Business

Paying wagesPaying taxesInvestment

Running costs of small businesses

Considerations

This topic is designed to give students a greater depth of experience in mattersrelating to mathematics, small business and commerce. It is intended that studentsbecome competent with the skills required for aspects of small business such aspaying wages and taxes, investments and running costs.

Students should be familiar with calculations on wages such as overtime, bonuspayments etc from the Consumer arithmetic topic of the core. They should be able toapply these skills to the situation of an employer calculating wages for employees.Students should also recognise that there are forms of taxation other than incometax that an employer has to calculate.

Examples given to students should reflect current practices in employmentconditions (number of hours in working week, overtime rates, rates for casuallabour) and in taxation.

Students should be competent in calculations involving the simple and compoundinterest formulae and with the use of the formula for depreciation. Examples thatreflect current interest rates and terms and conditions of repayment should beused.

Students should be aware of a range of different ways to invest money and be ableto calculate returns such as dividends for shares. Interest calculations could provideopportunities for the use of spreadsheets.

Students should be competent in reading tables of information and aware of thesituations in society in which insurance is applicable. It is not intended, however,that students look at insurance as an application of series. Questions on insuranceand superannuation costs need to be based on current rates. These may beobtained from relevant businesses.

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Content

i) Paying wages


• using completed time sheets or other methods, calculate earnings for a smallnumber of employees, including calculations involving overtime, leave loading,bonus payments, gratuities, allowances, lump sum payments

• make calculations involving deductions from wages such as superannuation andPAYE tax.

ii) Paying taxes


• make calculations involving payroll tax and provisional tax

• describe and calculate sales tax, fringe benefits tax, capital gains tax.

iii) Investment


• use simple and compound interest formulae to compare interest given byfinancial institutions to businesses

• calculate the effective flat rate of return on investments as a means to comparetwo different investments

• make calculations involving inflation

• describe and use the consumer price index

• calculate the dividend yield on shares.

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i) Paying wages

Students could:

◊ complete time sheets for a small number of employees

◊ use spreadsheets or other methods to calculate employee earnings

◊ calculate the cost of holiday leave loading for a business with a small number ofemployees

◊ investigate the cost involved in providing superannuation for employees

◊ use a PAYE table to calculate the tax payable on average weekly wages foremployees

◊ use spreadsheets to model superannuation.

ii) Paying taxes

Students could:

◊ calculate the amount of payroll tax to be paid after considering the total wagesbill for a fortnightly period

◊ research and report on the different types of current taxes for business

◊ calculate sales tax on goods such as cosmetics, books, jewellery etc

◊ investigate capital gains tax on assets relating to a business.

iii) Investment

Students could:

◊ research and report on the different types of investment available by comparinginterest rates for businesses and also by looking at the different types ofinvestment accounts available

◊ find the effective flat rate of interest when $6000 is invested at 5.25% perannum compounding monthly for a year

◊ compare inflation rates in different countries, considering the real value of acertain amount over a fixed time

◊ investigate how the consumer price index is calculated and its effect on the rateof inflation (continued)

203




Content

iv) Running costs of small businesses


• interpret tables of information to calculate cost of insurance for cars, buildings,goods and contents of premises and other forms of insurance for persons and/orproperty

• determine the cost of workers’ compensation

• describe and calculate variable and fixed costs in the running of a smallbusiness

• compare different costs involved in running a business (eg renting v buyingpremises, leasing v buying vehicles and equipment)

• calculate the depreciation of capital items and equipment.

204





iii) Investment (continued)

Students could:

◊ research dividends paid to investors and determine what percentage thisdividend is of (a) the face value of the share; (b) the current market price

◊ find the dividend yield if a dividend of 10% is paid on the face value of $2 ashare if the current market price is $8 per share

◊ look at the price of shares of a particular company and graph its ‘progress’ overa short period of time, determining whether buying the shares would be a‘good’ investment

◊ investigate, by using prepared tables, how much the amount repaid on a loanvaries with the frequency of the payments.

iv) Running costs of small businesses

Students could:

◊ compare the cost of leasing or purchasing business premises and equipment

◊ determine profit for a business after deducting all running costs, and expressprofit as a percentage of turnover (sales)

◊ investigate and report on the cost of insuring a car, a business premise, or theequipment for a business, obtaining and comparing information given byinsurance companies

◊ read tables to determine the cost of workers’ compensation for a small businessthat employs a number of people

◊ investigate and report on the variable and fixed costs (eg rent, payment andmaintenance of equipment, staffing costs, stock costs) for a small business (eg alawn-mowing business, a take-away food business, a shop/store, a factory withless than ten employees)

◊ investigate a situation such as: you are going to start a small business that involvesmaking a product to sell. Prepare a business report that discusses what type ofproduct you are going to manufacture, whether you are making this product athome or in a small factory, the production costs involved, labour costs, andexpected profits (which will be determined by the selling price of the product)

◊ calculate the depreciated value after three years of machinery originally costing$20 000, if the rate of depreciation is 15% per annum.

205



206



207


Option 4: Practical Applications of Measurement

SurveyingNavigation

Navigation on land

Considerations

Students should have some practical experience of the surveying methods they willencounter in this option. Other methods of surveying could be discussed, such asoptical, electronic and aerial methods, along with the appropriateness of such methodsand their orders of accuracy. The section on surveying requires students to be able toapply the sine, cosine and area rules from trigonometry and to work with bearings.This may need to be revised if students are not competent with this work.

