Mathematics - ARC · pedagogy for mathematics since the previous Mathematics 9–10 Syllabus, written in the early 1980s. The approach of this Mathematics 9–10 Syllabus reflects

MathematicsYears 9–10

Syllabus

Standard Course

Stage 5

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

© Board of Studies NSW 1996

Published by the Board of Studies NSWPO Box 460North Sydney NSW 2059AustraliaTel: (02) 9927 8111

ISBN 0 7310 7504 8

August 1996

96201

96201 1Intro Maths 9-10 9/7/96 3:12 PM Page b

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 13

Communication — The Role of Language 15

Collecting, Analysing and Organising Information 15

Using Technology 16

Working With Others and in Teams 16

Planning and Organising Activities 17

Teaching Strategies 17

Programming 18

Syllabus Structure 18

Summary of Years 9–10 Standard Course — Core 23

Summary of Years 9–10 Standard Course — Options 24

Outcomes 25

Assessment 34

Evaluation of School Programs 38

Content Mapped to Themes 40

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Standard Course Content — Core 51

Theme 1: Mathematics of Our Environment 53Theme 2: Mathematics Involving Food 65Theme 3: Mathematics in the Workplace 77Theme 4: Building Design 89Theme 5: Mathematics Involving Sport 105Theme 6: Mathematics in the Community 115Topic 1: Geometrical Facts, Properties and Relationships 131Topic 2: Pythagoras’ Theorem 143Topic 3: Chance 149Topic 4: Introductory Algebra 157

Standard Course Content — Options 169

1. Mathematics Involving Handcrafts 1712. Tourism and Hospitality 1833. Geometrical Patterns 1934. Trigonometry 1995. Further Geometry 2056. Further Number 2137. Further Measurement 2198. Further Algebra 227

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IntroductionMathematics 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

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

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Mathematics Years 9–10 Syllabus — Standard Course

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


RationaleMathematics 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 studentcommunication. 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.

7

Rationale


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. (p (i))

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 CoursesThe 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 these courses. The number of new conceptsand level of difficulty is less than in the Advanced course. The Intermediate courseis designed for those students who require more time than those doing theAdvanced course to develop their mathematical ideas, and for students who arestill developing a more abstract approach to mathematical thinking.

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 higher

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cognitive 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:

9

The Three Courses

Advanced Course

3/4 Unit

2 Unit

Mathematics inSociety

(2 Unit General)

Mathematics inPractice

IntermediateCourse

Standard Course


AimThe 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 stage ofmathematical development and in preparation for further study of Mathematics

• 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.

ObjectivesStudents 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

• knowledge, understanding and skills in Chance and data

• knowledge, understanding and skills in Algebra.

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Equity Principles and IssuesThis 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 and secondary educationsuited to the needs of the full range of students;

• to assess student achievement and award credentials of international standards to meet theneeds 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 the curriculum andparticipating 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 of sex stereotyping.Ibid.

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.Aboriginal world-views emphasise an intelligent responsiveness to the environmentthat is characterised by cooperation and coexistence 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.)

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Equity Principles and Issues


Effective approaches to teaching and learningThe 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 maths to the students’ lives, using materials as well as making explicitconnections between the concrete and the abstract, will help the students to gain afirmer understanding of new words and their meanings as well as their associatedconcepts. A variety of teaching methods, including group work, working in pairs,working outdoors and working with materials, helps to create an environmentconducive to learning.

The language of maths is often the same as everyday language. This can add tosome 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 issuesResearch 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.

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 that will complement and facilitate

good teaching practice• a wide range of applications, suggested activities and sample questions that

emphasise the use of relevant problems in the learning process• three courses in Stage 5 so that students’ mathematical needs can be more

appropriately met• a number of option topics that provide additional flexibility for catering for

student needs• suggestions for teaching strategies which include group work, discussion and

active participation• a range of suggested assessment methods and strategies through which to identify

students’ achievement within a range of modes.

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Solving ProblemsA 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 importance to society. Problem solving should encourage studentsto be systematic when recording information and to persevere.

Four important elements of problem solving are detailed below.

1. Understanding the problemTeachers 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 ensure

appropriate 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 solutionPlanning a solution involves categorising a problem and then knowing theappropriate procedures for that type of problem. Teachers can help students plansolutions by:

• facilitating students’ schema acquisition. This can be helped by reducing students’cognitive load through providing goal-free or open-ended questions, integratingtext with diagrams and encouraging the study and development of workedexamples

• discussing plans for solving problems• organising group activities in which students sequence plans for solving

problems.

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Solving Problems


This syllabus provides a number of sample questions that are open-ended. Suchquestions are also useful when assessing student achievement, since they allowstudents 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 befacilitated by students working in groups, listening to and discussing the responsesof others.

3. Implementing the plan to find the solutionTeachers 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 foundation of basic mathematical ideas on which tobuild their understanding

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

• encouraging competence with routine skills so that students can carry out thesolution phase of a problem.

It is intended that such strategies as those above would be encouraged throughoutthis course.

4. Looking backReflection 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:

• discussing errors and different solutions

• ensuring that students write up their work, including:

– a statement of the problem in their own words– all the necessary working– a statement of what has been discovered– some discussion of how the problem was solved– some of the ideas which may have helped along the way.

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Communication — The Role of LanguageThe Mathematics 9–10 Syllabus makes a significant contribution towards thedevelopment of the Key Competency, 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 within the theme Mathematics in the community.

Students’ command of language dramatically affects the quality of learning inschool mathematics classrooms. Students need to develop both a deepunderstanding of the meaning of mathematical vocabulary and facility incommunicating their understanding to others. This understanding will allow themto use the mathematics terms meaningfully, both inside and outside school. Beyondmathematical vocabulary, unravelling semantic structure places significantdemands on students’ problem-solving skills. For example, for the problem ‘Thereare twelve times as many sheep (s) as people (p), write this relationship in symbols’,many students will write ‘12s = p’. Students can lose the meaning of the wordsbecause of the sentence structure. They need to focus on semantic structure ratherthan a key-words approach. This syllabus supports the teaching of mathematics tolink learners’ personal worlds with their formal mathematical skills and theirformal mathematical language.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 InformationThis 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 key questions prior to investigation, to conduct aninvestigation and to make informal evaluations of the results of the investigation.

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Collecting, Analysing and Organising Information


Using TechnologyMathematics provides an opportunity for students to use materials and equipmentin a manner that constitutes a process and 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 computers is optional, but issuggested in the Applications, suggested activities and sample questions sections toenhance the teaching and learning of mathematics. Some schools may not haveaccess to these tools. It is important to recognise that this course can be taughtsuccessfully without the use of computers, but that the appropriate use of suchtechnology within this course will enhance students’ mathematical learning.

Technology also has a role in assisting students with special education needs togain access to the mathematics curriculum. A computer may assist students whohave a physical disability. For example, students who are unable to write may beable to use 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 TeamsExperience 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 andconfidence in mathematics. Group work should be carefully managed — studentsneed to be very clear about their tasks and each member of a group should begiven responsibility 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 on

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the statistical investigation or on shorter investigations. Such students can developtheir awareness that working with others requires them to establish group goalsand 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 ActivitiesThe statistical investigation also allows students to develop the capacity to plan andorganise their own activities. This involves the ability to set goals, establishpriorities, implement a plan, manage resources and time, and monitor one’s ownperformance.

Teaching StrategiesTo allow students to achieve the outcomes of this course, a range of teachingstrategies must be employed. If students are to improve their mathematicalcommunication, for example, they must have the opportunity to discussinterpretations, solutions, explanations etc with other students as well as theirteacher. They should be encouraged to communicate not only in writing but orally,and to use diagrams as well as numerical, algebraic and word statements in theirexplanations.

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 backgrounds 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 anotherstudent, or works independently. All arrangements have their place in themathematics classroom.

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Teaching Strategies


ProgrammingThere is no need for teachers to teach a whole theme at once. Rather, the contentof each of these courses, being spiral in nature, provides for the themes and topicsto be revisited and concepts developed further over time as students mature intheir understanding of mathematics. It is not intended that each theme or topicoccupy the same amount of time. Within each theme especially, many conceptsrelate to aspects of other themes and topics. This interrelatedness is fundamental tomathematics and should be included in students’ learning experiences. TheConsiderations at the beginning of each theme and topic discuss these relationshipsand identify areas where connections should be made. The incorporation of issuesraised in these considerations encourages students to view the course as a wholeand helps them to appreciate the interrelatedness of mathematics. The supportdocument that accompanies this syllabus provides further advice on programming,along with some sample formats for programs and sequences of teaching for eachcourse.

Syllabus StructureThe Mathematics Years 9–10 Syllabus: Standard Course, is divided into core andoptions as below:

The core is divided into six themes and four topics, each of which must be studiedby all students. All the content of the core is to be studied. The themes and topicsare:

Mathematics of our environmentMathematics involving foodMathematics in the workplaceBuilding designMathematics involving sportMathematics in the communityGeometrical facts, properties and relationshipsPythagoras’ theoremChanceIntroductory algebra.

As well as the core, eight options are provided, two of which are thematic.

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Core (160 hours minimum)(Themes and topics)

Options(40 hours minimum)


Teachers must ensure that at least 40 hours (indicative) are spent in the study of theoption topics included here.

Each theme, topic and option is introduced by Considerations which teachers shouldread before teaching sections of the course. The considerations raise issues relatedto the teaching and learning of the concepts and skills in the theme, topic oroption. The issues relate to:

• the syllabus aim and objectives

• the syllabus outcomes

• possibilities for integration with other themes, topics and subjects

• language development

• assumed knowledge and skills from Stage 4

• other specific aspects of the syllabus.

A set of grids has been included before the core content, which maps themathematical content to the themes. In the content, each theme is followed by asummary of the mathematical content of that theme.

Content in themes and topicsThe mathematical and thematic content statements (core topics and options)describe in detail what mathematics students should know, understand and be ableto do as a result of appropriate and relevant learning experiences facilitated by theteacher. These statements guide teachers on the extent and depth of treatmentexpected. The content provides the basis for the achievement of syllabus outcomesand includes the skills that students acquire as they undertake the learningexperiences described.

For each topic, the content statements have been grouped under subsections andhave been arranged somewhat sequentially. This does not imply that teachers mustfollow this particular sequence when teaching each section.

The thematic content has also been grouped under subsections. The extent anddepth of the mathematics to be treated within each theme has been furtheramplified by statements at the end of each theme. These statements ofmathematical content have been organised within the strands of Geometry, Number,Measurement, Chance and data and Algebra. Teachers may wish to teach a theme byfirst teaching some or all of the mathematical content and then reinforcing thiscontent within the context of the theme. Alternatively, teachers may wish to teachthe theme and emphasise the mathematical skills when appropriate. This does notimply that teachers must follow a particular sequence when teaching each theme.However students must be provided with the opportunities to acquire themathematical knowledge, understandings and skills in the context of each theme.

19

Syllabus Structure


Applications, suggested activities and sample questionsThe applications, suggested activities and sample questions on the right-hand pagesfacing the thematic and topic content are optional. They are suggestions forlearning experiences and relevant problems that will aid students in theirachievement of syllabus outcomes. They reflect current research on the teachingand learning of mathematics. They give a range of problem types andinvestigations to aid the teaching and learning process. The activities includedhighlight the relevance of mathematics. Their use within the teaching programfacilitates a problem-solving approach to student learning experiences. They arealso intended to provide teachers with a guide to the level of difficulty intended bythe syllabus. The list of suggestions provided is not intended to be exhaustive, noris it intended that students must experience every one of the activities andquestions listed. Teachers should choose those activities and questions that areappropriate for their students and will need to use additional applications, activitiesand questions to ensure that the students have broad experiences in mathematics.

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 course, but which some students and teachers maywish to use as extension or enrichment activities.

The following diagrams show the format of the syllabus.

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Theme 1: Mathematics of Our Environment

Sub-themes: A) Populations B) The School Environment C) The General Environment

Considerations

The final three pages of this theme detail themathematical content of the theme Mathematicsof our environment. This is intended to helpteachers identify the mathematical skills,knowledge and understandings which studentsshould acquire as they undertake this theme.

The teaching approach can be flexible …

Title page for themeSummary of themeConsiderations


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Syllabus Structure


Thematic contentA) PopulationsLearning experiences should provide studentswith the opportunity to:• use sampling techniques to extimate large

numbers, eg crowd size• set up number sequences reflecting

population growth, given certainassumptions (eg every minute, a bacteriumdivides into 2), and extend the patterns

• follow and construct rules for describingpatterns and sequences using naturallanguage

• describe simple patterns using naturallanguage and algebra

• interpret data from tables and graphs onanimal and/or human populations

• represent data on animal and/or humanpopulations using tables and graphs andmake predictions for their future.

• use simple rates to compare growth anddecline of populations.

Theme 1: Mathematics of Our EnvironmentMathematical content

Through the theme, Mathematics ofour environment, students shoulddevelop the ability to:

iv) make reasonable sketches ofsimple solids and their cross-sections

vi) construct models of simplesolids including when givendifferent views (front, back,top, side)

vii) represent three dimensionalsolids in two dimensionsusing basic conventions

ix) interpret scale drawings,using the scale to calculateactual lengths

x) choose appropriate scalesxi) make scale drawings and use

them to solve problems.

ii) locate and plot positions on anumber plane.


Applications, suggested activitiesand sample questions

A) PopulationsStudents could:• estimate crowd sizes from photographs• estimate the number of people at a local

sports field for a sporting event• estimate the number of bugs on a

windscreen by sampling• study numerical patterns related to asexual

population growth. They could organisenumbers into tables and graph thepopulation number against the number ofdivisions, eg the growth of bacteria as shownhere. Students could extend the pattern andattempt to describe the rule in naturallanguage.

• for different conditions of populationgrowth, eg the population grows by 10%each year, draw up a table of values for apopulation starting at 1000, graph thepopulation against the number of years anddescribe the graph …

Contentstatementsfor thetheme

Applications,suggestedactivitiesand samplequestions forthe theme

Mathematicalcontent

Geometry1) Drawing figures

and makingmodels

2) Arrangementsand locations


The coreThe core of essential learning has been designed for a minimum of 160 of theindicative hours for Stage 5. The core of the Standard course is composed of sixthemes and four topics. All students need to undertake the appropriatemathematical experiences through each of the themes and topics so that they haveample opportunity to achieve the outcomes of this course.

The optionsThe remainder of the time spent on Mathematics in Years 9–10 is to be taken upby the options component. Students should study the options for at least 40 hours.It is not 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 ofthe students. Two of the options are thematic in format, while the remaining six areorganised as topics. The options will give students experience in applications ofmathematics that are relevant to them, and also provide further preparation fortheir chosen course of Mathematics for Years 11 and 12.

Students completing the Standard course who intend to continue their study ofMathematics at 2 Unit General (Mathematics in Society) level in Stage 6 shouldstudy the following options as preparation for this course:

Option 4: TrigonometryOption 5: Further geometryOption 6: Further numberOption 7: Further algebra.

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

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Summary of Years 9–10Standard Course — Core

Themes

Topics

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Summary — Core

Theme 1: Mathematics of our environmentA) PopulationsB) The school environmentC) The general environment

Theme 2: Mathematics involving foodA) Food containersB) Food productionC) Food preparation and storageD) Purchasing

Theme 3: Mathematics in the workplaceA) EmploymentB) General skills for employeesC) Running a small business

Theme 4: Building designA) Style of buildingsB) Structure of buildings C) Plans and models of buildings

Theme 5: Mathematics involving sportA) Sporting venuesB) Sporting costsC) Performance in sport

Theme 6: Mathematics in the communityA) SchoolB) HomeC) Local and wider

Topic 1: Geometrical facts, properties andrelationships• Drawing geometrical figures• Angles• Triangles• Quadrilaterals• Congruent figures• Similar figures • Other figures

Topic 2: Pythagoras’ theorem

Topic 3: Chance• Informal concept of chance• Simple experiments• Probability

Topic 4: Introductory algebra• Patterns• Graphs• Algebraic manipulation• Equations• Formulae


Summary of Years 9–10Standard Course — Options

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Option 1 (thematic): Mathematicsinvolving handcrafts• General design• Patchwork• Cross-stitch• Boxes• Origami

Option 2 (thematic): Tourism andhospitality• Tourist attractions• Hospitality

Option 3: Geometrical patterns• Symmetry• Tessellations• Fractals

Option 4: Trigonometry• Right-angled triangles and trigonometry• Applications of trigonometry

Option 5: Further geometry• Angles• Quadrilaterals• Polygons• Solving geometrical problems

Option 6: Further number• Directed numbers• Index notation• Scientific notation• Applying the index laws

Option 7: Further measurement• Surveying• Navigation• Navigation on land

Option 8: Further algebra• Algebraic skills• Equations• Graphs of straight lines


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

achieved• focus on student growth and progress, and make informed judgements about

student 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 appropriate

knowledge 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 and select

teaching resources• focus upon the product as well as the process of teaching, thereby taking greater

responsibility 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 syllabus. As there issome commonality between the courses, there is also some overlap of theoutcomes for each course. In particular, there is overlap between the outcomes ofthe Intermediate and Standard courses. There is also overlap between theoutcomes of Mathematics 7–8 and those of the Stage 5 Standard course.

The outcomes for the core of the Standard course are organised under theobjectives for Values and attitudes, Working mathematically, Geometry, Number,Measurement, Chance and data, and Algebra. The outcomes reflect the mathematicalcontent of the themes and topics. The outcomes for the Geometry strand areincluded as an example following. These outcomes relate to the geometricalknowledge, understanding and skills that are developed through both the themesand the topics.