The work on navigation should be centred on navigational charts. Students should beencouraged to use the language of navigation (eg parallels of latitude, meridians oflongitude, great circles, poles, tropics, principal compass points, nautical miles andknots) appropriately. A globe can be used to show great and small circles andencourage understanding of navigation. This can be used in conjunction with a flatmap of the world. The distortions could be discussed. Students should plot journeys bylocating up to five positions. Back bearings and simple problems involving speed,distance and time should be encountered.

A practical approach to navigation on land would enable students to develop someorienteering skills through following a course in a park or in the school grounds.Teachers could make use of local SES personnel and class members who belong toScouts, Guides or orienteering clubs, or who are experienced in bushwalking andreading maps. Students could also investigate surveying equipment such as the GlobalPositioning System, which gives very accurate readings for latitude and longitude.



Content

i) Surveying


• choose and use the most appropriate method of measuring horizontal lengths

• draw a perpendicular to a fixed line using a variety of methods and understandthe geometrical justification of each method

• mark out right angles on the ground

• measure horizontal angles using available instruments such as an alidade and acompass

• use a traverse survey (offset method) to obtain the measurements necessary toconstruct a scale drawing of a shape, using the field notebook method to recordthe measurements

• use the radial method to construct a scale drawing of a shape, using both theplane table and compass to measure the angles

• find the perimeter and area of a shape using a scale drawing obtained from theoffset method and the radial method

• construct scale drawings from sets of measurements obtained by triangulationand calculate relevant lengths and angles from the drawings

• calculate the perimeter and area of a suitable shape using a sketch and theappropriate trigonometric formula.

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i) Surveying

Students could:◊ mark out a field in the school grounds and use a variety of techniques to survey

the field◊ use a variety of methods to make a scale drawing of an irregular shaped field

that has been marked out◊ use triangulation to find the length of an inaccessible boundary◊ use field book entries to mark out a shape on the ground◊ given a practical problem, decide on the most appropriate survey technique and

the best method to obtain perimeter and area ◊ answer questions like:

a) use these field book entries to make a scaledrawing and find the perimeter and area of theshape

b) this sketch is the result of a radialsurvey. Use trigonometry to findthe area and perimeter of the field

c) use a plane table and triangulation todraw a scale drawing of a field similarto the one shown

d) the area of a triangular field is 520 m2. Draw a diagram that illustrates theangles and lengths of a possible field.

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410m

440m

365m311°

237° 123°

046° N

270m

S

PQ

R

B46

24 3322 21

17 120A



Content

ii) Navigation


• recognise that Earth is approximately a sphere and identify the important partsof a sphere

• explain the use of parallels of latitude and meridians of longitude to determineand record the position of a point on Earth’s surface

• recognise a chart as a flat representation of the curved surface of Earth

• describe the advantages and disadvantages of using a Mercator Projection(where the lines of latitude and longitude appear at right angles)

• define and use units such as nautical mile and knot

• use the latitude scale of a chart to calculate distances (1° = 60 n miles on a greatcircle)

• locate a position on a chart given its latitude and longitude and determine thelatitude and longitude of a given point on a chart

• use the compass rose to solve problems involving bearings

• describe the difference between true north and magnetic north, and be able tochange true bearings to magnetic bearings and vice versa

• fix positions using a transit fix, a cross-bearing fix, and a vertical angle andcompass fix

• plot courses on a chart by locating a number of positions from suppliedinformation.

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ii) Navigation

Students could:

◊ from a wire model of Earth, identify the important parts of the sphere such as thecentre, radius, diameter, great circles and small circles

◊ relate the great circles and small circles on a sphere to the parallels of latitude andmeridians of longitude

◊ investigate the history of navigation and the use of various map projections

◊ use a globe and cylinder to investigate the Mercator Projection and consider thedistortions that occur

◊ draw a scale diagram to find the position and bearing of a boat from a set of writteninstructions

◊ convert distances in nautical miles to kilometres and knots to km/h and vice versa

◊ answer questions like:a) what distance would be represented by a latitude difference of 3°?b) convert a compass bearing of 160° to a true bearing if the magnetic variation is

8° Ec) find cities or towns that have the same longitude as Tamworthd) what is the difference in latitude between Bangkok and Newcastle?e) two cities have a difference in latitude of 20°. What could they be? How far

apart are these cities?f) find the city closest to the position 32°S, 142°E

◊ give a report on how the Endeavour was navigated to Botany Bay. Explain anydifferences in methods used for navigation today

◊ make up a course for a navigational chart to locate buried treasure at a particularposition.

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Content

iii) Navigation on land


• orient a topographical map by compass

• read a topographical map

• orient a map by physical features

• use a compass proficiently

• navigate a simple course

• set a simple course

• use nature’s compass (sun or stars) to find the four main compass points.

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iii) Navigation on land

Students could:

◊ name and describe the main parts of a compass

◊ explain what a compass bearing is, and set and follow one

◊ explain what a back bearing is, and calculate and check one

◊ pay attention to scale, legend, symbols and contour lines when reading atopographical map

◊ give a written or verbal description of an area after reading and interpreting atopographical map for the area

◊ understand and use grid references

◊ make a simple map

◊ set up and use simple courses around school grounds or local parks

◊ read and interpret pen and paper courses on grid paper with protractors

◊ have excursions to parkland where semi-permanent courses are set up

◊ trace out the following course at a park:1) walk 20m north2) walk 35 m on a bearing 120°3) walk 22 m on a bearing 200°4) take a bearing to the starting position and measure the distance back to this

position. They could then represent this course on paper

◊ obtain a local street map and describe how to travel from the school to the postoffice, giving grid references and compass bearings

◊ set up a simple orienteering course for other students to follow

◊ locate the direction south using the Southern Cross and the pointers.

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Option 5: Further Geometry

Further constructionsProofs of geometrical relationships involving triangles and quadrilaterals

Right-angled triangles and Pythagoras’ theoremCircles

Chord properties of circlesAngle properties of circles

Tangents and secantsProofs using circle theorems

Considerations

In this option, students should progress in their understanding of deductivereasoning and become more competent with formal proof, especially in relation tothe geometric theorems related to a circle. It may be necessary to review the coresection on Geometry, especially the sections on congruence, similarity andreasoning, along with the Years 7–8 work on constructions, if students have notdeveloped competence in these aspects. This option further develops students’facility with constructions involving angles and provides the opportunity forstudents to prove the validity of constructions using congruence.

As well as solving arithmetic and algebraic problems in circle geometry, studentsshould be able to reason deductively within more theoretical arguments. Diagramswould normally be given to students, with the important information labelled onthe diagram to aid reasoning. Students would sometimes need to produce a carefuldiagram from a set of instructions.

While students are not expected to reproduce ‘standard proofs’ by rote, theyshould encounter proofs to theorems and use the theorems related to circles toprove other geometric results. Students should have already begun to develop theability to reason deductively, and gained an appreciation of the importance ofgeometrical language and clear diagrams. They should have developed someclarity and conciseness in their arguments. Their arguments in circle geometryshould be logically developed and succinct, with correct use of geometric languageand symbols. Creative, correct arguments should be encouraged and discussed.Within the core of this course, students will have already encountered the proofthat angles in semi-circles are right angles.

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Some students may benefit from experimentation and empirical investigation ofparticular cases. Sketching, accurate construction, measurement by ruler andprotractor, and the use of templates, graphics calculators and geometrical softwareare all recommended.

Students should use appropriate terminology associated with circles andunderstand statements such as: ‘chord AB subtends ∠ APB at the circumference’and ‘∠ APB stands on the arc ACB’.

Students should be aware that more than one diagram may be needed to cover allthe possibilities encompassed by a general result (eg the angle at the centre of acircle is twice the angle at the circumference standing on the same arc). In thesecases, more than one proof may be needed to prove the general result.

If the theorems on triangles and quadrilaterals have been developed in the corethrough formal proof, the students have already covered some of the material inthis option. Some examples of proofs are provided in the support documentaccompanying this syllabus.

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Content

i) Further constructions


• construct angles of 22.5°, 15°, … , given the vertex and one ray

• explain, using congruence or theorems about special triangles andquadrilaterals, why the following constructions using straight edge andcompasses are valid:– perpendicular bisector of a given interval– bisector of a given angle– angles of 60°, 120°, 90°, 45°, 30°, 22.5°, 15°, …– copy of a given interval or angle – perpendicular to a given line from a given point off the line– line parallel to a given line through a given point not on the line.

ii) Proofs of geometrical relationships involving triangles and quadrilaterals


• prove theorems about triangles:

If two sides of a triangle are equal, then the angles opposite those sides are equal.

Conversely, if two angles of a triangle are equal, the sides opposite those angles areequal.

Each angle of an equilateral triangle is 60°.

• prove theorems about quadrilaterals:

The opposite angles of a parallelogram are equal.

The opposite sides of a parallelogram are equal.

The diagonals of a parallelogram bisect each other.


The diagonals of a rhombus bisect the vertex angles through which they pass.

The diagonals of a rectangle are equal.(continued)

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i) Further constructions

Students could:

◊ construct on a given interval AB, using straight edge and compasses, a rhombusABCD, in which angle A = 30°

◊ construct an isosceles triangle with one side 4 cm and equal angles 22.5°

◊ construct, using straight edge and compasses, a triangle with sides 5 cm, 4 cmand included angle 120°

◊ prove, using congruence, that the method for construction of the perpendicularto a line from a point away from the line is valid.

ii) Proofs of geometrical relationships involving triangles and quadrilaterals

Students could:

◊ prove, using congruence, that the altitude to the base of an isosceles triangle isan axis of symmetry of the triangle

◊ prove the theorem that the line through the midpoint of one side of a triangleparallel to another side bisects the third side

◊ prove that the intervals joining the midpoints of the three sides of a triangledissect the triangle into four congruent triangles, each similar to the originaltriangle

◊ prove that the midpoints of the sides of a quadrilateral are the vertices of aparallelogram

◊ develop tests for special quadrilaterals based on diagonals

(continued)

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Content

ii) Proofs of geometrical relationships involving triangles and quadrilaterals(continued)


• prove the tests for quadrilaterals:

If both pairs of opposite angles of a quadrilateral are equal, then it is aparallelogram.

If both pairs of opposite sides of a quadrilateral are equal, then it is aparallelogram.

If all sides of a quadrilateral are equal, then it is a rhombus.

iii) Right-angled triangles and Pythagoras’ theorem


• demonstrate understanding of, and apply, the converse of Pythagoras’ theorem:

If the square on one side of a triangle equals the sum of the squares on the othertwo sides, then the angle between these other two sides is a right angle.

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ii) Proofs of geometrical relationships involving triangles and quadrilaterals(continued)

Students could:

◊ explain why the constructions in the concentric circles below produce aparallelogram and a rhombus

◊ (E) prove that the three medians of a triangle are concurrent, and that the pointof intersection trisects each median.

iii) Right-angled triangles and Pythagoras’ theorem

Students could:

◊ use Pythagoras’ theorem, and trigonometry if necessary, to find the areas ofregular polygons of given side length or inscribed in a given circle

◊ use the converse of Pythagoras’ theorem to develop an alternative method ofconstructing a right angle using straight edge and compasses, given the vertexand one ray

◊ (E) prove the converse of Pythagoras’ theorem by proving that if ABC is atriangle in which a2 + b2 = c2, then ∠ C = 90° (construct another triangle XYZ inwhich YZ = a, XZ = b and YZ ⊥ XZ, then prove XY = c so that ∆ABC ≡ ∆XYZ)

◊ (E) find alternative proofs of Pythagoras’ theorem using contrasting methodsbased on algebra, or on dissection of areas, or on geometry

◊ (E) read about Pythagoras’ theorem in the ancient world.