25

Outcomes


26


The mathematical content of the themes from the core and options has beenidentified within the grids in the section Content mapped to themes. This links themathematical content for Geometry, Number, Measurement, Chance and data andAlgebra to each theme.

The outcomes for the options of the syllabus have been organised under eachoption. Students will work towards the achievement of the relevant outcomes fromthe options or parts of options which they study. It is intended that most studentsundertaking the Standard course will achieve most of the course outcomes by theend 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 across the themes and topics.

The outcomes statements for the Mathematics Stage 5 (Years 9–10) Standardcourse are included in the following pages.

Outcomes

A student:

• recognises and sketches 3D objects in various orientations• uses geometric techniques and tools to construct angles,

lines and 2D figures • interprets and makes scale drawings and uses scale factors

to solve problems• reads, interprets and makes simple maps and plans using

distance, direction, coordinates and scales• uses and draws network diagrams to represent the order of

events and paths between locations• recognises and names common geometrical figures and

solids and their parts• identifies and uses geometrical facts, properties and

relationships to solve problems relating to angles, lines andtriangles

• recognises symmetry in two dimensional shapes• uses transformations to draw geometrical patterns• recognises congruent figures as those which can be

superposed through a series of transformations.

Objectives

Students will developknowledge,understanding andskills in:

• Geometry.

Knowledge, Understanding and Skills


Objectives and OutcomesStandard Course

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Outcomes

Outcomes

A student:

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

• demonstrates a positive response to the use ofmathematics as a tool in practical situations

• shows an interest in and enjoyment of the pursuit ofmathematical knowledge

• demonstrates the confidence to apply mathematics andto seek and gain knowledge about the mathematics theyneed from a variety of sources

• shows a willingness to work cooperatively with othersand to value the contributions of others

• appreciates the importance of visualisation whensolving problems

• shows a willingness to take risks when workingmathematically

• shows a willingness to persist when solving problemsand to try different methods

• uses mathematics creatively in expressing new ideasand discoveries

• appreciates that conventions, rules about initialassumptions, precision and accuracy enable informationto be communicated effectively

• 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.

Objectives

Students will develop:

• appreciation ofmathematics as anessential andrelevant part of life.

Values and Attitudes


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Objectives and OutcomesStandard Course — Core

Outcomes

A student:

• estimates the results of calculations and checks thereasonableness of results

• uses appropriate technology effectively to assist in thesolution of problems

• selects and uses appropriate mathematical techniqueseffectively

• interprets and uses mathematical information presentedin a variety of forms (ie diagrams, text, tables, symbols)

• uses appropriate problem-solving strategies whichinclude selecting and organising key informationand breaking the problem into smaller parts

• interprets the results of problem solutions indifferent contexts

• plans, carries out and reports on a statisticalinvestigation with guidance

• communicates mathematical knowledge andunderstanding, using mathematical terms.

Objectives


• Workingmathematically.



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Outcomes


Outcomes

A student:

• recognises and sketches 3D objects in various orientations• uses geometric techniques and tools to construct angles,

lines and 2D figures • interprets and makes scale drawings and uses scale

factors to solve problems• reads, interprets and makes simple maps and plans using

distance, direction, coordinates and scales• uses and draws network diagrams to represent the order

of events and paths between locations• recognises and names common geometrical figures

and solids and their parts• identifies and uses geometrical facts, properties and

relationships to solve problems relating to angles,lines and triangles

• recognises symmetry in two dimensional shapes• uses transformations to draw geometrical patterns• recognises congruent figures as those which can be

superposed through a series of transformations.

A student:• uses negative numbers in practical situations• chooses and sequences arithmetic operations to solve

problems• selects and uses appropriate mental, written or calculator

techniques to perform a variety of operations involvingfractions, decimals, percentages and positive integers

• interprets and uses written and graphical informationto solve problems related to consumer arithmetic

• understands and uses different representations of numbersincluding fractions, decimals, percentages and words

• rounds numbers appropriately to a desired degree of accuracy• interprets and uses positive integer powers and square roots• interprets and uses ratios and rates to solve simple problems.

Objectives


• Geometry.


• Number.




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Outcomes

A student:

• uses a variety of techniques and tools to measure andcompare quantities, including angles

• selects and uses appropriate common units and convertsbetween measures

• estimates measurements appropriately in various contexts• finds the perimeters and areas of triangles, quadrilaterals,

circles and simple composite figures, including usinggiven formulae

• calculates the surface area of solid shapes whose facesare rectangles and triangles

• uses formulae to find the volume of right prisms andcylinders and solves simple problems involvingvolume and capacity

• estimates, measures and calculates time and usestimelines and timetables to solve problems

• uses Pythagoras’ theorem to solve problems.

A student:

• investigates a problem by posing suitable questions andplanning data collection

• organises and displays collected data in a variety of ways• finds measures of location and spread from sets of scores• interprets data represented in tables and graphs • draws informal conclusions from data displays

and summary statistics• places informal expressions of chance on a scale of 0 to 1• performs simple chance experiments and uses these

experiments to estimate probabilities • solves simple probability problems.

Objectives


• Measurement.


• Chance and data.




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Outcomes

Outcomes

A student:

• identifies, describes and extends number patterns• represents patterns in symbols and extends patterns

by substitution• interprets a variety of graphs including travel, step and

conversion graphs, and uses them to solve problems• draws graphs to represent relationships, given

descriptions or tables of values• recognises and uses conventions to manipulate simple

algebraic expressions• substitutes into given formulae and evaluates the

resulting expression• solves simple linear equations and those arising

from substitution into formulae.

Objectives


• Algebra.



Objectives and OutcomesStandard Course — Options

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Outcomes

A student:• applies their knowledge and skills,

particularly in geometry, to investigatepatterns, make designs and solveproblems related to handcrafts.

A student:• applies their knowledge and skills,

particularly in number and measurement,to solve problems related to touristattractions and hospitality.

A student:• identifies objects as symmetrical about a

plane, a line or a point• constructs a variety of tessellations • identifies shapes and transformations used

in tessellations• uses a variety of methods to produce

simple fractals.

A student:• identifies the appropriate trigonometric

ratio to use for a problem• uses trigonometry to solve practical

problems involving right-angled triangles.

A student:• identifies angles related to parallel lines

and uses relationships between the anglesto solve problems

• describes angle and side properties ofquadrilaterals and polygons

• applies geometrical facts, properties andrelationships to solve geometricalproblems relating to angles, triangles,quadrilaterals and polygons, givingreasons.

Objectives



• Geometry• Number• Measurement• Chance and data• Algebra.

Options

Mathematicsinvolvinghandcrafts

Tourism andHospitality

Geometricalpatterns

Trigonometry

Furthergeometry



Objectives and OutcomesStandard Course — Options

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Outcomes

Outcomes

A student:• operates with directed numbers

• demonstrates understanding of indices(integral and , ) and applies the index

laws to solve numerical problems• converts between ordinary and scientific

notation• performs operations in scientific notation

and solves related problems.

A student:• uses a traverse survey method to take

measurements • constructs scale drawings from sketches

and practical exercises• calculates areas and perimeters from scale

drawings• uses the parallels of latitude and

meridians of longitude to locate positions• identifies and uses the main features of a

topographical map.

A student:• recognises and uses conventions to

manipulate and simplify algebraicexpressions

• uses a variety of techniques to solve linearequations and solves related problems

• graphs equations of the form y = mx + b• recognises equations, eg y = 2x + 3, as

representing a straight line and interpretsthe coefficients 2 and 3 appropriately.

13

12

Objectives



• Geometry• Number• Measurement• Chance and data• Algebra.

Options

Furthernumber

Furthermeasurement

Furtheralgebra



AssessmentAssessment 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 Measurement wouldrequire 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.

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, Measurement,Chance and data and Algebra.

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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 assignmentsetc.)

• 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 andskills

– it should allow for valid judgements to be made of the students’ achievements– will the task be commented upon, graded, and/or marked?

• 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.

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Assessment


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)

• 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

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• 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 provides advice on the adjustment of assessment strategies forspecial needs students. This document will be very useful for teachers who havestudents with physical disabilities in their class.

Assessment for the School CertificateOne 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 tasksthat will allow decisions to be made on students’ achievement of their Mathematicscourse in relation to other students in their school who are studying the samecourse. The tasks need to validly discriminate between students. They must bebased on the relevant course of the Mathematics 9–10 syllabus and could employ avariety 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.

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

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Assessment


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.

Evaluation of School ProgramsA 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 student motivationand involvement in mathematics.

Schools should conduct effective ongoing evaluation that addresses questions suchas:

• 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?

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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 oncethroughout its period of implementation.

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Evaluation of School Programs


Content Mapped to Themes

40


1) Drawing figures and making modelsThrough the themes, students should develop the ability to:

i) use appropriate geometric instruments to draw angles, parallel and perpendicular lines, and common plane shapes

X X X

ii) construct squares, rectangles and triangles given dimensionss X X X

iii) use appropriate geometric instruments to draw circles, arcs and sectorss X X

iv) make reasonable sketches of simple solids and their cross-sections X X X

v) identify and draw nets of simple solids X X

vi) construct models of simple solidsincluding when given different views(front, back, top, side)

X X X

vii) represent three dimensional solids in twodimensions using basic conventions X X X

viii) sketch common solids and models from different views (top, side, front etc) X X

ix) interpret scale drawings, using thescale to calculate actual lengths X X X X

x) choose appropriate scales X X X Xxi) make scale drawings and use them to

solve problems. X X X X X

E F W B S C H TN O O U P O A OV O R I O M N UI D K L R M D RR P D T U C IO L I N R SN A N I A MM C G T FE E Y TN ST

Geometry

Options


41


2) Arrangements and locations

Through the themes, students should develop the ability to:

i) locate positions on street directories, mapsand grids X X X X X

ii) locate and plot positions on a number plane X Xiii) read, interpret and make simple maps and

plans using distance, direction, coordinatesand scales

X X X

iv) use and draw network diagrams to representthe order of events and paths betweenlocations.

X X

3) Geometrical facts, properties and relationshipsThrough the themes, students should develop the ability to:

i) recognise and name common plane figures(types of triangles, quadrilaterals andpolygons, circles, ellipses)

X X X

ii) describe and use properties of triangles(angle sum, rigidity) X

iii) recognise and name parts of a circle(centre, radius, diameter, chord, sector,tangent, circumference, arc, semi-circle)

X

iv) recognise and name common solids(prisms, cylinders, pyramids, cones andspheres)

X X X

v) recognise and name horizontal, vertical,parallel and perpendicular lines and planesand skew lines

X

vi) recognise congruent shapes as those able tobe superposed X

vii) recognise similar figures as those involvingenlargements or reductions of figures X X

viii) calculate lengths of similar figures using theenlargement or reduction factor. X X


Geometry

Options


42


4) Movements and transformations


i) reflect, rotate and translate simple shapes X X

ii) recognise and draw geometric patterns usingrotations, reflections and translations X X

iii) recognise and draw geometric patterns usingtessellations (both regular and semi-regular) X X

iv) recognise symmetry in two-dimensionalshapes X X

v) give the position of the axes of symmetry intwo-dimensional symmetrical figures. X X


Geometry

Options


43


1) Number skillsThrough the themes, students should develop the ability to:

i) read, write and order integers, money,decimals, percentages and commonfractions, including the use of inequalitysymbols (≤, ≥, <, >)

X X X X

ii) choose and sequence arithmetic operationscorrectly to solve a problem X X X X X X

iii) estimate results of operations with integers,money, decimals, percentages and commonfractions, being alert to unreasonableanswers

X X X X X X X

iv) choose the appropriate mental, written orcalculator technique to perform calculations X X X X X X X

v) mentally perform simple numericalcalculations with whole numbers, moneyand common fractions (eg 35 + 15, 84 ÷ 4,23 × 5, + , of 39)

X X X

vi) calculate with whole numbers and money X X X X X Xvii) use a calculator efficiently to perform

operations with money, decimals,percentages and common fractionsincluding converting between them

X X X X X X X

viii) mentally give conversions for commonfractions, decimals and percentages(eg 0.5 = = 50%)

X X X

ix) round up or down appropriately to adesired degree of accuracy X X X X X X

x) interpret and use positive integer powers X Xxi) interpret and use square roots Xxii) read and interpret tables and graphs related

to consumer information X

xiii) apply numerical skills to problems includingthose related to the consumer. X X X X X X

12

13

12

12


Number

Options


44


2) Ratio and rateThrough the themes, students should develop the ability to:

i) use a ratio to describe the relationshipbetween two quantities X X X

ii) simplify ratios Xiii) divide a quantity into a given ratio Xiv) use a ratio to calculate one quantity from

another X X X X

v) use the unitary method to solve ratioproblems X X

vi) use a familiar rate to describe therelationship between two quantitieswhich are directly proportional

X X X

vii) simplify rates X X Xviii) use rates to make comparisons X X X X Xix) calculate rates and use them to solve

problems. X X X X X


Number

Options


45


1) Choosing unitsThrough the themes, students should develop the ability to:

i) give reasonable estimates for the size ofstandard units (mm, cm, m, km, g, kg,mL, L)

X X

ii) select appropriate units to measure length,area, temperature, mass, volume, capacityand time

X X X X X

iii) convert between metric units of length,mass and capacity. X X X X X

2) MeasuringThrough the themes, students should develop the ability to:

i) select and use appropriate measuringinstruments to measure lengths, mass,capacity and angles

X X X X X X

ii) read a variety of scales on standardmeasuring instruments to the nearestmarked graduation

X X X X X X

iii) decide on an appropriate level of accuracyfor measurements which are to be taken X X X X X X

iv) appreciate, through drawing or building,the size of the standard units of area andvolume (cm2, m2, cm3, m3).

X

3) EstimatingThrough the themes, students should develop the ability to:

i) use their knowledge of the size of familiarobjects to make estimates X X X X X

ii) make estimates (in standard units) of length,perimeter, angle, area, mass, volume andcapacity and check by measuring

X X X X X X X X

iii) decide whether an estimate is reasonable. X X X X X X X


Measurement

Options


46


4) PerimeterThrough the themes, students should develop the ability to:

i) find the perimeter of plane shapes X X Xii) calculate the circumference of a circle

given the radius or diameterCircumference = π × diameter, C = πd.

X X

5) AreaThrough the themes, students should develop the ability to:

i) find the area of plane shapes by countingsquare units X

ii) calculate the area of squares, rectanglesand triangles X X X X X

iii) calculate the area of quadrilaterals(parallelogram, trapezium, rhombus) X X XParallelogram: Area = base × perpendicularheight, A = bhTrapezium: Area = half the sum of the parallelsides × perpendicular height, A = ( )h

Rhombus: Area = half the product of thediagonals, A = xy

iv) calculate the area of a circle given thediameter or the radiusArea of a circle = π × radius squared, A = πr 2

X X X X

v) find the area of composite figures which canbe divided up into rectangles, trianglesand/or circles.

X X X

6) Volume and capacityThrough the themes, students should develop the ability to:

i) find the volume of solids by counting cubicunits X

ii) calculate the volume of rectangular andtriangular prismsVolume of a prism = area of the base × verticalheight, V = Ah

X X X X X

12

a + b2


Measurement

Options


47


6) Volume and capacity (continued)Through the themes, students should develop the ability to:

iii) calculate the volume of a cylinderVolume of a cylinder = π × radius squared ×height, V = πr 2h

X X

iv) recognise that a container with a volume of1000 cm3 holds 1 litre X X

v) solve problems involving volume and capacity. X X7) Surface areaThrough the themes, students should develop the ability to:

i) find the surface area of rectangular andtriangular prisms and composite figures X Xinvolving these.

8) TimeThrough the themes, students should develop the ability to:

i) place a series of events in order X Xii) estimate and measure time and duration

of time X X X X

iii) add times and make calculations involvingtime differences X X X

iv) use timelines and a range of types oftimetables to solve problems X X X

v) solve simple problems involving time zoneswithin Australia X

vi) plan an event which must satisfy a set oftime constraints. X X

9) Pythagoras’ theoremThrough the themes, students should develop the ability to:

i) use Pythagoras’ theorem to find a side of aright-angled triangle given the length of Xtwo other sidesIn a right-angled triangle, the hypotenusesquared = the sum of the squares of the othertwo sides (c 2 = a 2 + b 2)

ii) use Pythagoras’ theorem to solve practicalproblems. X


Measurement

Options


48


1) Collecting and organising data


i) pose suitable questions which can beanswered through data collection X

ii) design and refine a simple survey Xiii) recognise the difference between a

sample and the population X

iv) plan how data will be collected Xv) record data using tally marks or

organised lists X

vi) collect data as consistently and fairly aspossible X

vii) check raw data for obvious and gross errors X

viii) organise data into frequency tables usinggrouped intervals where appropriate(groupings provided).