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OO



Specific content outcomes

iv) Circles


• identify and name parts of a circle (centre, radius, diameter, circumference,sector, arc, chord, secant, tangent, segment, semi-circle)

• use terminology associated with circles such as subtend, standing on the samearc, angle at the centre, angle at the circumference, angle in a segment

• identify the arc on which an angle at the centre or circumference stands

• demonstrate understanding that at any point on a circle, there is a uniquetangent to the circle, and that this tangent is perpendicular to the radius at thepoint of contact.

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iv) Circles

Students could:

◊ draw a circle using LOGO software and describe how the process works

◊ find the centre of a circle using as many methods as possible and explain themethods

◊ draw a tangent from an external point to a circle using a variety of methods, eg:– using the right-angled corner of a set square– connecting the external point to the centre, bisecting this line and

constructing a semi-circle to intersect the original circle at the tangent point

◊ draw a series of different quadrilaterals in circles and investigate the size of theangles

◊ investigate the inscription of polygons (eg why a hexagon can be drawn in acircle by marking six lengths equivalent to the radius around the circumference)

◊ investigate which quadrilaterals can always have a circle drawn through theirvertices, starting with a square

◊ construct the smallest square that contains a given circle.

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Specific content outcomes

v) Chord properties of circles


• prove and apply the following theorems:

Chords of equal length in a circle subtend equal angles at the centre and areequidistant from the centre.

The perpendicular from the centre of a circle to a chord bisects the chord.

Conversely, the line from the centre of a circle to the midpoint of a chord isperpendicular to the chord.

The perpendicular bisector of a chord of a circle passes through the centre.

Given any three non-collinear points, the point of intersection of the perpendicularbisectors of any two sides of the triangle formed by the three points is the centre ofa circle through all three points.

When two circles intersect, the line joining their centres bisects their common chordat right angles.

vi) Angle properties of circles


• prove and apply the following theorems:

The angle at the centre of a circle is twice the angle at the circumference standingon the same arc.

Angles at the circumference, standing on the same arc, are equal.

The opposite angles are supplementary in cyclic quadrilaterals.

If opposite angles of a quadrilateral are supplementary, the quadrilateral is cyclic.

An exterior angle at a vertex of a cyclic quadrilateral is equal to the interioropposite angle.

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v) Chord properties of circles

Students could:

◊ answer questions like: in the circle,centre O, two chords LM and MNare drawn. OX = OY as shown.MN = 14 cm. Find the length of LX(give reasons for your answer)

◊ find the radius of a circle in which a chord of length 14 cm:a) is 10 cm from the centre b) subtends an angle of 70° at the centre

◊ construct, using straight edge and compasses, the centre of a given circle andthe circumcentre of a given triangle

◊ prove that two chords have equal length if they subtend equal angles at thecentre, or if they are equidistant from the centre

◊ prove that if two circles intersect, then the interval joining their centres is theperpendicular bisector of their common chord

◊ (E) prove that the perpendicular bisectors of the sides of a triangle areconcurrent, the point of intersection being the centre of the circumcircle of thetriangle.

vi) Angle properties of circles

Students could:

◊ prove that the angle at the centre of a circle is twice the angle at thecircumference standing on the same arc, using the diagrams below

◊ prove that four points A, B, C and D are concyclic if the opposite angles of thequadrilateral ABCD are supplementary

(continued)

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O

X Y

M

L N

O

A B

CO

A B

C

O

C

A B



Content

224





vi) Angle properties of circles (continued)

Students could:

◊ prove that is the diameter of the circumcircle of ∆ABC and hence provethe sine rule

◊ answer questions like:a) ABCD is a cyclic quadrilateral in a circle, centre

O. If ∠ ADC = 138°, find the size of ∠ AOC,stating brief reasons for your answer

b) in the diagram, ∠ AOB = 116°. Find the size of∠ OBC, giving reasons

◊ prove that an exterior angle of a cyclic quadrilateral is equal to the interioropposite angle

◊ complete, from given diagrams, the other two cases of the proof of the theoremabout angles at the centre and at the circumference

◊ (E) prove the converses of each of the given theorems.

asin A

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O

D

138°

B

A C

O

116°

C

A B



Content

vii) Tangents and secants


• prove and apply the following theorems related to tangents and secants:

A tangent to a circle is perpendicular to the radius drawn to the point of contact.

The angle between a tangent and a chord drawn to the point of contact is equal tothe angle in the alternate segment.

The two tangents drawn to a circle from an external point are equal in length.

When two circles touch, their centres and the point of contact are collinear.

The products of the intercepts of two intersecting chords of a circle are equal.

The product of the intercepts of two intersecting secants to a circle from an externalpoint are equal.

The square of a tangent to a circle from an external point equals the product of theintercepts of any secant from the point.

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vii) Tangents and secants

Students could:

◊ answer questions like: AD is a tangentto a circle, centre O. C is a point on thecircle and AC cuts the circle at B suchthat AB = 2 cm and BC = 5 cm. Findthe length of AD

◊ construct, using straight edge and compasses, the tangent to a given circle at agiven point on the circle, the tangents to a given circle from a given externalpoint, and the circle with given centre and tangent

◊ find how long the tangents will be from P to the circle, if a point P is 20 cmfrom the centre of a circle of radius 6 cm

◊ prove that when two circles touch externally, the tangents to the two circlesfrom any point on the common tangent are equal

◊ prove that when two circles touch internally or externally, the interval joiningtheir centres is perpendicular to their common tangent

◊ (E) assuming that at each point of a circle there is a tangent to the circle, andthat every tangent lies entirely outside the circle apart from the point of contact,prove by contradiction that this tangent is perpendicular to the radius at thepoint of contact, and is therefore unique

◊ (E) prove that the three angle bisectors of a triangle are concurrent, and thattheir point of intersection is the centre of the incircle of the triangle

◊ (E) prove that if three differently sized coins are laid flat on a table, eachtouching the other two, then the three common tangents are concurrent.

227


D

A�B

C



Content

viii) Proofs using circle theorems


• recognise when a deductive exercise requires the circle theorems, identify theappropriate theorems and use them to prove other results in geometrical figures.

228





viii) Proofs using circle theorems

Students could:

◊ answer questions like:a) AB and CD are two chords of a circle,

meeting at the point X. Show that trianglesAXC and BXD are similar and hence provethat AX × XB = CX × XD

b) PQRS is a cyclic quadrilateral. Side PQ hasbeen produced to T so that PTRS is aparallelogram. Prove that RQT is an isoscelestriangle

c) A, B and C are three points on a circle whereAC = BC. CD is a tangent to the circle. Provethat AB || CD.

d) in the diagram, O is the centre of twoconcentric circles. ABCD is a straight line.Prove that AB = CD

e) AT is a tangent and is parallel to BP. ACP is astraight line. Prove that ∠ ABP = ∠ ACB.

229


C

AX

D

B�

P

S

T

R

Q

C

A B

OA B C D

C

A

B P

T


230



Option 6: Curve Sketching and Polynomials

Curve sketchingPolynomials

Sketching polynomials

Considerations

Students need a firm grounding in curve sketching if they are to proceedsuccessfully to 2 and 3 Unit Mathematics in Year 11. This option is designed toencourage students to investigate the transformations (vertical and horizontaltranslations and reflections) that are made to a curve by the inclusion of a constantterm in the appropriate place in the original equation. In this way, the optionrelates to the transformations studied in Years 7 and 8.

Mathematical templates, graphics calculators and graphics software on computers canfacilitate exploration of the effects of the inclusion of a constant in different positions.

The second section of this option concerns polynomials. Students need to be ableto use the notation P (x) and P (a) and to recognise polynomials from a list ofexpressions. They should see the connection between polynomials and number,especially through their relation to place value if the variable is replaced by 10.Students should have a clear understanding of the difference between a polynomialexpression and a polynomial equation. Division of polynomials would usually berestricted to division by a linear expression, although students could be takenfurther here if appropriate. Students should be able to develop the remaindertheorem intuitively through experimentation with quadratic and simple cubicpolynomials. While they need not reproduce the proof of the remainder theorem,they should understand the proof and see the connection to the factor theorem.

In the third section of this option, connections are made between the initial sectionon curve sketching and polynomials. Students should be able to factor apolynomial of degree ≤ 4 and sketch its graph. They should be able to sketchvariations of this polynomial, ie y = –P (x), y = P (–x), y = P (x) + c, y = aP (x).Graphics calculators or graphics software on computers can be a very useful toolfor exploring the different graphs.

Teachers may choose their own order of treatment in this option as appropriate.For example, teachers may wish to do the curve sketching section using functionnotation. In this case, they could complete the section on Functions (Option 1), orparts thereof, first.

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Content

i) Curve sketching


• graph equations of the form y = ax3 and describe the effect on the graph fordifferent values of a

• graph equations of the form y = ax3 + d and describe the effect on the graph ofdifferent values of the constant d

• graph equations of the form y = a(x – r)(x – s)(x – t), relating the zeros of theequation to the x-intercepts and relating the constant term to the y-intercept

• graph a variety of equations of the form y = axn for n > 0, describing the effectof n being odd or even on the shape of the curve

• by vertical transformations, graph curves of the form y = axn + k from curves ofthe form y = axn

• by horizontal transformations, graph curves of the form y = a(x – r)n after firstgraphing curves of the form y = axn

• establish the equation of the circle centre (h,k), radius r, and graph equations ofthe form (x – h) 2 + (y – k) 2 = r 2

• recognise and describe the algebraic equations that represent circles

• find the centre and radius of a circle whose equation is in the form x 2 + gx + y 2 + hy = c, by completing the square

• find the points of intersection of a line with a parabola, circle or hyperbola,graphically and algebraically.

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Option 6 : Curve Sketching and Polynomials


i) Curve sketching

Students could:

◊ sketch y = x3 and consider questions like:a) what happens to the value of y as x becomes very large or very small?b) what happens near and at x = 0?c) is there a y value for every x value? Is there an x value for every y value?