X X


Chance and Data

Options


49


2) Displaying, summarising and interpreting dataThrough the themes, students should develop the ability to:

i) display data in dot plots and stem-and-leafplots X X X

ii) display data in frequency histograms andpolygons choosing appropriate scales forthe axes

X X X X X

iii) put data in order and find the mean, mode,median and range for a small set of scores X X X

iv) use a scientific calculator to find the mean( ) of a set of scores X X X X

v) use a stem-and-leaf plot to find the medianand range of a set of scores X X X

vi) use fractions and percentages to summarisedata X X X

vii) describe briefly the conclusions that can bedrawn from the summary statistics (mean,mode, median and range)

X X X X

viii) read and interpret information representedin tables and graphs (picture, divided bar,sector, column, line and frequencyhistograms and polygons )

X X X X X X

ix) interpret displays which show two sets ofdata and make comparisons X X

x) identify graphs that give misleadinginformation. X X

x


Chance and Data

Options


50


1) PatternsThrough the themes, students should develop the ability to:

i) identify patterns in number sequences andextend the pattern X

ii) explain and describe patterns using naturallanguage X

iii) construct simple rules to describe patternsusing natural language X

iv) recognise that pronumerals may be used tostand for numbers. X

2) GraphsThrough the themes, students should develop the ability to:

i) draw and interpret travel, step andconversion graphs X

ii) use a table of values to represent alinear relationship X

iii) extend a table of values by substitutinginto a rule X

iv) choose appropriate scales on the verticaland horizontal axes when drawing graphs X

v) sketch graphs to represent relationships, egone set of values is always twice the other. X

3) FormulaeThrough the themes, students should develop the ability to:

i) substitute into given formulae, evaluatingthe subject of the formula X X X X X

ii) solve linear equations arising fromsubstitution into formulae, eg P = 2l + 2b X X X

iii) substitute into simple formulae and solvea resulting simple equation, eg s = , X X X X X X

C = πd, c 2 = a 2 + b 2.

dt


Algebra

Options


51

Standard Course — Core

Standard Course

Content — Core

Themes

1. Mathematics of Our Environment

2. Mathematics Involving Food

3. Mathematics in the Workplace

4. Building Design

5. Mathematics Involving Sport

6. Mathematics in the Community

Topics

1. Geometrical Facts, Properties and Relationships

2. Pythagoras’ Theorem

3. Chance

4. Introductory Algebra

96201 2Core Maths 9-10 9/7/96 3:21 PM Page 51

52




Sub-themes: A) Populations B) The school environmentC) The general environment

ConsiderationsThe final three pages of this theme detail the mathematical content of the themeMathematics of our environment. This is intended to help teachers identify themathematical skills, knowledge and understandings that students should acquire asthey undertake this theme.

The teaching approach can be flexible — teachers could introduce the theme byfirst teaching the mathematical content and then reinforcing this through teachingthe thematic content and the applications of the theme. Alternatively, teacherscould use the theme to identify areas where students need further skilldevelopment and spend time as required developing such skills. Teachers may wishto use a combination of such teaching approaches as appropriate to their class.

Mathematics of our environment emphasises measurement skills. Aspects of Geometry,Number, Chance and data and Algebra are also included in the sub-themes ofPopulations, The school environment and The general environment. Teachers may wish toteach one or two of the sub-themes together, move on to another theme and returnto this theme later, to ensure variety for their students.

Students should be competent with simple algebra for the sub-theme Populations,and it may be necessary for students to complete the topic on Introductory algebrabefore beginning this section of the theme.

If students are in a non-school environment, homes or other buildings can besubstituted for school buildings and classrooms as appropriate. The important thingis that students are estimating, measuring and calculating areas and volumes for afamiliar environment, and that they can make comparisons between measurementsof different rooms or buildings.

53




Thematic content

A) PopulationsLearning experiences should provide the opportunity for students to:

• use sampling techniques to estimate large numbers, eg crowd size

• set up number sequences reflecting population growth, given certain assumptions(eg every minute, a bacterium divides into 2), and extend the patterns

• follow and construct rules for describing patterns and sequences using naturallanguage

• describe simple patterns using natural language and algebra

• interpret data from tables and graphs on animal and/or human populations

• represent data on animal and/or human populations using tables and graphs andmake predictions for their future

• use simple rates to compare growth and decline of populations.

54




Applications, suggested activities and sample questions

A) PopulationsStudents could:

◊ estimate crowd sizes from photographs

◊ estimate the number of people at a local sports field for a sporting event

◊ estimate the number of bugs on a windscreen by sampling

◊ study numerical patterns related to asexualpopulation growth. They could organisenumbers into tables and graph the populationnumber against the number of divisions, eg thegrowth of bacteria as shown here. Studentscould extend the pattern and attempt todescribe the rule in natural language.

◊ for different conditions of population growth, eg the population grows by 10%each year, draw up a table of values for a population starting at 1000, graph thepopulation against the number of years and describe the graph

◊ discuss the future situations of a specific animal population given currentavailable data

◊ contribute to or collaborate in the organisation of data into diagrams and tablesto help answer questions about animal populations

◊ collect, analyse and interpret data from published material about animalpopulations

◊ produce a report on their findings and conclusions from consideration of aspecific animal population

◊ choose a particular animal, eg the kangaroo, and investigate its populationgrowth. They could collect data about numbers presently living, graph thesenumbers and extrapolate the graph to show the population growth over thenext 50 years. They could make predictions about its viability as a population.They could investigate the difference made by different rates of populationgrowth (this could be done using a spreadsheet).

◊ investigate the population of koalas in NSW, drawing a graph to represent theirpopulation size over the past fifty years and on a map indicate the pockets ofpopulations still living and make predictions about the size of the koalapopulation in ten and twenty years time

(continued)

55


No of divisions

No of bacteria

1

2

3

2

4

1

8



Thematic content

B) The school environmentLearning experiences should provide the opportunity for students to:

• make reasonable estimates of lengths and perimeters in the school environmentand use measuring tools to find actual lengths and perimeters

• make a square metre and use it to estimate and subsequently calculate thenumber of square metres of floor space in the classroom

• measure lengths accurately and work out floor areas of different rooms or partsin the school

• make a cubic metre and use it to estimate and subsequently calculate the volumeof the classroom

• compare the volume of different classrooms

• create simple scale models of school buildings

• draw different views of buildings (eg side, top, front, back views).

56





A) Populations (continued)

Students could:

◊ use information on the rabbit population in Australia over the past 100 years toprepare a report on the efforts to control rabbits during this time, concentratingon the statistics of the population

◊ visit the zoo or invite a zoo keeper to the school to talk about animalpopulations and their requirements.

B) The school environmentStudents could:

◊ consider the ratio of student passive areas to active areas, or the ratio of builtareas to open areas

◊ estimate and measure parts of the school buildings and grounds, find the areasof floor/paved/grass, compare their ratios and determine the average area ofeach for each student

◊ from newspaper or cardboard, make a square metre. Investigate how manypeople can stand on one square metre. Find the area of floor space in theclassroom and the average amount of space per student. This average spacecould be modelled using one person and a number of square metres

◊ using rods and corner pieces, build a cubic metre. Use this to estimate thevolume of air in the classroom and then calculate the volume of air in theclassroom. Calculate the average amount of air per student and compare thisvalue to the value for other classes and classrooms.

◊ survey the amount of recyclable and non-recyclable waste produced each day

◊ create models of simple school buildings, eg shade structures, drawing the top,side, back and front views.

57




Thematic content

C) The general environmentLearning experiences should provide the opportunity for students to:

• recognise the Fibonacci sequence and relate it to similar arrangements in nature

• extend the Fibonacci sequence and describe the rule in words

• use a calculator and tabulation to work out the ratio of pairs of successive termsin the Fibonacci sequence

• determine an approximation for π from the circumference and diameter ofobjects with a circular cross-section

• use the formula (C = πd ) to find the circumference of circular objects, eg thegirth of a tree after estimating its diameter

• use the formula (C = πd ) to find the diameter of circular objects after measuringthe circumference, eg find the diameter of trees or power poles after measuringthe girth (the girth of a tree is usually measured 1.3 m up from the base of thetree)

• substitute into formulae related to aspects of the environment and solve anyresulting equation

• estimate and find heights of objects within the environment, eg trees, flagpolesetc (using a shadow stick and similar triangles, and/or a clinometer)

• interpret data on an aspect of the environment, eg management of forests or useof trees for paper or furniture, identify any misleading statistics, displayinformation and present a brief report on the aspect investigated.

58





C) The general environmentStudents could:

◊ study Fibonacci patterns in the arrangement of petals of flowers, spirals of pinecones or pineapples etc

◊ represent the patterns found in leaves, stems and flower petals pictorially andnumerically, and describe the pattern using natural language

◊ make sketches of the patterns found in sunflowers

◊ develop the Fibonacci sequence for the first few terms, describe its rule in wordsand find further terms in the sequence

◊ consider other ways the Fibonacci sequence can be obtained, eg terms involvedin the number of different ways one can walk up 1, 2, 3, 4, . . . stairs taking 1 or2 steps at a time

◊ find the ratio of the 5th term to the 4th term, and the 6th term to the 5th etc

◊ compare the ratios of pairs of successive terms, approximating the ratio ingeneral

◊ estimate the girth of trees or electricity poles from the school yard or local park

◊ measure the girth of trees or electricity poles around the school or local parkand calculate their diameter

◊ use clinometers or shadow sticks and scale drawings to find the height of treesor flagpoles and consider whether the results are reasonable

(continued)

59




Thematic content

60





C) The general environment (continued)

Students could:

◊ find reasonable approximations for the areas of different leaves of plants ortrees

◊ use formulae related to the environment, eg:

a) to compare the growth rate of plants, the following formula can be used:

growth rate =

b) to find the volume of wood in a tree, the following formula can be used:

Volume (of log) = (0.4724d 2h + 9.86) cm3, where d = diameter of trunk incentimetres calculated 1.3 m up from the ground and h = height of trunkmeasured from 1.3 m above the ground to the lowest limb

◊ answer questions like:

it is known that the trunk of a huon pine tree increases in diameter by 12 cmevery 100 years. Find the increase in girth. Set up a table to show the increasein diameter over 1000 years, if the original diameter is 10 cm. Draw a graphshowing this information

◊ investigate the tree/plant population of the school/local area, draw a roughsketch of the area and find the average number of trees/plants per hectare

◊ investigate the management of forests or orchards, including the rate of growthof trees and required time to reach maturity.

height in cmage in days

61




Mathematical content

62


Geometry

1) Drawing figuresand making models

2) Arrangements andlocations

Number

1) Number skills

2) Ratio and rate

Measurement

1) Choosing units

Through the theme, Mathematics of our environment,students should develop the ability to:

iv) make reasonable sketches of simple solids and their cross-sections

vi) construct models of simple solids including when givendifferent views (front, back, top, side)

vii) represent three dimensional solids in two dimensions usingbasic conventions

ix) interpret scale drawings, using the scale to calculate actuallengths

x) choose appropriate scales

xi) make scale drawings and use them to solve problems.

ii) locate and plot positions on a number plane.

iii) estimate results of operations with integers, money,decimals, percentages and common fractions, being alert tounreasonable answers

x) interpret and use positive integer powers

xiii) apply numerical skills to problems including those related tothe consumer.

i) use a ratio to describe the relationship between twoquantities

iv) use a ratio to calculate one quantity from another

vi) use a familiar rate to describe the relationship between twoquantities which are directly proportional

vii) simplify rates

viii) use rates to make comparisons.

ii) select appropriate units to measure length, area,temperature, mass, volume, capacity and time.




63


Measurement (continued)

2) Measuring

3) Estimating

4) Perimeter

5) Area

6) Volume andcapacity


i) select and use appropriate measuring instruments tomeasure lengths, mass, capacity and angles

ii) read a variety of scales on standard measuring instrumentsto the nearest marked graduation

iii) decide on an appropriate level of accuracy formeasurements which are to be taken

iv) appreciate, through drawing or building, the size of thestandard units of area and volume (cm2, m2, cm3, m3).

i) use their knowledge of the size of familiar objects to makeestimates

ii) make estimates (in standard units) of length, perimeter,angle, area, mass, volume and capacity and check bymeasuring

iii) decide whether an estimate is reasonable.

i) find the perimeter of plane shapes

ii) calculate the circumference of a circle given the radius ordiameter Circumference = π × diameter (C = πd).

i) find the area of plane shapes by counting square units

ii) calculate the area of squares, rectangles and triangles

iv) calculate the area of a circle given the diameter or the radiusArea of a circle = π × radius squared (A = πr 2).

i) find the volume of solids by counting cubic units

ii) calculate the volume of rectangular and triangular prismsVolume of a prism = area of the base × vertical height (V = Ah).




64


Chance and Data

2) Displaying,summarising andinterpreting data

Algebra

1) Patterns

2) Graphs

3) Formulae


ii) display data in frequency histograms and polygons choosingappropriate scales for the axes

vi) use fractions and percentages to summarise data

vii) describe briefly the conclusions that can be drawn from thesummary statistics (mean, mode, median and range)

viii) read and interpret information represented in tables andgraphs (picture, divided bar, sector, column, line andfrequency histograms and polygons)

x) identify graphs that give misleading information.

i) identify patterns in number sequences and extend thepattern

ii) explain and describe patterns using natural language

iii) construct simple rules to describe patterns using naturallanguage

iv) recognise that pronumerals may be used to stand fornumbers.

ii) use a table of values to represent a linear relationship

iii) extend a table of values by substituting into a rule

iv) choose appropriate scales on the vertical and horizontalaxes when drawing graphs

v) sketch graphs to represent relationships, eg one set of valuesis always twice the other.

iii) substitute into simple formulae and solve a resulting simple

equation, eg s = , C = πd, c 2 = a 2 + b 2.dt


Theme 2: Mathematics Involving Food

Sub-themes: A) Food containers B) Food productionC) Food preparation and storage D) Purchasing

ConsiderationsThrough a quantitative examination of food and how it is stored, students shoulddevelop an appreciation of the cost and amount of food needed by families andsociety in general, and the importance of preparing and storing food carefully.

The final three pages of this theme detail the mathematical content of the themeMathematics involving food. This is intended to help teachers identify themathematical skills, knowledge and understandings that students should acquire asthey undertake this theme.


Skills related to Number and Measurement are particularly represented in this theme,while Geometry, Chance and data and simple Algebra are also represented, but to alesser extent. Students need to be able to work with formulae in this theme and socompetence with the topic Introductory algebra is required.

65




Thematic content

A) Food containers Learning experiences should provide the opportunity for students to:

• estimate the capacity of a variety of food containers

• estimate the mass of the contents of food containers

• measure a variety of packages (prisms and cylinders) and calculate their volumegiven appropriate formulae

• find the area of the surfaces of packages which are right prisms

• measure accurately the mass of food containers and their contents.

B) Food productionLearning experiences should provide the opportunity for students to:

• use a map to locate centres of local or regional food production

• calculate the rate of food production including aspects such as the yield perhectare, or the rate of processing in a factory

• display prepared data relating to food production in graphs

• interpret data displayed in tables and graphs on local or global food production

• make comparisons between localities or countries concerning populations andfood or beverage production and consumption.

• use networks to show export/import and transportation routes.

66





A) Food containersStudents could:

◊ given a variety of different food containers, estimate and calculate their volume,comparing this to the volume quoted on labels

◊ consider the settling of contents and calculate the size of packages for cereal andthe number of servings in different packages

◊ recognise that different packages could hold the same volume and, byconstructing packages if necessary, find two packages which have differentdimensions but the same volume

◊ check by measuring and calculating whether the information shown on theoutside of packages relating to volume and mass is accurate.

B) Food production Students could:

◊ use coordinates to locate positions on grids and maps in investigating global andlocal production

◊ visit producers of bulk food (eg canneries, egg board, chicken growers, milkproducers), or obtain information from pamphlets on the amount of foodproduced in different time periods

◊ research and report on production of a particular food (eg wheat, fish, beef,chicken, oranges) in the local region, considering aspects such as:

– cropping rates and yield– amounts produced or collected– production rates for factories– methods of distribution– distribution networks

◊ collect data on food produced or collected locally, eg the amount of a particulargrain, the amount of milk produced by a dairy farm over a year. Draw graphsto illustrate the information and discuss possible explanations for any variationsin production.

◊ compare food requirements and food production of Australia to that of othercountries

◊ investigate the world consumption and production of rice or wheat or corn,calculating the average kilogram consumption per person for different countries.

67




Thematic content

C) Food preparation and storageLearning experiences should provide the opportunity for students to:

• use a variety of measuring devices to measure the volume of liquid and the massof food quantities

• make calculations involving the percentage of waste in food preparation

• calculate average costs per serving and per kilogram

• compare costs of different foods, eg fresh vs processed food, home cooked vs arestaurant meal

• use ratios and the unitary method to vary quantities given in recipes

• plan and sequence events for cooking a meal

• solve time problems in relation to rates of cooking

• make calculations with negative numbers in relation to temperatures of storagefacilities

• use formulae applicable to food preparation and storage

• interpret and display data on the deterioration of the quality of food duringstorage.

68





C) Food preparation and storageStudents could:

◊ visit a fast food outlet and/or food processing plant to observe food storage andpreparation, then prepare a report detailing the amount of wastage

◊ allow for wastage and convert the purchase cost per kilogram to the true costper kilogram of the useable quantity


a) Michael paid $9 per kilogram for a large piece of meat. After he trimmed thefat and removed the bone he found each kilogram of meat he had purchasedproduced 750 grams of useable meat.

i) What percentage of the meat was wasted?

ii) What was the price of the meat per useable kilogram?

b) ice-cream is to be kept at –4°C, but the freezer’s temperature is –1°C. Is thefreezer too warm or too cool, and by how much?

c) a freezer’s temperature drops from 0°C by 8° and then rises by 3°. What is itstemperature then?

◊ compare the cost of lunch at a restaurant to the cost of the same lunch preparedat home

◊ compare the cost of tinned or frozen fruit or vegetables to the cost of the samequantity of fresh produce.