◊ sketch y = x3 and hence sketch the curves y = x3 – 3 and y = (x – 3)3

◊ by considering the zeros of the equation and the y-intercepts, sketch y = (x – 2)(x + 3)and sketch y = (x – 2)(x + 3)(x – 5)

◊ sketch the curve y = x(x – 3)(x + 3)

◊ compare the sketches of the curves y = x3, y = x4, y = x5 and y = x6 using agraphics calculator or graphing package, and describe the difference betweenthe graphs whose equations have an even index and those which have an oddindex

◊ from the sketch of the curve represented by y = x4, sketch the curvesrepresented by y = (x – 2)4, y = (x – 1)4, y = (x + 4)4, y = (x – 1.5)4 and describethe effect on the graph of y = x4 of the constant a when y = (x – a)4

◊ by translation or using Pythagoras’ theorem, investigate the equation thatrepresents all points that are 2 units from the point (2, –3)

◊ recognise equations like x2 + y2 = 16, (x – 3)2 + (y + 5)2 = 25, x2 + 6x + y2 – 4y = 3as equations of circles and find the centre and radius for each

◊ find the x- and y-intercepts of circles

◊ find the points of intersection of curves such as:a) y = x – 2 and y = x2 – 4b) x2 + y2 = 25 and x + y = 7c) xy = 2 and y = 4 – x.

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Content

ii) Polynomials


• recognise a polynomial expression anxn + an–1 xn–1 + … + a1x+ a0 and use theterms degree, leading term, leading coefficient, constant term and monicpolynomial appropriately

• describe how polynomials show generalised number patterns

• use the notation P (x) for polynomials and P (c) to indicate the value of P (x) at x = c

• add and subtract polynomials and multiply polynomials by linear expressions

• divide polynomials by linear expressions to find the quotient and remainder,expressing the polynomial as the product of the linear expression and anotherpolynomial plus a remainder, eg P (x) = (x – a)Q (x) + c

• verify the remainder theorem and use it to find factors of polynomials

• use the factor theorem to factorise certain polynomials completely, ie if (x – a) isa factor of P (x), then P (a) = 0

• use the factor theorem and long division to find all zeros of a simple polynomialand hence solve P (x) = 0 (degree ≤ 4).

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ii) Polynomials

Students could:

◊ relate a polynomial equation, such as P (x) = 5x 4 + 2x 3 + x 2 + 4 to place valueby writing P (10) = 5 × 104 + 2 × 103+ 102 + 4 = 52 104

◊ answer questions like:a) for P (x) = 3x 3 – 7x + 2, find P (2), P (0)b) if P (x) = x + 1 and Q (x) = 3x4 – 5x + 2, find:

i) P (x) + Q (x)ii) Q(x) – P (x)iii) P (x) × Q(x)iv) Q(x) ÷ P (x)

c) divide 3x 4 – 5x + 2 by (x – 5) and hence find the remainder. Write 3x 4 – 5x + 2 as the product of (x – 5) and another polynomial plus the remainder, and verify the remainder theorem

d) if x 3 + ax + b is divisible by both x + 2 and x – 3, find the values of a and b

◊ solve a problem like: find an expression for the volume of a box if the box hasa square base and a height that is 5 cm longer than the length of the base. If thevolume is to be 28 cm3, find a polynomial equation that represents thisinformation, and find the dimensions that would give this volume

◊ show that x – 3 is a factor of P (x) = x 3 – 4x 2 + x + 6

◊ find all the factors of the polynomial x 3 – 2x 2 – 5x + 6

◊ solve equations like x 2 (x – 3) = 6x – 8.

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Content

iii) Sketching polynomials


• recognise linear, quadratic and cubic expressions as examples of polynomialsand sketch the graphs of quadratic, cubic and quartic polynomials by factorisingand finding the zeros

• determine the significance of single, double and triple roots of the polynomialequation on the shape of the curve

• use the leading term, the roots of the equation and the x- and y-intercepts tosketch polynomials

• state the number of zeros that a polynomial of degree n can have

• determine the importance of the sign of the leading term of the polynomial onthe behaviour of the curve as x → ±∞

• use the sketch of y = P (x) to sketch y = –P (x), y = P (–x), y = P (x) + c, y = aP (x).

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iii) Sketching polynomials

Students could:

◊ answer questions like:a) solve 3(x + 1)2 (x – 2)(2x – 7) = 0b) solve x 3 – x2 – 10x – 8 = 0

◊ using a graphics calculator or otherwise, investigate the effect on the graph of apolynomial of adding, subtracting or multiplying a polynomial by a constant

◊ sketch:y = (x – 2)(x + 1)2

y = – (x – 2)2 (x + 1)y = (x – 2)3 (x + 1)y = x (x – 2) (x + 1)(x – 5)

◊ sketch y = – x3 + 4x2 – x – 6 by first finding the zeros

◊ given P (x) = (x – 3)2 (x + 1)3, sketch the curve of y = P (x)

◊ answer questions like:a) can a cubic equation have no roots? Give examples of cubic equations that

have one, two, or three rootsb) a polynomial of degree 3 has a double root at 2 and a single root at –1. Write

an equation that could represent this polynomial and sketch the curve forthis equation

c) y = P (x) is represented in thediagram opposite. Use it to helpyou sketch the following:

y = P (x) + 1y = P (x) – 2y = 3P (x)y = –P (x)y = P (–x)y = P (x)

(continued)

12

237


-2

-4

3

y

x



Content

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iii) Sketching polynomials (continued)

Students could:

◊ using a graphics calculator or otherwise, sketch polynomials of odd and evendegrees and investigate the relationship between the number of zeros and thedegree of the polynomial

◊ produce a sketch of a polynomial and describe the key features of their sketchesto other students, so that students can reproduce the curves without having seenthem

◊ using a graphics calculator or otherwise, find the maximum value of apolynomial

◊ using a graphics calculator or otherwise, sketch y = x3 + x2 + 1 and y = x3 + x2 + x + 1 and discuss the similarities and differences in the graphs.Students might consider the extent of the similarities and differences when x is avery small negative value or a very large positive value.

239



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Option 7: Functions and Logarithms

FunctionsLogarithms

Considerations

Throughout the core, students have had experience with developing rules andsketching graphs of relationships, but have not used function notation orconsidered the definition of a function. Functions could be introduced byconsidering a function ‘machine’ (which assigns to every allowable input value x aunique output value y), or a set of ordered pairs where for each x value there isonly one y value. Through this option, students become familiar with functionnotation and the idea of a variable, initiated in the core in the Algebra strand. It isnot intended that formal definition of domain or range be included at this stage,however students should be able to discuss which values of x are valid inputs andwhich values of y are possible outputs for a given equation.

In the Further reasoning in algebra section of the syllabus, students have had experiencein choosing a graph that represents a physical phenomenon and describing the keyfeatures of the graph. By starting with practical situations, students should developthe ideas of dependent and independent variables, and the values of the variables forwhich the relationship makes sense. Experience in defining the variables anddescribing the features of a variety of graphs of functions will provide students withan opportunity to gain further understanding of variables and functions.

Students could explore inverse functions initially by considering the result ofexchanging the variables in linear functions. Graphics calculators and graphicssoftware on computers could facilitate this exploration. Once students see that theresulting graph is a reflection of the original graph through the line y = x, and thatthe inverse function undoes what the original function did, they might explore theinverses of other monotonic functions.

It is vital that students understand that logarithms are indices, and relate thelogarithm laws to the index laws. Students have had experience in reading andinterpreting graphs drawn using indices in the Measurement strand. This optionshould be related back to their previous experience of Richter scales, decibels andpH scales as examples of where logarithmic scales are used in practical situations.

The graph y = logax should be related to the graph of y = ax so that studentsunderstand that the exponential function is the inverse function for the logarithmicfunction and the logarithm of the exponential function gives the original function.

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Content

i) Functions


• sketch graphs to model relationships that occur in practical situations anddescribe the relationship between the variables represented in the graph

• define a function as a rule or relationship where for each input there is only oneoutput, or that associates every member of one set with exactly one member ofa second set

• use the vertical line test on a graph to decide whether it represents a function

• use the notation f (x) to describe functions

• translate a problem in words to an expression in function notation whereappropriate

• use f (c) notation and replace variables in a function by a constant, and evaluatethe result

• substitute an algebraic term into a function and write an expression for theresult in simplest form

• find the permissible x and y values for a variety of functions (including straightlines, parabolas, exponentials and hyperbolas)

• determine the inverse functions for a number of linear functions and recognisetheir graphs as reflections through y = x

• explain the conditions for a function to have an inverse function

• given the graph of y = f (x), sketch y = f (x) + k and y = f (x – a).

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i) Functions

Students could:

◊ consider a range of graphs that represent practical situations, describing whatthe two variables are and what the graph indicates about the variables

◊ decide whether the graphs below represent functions and explain why

◊ decide whether a straight line graph is a function always, sometimes or never

◊ answer questions like:a) for f (x) = 2x – 3, find the following:i) f (2) ii) f (–1) iii) f (a + 1) iv) f (3a) v) f (a 2)b) the distance fallen by a stone dropped from a cliff is a function of time since

being dropped. The function is given by f (t) = 4.9t 2, where t is the time inseconds and the distance f (t) is in metres. Describe the variables involvedand the relationship between them. How far has the stone fallen after 2seconds? What is meant by f (1.5)? Describe the type of function this is

c) for the equation y = 3x + 6, interchange the x and y variables and form a newequation with y the subject. Sketch both straight lines. What do you noticeabout the two graphs? Describe any other interesting features

d) find the inverse function for linear functions, eg f (x) = 5x, f (x) = 3 – x, f (x) =

(continued)

x – 32

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price

size height time radius

mass value volume

y

x

y

x

y

x



Content

ii) Logarithms

The logarithm of a number to any base is the index when the number is expressed as a powerof the base. Alternatively, ax = y ⇔ logay = x where a > 0.


• define logarithms as indices and translate index statements into equivalentstatements using logarithms and vice versa

• use the language of logarithms accurately

• deduce the following laws of logarithms from the laws of indices:

logax + logay = loga(xy)

logax – logay = loga( )

logaxn = n logax(continued)

xy

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i) Functions (continued)

Students could:e) decide which graphs represent functions from the graphs illustrated. What x

values are possible inputs and what y values are possible outputs? Whichgraphs would have inverse functions?

◊ discuss the restrictions that need to be placed on quadratic functions so thatthey have an inverse function, by first considering why y = x 2 does not have aninverse function

◊ sketch the graph of f (x) = x 2 for x > 0, and its inverse

◊ investigate the inverse function of y = x 3.

ii) Logarithms

Students could:

◊ research the historical use and application of logarithms

◊ construct a simple slide-rule using logarithmic graph paper

◊ translate index statements such as those following into statements using logarithms:

9 = 32, = 2–1, = 8

◊ evaluate expressions as in the following:log28 log813log1025 + log1043log102 + log10(12.5)log218 – 2log23

(continued)

43

212

245


y

x3

y

x-2 2

4y

x

(1,1)

(-1,-1)



Content

ii) Logarithms (continued)


• establish the following results:

logaax = x

logaa = 1

loga1 = 0

loga( ) = – logax

• apply the laws of logarithms to simple calculations

• draw the graphs of y = ax and y = logax for a > 0

• recognise y = ax and y = logax as inverse functions

• solve simple equations that contain exponents.