◊ change proportions to cater for varying numbers of people, eg adjust a smallquantity luncheon recipe in order to provide lunch for the whole class

◊ calculate cooking times for different masses of food after reading instructions

◊ graph the times for cooking different quantities of meat and use the graph topredict times for very large or very small amounts

◊ consider best-by-dates, calculating and comparing the shelf-lives of a variety ofdifferent food products

(continued)

69




Thematic content

70





C) Food preparation and storage (continued)

Students could:


a) Chantelle compared two similar cartons of yoghurt. The ‘best-by’ dates onthem were 26 April and 3 May. How many days longer will the secondcontainer last than the first?

b) the amount of wastage (W ) produced in the preparation of cauliflower canbe approximated using the formula W = K, where K is the weight of the

cauliflower in kilograms. Jenny bought acauliflower weighing 1.2 kg. What amount, in grams, will be wasted?

c) when a cook used a spoon he had licked tostir a soup, he introduced bacteria into thesoup. The graph shows the amount ofbacteria in the soup at different times. Howmany units of bacteria were in the soup onehour after the cook licked the spoon?

d) the number of days (D) fresh milk will keep at different temperatures (t°C)

above freezing can be calculated using the formula D = . How long will

milk keep if it is stored at 3°C? How many days longer will milk keep at 1°Cthan 3°C?

◊ as a practical project, plan the arrangement of food in a pantry or freezerconsidering best-by-dates, height of shelving, type of food etc.

6t + 1

310

71


20

12

3

40Minutes

Million units

60



Thematic content

D) PurchasingLearning experiences should provide the opportunity for students to:

• calculate costs related to food purchasing, eg given that an amount of a foodproduct has a certain cost, find the cost for a different amount of the same food

• estimate costs related to food purchasing (will $10 be enough?)

• compare costs of food items, considering for example different types of stores,different brands of the same item of food

• make calculations to determine ‘best buys’.

72





D) PurchasingStudents could:

◊ make calculations related to purchasing food, eg find the cost of 1 kg of

tomatoes at $2.89 per kg, or if 2.5 kg of oranges cost $3, how much is this perkg?

◊ given the cost of a number of items of food, work out the cost of a single itemand other amounts, eg if 5 mangos cost $7.50, how much for 1 and how muchfor 8?

◊ compare costs of the same article at different types of shops, (eg supermarkets,convenience stores, corner shops) or of generic brands to other brands

◊ investigate whether it is generally more economical to buy items in larger sizesthan in small or medium sizes

◊ monitor the cost of particular varieties of fruit, vegetables or meat over a smalltime period, present the information graphically and draw conclusions aboutwhich item varies the most and the least in price. Discuss whether the trends inthe data are likely to continue and possible reasons for the variation.

◊ calculate the average amount of money per person spent on food for a familyover a week. Consider whether the average cost would reflect the actual cost offood which each person in a family would eat.

◊ calculate the average cost per person for food for a school event, eg a sausagesizzle

◊ visit a supermarket to obtain information for analysis, eg: queuing time, varietyof items purchased, total cost of a ‘trolley of shopping’, quantity of conveniencefood bought, most popular shopping times, layout of products, marketingtechniques

◊ invite a supermarket manager to speak to the class about topics related to thefood purchasing business, eg staffing, ordering, keeping track of stock, bar codesetc.

12

73





74


Geometry


Number

1) Number skills

2) Ratio and rate

Through the theme, Mathematics involving food, studentsshould develop the ability to:

i) locate positions on street directories, maps and grids

iv) use and draw network diagrams to represent the order ofevents and paths between locations.

i) read, write and order integers, money, decimals,percentages and common fractions including the use ofinequality symbols (≤, ≥, <, >)

ii) choose and sequence arithmetic operations correctly tosolve a problem


iv) choose the appropriate mental, written or calculatortechnique to perform calculations

v) mentally perform simple numerical calculations with wholenumbers, money and common fractions (eg 35 + 15, 84 ÷ 4,23 × 5, + , of 39)

vi) calculate with whole numbers and money

vii) use a calculator efficiently to perform operations withmoney, decimals, percentages and common fractionsincluding converting between them

ix) round up or down appropriately to a desired degree ofaccuracy



v) use the unitary method to solve ratio problems

ix) calculate rates and use them to solve problems.

13

12

12




75


Measurement

1) Choosing units

2) Measuring

3) Estimating


7) Surface area

8) Time


ii) select appropriate units to measure length, area,temperature, mass, volume, capacity and time

iii) convert between metric units of length, mass and capacity.



iii) decide on an appropriate level of accuracy formeasurements which are to be taken.




ii) calculate the volume of rectangular and triangular prismsVolume of a prism = area of the base × vertical height, V = Ah

iii) calculate the volume of a cylinderVolume of a cylinder = π × radius squared × height, V = πr 2h

iv) recognise that a container with a volume of 1000 cm3 holds1 litre.

i) find the surface area of rectangular and triangular prismsand composite figures involving these.

i) place a series of events in order

ii) estimate and measure time and duration of time

iii) add times and make calculations involving time differences

iv) use timelines and a range of types of timetables to solveproblems.




76


Chance and Data


Algebra

3) Formulae



iv) use a scientific calculator to find the mean ( ) of a set ofscores

viii) read and interpret information represented in tables andgraphs (picture, divided bar, sector, column, line andfrequency histograms and polygons).

i) substitute into given formulae, evaluating the subject of theformula

ii) solve linear equations arising from substitution intoformulae, eg P = 2l + 2b



x


Theme 3: Mathematics in the Workplace

Sub-themes: A) Employment B) General skills for employeesC) Running a small business

ConsiderationsThe final three pages of this theme detail the mathematical content of the themeMathematics in the workplace. This is intended to help teachers identify themathematical skills, knowledge and understandings that students should acquire asthey undertake this theme.


It is not intended in this theme that students calculate tax using a sliding tax scale.Rather students should read and interpret tables including those related to theamount of tax paid for different taxable incomes.

This theme covers the general knowledge and mathematical skills required inaspects of work. It covers finding employment, running a small business, andgeneral mathematical skills for employees. Number, Measurement and Chance anddata are the main strands featured, along with simple Algebra to a lesser extent. Thealgebra in this theme involves substitution and evaluation of formulae, studentswould need to be competent with the Introductory Algebra topic for this.

The skills in the sub-theme of Running a small business could be developed byhaving students set up an imaginary business or conduct a fund raising activitywithin the school. It could be useful to liaise with teachers of Commerce on suchactivities since they may use similar activities, although with a different focus andwithout the emphasis on mathematics. Liaison could facilitate cooperative planningand ensure that the activities are not in conflict for the students in common.

77




Thematic content

A) EmploymentLearning experiences should provide the opportunity for students to:

• calculate earnings including: wages, salary, commission, bonus and payment bypiece and recognise the differences between these different forms of income

• calculate income earned in casual and part-time jobs, considering agreed ratesand special rates for Sundays and public holidays

• perform calculations related to payslips and time sheets

• compare the mean and median wage for individuals (20 or less), indicating whichgives the better picture of the wage for the group

• read and interpret tables related to income, eg PAYE tax tables

• calculate net income after considering deductions such as taxation,superannuation

• prepare a budget for a given income, considering such expenses as rent, food,transport etc

• calculate and make comparisons involving different employment conditions fordifferent careers, considering for example, employment rates, prospects,payment, conditions, and qualifications needed for different careers

• graph data on careers, eg employment rates and payment, deciding upon themost appropriate graph and summarising the results shown by the graph

• interpret graphs and other data on employment and make comparisons betweendata displays, eg the changing nature of the workforce, the incidence of peoplechanging careers or the participation rates for men and women in differentcareers.

78





A) EmploymentStudents could:


a) a data entry clerk earns $28 000 per annum.

i) find how much will be earned in a fortnight and in a week

ii) use a PAYE table to find the amount of tax paid in a year

iii) find the net income

iv) calculate the percentage of tax paid on this income

b) Jo the shearer earns $168.66 per 100 sheep shorn.

i) How much is this per sheep?

ii) Jo shears 2100 sheep in 4 weeks. How much will Jo earn?

iii) Jo receives a vehicle allowance of 20 cents per km for travel to and fromthe shearing shed. Jo travels 19 km each way, five days a week for thisjob. Calculate the travel allowance

◊ compare mean weekly earnings to median weekly earnings for a group ofteenagers working at part-time jobs and discuss which gives the best predictionof the wages for the group

◊ verify a variety of pay slips and time sheets for accuracy

◊ use newspapers to collect information about a number of careers, and read andinterpret advertisements in the positions vacant section

◊ compare different rates of pay for the careers of their choice, considering totalpay per year for different methods of payment (wages, salary, piece work etc)

◊ investigate employment rates for different careers, interpreting publishedinformation including graphs

◊ examine differences in earnings by gender and the participation rates indifferent careers

◊ choose a job from the positions vacant section of the newspaper which theywould like to apply for and plan their application. Students could use a streetmap to find where the job is located and investigate ways of getting there bypublic transport, calculate the weekly and annual wage and the amount of taxand medicare to be paid and investigate the prospects of the job and thetraining and/or qualifications needed.

(continued)

79




Thematic content

B) General skills for employeesLearning experiences should provide the opportunity for students to:

• work with money, giving change, estimating costs, rounding amounts• mentally perform simple numerical calculations involving whole numbers and

simple fractions• use a calculator efficiently to work out more difficult numerical problems

involving whole numbers, fractions, decimals and percentages• calculate discounts given the percentage of the discount• calculate the percentage rate of discount given the amount of discount• solve simple ratio and rate problems• give estimates for the sizes of the standard units of length, capacity, mass, time• use measuring instruments to efficiently measure length, capacity, mass• make reasonable estimates of measurements • substitute into simple given formulae and evaluate the resulting algebraic

expressions• read and use tables and graphs.

80





A) Employment (continued)

Students could:

◊ give a short presentation about part-time jobs, including rates of pay, anypenalty rates received and associated banking matters

◊ calculate their weekly/yearly expenses considering the difference between livingat home and living away from home

◊ use a prepared spreadsheet or table to chart a superannuation policy over yearsof contributions given different rates of interest

◊ represent their life on a time line indicating the years in which they think theywill work and then make comparisons with statistics on length of working lifeetc to see how realistic their expectations were

◊ consider statistics from the Australian Bureau of Statistics on occupations inAustralia over the past 20 years

◊ compare rates of male/female earnings, full-time/part-time work, differences intypes of employment over the years (white collar, blue collar, service industryworkers, farm workers etc).

B) General skills for employeesStudents could:

◊ estimate the size of the result of calculations, eg having purchased three itemsworth $3.80, $2.99 and $1.60, realise that a $10 note would cover the cost andthat the change will be about $1

◊ make approximations, eg the result of 8 × 84.4 is between 8 × 80 and 8 × 90 orbetween 640 and 720 to check the reasonableness of answers

◊ answer questions like: an advertisement for a radio cassette player offers $100off, so that the discounted price is $279. What percentage discount is this?

◊ use calculators to work out amounts of discounts, eg if there is 10% off themarked price of $37, what will be the discounted price?

(continued)

81




Thematic content

C) Running a small businessLearning experiences should provide the opportunity for students to:

• calculate the costs involved in employing people in a business (eg agreed rates,payroll tax, maternity leave, long service leave, holiday leave loading)

• perform calculations related to pricing (cost price, selling price, markup, margin,markdown, trade discounts)

• perform calculations relating to financing the business (eg ratio of investment bypartners, cost of buying initial stock, investment loans)

• perform calculations relating to banking records (eg minimum monthly balances,overdrafts, simple interest)

• perform calculations related to the costs of running a business (eg rentingpremises, insurance, purchase of stock and equipment).

82





B) General skills for employees (continued)

Students could:

◊ use fractions, decimals and percentages in the retail context, eg

a) find the cost of 1.75 m of timber at $6.95 per metre

b) a measuring cup of lawn food (about 20 g) is needed to fertilise each metresquared of lawn. Calculate the cost of fertilising a rectangular lawn that is130m × 85m if the lawn food comes in 10 kg packets costing $9.80 each

c) a 4 litre tin of paint covers 64 m2. How much paint is needed to cover arectangular wall which is 30 m by 8 m?

◊ develop a clear concept of the size of the main units (eg a metre is about thelength of a long pace, a centimetre is about the size across the little finger, alarge car windscreen is around a square metre, a kilogram is two tubs (thecommon size) of margarine, reasonable walking speed is 5 km/h) in order to aidestimation skills

◊ use measuring equipment (ruler, tape measure, trundle wheel, measuring jugscalibrated in millilitres and litres, balance beam, scales calibrated with 10g, 100gand 1 kg, digital clock and analog clock, protractor, clinometer and directionalcompasses) to increase their competence with practical measuring.

C) Running a small businessStudents could:

◊ answer questions like: a business pays a total of $120 000 in salaries annuallyand the employer’s superannuation contribution is 3%. Find the cost of thiscontribution.

◊ investigate how the staffing needs of different types of businesses can be met, egcasual vs full-time employees, rosters, peak times, seasonal variations

◊ calculate the percentage markup required for appropriate margins in differentretail stores

◊ answer questions like: a shop owner marks everything up by 30% and the costprice of an article is $72. What is the selling price and the value of the profit onthis article?

(continued)

83




Thematic content

84





C) Running a small business (continued)

Students could:

◊ organise a fundraising activity at school to experience real ‘profit and loss’

◊ answer questions like: two partners invested in a business in the ratio 4:9. Thesmaller investment was $12 000. What was the other investment?

◊ use a spreadsheet or tables of compound interest to compare interest, egcompare the interest earned on $10 000 invested at 8% simple interest perannum for three years to $10 000 invested at 8% per annum compoundedannually over the same time period

◊ compare the cost of purchasing, leasing and renting a photocopier or computer

◊ set up and/or use spreadsheets for running a business, controlling stock,investigating sales

◊ find the cost of purchasing a business from a newspaper advertisement,including rental, stock and staffing costs, and calculate the takings that would beneeded to cover this

◊ (E) set up an imaginary company to produce a particular product, consideringaspects such as: raising capital, paying interest, partnerships, advertising andmarketing the product, records and calculations of banking transactions,maintenance of financial records

◊ (E) given an imaginary sum of money ($5000), purchase shares in two or threecompanies, estimate the gains to be had over two months, chart the actualprofit/loss obtained, compare to the profit/loss which could have been made ifother shares had been bought. Set the information up on a spreadsheet and usecomputer-generated graphs to predict gains or losses over a 1 year period.

85





86


Number

1) Number skills

2) Ratio and rate

Through the theme, Mathematics in the workplace, studentsshould develop the ability to:





v) mentally perform simple numerical calculations with wholenumbers, money and common fractions (eg 35 + 15, 84 ÷ 4,23 × 5, + , of 39)



viii) mentally give conversions for common fractions, decimalsand percentages (eg 0.5 = = 50%)


xii) read and interpret tables and graphs related to consumerinformation



ii) simplify ratios

iii) divide a quantity into a given ratio


v) use the unitary method to solve ratio problems

viii) use rates to make comparisons


12

13

12

12




87


Measurement

1) Choosing units

2) Measuring

3) Estimating


i) give reasonable estimates for the size of standard units (mm,cm, m, km, g, kg, mL, L)












88


Chance and Data


Algebra

3) Formulae


i) display data in dot plots and stem-and-leaf plots


iii) put data in order and find the mean, mode, median andrange for a small set of scores


v) use a stem-and-leaf plot to find the median and range of aset of scores




ix) interpret displays which show two sets of data and makecomparisons

x) identify graphs that give misleading information.




equation, eg s = , C = πd, c 2 = a2 + b2.dt

x


Theme 4: Building Design

Sub-themes: A) Style of buildings B) Structure of buildingsC) Plans and models of buildings

ConsiderationsThe final three pages of this theme detail the mathematical content of the themeBuilding design. This is intended to help teachers identify the mathematical skills,knowledge and understandings that students should acquire as they undertake thistheme.


Building design emphasises geometrical skills. Aspects of Number, Measurement andAlgebra are also included in the three sub-themes. Teachers may wish to teach oneor two of the sub-themes together, move on to another theme and return to thistheme later, to ensure variety for their students.

Students should be competent with the topic Introductory algebra for this theme,since they need to be able to use formulae related to building.

This theme also requires students to be able to work with Pythagoras’ theorem —students could have completed the topic on Pythagoras before the sub-themeStructure of buildings or study Pythagoras’ theorem within this sub-theme.

The dimensions of building materials, eg timber, are usually quoted in millimetres.Diagrams in this theme have measurements in millimetres, unless otherwiseindicated.

89




Thematic content

A) The style of buildingsLearning experiences should provide the opportunity for students to:

• recognise, name and draw horizontal, vertical, parallel and perpendicular linesand planes, and skew lines in buildings and drawings

• recognise parallel lines in drawings of buildings

• recognise and locate axes of symmetry in building facades

• observe and recognise the occurrence of common shapes and solids in buildings

• locate and draw geometric shapes, patterns and tessellations in buildings

• describe geometric patterns and tessellations in terms of rotations, reflections andtranslations

• design and draw simple geometric patterns and tessellations suitable for buildingsusing rotation, reflection or translation (or a combination of these) of geometricshapes

• recognise congruent and similar shapes in buildings.