1x

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ii) Logarithms (continued)

Students could:

◊ solve equations involving indices, eg 2t = 8 and 4t+1 =

◊ use guess and check or other appropriate methods to solve equations like 3x = 5

◊ evaluate the following expressions:

log416

log5(25 )

log9( )

◊ simplify expressions using logarithm laws, eg 5logaa – logaa4

◊ expand expressions using logarithm laws, eg loga( ), logy

◊ solve equations involving logarithms, such as the following:log273 = xlog4x = –2logx144 = 4

◊ relate logarithms to practical scales that use indices (Richter, decibel and pH)

◊ use the graph of y = log10x to find the value of expressions like log103 and tosolve equations like log10x = – 0.6

◊ compare the graphs of y = 2x and y = log2x, relating the graphs to earlier workdone on inverse functions

◊ interchange the x and y variables in the equation y = 3x and make y the subjectof the equation, taking logs to base 3 of both sides. They could discuss thesignificance of the result in relation to inverse functions and describe in generalterms the inverse function of y = ax

◊ estimate the value of expressions like log38 from the graph y = log3x

◊ sketch y = 2x , y = log2x, y = 3x and y = log3x on the same set of axes andcompare the graphs. What similarities and differences are there?

x(a + b)y2

x2

y

127

5

1

8 2

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248



Option 8: Modelling

Considerations

In this option, students will apply mathematics to everyday situations within theirexperience and to unfamiliar situations.

Mathematical modelling is the process of setting up a model of a situation in orderto formulate a mathematical description; finding and interpreting the mathematicalsolution; and relating it back to the original context.

The modelling process can be structured in a number of ways:

• The class can be given the situation and the teacher directs the process,selecting the model to be used (known as structured modelling).

• The class can be divided into groups and allowed to work through the process,choosing the mathematics to be used. The reporting session would require eachgroup to justify its choice of model and to evaluate the model’s strengths andweaknesses (known as open modelling).

Some teachers may use a combination of open modelling and a more structuredapproach. In some instances, special techniques may be required for the solution ofa modelling problem.

Students should become familiar with the modelling process. The emphasis is notupon the learning and application of models but upon the process through whichmodels are created and modified. It is very important that the assumptions madein setting up the model are discussed and evaluated.

The modelling process

249


Real-worldproblem

Makeassumptions

Formulatemathematicalproblem

Verify themodel

Interpret thesolution

Solve themathematicalproblem

Producereport, explain,predict etc


Option 8: Modelling

Content


• construct a model of a real situation

• identify the assumptions made in developing the model

• formulate a mathematical description of the situation

• use the mathematical formulation of the situation to obtain one or moresolutions

• interpret the mathematical solution

• relate the solution to the real situation

• evaluate and modify the mathematical model used (and repeat the modellingprocess)

• prepare and present a report on the situation considered and the modellingprocess used

• use the modelling process to investigate everyday situations within theirexperience, eg typical calculus problems without calculus

• use the modelling process to investigate unfamiliar situations such as rates ofchange, using algebra and spreadsheets (precalculus) or constructing indexes bycombining specific databases

• use the modelling process to investigate unfamiliar situations that requirespecific modelling techniques such as linear programming, logarithms orqueuing, scheduling and bin packing.

250



Option 8: Modelling


General modelling problems

Students could:

◊ investigate optimisation problems without calculus where they apply previousknowledge to unfamiliar situations, eg a farmer has 100 m of fencing and wishesto fence off a corner of a rectangular field. What is the maximum area that canbe enclosed by the fence?

◊ investigate the conditions (eg speed and number of car lengths between eachcar) to maximise the traffic flow through a tunnel

◊ investigate the timing of temporary traffic lights that would provide the mostefficient flow of traffic during roadwork on a bridge

◊ given the area of land and other constraints, design a parking station to providethe maximum number of spaces for cars

◊ design the most efficient route for paper delivery

◊ construct indexes by combining specific databases, appreciate the difficulties ofcombining databases and investigate various statistical procedures for weightingand combining various databases

◊ solve problems like: a) a new street of nine houses has just been built. A post box is to be situated in

the fairest place. Where should it be placed?b) how many days would it take for Lake Eyre to dry out?c) which is more economical, buying or renting the house in which you are

living?d) using UNESCO databases for calorie intake, life expectancy and health care,

construct an index to measure the quality of life of the countries listed in thedatabases.

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Option 8: Modelling

Content

252



Option 8: Modelling


Problems arising from modelling requiring specific techniques

Bin packing

Students could:

◊ investigate a way of storing a large number of files (of given sizes all less than1.4 Mb) on a series of floppy disks that store 1.4 Mb each

◊ answer a question like: ‘what would be the minimum number of keyboardoperators, all having equal skill, that must be hired to type (within six hours) aseries of projects requiring the following times (in mins): 30, 60, 90, 60, 60, 30,90, 90, 30, 60, 30, 30, 60, 60, 30, 30?

Queuing

Students could:

◊ investigate or simulate queuing patterns in a shop and determine the number ofcashiers required for efficient movement of customers past the cash registers.

Linear programming

Students could:

◊ use linear programming to identify and find the points of intersection of a set oflinear graphs using algebraic or graphical means, identify the half planecorresponding to a linear inequality, recognise the region determined by a set oflinear inequalities and use linear programming to find the solution to twovariable problems

◊ solve a problem like: a company makes two products, A and B. It must produceat least 300 of A and 200 of B each week to keep its staff fully employed. Itcannot produce more than 600 of product A or 500 of product B in any givenweek, and its total production of A and B cannot exceed 1000 items per week.If there is a $3 profit on A and $5 on B, how many of each product should itproduce each week to maximise its profits?

Logarithms

Students could:

◊ use semi-log paper to graph the world population over a number of years andpredict the population in the future.

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mathematics - arc · 2003-03-31 · philosophy continues that of the nsw mathematics k–6 and...

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