90





A) The style of buildingsStudents could:

◊ find examples of horizontal, vertical, skew, parallel and perpendicular lines inthe classroom

◊ study the facades of local buildings, eg churches, school buildings, shops andoffices, identifying and naming geometric features such as types of lines, planeshapes and common solids

◊ study well-known buildings, eg Sydney Opera House, Sydney Tower, AustraliaSquare, Academy of Science (Canberra), Parthenon, Tower of Pisa, Taj Mahal,Geodesic Dome (Montreal), Olympic Hall (Tokyo), Guggenheim Museum (NewYork), identifying geometrical shapes, transformations, tessellations andsymmetry

◊ mark a walk on a map of the local area to view some buildings of interest

◊ examine the work of specific architects, eg Francis Greenway, Walter BurleyGriffin, Marion Griffin, Louis St John, Frank Lloyd Wright, Phillip Cox, GlennMurcutt, John Andrews, John Verge, Harry Seidler, considering similarities anddifferences in the style of buildings which they have designed and identifyingany special geometric features of the buildings


a) from a sketch representing a garage,identify any horizontal, vertical andoblique lines and planes whichwould occur on the actual garage

b) in this picture of a house, draw inthe axis of symmetry

(continued)

91




Thematic content

B) Structure of buildingsLearning experiences should provide the opportunity for students to:

• substitute into simple building formulae and evaluate the resulting expression• substitute into simple building formulae and solve the resulting equation• estimate and calculate heights of buildings using similar triangles and ratios• calculate heights of buildings using scale drawings and the angle of elevation• estimate the length, area and volume of materials needed for a simple structure

or part thereof, eg a roof of a cubby house or the amount of concrete for footings• calculate the amount of materials required for a simple structure, including

finding the perimeter and area of plane shapes such as the roof, and the volumeof solids such as the footings, and calculate the cost of the materials

• recognise that triangles are used to ensure rigidity in buildings • use a variety of methods to test squareness in structures, including using

diagonals and Pythagoras’ theorem• use Pythagoras’ theorem to solve problems involving buildings.

92





A) The style of buildings (continued)

Students could:

◊ draw in axes of symmetry from pictures or photos of houses and buildings◊ given an incomplete diagram of a building or pattern which has one or more

axes of symmetry, complete the diagram◊ from a picture of the Parthenon, name at least three geometric shapes which can

be seen◊ go on a walk of the local area to observe geometric features in the design of the

buildings and observe and draw tessellations in paving and flooring◊ observe roof styles, eg gable, hip, flat, skillion and discuss and sketch the 2D

and 3D shapes which can be seen◊ draw a variety of tessellations of rectangular and/or triangular paving bricks ◊ use a graphics package to reproduce tessellated paving patterns◊ describe the movements needed to superpose one shape to another from a

picture of a building which contains congruent shapes◊ produce a tessellation which uses regular geometric shapes and have another

student try to describe the transformations used. Discuss whether there is morethan one way to produce the same design.

B) Structure of buildingsStudents could:

◊ use formulae like the following:

a) timber quantities required for flooring:

L = , where L is the number of linear metres of floorboards needed, A is

the area of floor to be covered in m2 and C is the effective cover, ie widthuppermost in metres, of each floorboard

b) volume of concrete for footings = area of the cross-section × thickness

c) no. of bearers = + 1

(continued)

width of roomspacing

AC

93




Thematic content

94





B) Structure of buildings (continued)

Students could:


a) calculate the number of linear metresof flooring required for the followingfloor using the floorboard drawn,given the dimensions:

b) calculate the volume of concreterequired for a driveway which is tobe 300 mm thick. If it costs $125 fora cubic metre of concrete, calculatethe cost of the concrete for thedriveway

◊ use shadow sticks and scale drawings to determine the height of a localbuilding, flagpole, windmill or tower

◊ make and/or use a plumb bob to test if walls are vertical and a spirit level totest whether walls/surfaces are horizontal or vertical

◊ use a clinometer, trundle wheel and a scale drawing to work out the height of alocal building

◊ make a scale drawing using aconvenient scale, use it to calculate theheight of the building, eg:

◊ estimate the number of bricks needed for a particular wall and check the answerby calculation

◊ answer questions like: a) a house is to have six windows measuring 1.5 m × 1.8 m, one window

1 m × 1 m and one window 0.5 m × 1 m. If the window frames are to bewooden, calculate the length of wood required for the frames

b) a garden shed is in the shape of a rectangular prism with height 2 m, length2.3 m and width 1.4 m. Draw a diagram to represent the shed. The door ofthe garden shed has dimensions 0.5 m × 1.7 m. Calculate the area of thedoor and the surface area of the walls.

(continued)

95


0.1m

floorboard cross section

4.5m

3m

floor plan

28m36°

25m0.5m

0.5m



Thematic content

96





B) Structure of buildings (continued)

Students could:

◊ design and cost a simple structure, eg cubby house, garden shed, pergola, carpark, dog kennel

◊ investigate the use of triangles to ensure rigidity in buildings and visit a homebuilding site to see where these shapes are used in the frame

◊ investigate how builders use Pythagoras’ theorem to ensure squareness andrigidity of structures

◊ investigate the Egyptians’ use of rope and Pythagoras’ theorem to make rightangles in buildings

◊ a building 7 m wide is to have a new gable roof with a maximum rise of 1metre. Use Pythagoras’ theorem to calculate the length of the sloping edge ofthe roof.

97




Thematic content

C) Plans and models of buildingsLearning experiences should provide the opportunity for students to:

• use appropriate geometrical tools to draw representations of objects includingcircles, arcs, sectors, angles, parallel lines, perpendicular lines, squares, rectanglesand triangles of given dimensions

• make reasonable sketches of buildings which contain solids such as prisms,cylinders, pyramids and cones

• identify nets of simple solids (prisms, cylinders and pyramids)

• draw nets of simple solids from 3D models

• sketch cross-sections of prisms, cylinders, pyramids and cones

• build models of solids, including those using cubes given front, back, top andside views of 3D figures

• draw solids on isometric grid paper

• given a simple model draw front, top and side views

• choose an appropriate scale from a selection of scales

• make simple scale drawings of a floor plan of a building

• read simple site and house plans and use scales to calculate actual lengths aftermeasuring lengths on the plans

• measure angles from scale drawings and realise that the size of angles remainsconstant between an object and its image under enlargement or reduction

• locate north on site plans and orient site plans accordingly

• draw, read and use simple elevations and sections.

98





C) Plans and models of buildingsStudents could:

◊ draw the net of a tetrahedron and/or an Egyptian Pyramid and discuss theshape of the cross-sections of the solid

◊ build a model, using cubes, of the figure with these views:

◊ make simple models of solids/buildings (eg school, house) and represent themodels on grid paper or isometric paper

◊ build a model using limited budget/resources

◊ answer questions like: draw the solidrepresented by the views indicated in thediagram

◊ draw a scale plan of part of the school or a house including elevations andcross-sections

◊ use graphics packages to draw simple site or floor plans

◊ measure lengths on plans and convert to actual size and realise that angle sizesare constant

◊ from a site plan work out the dimensions of the block of land

◊ measure the angle of pitch on the scaledrawing of a roof, eg:

◊ sketch three different elevations of a house or building from pictures, a modelor the actual building

◊ use an elevation or section to find the height of the roof above the ground orthe dimensions of windows, doors or other features

(continued)

99


front back left right top

top side front



Thematic content

100





C) Plans and models of buildings (continued)

Students could:

◊ answer questions like: draw three possible elevations of a building such as theones here:

◊ decide which would be the appropriate scale to use for a house plan: 1:20,1:100, or 1:1000

◊ make framework models of solids/buildings with pipe cleaners or straws andthen sketch the models.

101





102


Geometry

1) Drawing figuresand makingmodels

2) Arrangementsand locations

3) Geometrical facts,properties andrelationships

Through the theme, Building design, students should develop theability to:

i) use appropriate geometric instruments to draw angles,parallel and perpendicular lines, and common plane shapes

ii) construct squares, rectangles and triangles given dimensions

iii) use appropriate geometric instruments to draw circles, arcsand sectors


v) identify and draw nets of simple solids


vii) represent three-dimensional solids in two dimensions usingbasic conventions

viii) sketch common solids and models from different views (top,side, front etc)





iii) read, interpret and make simple maps and plans usingdistance, direction, coordinates and scales.

i) recognise and name common plane figures (types oftriangles, quadrilaterals and polygons, circles, ellipses)

ii) describe and use properties of triangles (angle sum, rigidity)

iv) recognise and name common solids (prisms, cylinders,pyramids, cones and spheres)

v) recognise and name horizontal, vertical, parallel andperpendicular lines and planes and skew lines

vi) recognise congruent shapes as those able to be superposed

vii) recognise similar figures as those involving enlargements orreductions of figures

viii) calculate lengths of similar figures using the enlargement orreduction factor.




103


Geometry (continued)

4) Movements andtransformations

Number

1) Number skills

2) Ratio and rate

Measurement

1) Choosing units


i) reflect, rotate and translate simple shapes

ii) recognise and draw geometric patterns using rotations,reflections and translations

iii) recognise and draw geometric patterns using tessellations(both regular and semi-regular)

iv) recognise symmetry in two-dimensional shapes

v) give the position of the axes of symmetry in two-dimensional symmetrical figures.





x) interpret and use positive integer powers

xi) interpret and use square roots.


iv) use a ratio to calculate one quantity from another.






104



2) Measuring

3) Estimating

4) Perimeter

5) Area


7) Surface area

9) Pythagoras’theorem

Algebra

3) Formulae








i) find the perimeter of plane shapes.

ii) calculate the area of squares, rectangles and triangles.

ii) calculate the volume of rectangular and triangular prismsVolume of prism = area of base × vertical height, V = Ah

v) solve problems involving volume and capacity.

i) find the surface area of rectangular and triangular prismsand composite figures involving these.

i) use Pythagoras’ theorem to find the side of a right-angledtriangle given the length of two other sidesIn a right-angled triangle, the hypotenuse squared = the sum of thesquares of the other two sides (c 2 = a 2 + b 2)

ii) use Pythagoras’ theorem to solve practical problems.





Theme 5: Mathematics Involving Sport

Sub-themes: A) Sporting venues B) Sporting costs

C) Performance in sport

ConsiderationsThe final three pages of this theme detail the mathematical content of the themeMathematics involving sport. This is intended to help teachers identify themathematical skills, knowledge and understandings that students should acquire asthey undertake this theme.


Mathematics involving sport uses skills mainly from Measurement, Number and Chanceand data. Aspects of Geometry and Algebra are also included in the three sub-themesof Sporting venues, Sporting costs and Performance in sport. Teachers may wish to teachone or two of the sub-themes together, move on to another theme and return tothis theme later, to ensure variety for their students.

This theme could encompass any sports or activities in which the students have aninterest. Teachers may wish to consider individual and/or group project workwithin this theme in order to cater for differing student interest.

It may be necessary for students to complete the topic on Introductory algebra beforecommencing the Performance in sport sub-theme because of the use of formulae inthis section.

Any measurement formulae required should be given to students within thequestion, as will be the case in external examinations for this course.

105




Thematic content

A) Sporting venuesSporting venues could include tracks, fields and courts.

Learning experiences should provide the opportunity for students to:

• give reasonable estimates for the size of standard units of length and area (metre,kilometre, metre squared, hectare)

• estimate and measure the dimensions of a sporting venue

• identify the geometric shapes contained in sporting venues

• produce scale drawings of sporting venues including those which are based ongeometrical shapes

• calculate the dimensions of a sporting venue from a scale drawing

• find the perimeter and area of sporting venues

• calculate the cost of laying a surface for a sporting venue, eg a tennis court.

106





A) Sporting venuesStudents could:

◊ estimate the dimensions of a netball court or football field marked out at schooland then check the dimensions by measurement

◊ draw a scale drawing of a court (eg squash court) or field at the school or in thelocal area

◊ given the dimensions of a sporting field, mark out the field by measuring, usinga reasonable technique for forming right-angles


the diagram here represents a basketball court with some of the dimensionsindicated.

i) list the geometric shapes which can be found in thediagram

ii) represent this court using a scale diagram iii) calculate the perimeter of the court iv) find the length of tape needed to mark out the

perimeter lines of the court, the centre line and thecentre circle

v) calculate the area of the courtvi) the basketball court needs to have a space 2 m wide all around it between

the perimeter lines and the spectators. Calculate the total area needed for abasketball court

◊ draw scale diagrams of throwing areas, eg areas for discus or shot-put

◊ calculate distances around a runningtrack shaped as the one in the diagram(ie semi-circular ends), using theformula Circumference = π × diameter,C = πd. Find the area of the track andcalculate the cost of surfacing the trackat a cost of $27 per square metre

◊ calculate the distance around a running track for different lanes and investigatewhether the distance allowed for a staggered start is accurate

◊ investigate the size, shape and mass of various throwing objects, eg shot-put,discus, hammer.

107


1m28m

15m

100m

50m 64m



Thematic content

B) Sporting costsLearning experiences should provide the opportunity for students to:

• calculate the cost of participating in different sports considering such aspects asthe cost of equipment, uniforms, fees for use of venues, transport costs etc

• compare the costs of participating in different sports.

C) Performance in sportLearning experiences should provide the opportunity for students to:

• estimate time, and measure time using a stop watch for a sporting event,interpreting the display on the stop watch

• given performance information for athletes, make speed, distance and timecalculations for competitors

• compare the performance of competitors by estimating and calculating aspectssuch as:

– rates, eg run rate, strike rate– percentages, eg goals from attempts– improvement over a number of years in the time for a particular event, eg

marathon • organise and display data relating to sport using frequency tables and graphs

• use the mean, mode, median and range to summarise the data

• interpret tables and graphs which represent information concerning sport andmake reasonable conclusions

• interpret displays of data to make comparisons relating to sporting performanceor participation

• use formulae to make calculations related to a variety of sporting statistics and/orscores

• report on the mathematical aspects of a chosen sport.

108





B) Sporting costsStudents could:

◊ choose a particular sport and investigate the cost of participating in it, eg– cost of uniform– cost of special equipment– membership fees– cost of using the venue etc

◊ compare the fees and conditions of membership for different gymnasiums orgolf clubs, etc.

C) Performance in sportStudents could:

◊ estimate the time it would take to walk or run 100 metres then check theestimates by measuring the time taken for these activities

◊ interpret the times shown on a stopwatch, eg read [1:03:46] as 63.46 seconds◊ calculate speed and average length of pace for a 100 m running race◊ make comparisons between runners, considering average speed, reaction time,

length of pace etc◊ investigate swimming and athletics, and

– time races and read measuring equipment at swimming and/or athleticscarnivals

– calculate speed for various strokes and/or races over short or long distances◊ make speed comparisons in km/h for different events, eg sprinting 100 m,

running the 10 000 m event, swimming 200 m etc◊ answer questions like:

a) an athlete runs at 21 km/h. Calculate the number of metres travelled in 15minutes

b) two cricketers had the following statistics: 262 runs from 362 balls faced and400 runs from 628 balls. Who had the better run rate?

◊ compare the scoring rates for two cricket teams, eg consider run rate, strike rate,fall of wickets etc

◊ compare the percentage success rates for various competitors, eg percentage ofgoals scored from attempts (continued)

109




Thematic content

110





C) Performance in sport (continued)

Students could:

◊ answer questions like: the scores for a diver in a diving competition were 6.5, 7,7.5, 5, 7, 8.5, 6. Calculate the range, median score and average for these scores.Before the judges calculate the average for the dive they remove the top andbottom score. Find this average and compare to the first average

◊ graph the winning times of the 400 m running race for the Olympics over anumber of games, predicting the results for the next Olympics

◊ display the winning times for the 100 m freestyle for men and women andcompare the performances

◊ run a mini-Olympics in the classroom or outside using cut-down versions ofequipment (eg a straw for a javelin). Record the results on a spreadsheet, usingthe spreadsheet to total scores for houses and individuals and give interimresults.

◊ work with formulae related to sport, eg:

Figure skating: Final mark is based Cricket: Strike rate on 3 parts of the competition: SR = R ÷ B × 100i) compulsory figures (CF ) SR = strike rate, R = runs scored,ii) a short program (SP ) B = balls facediii) free skating (FS ) Career average

0.06 × CF – standing in CF CA = TR ÷ NO+ 0.04 × SP – standing in SP CA = career average, TR = total runs,+ FS – standing in FS NO = number of times given out

Netball: Success rate Basketball: Individual point scoresSR = G ÷ A × 100 2B + F = PSR = success rate, G = number of goals, B = number of basketsA = number of attempts F = number of successful free throws

P = number of points

◊ choose a particular sport and investigate mathematical aspects of it, eg speed ofcompetitors, track/field shape and size, rules of competition, instruments usedfor measuring, methods of scoring, records in the competition, past statistics ofperformances, the participation rate for the sport, whether it is increasing ordecreasing in popularity.

111





112


Geometry



Number

1) Number skills

Through the theme, Mathematics involving sport, studentsshould develop the ability to:



iii) use appropriate geometric instruments to draw circles, arcsand sectors





iii) recognise and name parts of a circle (centre, radius,diameter, chord, sector, tangent, circumference, arc, semi-circle)

iv) recognise and name common solids (prisms, cylinders,pyramids, cones and spheres).












113


Number (continued)

2) Ratio and rate

Measurement

1) Choosing units

2) Measuring

3) Estimating

4) Perimeter

5) Area



vii) simplify rates viii) use rates to make comparisonsix) calculate rates and use them to solve problems.

i) give reasonable estimates for the size of standard units (mm,cm, m, km, g, kg, mL, L)








i) find the perimeter of plane shapesii) calculate the circumference of a circle given the radius or

diameter Circumference = π × diameter, C = πd.


iii) calculate the area of quadrilaterals (parallelogram,trapezium, rhombus), Parallelogram: Area = base ×perpendicular height, A = bh

Trapezium: Area = half the sum of the parallel sides ×

perpendicular height, A = ( )h

iv) calculate the area of a circle given the diameter or the radiusArea of a circle = π × radius squared, A = πr2

v) find the area of composite figures which can be divided upinto rectangles, triangles and/or circles.

a + b2

Rhombus: Area = half the product of the diagonals, A = xy12




114



8) Time

Chance and Data

1) Collecting andorganising data


Algebra

3) Formula


ii) estimate and measure time and duration of time.

viii) organise data into frequency tables using grouped intervalswhere appropriate (groupings provided).








ix) interpret displays which show two sets of data and makecomparisons.



equation, eg s = , C = πd, c2 = a2 + b2.dt

x


Theme 6: Mathematics in the Community

Sub-themes: A) School B) Home C) Local and wider

ConsiderationsThe final three pages of this theme detail the mathematical content of the themeMathematics in the community. This is intended to help teachers identify themathematical skills, knowledge and understandings that students should acquire asthey undertake this theme.


Mathematics in the community involves mainly numerical and statistical skills. Aspectsof Geometry, Measurement and Algebra are also included in the three sub-themes ofSchool, Home and Local and wider communities. Teachers may wish to teach one ortwo of the sub-themes together, move on to another theme and return to thistheme later, to ensure variety for their students.

The sub-theme on School provides the opportunity for a statistical investigation.Depending on the size of the school population, students may investigate a class orseveral classes. The number of scores should be kept to less than 30 to help avoidconfusion and arithmetic mistakes. Students should be encouraged to carefullyconsider the question(s) they wish to investigate and how the data will be collected.They may decide to design and use a simple survey for this purpose. Some aspectsof data display, eg stem-and-leaf plots, will need to be taught before students canbegin their investigation. Student should be encouraged to explore different waysof displaying the data and to consider how different displays can change theimpression of the data. Graphs chosen could include histograms, polygons, picture,divided bar graphs, sector, line and column graphs. Further information on theinvestigation is available within the accompanying support document.

While students will be expected to calculate summary statistics (mean, mode,median and range) for relatively small data sets, they will not be required to usecumulative frequency in order to find the median. They could use their stem-and-leaf plot to find the median. If data needs to be organised using grouped intervalsstudents should be assisted with the groupings. Students should be able to interpretdata in tabular and graphical form, including data from newspapers, and drawinformal conclusions.

Students should be competent with the topic Introductory algebra for this theme sincethey will be working with formulae. It is not intended that students will use thecompound interest formula, rather they would find amounts using percentagesover a small number of periods (usually two).

If students are in a non-school environment, families and/or friends can besubstituted for school communities as appropriate. The important thing is thatstudents undertake a statistical investigation within a familiar environment.

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Thematic content

A) SchoolLearning experiences should provide the opportunity for students to:

• use fractions, decimals and percentages to compare (by estimation andcalculation) amounts of time spent on each subject

• compare the amount of time allocated to different activities within the day, egsubjects, sport, travel etc

• plan an event, eg a school excursion, which satisfies a set of time constraints

• read and interpret maps of the local area, including interpreting directions usingcompass points and three-figure bearings

• plan and carry out a statistical investigation concerning students at the school orpeople in the community

– define and clarify the question which is to be answered– plan how the data will be collected including whether a sample or population

is to be used and designing a simple questionnaire if necessary– collect the data consistently and use tally marks or organised lists to record the

data– check the data for obvious errors – organise the collected data into frequency tables, using grouped intervals (with

guidance) as appropriate– display collected data in dot plots and stem–and–leaf plots– display collected data in frequency histograms, polygons or other graphs,

choosing appropriate scales for the axes– find the mean, median, mode and range of the set of scores– use appropriate statistical measures such as mean, mode, median and range,

along with fractions and percentages, to summarise the data– report orally or in writing about the data collected, drawing informal

conclusions about the results.

116





A) SchoolStudents could:

◊ calculate the amount of time or average amount of time spent at school on eachsubject per week


a) what fraction of the school day is taken up by recess and lunch?

b) what percentage of the school week is spent on sport?

c) what is the ratio of time spent on English to that spent on History orGeography?

d) which subject(s) take up the most time/least time in the school week?

e) a student studies both French and Japanese, each for six 40 minute periodsevery week. There are 40 weeks in the school year. How much time will theyspend on these languages in one school year?

◊ plan time for answering a test, considering the amount of time which should bespent on each part from the marks assigned to each part

◊ draw up a study timetable

◊ plan an excursion to a local museum for which only three hours is available.They could write up the proposed agenda and estimate times for differentactivities, eg travel to and from the museum, introductory talk, viewing time etc

◊ identify the coordinates of the positions of the school, their home and majorpoints of interest in the locality on a map of the local area

◊ sketch on a map of the locality the route taken from home to school

◊ from a map of the local area find points which are north, north-west, east etc ofthe school

◊ using a directional compass find the bearings (three-figure) of a number ofobjects in the school playground from a fixed point

◊ mark on a map of the local area two points of interest and give the three-figurebearing of one point from another

◊ mark areas on a local map to indicate major areas of residence for students intheir class

(continued)

117




Thematic content

118





A) School (continued)

Students could:

◊ decide upon a statistical investigation to be undertaken relating to students atthe school or people in the community. Examples of such investigations couldinclude: physical and/or personal characteristics (height, reaction time, hand-span, shoe size, resting pulse rate, time travelling to school, favourite food,sport, musical band, TV program, hobby, time spent studying, etc) or other (useof school library or canteen). They could:

– decide how the information could be collected, eg each group of studentscollecting information on a different characteristic or students designing asurvey which could be distributed and then responses collated

– decide on the target population which they are to investigate

– discuss factors which may affect the consistency of the data, eg lack ofrepresentative nature of the chosen group

– display measurement data, eg height in dot plots or stem–and–leaf plots togain an impression of the data (this would only be appropriate for a data setof 30 or less — if the population surveyed is larger than this a particularsample could be selected for more detailed analysis)

– calculate the mean, mode, median and range for a particular sample andmake summary statements about the information

– display the data using frequency histograms, polygons or other graphs (ifclass intervals are required they should be provided)

– create a poster or give a report which represents information about theresults of the investigation

◊ compare information about the school population from two or more differentyears to see if any trends are emerging, eg increasing number of girls, decliningnumber of senior students etc.

119




Thematic content

B) HomeLearning experiences should provide the opportunity for students to:

• calculate and compare the cost of purchasing household items, eg sofas,televisions, using different methods of payment including cash, lay-by, loan andbuying on terms

• find the value of household items considering appreciation or depreciation over asmall number of time periods

• estimate the amount of water used for particular circumstances in the home, eg tofill a bath, water the garden

• calculate the capacity of containers, eg water tanks, in the shape of prisms andcylinders

• perform calculations involving water considering such aspects as the volume andrate of water discharge

• read and interpret gas, electricity and water bills including checking theircorrectness by calculation and finding the interest payable on any overdueaccounts

• read and interpret telephone bills calculating and comparing the cost of phonecalls at different rates.

120





B) HomeStudents could:

◊ perform calculations involved in the purchase of household items using cash,lay-by, debit and/or credit cards, discounts etc using current figures fromadvertisements

◊ plan a budget to save the deposit for a car

◊ investigate the cost of buying a car considering aspects such as:

– interest rates (flat rate and reducible using tables)

– term payment

– registration and insurance costs

– running costs, eg petrol, maintenance

◊ answer questions like: a new computer depreciates at a rate of 30% per year. Itcosts $2000 new. What will it be worth in two years?

◊ find out the usual rate of depreciation for a new car and calculate how much anew family car will be worth in three years’ time

◊ compare the depreciation rates of new cars to the depreciation rate for ten-year-old cars, or the depreciation rates of farm machinery

◊ after determining the rate at which a hose discharges water by considering thetime taken to fill a bucket, estimate the amount of water discharged by the hosein 20 minutes

◊ after surveying the water usage of members of the family and using knowndischarge rates (eg the amount of water used for a shower etc), estimate thewater usage of a family during one week. Calculate the average water usage perperson. Calculate the cost of this amount of water. Use the average water usageper person to estimate the water cost for the family over a given time period.Compare the amount shown on water bills with the estimated cost of the waterfor the family.

◊ estimate and calculate the amount of water wasted in a year if a person alwaysleaves the tap running while cleaning their teeth

◊ calculate the amount of water collected by a rectangular flat roof, or a circulardam for different amounts of rainfall

(continued)

121




Thematic content

122





B) Home (continued)

Students could:


a) calculate the capacity of a cylindrical tank which has diameter 6 m andheight 3 m

b) a swimming pool in the shape of a rectangular prism has the followingdimensions: length 7.6 m, width 3.6 m and depth 1.2 m. Draw a diagram torepresent the pool and this information. The pool is filled to a depth of 1 m,calculate the volume of the pool in cubic metres. 1 m3 holds 1 kilolitre (or1000 L) of water, how many kilolitres will be needed to fill the pool? If thecost of 1000 litres of water is 65 cents, how much will it cost to fill the pool?

c) a water spray, which sprays over a circular area, has an output of 75 litresper hour. Calculate the volume of water used if the spray is on for 2 hours.Find the rate of water flow in litres per minute for this water spray.The radius of the area covered is 2.8 m. Find the amount of area covered bythe water spray.

◊ answer questions like: a person receives a water rate bill for $160. They pay thebill one month late and the interest on overdue accounts is 9.5% per annum.How much extra do they need to pay to settle the account?

◊ examine water, electricity and gas bills, discuss the information given and checkthe calculations, making comparisons between the amounts used for differenthouseholds and for different time periods

◊ draw a graph representing the economy rates for phone calls over varyingdistances or for different periods of time

◊ answer questions like: an STD phone call of 3 minutes at the night time ratecosts 92 cents. The cost of the same phone call at the economy rate is 61 cents.Calculate the percentage saved by ringing when the economy rate applies.

12

123




Thematic content

C) Local and widerLearning experiences should provide the opportunity for students to:

a) draw and interpret step graphs representing postal charges

b) use conversion graphs or tables to change units, eg Australian dollars intoforeign currency and vice versa

c) perform calculations which involve time zones within Australia

d) draw and interpret travel graphs relating to journeys

e) read and interpret road safety statistics presented in tables and graphsconsidering, for example, different times of day for accidents, different types ofvehicles involved in accidents, speed, blood alcohol level, age/sex of drivers,comparisons between localities or countries

f) compare stopping distances of vehicles travelling at different speeds

g) calculate blood alcohol content using formulae and consider the effect ofalcohol on reaction time.

124





C) Local and widerStudents could:

◊ compare the cost of various ways of sending parcels, eg express, surface,ordinary mail


a) the graph indicates the cost ofmailing periodicals to addresseswithin the state. A firm wishes tosend 9 letters each weighing 30 g,135 letters each weighing 65 g and30 letters each weighing 350 g tointrastate addresses. Use the graphprovided to calculate the cost of thismail-out.

b) the graph indicates the approximaterelationship between the Australiandollar and the English pound on aparticular day.

i) how much will 400 Australiandollars be worth in £ sterling?

ii) when Patty returned from aholiday in England she had £150.Use the graph to find out howmuch this would be in Australiandollars

c) the table here shows the conversionfrom temperature in degreesFahrenheit to degrees Celsius.

i) water boils at 100°C. What is this in Fahrenheit?

ii) estimate 100°F in Celsius

(continued)

125


100 200 300 400 500

3040506070

postage costs for large letters intrastate

cents (c)

100 200 300( $ ) Australian dollars

400 500

50100150200250( £ )

pounds sterling

°F

°C

32°

0°

68°

20°

122°

50°

167°

75°

212°

100°



Thematic content

126





C) Local and wider (continued)

Students could:

◊ for a travel graph such as the oneindicated, act out the journey

◊ discuss the time zones in Australia and the differences in times during summer

◊ work out the time in each state if it is 2 pm in NSW (Eastern Standard Time)

◊ use an aeroplane timetable to work out the time taken and average speed for aplane travelling from Sydney to Perth, allowing for the time difference

◊ survey traffic in the local area and prepare a report which summarises the typeand frequency of traffic in local streets. Identify any traffic problems and, on amap of the locality, design alternative routes for heavy traffic.

◊ experiment to find the reaction times of students in the class, eg using a ruler

◊ use the formula D = ut + , where D = stopping distance, u = original speed

in m/s and t = reaction time in seconds, to calculate the stopping distance ofvehicles travelling at different speeds and compare these distances for varyingreaction times and speeds

◊ answer questions like: the blood alcohol content (BAC ) can be approximatedusing the formula: BAC = N × A × 0.0012, where N = the number of drinks, andA = the amount of alcohol in mL. Calculate the BAC of a person who consumesthree (120 mL) glasses of wine (containing 12% alcohol). Is this person over theBAC limit of 0.05?

u2

14

127


10 20 30time in seconds

40 50

51015distance

in meters




128


Geometry


Number

1) Number skills

2) Ratio and rate

Measurement

1) Choosing units

3) Estimating

Through the theme, Mathematics in the community, studentsshould develop the ability to:


iii) read, interpret and make simple maps and plans usingdistance, direction, coordinates and scales.




vi) calculate with integers and money


viii) mentally give conversions for common fractions, decimals

and percentages (eg 0.5 = = 50%)




vii) simplify rates



iii) convert between metric units of length, mass and capacity



12




129




8) Time

Chance and Data

1) Collecting andorganising data


ii) calculate the volume of rectangular and triangular prismsVolume of a prism = area of the base × vertical height, V = Ah

iii) calculate the volume of a cylinder Volume of a cylinder = π ×radius squared × height, V = πr 2h

iv) recognise that a container with a volume of 1000 cm3 holds1 Litre

v) solve problems involving volume and capacity.



iv) use timelines and a range of types of timetables to solveproblems

v) solve simple problems involving time zones withinAustralia.

vi) plan an event which must satisfy a set of time constraints.

i) pose suitable questions which can be answered through datacollection

ii) design and refine a simple survey

iii) recognise the difference between a sample and thepopulation

iv) plan how data will be collected

v) record data using tally marks or organised lists

vi) collect data as consistently and fairly as possible

vii) check raw data for obvious and gross errors

viii) organise data into frequency tables using grouped intervalswhere appropriate (groupings provided).




130


Chance and data(continued)


Algebra

2) Graphs

3) Formulae










i) draw and interpret travel, step and conversion graphs.




equation, eg s = , C = πd, c2 = a2 + b2.dt

x


Topic 1: Geometrical Facts, Properties andRelationships

Drawing geometrical figures; Angles, Triangles, Quadrilaterals,Congruent figures, Similar figures, Other figures

ConsiderationsGroup learning methods, practical activities, written assignments and projectsshould be an integral part of the format in which geometry is taught. Studentsshould discuss their findings, realise the need for reasons and begin to learn to givea reason for their answers, at least informally.

It is important that students can visualise three dimensional objects in differentorientations and draw possible ‘other views’ of the object. Basic conventions shouldbe followed in drawing representations, eg dotted lines represent edges of faceswhich cannot be seen, strokes to represent equal sides etc.

Mathematical templates and geometrical computer software are additional toolsthat are very useful in helping to draw geometrical figures.

Students will have met many of the geometrical facts and relationships in Years7–8. These should be reviewed using a practical approach, ie drawing includingusing templates, cutting out, folding, matching etc. The extent of time spent on thistopic would depend on students’ competence from Years 7–8. In this topic thetreatment of quadrilaterals is informal, with students developing an idea of theshape of different quadrilaterals and being able to identify them, rather than usingformal definitions.

An emphasis on the development and use of appropriate language is important.Students should be encouraged to give a reason for their results to simplenumerical questions which require only one step in reasoning. Such justifications orreasons could be given orally or written as a single word or phrase. Students arenot expected to give formal proofs of results. They should, however, beencouraged to make their reasons concise and clear. Discussion of different studentresponses could help this development.

Extensive links can be made with artistic design, technical drawing and thebuilding industry. The theme Building design also includes some geometricalknowledge and skills, though not to the depth indicated within this topic. There ismore emphasis here on careful constructions of shapes using ruler and protractor,and compasses for circles. If this topic is taught before Building design, students willprobably need to spend less time in the theme practising their drawing skills. If thisis not the case, their experience of drawing figures in Building design will aid themore formal drawing expected here.

131



Topic 1: Geometric Facts, Properties and Relationships

Content

i) Drawing geometrical figuresLearning experiences should provide students with the opportunity to:

• use appropriate geometric instruments to draw angles

• use appropriate geometric tools to draw parallel and perpendicular lines

• construct triangles given their dimensions using ruler, compasses and protractor

• construct quadrilaterals given their dimensions using a ruler, compasses andprotractor

• use compasses and ruler to draw circles, arcs, and sectors

• inscribe regular polygons in circles (equilateral triangles, squares, hexagons,octagons)

• make reasonable sketches of simple solids and their cross-sections

• identify and draw nets of simple solids

• sketch common solids from different views (top, side, front etc)

• draw and label diagrams from a set of simple specifications or a copy of thediagram.

132





i) Drawing geometrical figuresStudents could:

◊ draw a pair of parallel lines which are 2 cm apart, using a ruler and a set square

◊ investigate the most efficient way(s) of checking that two sides of an object areparallel

◊ construct a variety of figures given their dimensions, eg a square of side 5 cm, arectangle of sides 7.5 cm and 4 cm, and a triangle with sides 6 cm, 4 cm and 3 cm

◊ in a circle of diameter 7 cm, draw an equilateral triangle and a hexagon.Measure the sides of these polygons.

◊ represent three-dimensional objects in two dimensions using the appropriateconventions, eg drawing those which are further away smaller, making parallellines come closer together, using ellipses for circles etc

◊ explain the features of a shape or solid to someone who cannot see the shape sothat they could draw an accurate version of the figure

◊ draw a simple diagram from verbal instructions

◊ try to draw triangles with a variety of lengths of sides including one withdimensions of 2, 2 and 4 cm. Develop a ‘rule’ for the lengths of sides oftriangles so that they can be drawn.

◊ make and interpret isometric drawings of solids


a) make drawings to represent solids, egdraw the top, front, side elevations forthe ‘building’ shown here:

b) without measuring, write a set ofgeometric instructions that would enable another person to constructdiagrams similar to these shown below:

◊ draw and label an isosceles triangle ABC, where AB = AC.

133


A B

D C



Content

ii) AnglesLearning experiences should provide students with the opportunity to:

• estimate the size of angles

• recognise and name angle types (acute, right, obtuse, straight, reflex, revolution)

• identify vertically opposite angles and recognise that they are equal

• recognise that adjacent angles adding to 90° form a right angle

• recognise that adjacent angles adding to 180° form a straight angle

• recognise that adjacent angles adding to 360° form a revolution

• recognise and name vertically opposite angles.

iii) Triangles Learning experiences should provide students with the opportunity to:

• recognise and name types of triangles:

A scalene triangle has no sides equal and no angles equal.

An isosceles triangle has two sides equal and the angles opposite the equal sidesare equal.

An equilateral triangle has three equal sides and three angles equal to 60°.

• state and use the property of triangles

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

• recognise that triangles are rigid

• draw axes of symmetry on different triangles

• use the properties of equilateral and isosceles triangles to solve numericalproblems

• give a reason for solutions to simple numerical problems related to triangles.

134





ii) AnglesStudents could:

◊ estimate the size of a variety of angles, eg the ones shown here:

◊ draw and name angles of 25°, 67°, 127°, 90°, 240°, 180°, 360°, 175° … using aprotractor

◊ from diagrams such as these,identify any angles that add to 90°,180° or 360°, or any verticallyopposite angles. Find the size of theangles marked by the pronumerals.

iii) TrianglesStudents could:

◊ generate and classify triangles that satisfy a given condition, eg producetriangles that have two sides equal


a) cut off each corner of an equilateral triangle, as far as the midpoints of thesides. What shape is left over? How could the three ‘corners’ be reassembledto make another triangle?

b) an isosceles triangle has one side 10 cm and one angle 25°. What might thetriangle look like?

c) find the size of the angles marked x inthese diagrams and write a sentence toexplain how you found the size of themarked angle.

135


40° x°

y° y°y°x°z° 100°

45°30°x°x°

115°30°

50°60°x°

x°

x°



Content

iv) Quadrilaterals Learning experiences should provide students with the opportunity to:

• recognise and name types of quadrilaterals: kite, trapezium, parallelogram,rhombus, rectangle, square

• draw axes of symmetry on different quadrilaterals.

136





iii) Triangles (continued)

Students could:


a) write down all you can about this triangle

b) find the size of the angle marked x in thediagram below. What type of triangle is this?Explain your reason.

iv) QuadrilateralsStudents could:

◊ develop a set of questions that could be used to classify quadrilaterals

◊ classify the following quadrilaterals:


a) cut off each corner of a square, as far as the midpoints of the edges. Whatshape is left over? How could the four ‘corners’ be reassembled to makeanother square?

b) take two squares and place the second square centred over the first squarebut at a 45° angle. What is the intersection of the two squares?

137


55°

35°

x°

72°36°



Content

v) Congruent figures Learning experiences should provide students with the opportunity to:

• recognise that figures are congruent if they can be superposed through acombination of rotations, reflections and translations

• identify congruent two-dimensional shapes from a set of shapes by superposition

• recognise that the size of areas, matching sides and angles are preserved incongruent figures.

vi) Similar figuresLearning experiences should provide students with the opportunity to:

• recognise the relationship between the scale factor and the ratio of matchingsides of similar figures

• recognise that shape, angle size and the ratio of matching sides are preserved insimilar figures

• identify similar two-dimensional figures from a set of shapes

• find the scale factor of enlargements

• calculate dimensions of similar figures using the enlargement or reduction factor.

138





v) Congruent figuresStudents could:

◊ from a set of plane figures, select those that are congruent by placing one shapeon top of another

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

◊ fold and cut plane shapes in various ways to form a given number of congruentparts

◊ consider the effect of reflections, rotations and translations on congruent figures

◊ decide whether figures are congruent andwhich transformations might have beenperformed to convert one figure toanother, eg for the figures here.

vi) Similar figuresStudents could:

◊ find examples from real life that involve enlargement or reduction, eg maps,photos, models, and use them to calculate lengths

◊ find the scale factor of enlargement and/or reduction for cartoons or logos

◊ consider the triangle tessellation in the diagram.Find the scale factor between triangle ABC andDEF. Investigate other pairs of similar triangles.

◊ enlarge triangles, eg a 3, 4, 5 triangle to a 6, 8, 10 triangle, or to 9, 12, 15 or 4.5,6, 7.5 etc, and verify that angle size and shape remain constant. Find the scalefactors for each enlargement.

(continued)

139


1 2 21

A D

B

E F

C



Content

vii) Other figuresLearning experiences should provide students with the opportunity to:

• recognise and name polygons, circles, ellipses

• recognise and name parts of a circle (centre, radius, diameter, chord, sector,tangent, circumference, arc, semi-circle)

• divide composite shapes into common geometrical plane figures

• recognise and name common solids (prisms, cylinders, pyramids, cones andspheres).

140





vi) Similar figures (continued)

Students could:


a) the rectangles in the diagram aresimilar. What is the scale factor? Whatis the width of the large rectangle?

b) decide which of the triangles in thediagram are similar and find thefactor of reduction or enlargement.

vii) Other figuresStudents could:

◊ dissect plane shapes into common geometrical figures, eg:

◊ predict which hexominoes can be used as nets for a cube; or colour the net of acube using different colours for the faces and predict how the colours wouldmeet on the cube

◊ identify and name geometrical solids from their environment

◊ identify and name polygons from a set of figures

◊ use a template of regular polygons to help determine which regular polygonstessellate

◊ draw a circle using the software LOGO and describe how the process works

◊ find the centre of a circle using any method.

141


6cm

3.5cm

24cm

?

1

1 1.5

2

3

4

5

12


142



Topic 2: Pythagoras’ Theorem

ConsiderationsPythagoras’ theorem is a very important theorem in mathematics. There should besome consideration of its use in the history of building. Students can be introducedto surds through Pythagoras’ theorem. Operations with surds are not part of thiscourse; however students should see the need for surds through results obtainedusing Pythagoras’ theorem.

The theme Building design uses Pythagoras’ theorem. The theorem could be taughtbefore beginning the theme, or alternatively taught as a topic within the theme,once the need has arisen.

Students should become competent in using Pythagoras’ theorem to solve realworld problems such as those experienced in the theme Building design, and inproblems involving construction, navigation, orienteering and art. In general therelevant diagram involving a right-angled triangle would be given to the students.In an external examination the theorem would be given as shown on the followingpage.

143




Content

Learning experiences should provide students with the opportunity to:

• identify the hypotenuse in a right-angled triangle as the longest side

• establish the relationship between the lengths of the sides of a right-angledtriangle in a practical way

• use Pythagoras’ theorem to find the length of the hypotenuse of a right-angledtriangle given the length of the two other sides

In a right-angled triangle, the hypotenuse squared equals the sum of the squares of the othertwo sides (c2 = a2 + b2).

• use Pythagoras’ theorem to find the length of the side of a right-angled trianglegiven the length of the hypotenuse and another side

• use Pythagoras’ theorem to establish that a triangle has a right angle

• use Pythagoras’ theorem to solve practical problems involving right-angledtriangles.

144





Students could:◊ point out the hypotenuse in right-angled triangles that are in a variety of

orientations

◊ verify Pythagoras’ theorem by measuring the sides of right-angled triangles andcalculating

◊ draw squares on the sides of right-angled triangles and show that the sum of theareas of the squares on the two smaller sides equals the area of the square onthe largest side

◊ given the two smaller sides of a right-angled triangle, find the length of thehypotenuse using Pythagoras’ theorem,eg answer questions like: find the valuesof x in the triangles:

◊ using knotted rope, make a 3, 4, 5 triangle and show that it has a right angleopposite the longest side

◊ discuss how builders use Pythagoras’ theorem

◊ research the history of the development and use of this theorem in othercultures

◊ research Pythagoras and write a short report about Pythagorean times

◊ find the length of the diagonal of a television screen by

a) measuring the diagonal,

b) measuring the sides and using Pythagoras’ theorem to find the diagonal.Compare the lengths of the sides of the screens for different sized sets, giventhat the advertised size of television sets refers to the diagonal.

(continued)

145


7cm

x cm5cm

x cm

6cm 8cm



Content

146





Students could (continued):◊ solve practical problems like:

a) find the distance above the ground ofa kite given the length of the string,the horizontal distance and thedistance between the person’s handholding the string and the ground

b) a farmer needs to brace a rectangulargate measuring 2 m by 3 m with a pieceof timber placed diagonally. Find thelength of timber needed for the brace.

◊ (E) find the value of x in cm below, given that the dimensions of the rectangleare length 5 cm and width 3 cm.

What dimension is not needed to work out x?

147


2m

3m

xx

15m1.2m

h m

30m


148



Topic 3: Chance

Informal concept of chance, Simple experiments, Probability

ConsiderationsIt is important that students develop an understanding of the language of chanceby consideration of chance words used in everyday life. Awareness of the use ofprobability in the media and for industries such as insurance is important. Teachersshould be mindful that this may be the first time in which students haveencountered the concept of chance in mathematics.

Students develop their understanding of probability through experiments. Simplegames of chance should be considered and students should be aware of thechances of winning in the short and long term.

There are many misconceptions about probability, eg some students may thinkthat, after tossing a fair coin six times and getting six tails, the probability ofgaining a head on the next toss would be greater than . Such ideas should bediscussed and clarified.

Simulations of experiments using computers enable many trials to be done quickly.Students can then see that the experimental probability comes closer to thetheoretical probability as the number of trials increases.

The probability of compound events is beyond the scope of this course.

12

149



Topic 3: Chance

Content

i) Informal concept of chanceLearning experiences should provide students with the opportunity to:

• distinguish between possible and impossible events

• distinguish between certain and uncertain events

• compare familiar events and order them from least likely to most likely tohappen

• use the 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.

ii) Simple experimentsLearning experiences should provide students with the opportunity to:

• 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

• recognise that probability estimates become more stable as the number of trialsincreases

• assign probabilities to simple events by reasoning about equally likely outcomes.

12

150



Topic 3: Chance


i) Informal concept of chanceStudents 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 of these events happening from a list of chance words such ascertain, probable, impossible, not very likely etc

◊ investigate the use of chance language in the printed media. Collect chancewords and organise them from most likely to least likely and then assign thewords an associated probability between 0 and 1

◊ comment on statements of chance from newspapers or magazines

◊ by considering the text of different authors, investigate the frequency ofcommon words used by them, eg words like ‘a’ and ‘that’. Can this methoddistinguish between different authors?

◊ describe events which would have a probability of , 0 and 1.

ii) Simple experimentsStudents could:

◊ investigate chance situations, eg throwing a die, tossing a coin, drawing a cardfrom a pack, spinning a wheel, tossing thumb tacks, drawing discs from a bag,matching names and pictures

◊ prepare an organised list of the sample space for the experiment

◊ estimate the relative frequency of an event by performing a series of trials andrecording the number of times the event occurred. Use the result to predict therelative frequency of the event in the future, eg tossing a coin a large number oftimes and then using the observed proportion to predict the chance of heads inthe 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

◊ discuss the idea of randomness and decide whether all results are equally likelyfor an experiment, that is that each result has an equal chance of happening

(continued)

12

151



Topic 3: Chance

Content

iii) Probability Learning experiences should provide students with the opportunity to:

• express probabilities using fractions, decimals and percentages

• use published data to assign probabilities to events

• solve simple probability problems

• recognise complementary events

• solve simple probability problems by reasoning about complementary events.

152



Topic 3: Chance


ii) Simple experiments (continued)

Students could:

◊ use random generators (coins, dice, spinners, cards) to develop the notion ofequally likely events and simulate probability situations

◊ make a spinner which has the two colours, red and blue, where the result willbe blue 3 out of 5 times on average

◊ design a four-coloured circular spinner which would give one colour twice thechance of being chosen as any one of the other colours

◊ use technology to generate random numbers and simulate probabilityexperiments

◊ for randomly chosen telephone numbers from their local area, decide what isthe relative frequency that the last digit of the number is 9

◊ estimate the probability of an event by considering the relative frequency ofevents

◊ from a bag containing an unknown number of red and green counters, estimateby sampling the probability of drawing a red counter

◊ 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.

iii) Probability Students could:

◊ answer questions like: the counters pictured areplaced into a bag. Find the probability of drawinga black counter on the first draw

◊ 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: a letter ischosen at random from the word MATHEMATICS. Swap questions withanother person in the group. Once the questions are completed, they coulddiscuss the solutions

(continued)

153



Topic 3: Chance

Content

154



Topic 3: Chance


iii) Probability (continued)

Students could:

◊ 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 at any instant 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?


a) a CD collection contains the following CDs: 20 classical, 5 jazz, 30 rock and5 country. If a CD is selected at random, find the probability that it is:

i) jazz

ii) country or classical

iii) neither rock nor country

b) what is the chance of winning a prize in the $2 lottery?

c) a fair coin is tossed four times, and a tail is shown each time. What is theprobability that the next toss shows a head?

◊ play a game such as: two players each take it in turns to toss two dice. Person Ascores a point if the total of the two dice is 3, 4 or 5. Person B scores a point ifthe total of the two dice is 2, 11 or 12. The player who reaches 10 points first isthe winner. Discuss whether the game is fair and consider ways of changing therules to make it fair

◊ consider a game involving chance and answer questions on probability such asthe following:

a) in a game where a 6 must be rolled on a dice before starting, what is theprobability of starting the game on the first roll of the dice?

b) in Scrabble, find the chances of drawing a ‘Z’ from the tiles on the first draw.

13

155



156



Topic 4: Introductory Algebra

Patterns, Graphs, Algebraic manipulation, Equations, Formulae

ConsiderationsAlgebra is developed through the use of patterning and the need to generalise todescribe the pattern that has been developed. The ability to describe a pattern intheir own language as well as symbolically will assist students in theirunderstanding of algebra. Students need to develop an understanding thatpronumerals stand for numbers rather than items (eg a could stand for the numberof apples, but not the apples themselves). The theme Mathematics of our environmentalso provides an opportunity for consideration of patterns and graphs.

Some students may have had considerable experience building and describingpatterns using materials. In this case and where students are competent withdescribing simple patterns in words and symbols it would be only necessary toreview this briefly.

Students need to have some experience with informal graphs before moving on tothe straight line graph. It is important that students have experience with graphingthat relates to their daily lives, eg students could graph their hunger levels duringthe day or the distance they are from their bed in the morning as they get readyfor school. Students need to have the opportunity to match a graph with anappropriate statement, draw an appropriate graph to match a statement andinterpret information presented in a graphical form. Students often havemisconceptions about graphs; for example, they may consider that the shape of thegraph represents a physical feature such as a hill or stairs. These misconceptionsneed to be addressed. Group work on graphs using class data could providestudents with the opportunity to improve their understanding of graphs.

Students need to be able to apply simple algebraic conventions, but it is notintended that they spend a large amount of time on more extensive and difficultalgebraic manipulations.

The use of formulae occurs frequently throughout the themes, eg Building designand Mathematics involving sport, and it may be appropriate to teach students to solvesimple equations before they work on these themes.

157




Content

i) PatternsLearning experiences should provide students with the opportunity to:

• identify patterns in number sequences and extend the pattern

• explain and describe patterns using appropriate language

• replace written sentences describing patterns by algebraic expressions

• construct rules to describe simple patterns using appropriate language

• recognise that pronumerals may be used to stand for numbers

• extend a pattern by substituting into a rule in words or symbols

• translate oral or written statements into algebraic statements and vice versa.

158





i) PatternsStudents could:

◊ extend number sequences such as 2, 4, 6, ...; 3, 5, 7, ...; 100, 90, 80, ...; or 1, 4,9, .... Describe the pattern in words and write a rule to describe how the nextvalue in the sequence is found. Use these rules to find the 10th, 20th and 40thnumber

◊ for a geometrical pattern made withmatchsticks such as the one shown, extendthe pattern. Complete a table such as theone shown and write a rule in words todescribe the pattern. Check whether therule works for further shapes by makingthe 10th step and write the pattern inpronumerals. Draw a graph of thenumber of triangles to the number ofmatchsticks and describe the graph

◊ extend a geometric pattern such as theone illustrated and explain in their ownwords how to calculate the number ofmatchsticks needed to make a particularnumber of squares

◊ for a geometric or number pattern, describe the rule in words and then insymbols

◊ answer questions like: the number of matchsticks needed to build a particularpattern is given by the rule N = 2n + 1, where N is the number of matchsticksand n is the number of triangles. Write this rule in your own words and makeand/or draw the pattern. How many matchsticks would be needed to build 8triangles? If there were 99 matchsticks, how many triangles would there be?

◊ for a particular rule expressed symbolically, find the first five terms of thesequence

◊ write the following as algebraic expressions: five times a number, three times anumber minus six, 5 more than twice n, 7 less than a

◊ write an algebraic expression for: think of a number, multiply it by two andthen add seven

(continued)

159


No of triangles 1 2 3 4 5

No of matches



Content

ii) GraphsLearning experiences should provide students with the opportunity to:

• sketch informal graphs to model familiar events, eg noise level within theclassroom during the lesson

• ‘tell the story’ shown by a graph by describing how one quantity varies with theother

• use the relative position of two points on a graph, rather than a detailed scale, tointerpret information

• choose appropriate scales on the vertical and horizontal axes when drawinggraphs

• compare graphs of the same simple situation and decide which one is the mostappropriate and explain why

• interpret travel, step and conversion graphs

• locate and plot positions on a number plane

• use a table of values to represent a linear relationship

• extend a table of values by substituting into a rule

• draw graphs to represent relationships, eg one set of values is always twice theother.

160





i) Patterns (continued)

Students could:

◊ translate expressions like 5 + x, 4b, 3m + 4, 5 – 2a into words

◊ work in pairs or small groups to explain expressions by using flowcharts, eg:

ii) GraphsStudents could:

◊ draw qualitative graphs of mood swings during a grand final football or netballmatch from different points of view

◊ consider the depth of water in various shaped tanks which are being filled at aconstant rate, write a story and sketch the graph of the situation (this might bedone by experimentation)

◊ consider questions like:

a) the graph shows the hours of sleep and age offour students. Who is the oldest? Who is theyoungest? Who needs the most sleep? Whoneeds the least sleep? Which children needthe same amount of sleep?

b) a child climbs a mountain at a steady speed and then starts to run down themountain. Choose the graph that matches this situation from the graphsbelow:

(continued)

161


Input First step Output

3 4 × 3 (4 × 3) ÷ 3

6 4 × 6 (4 × 6) ÷ 3

2 4 × 2 etc × 4b 4b ÷ 34b3

Age

Hours of sleep

AC

B

D

Speed

Time

Speed

Time

Speed

Time

Speed

Time



Content

162





ii) Graphs (continued)

Students could:

◊ interpret travel graphs, eg compare two sketch graphs (which do not havedetailed scales on axes) each of which purport to show how far a person is fromhome at various times after they leave school and walk up the hill towardshome

◊ interpret step graphs of situations such as: phone rates, taxi fares, parking rates

◊ interpret conversion graphs for situations such as: comparison of Australiancurrency to overseas currency, hours of work to weekly pay, m/s to km/h etc

◊ explain why this graph could not be atravel graph showing distance inkilometres and time in hours

◊ explain what is happening to the speedof the car in the graph

◊ consider a graph such as the one shown,complete a table of values from the graph,extend the table, find a rule in words andsymbols that describes the graph.

163


km

hours

d (km)

t (hours)

y

x

1

(1,2)(2,3)

(3,4)



Content

iii) Algebraic manipulationLearning experiences should provide students with the opportunity to:

• represent algebraic expressions using concrete materials

• recognise and use simple algebraic conventions (eg 2a = 2 × a = a + a,a × a = a2, a ÷ 3 = , 5a – a = 4a, a ÷ a = 1, a – a = 0)

• recognise like algebraic terms and collect like terms to simplify expressions

• substitute into simple expressions and evaluate the result.

iv) EquationsLearning experiences should provide students with the opportunity to:

• use algebraic symbols to write simple linear equations from a writtendescription

• decide whether a suggested value could be a solution to a simple linear equationby substitution

• solve linear equations to the level of difficulty reflected by equations similar to3a – 7 = 9.

a3

164





iii) Algebraic manipulationStudents could:


a) in the diagram there is a matchboxcontaining n counters and some extracounters. Use counters and matchboxesto build the expressions listed and drawdiagrams to record the answers:

i) 2n

ii) 3n +1

iii) n + n + 3

b) if n = 9, find the number of counters in each of the expressions above

c) three matchboxes contain an unknown but equal number of matchsticks.There are some additional loose matchsticks. Use these to represent algebraicexpressions such as the following: 3n + 4, 2n + 1, 2(n + 3), 3(n + 1) – 2,where n is the number of matchsticks contained in each box

d) Jie saves $12 per week. How much will he have after 6 weeks? How muchwill he have after n weeks?

◊ simplify expressions like: x + x + x, y – y, z × z, w ÷ w, 2a + 4 + 3a, 3a × 2 × 4,

3c – 2c, , 3v ÷ v.

iv) EquationsStudents could:

◊ decide on the most appropriate equation in symbols to represent a wordproblem, eg 4 more than twice a number is 20, find the number

◊ solve equations like: 3x = 21, 4 + y = 15, = 8, 25 – b = 12, 2c + 7 = 14,5d – 34 = 49

(continued)

a6

6a2

165




Content

v) FormulaeLearning experiences should provide students with the opportunity to:

• substitute into given formulae, evaluating the subject of the formula

• solve linear equations arising from substitution into formulae, eg P = 2l + 2b

• substitute into simple formulae and solve a resulting simple equation, eg s = ,C = πd, c 2 = a 2 + b 2.

dt

166





iv) Equations (continued)


a) Adam earns $5 for the first hour of babysitting and $4 for each hour afterthat. Write an equation to represent this information. If he earned $29 for ababysitting job, how many hours did he work for?

b) a student solved the equation 2a – 6 = 30, and got the answer a = 12. Wasthis correct? Explain your answer

c) is x = 15 the solution for the equation 3x – 21 = 24?

d) Chris saves $9 each week and has a total of $126. For how many weeks hasshe been saving?

v) FormulaeStudents could:


a) a car travelled at a speed (S ) of 85 km/h for 3 hours (T ). Using D = S × T,find distance (D )

b) if L = 20 – 3D, find L if D = 6

c) if A = 4B + 5C, find B if A = 60 and C = 4

d) if c 2 = a 2 + b 2, find c if a = 5 and b = 7

e) if s = , find d if s = 80 and t = 4

f) if C = (F – 32), find C when F = 40, 60, 100

g) if A = (x + y), find A if x = 4 and y = 9.12

59

dt

167



168


96201 3Optns Maths 9-10 9/7/96 3:30 PM Page 168

Standard Course

Content — Options

1. Mathematics Involving Handcrafts (Theme)

2. Tourism and Hospitality (Theme)

3. Geometrical Patterns

4. Trigonometry

5. Further Geometry

6. Further Number

7. Further Measurement

8. Further Algebra

169

Standard Course — Options


170



Option 1: Mathematics Involving Handcrafts

General design, Patchwork, Cross-stitch, Boxes, Origami

ConsiderationsThis option is structured like the themes in the core, with the mathematical contentidentified in the final two pages. It emphasises geometrical knowledge and skillsand requires students to be competent in drawing geometrical figures and tounderstand and know the geometrical facts, properties and relationships, andmovements and transformations.

The option also includes significant work on shape and measurement, and givesstudents further practice in constructing simple shapes given specifications andmeasuring accurately. It also involves estimating sensibly and calculating areas oftwo-dimensional figures, nets and drawings of three-dimensional shapes.

It is important that students use appropriate terminology and develop clarity andconciseness so that they are able to describe spatial patterns and shapes effectivelyto others. Some calculation of costs has been included.

In studying this option it is intended that students involve themselves in practicalactivities. While they may not actually make a patchwork quilt or a cross-stitchpiece, the emphasis is on them designing and planning the making of suchhandcrafts and being able to interpret and report on their design using languagethat includes mention of the geometrical shapes, symmetry and transformations.Students could, however, undertake a practical project and this may be anopportunity for liaison with another Key Learning Area such as Technological andApplied Studies. In this case, students would go through the process of designing,planning, making and reporting on their process.

171



Theme: Mathematics Involving Handcrafts

Thematic outcomes

A) General designLearning experiences should provide the opportunity for students to:

• calculate the amount and cost of material required to make an item, eg string artdesign, jigsaw puzzle, rectangular bag, (remembering seam allowances)

• draft a pattern for the item

• given the pattern pieces for toys, estimate the perimeter and area of irregularshapes and the total amount of material needed to make toys

• estimate the cost of materials for making toys of different sizes

• measure pattern pieces which are regular shapes and calculate the area of theshapes

• measure pattern pieces and calculate the areas of pieces which are simplecomposite shapes.

172





A) General designStudents could:

◊ make a craft item, eg rectangular bag, string art design on wood with nails,jigsaw puzzle etc


a) calculate/approximate the amount of material required to make the bearsketched below

b) if the calico you choose to make the bear is 115 cm wide and costs $2.99 ametre, how much will the bear cost to make?

173


6.5cm

BODY CUT 2

A E

CE

NT

RE

FR

ON

TCE

NT

RE

BA

CK

ARM CUT 4

L N

EAR CUT 4 INTERFACE

P O

LEG CUT 4

J

H I

SOLE CUT 2

H I

HEAD GUSSET CUT 1 CG

EE

DD

HEAD CUT 2

CF

GE

D

EAR



Thematic outcomes

B) PatchworkLearning experiences should provide the opportunity for students to:

• reproduce, on grid paper, patterns involving squares, rectangles and right-angledtriangles

• identify squares, rectangles and different triangles in patchwork designs

• using grid paper, enlarge and reduce patterns involving squares, rectangles andright-angled triangles

• design and draw patterns, involving squares, rectangles and right-angled triangles,using grid paper

• use shading/colour and symmetry to produce variations of the one design

• locate axes of symmetry in a design

• describe the patchwork pattern, using appropriate terminology for symmetry andtransformations

• measure pattern pieces made up of right-angled triangles and/or squares andrectangles

• make accurate templates of pattern pieces

• find areas including seam allowances of pattern pieces

• calculate the amount and cost of materials required to complete a design/project.

174





B) PatchworkStudents could:

◊ make a 20 cm or 30 cm patchwork block to go on the front of their bag

◊ collect, then reproduce, enlarge or reduce patterns of common, but simple,patchwork designs, eg Ohio Star, Eight-Pointed Star, Churn Dash, Log Cabin,Windmill, Jacob’s Ladder, Bear’s Tracks

◊ investigate Amish patchwork and some of their traditional patterns

◊ visit a local quilt exhibition and sketch some of the designs used


a) on the patchwork block shown (Crosses andLosses), mark in all axes of symmetry

b) copy the patchwork block shown (Ohio Star)onto 1 cm grid paper, so that it measures 12 cmby 12 cm

c) For a patchwork block such as the one shown in b), calculate:

i) the area of each type of material needed to make a patchwork blockmeasuring 12 cm × 12 cm excluding seam allowances

ii) how many 12 cm × 12 cm patchwork blocks would be needed to makethe quilt if it is to measure 1.2 m by 1.56 m

iii) the total cost of the quilt if each type of material used costs the sameamount per metre and total cost of the material that makes up the majorpart of the pattern is $90.00.

175


x x x x xx x x xx x xx xx

xxxxxxxxxxxxxxx



Thematic outcomes

C) Cross-stitchLearning experiences should provide the opportunity for students to:

• interpret grids/graphs used in cross-stitch

• using graph paper, mark out a design, eg the letters of the alphabet, in crosses

• using graph paper, devise a simple design to be cross-stitched

• recognise symmetry in cross-stitch patterns

• explain a pattern to someone else using appropriate terminology for two-dimensional shapes, symmetry and transformations.

D) BoxesLearning experiences should provide the opportunity for students to:

• recognise and name the three-dimensional shapes of boxes (square, rectangularand triangular prisms)

• identify nets of different prisms (open and closed prisms)

• draw nets of open prisms

• make boxes from nets of open prisms

• measure different rectangular and triangular prisms and calculate their volumes (V = Ah).

176





C) Cross-stitchStudents could:

◊ collect cross-stitch patterns for different alphabets and compare with their owndesign

◊ collect and analyse cross-stitch patterns of simple pictures

◊ make a simple bookmark or placemat using the cross-stitch design they devised,explaining their design to a peer or to the class

◊ using a cross-stitch alphabet, or a photocopy of one from a book, mark in anyaxes of symmetry for the letters

◊ decide whether all alphabets prepared for cross-stitching would have the sameaxes of symmetry and explain why.

D) BoxesStudents could:

◊ use cardboard to make a box, with or without a lid, to use for a special purpose,eg jewellery, stamps, letters

◊ collect boxes which are prisms (including triangular, square, rectangular prisms),and match a series of nets to the appropriate box

◊ cut the top off a clean milk carton as shown below, and make a net fromcoloured paper to cover the carton allowing a 1 cm edging around the top

◊ collect a series of different-sized milk cartons, measure the dimensions (ignoringthe top section) and find the volume of the cartons in cubic centimetres

◊ cut the net of an open cube out of cardboard; also cut another net of an opensquare prism with base dimensions 5 mm larger and height 1.5 cm greater;paste both nets onto the wrong side of coloured paper, leaving a 1 cm turningaround all edges; then cut out and glue together to make a box with a lid.

177




Thematic outcomes

E) OrigamiLearning experiences should provide the opportunity for students to:

• follow given instructions to use paper-folding techniques in making a variety ofsimple paper animals and/or objects

• describe how to make a simple origami shape, using concise and clear language.

178





E) OrigamiStudents could:

◊ make a mobile of shapes made from paper using origami to decorate theclassroom

◊ demonstrate the making of a particular origami animal to the class

◊ investigate the history of origami

◊ follow verbal instructions given by the teacher to make an origami crane

◊ demonstrate to a peer how to make an origami animal/object of their choice.

179





180


Space




4) Movements andtransformations

Through the theme, Mathematics involving handcrafts,students should develop the ability to:




v) identify and draw nets of simple solids


vii) represent three-dimensional solids in two dimensions usingbasic conventions

viii) sketch common solids and models from different views (top,side, front etc)



ii) locate and plot positions on a number plane.


iv) recognise and name common solids (prisms, cylinders,pyramids, cones and spheres)

vii) recognise similar figures as those involving enlargements orreductions of figures

viii) calculate lengths of similar figures using the enlargement orreduction factor.

i) reflect, rotate and translate simple shapes

ii) recognise and draw geometric patterns using rotations,reflections and translations

iii) recognise and draw geometric patterns using tessellations(both regular and semi-regular)

iv) recognise symmetry in two-dimensional shapes

v) give the position of the axes of symmetry in two-dimensional symmetrical figures.




181


Number

1) Number skills

Measurement

2) Measuring

3) Estimating

5) Area


Through the theme, Mathematics involving handcrafts,students should develop the ability to:




vii) use a calculator efficiently to perform operations withmoney, decimals, percentages and common fractionsincluding converting between them.




ii) make estimates (in standard units) of length, perimeter,angle, area, mass, volume and capacity and check bymeasuring.


iii) calculate the area of quadrilaterals (parallelogram,trapezium, rhombus): Parallelogram: Area = base ×perpendicular height, A = bhTrapezium: Area = half the sum of the parallel sides ×


Rhombus: Area = half the product of the diagonals,A = xy

iv) calculate the area of a circle given the diameter and/or theradius (formula given);Area of a circle = π × radius squared; A = πr 2

v) find the area of composite figures that can be divided upinto rectangles, triangles and/or circles.

ii) calculate the volume of rectangular and triangular prismsVolume of a prism = area of the base × vertical height, V = Ah.

12

a + b2


182



Option 2: Tourism and Hospitality

Tourist attractions, Hospitality

ConsiderationsHospitality institutions include hotels, motels, guest houses, cruise liners,restaurants and fast food restaurants. (This could also include non-commercial‘hospitality’: prisons, hospitals, boarding schools.)

This option is set up similarly to the themes in the core. The mathematical contenthas been identified in the final two pages. Teachers may choose to introduce thisoption by teaching the content first and then using the context of Tourism andhospitality to apply these mathematical skills. Alternatively, teachers might teach theskills through the context of Tourism and hospitality and provide the opportunity forstudents to practise the necessary mathematical skills as required.

183




Thematic outcomes

A) Tourist attractionsLearning experiences should provide the opportunity for students to:

• interpret maps of tourist attractions, describing the location of features and thepaths between them

• draw network diagrams to show the paths between locations at tourist attractions

• sketch a map of a tourist attraction

• from a map of a tourist attraction, use the scale to find the actual distancesbetween features

• estimate and/or calculate the area of a tourist attraction from a map using thescale

• plan a daytrip to a tourist location by considering and calculating aspects such aspublic transport links, modes of transport, opening times, estimated arrival anddeparture times

• calculate the total cost and cost per head of a day trip to a tourist location byconsidering fares, entry and other costs

• make calculations on the costs involved in running a tourist attraction, eg staffing,equipment costs, amount to be charged to ensure that costs are covered etc.

184





A) Tourist attractionsStudents could:

◊ take a map of a tourist attraction, eg Taronga Zoo, and plan the route to betaken during a visit, describing the features to be passed on the way

◊ estimate distances from a map or sketch of a tourist location andwalking/riding/driving times around the location using scales, describe featuresand routes from the map

◊ visit a park, caravan park or camping ground to examine the layout of the parkand then draw a sketch plan of the layout

◊ redesign, by drawing a simple map of the proposal and presenting a report, alocal park to include improved facilities for visitors; eg play equipment,barbecue area, skateboard ramp, swimming pool, walking trail or jogging track,cycle track

◊ from a map of a park, find its perimeter and area of land

◊ design a car park that would hold themaximum number of cars for the block ofland illustrated

◊ plan a hypothetical tourist attraction and:

– decide the best place to locate the car park, gates etc

– design a car park

– plan queuing facilities

– decide opening times

– estimate the length of stay

– produce maps and directions

– design and produce publicity brochures

(continued)

185


220m

40m

350m

450m



Thematic outcomes

186





A) Tourist attractions (continued)

Students could:

◊ plan a class trip for three days by bus to a city, eg Canberra, and:

– survey the class to ascertain what the majority of class members want to seein Canberra

– draw up a flowchart of times of departure, arrival, activities and visits inCanberra, meals etc

– calculate costs for transport, accommodation, meals etc while away

– calculate the cost for each student, considering the number of students whowill attend and work out the minimum number of students to make the tripviable

– obtain a map of the city and mark the planned routes to be taken throughthe city, and the place of accommodation

– revise the plan of activities to ensure the most economical use of transport

◊ visit a tourist attraction and:

– take note of facilities including car parks, toilets, food outlets, queuingarrangements

– investigate staffing levels

– investigate number of patrons and length of stay

– estimate gate takings and other revenue

– produce maps and network diagrams of the tourist attraction.

187




Thematic outcomes

B) HospitalityLearning experiences should provide the opportunity for students to:

• plan the organisation of a social event

• make calculations involving accounts or bills for events, eg meals at a restaurant,catering for a function

• calculate wages from an employee’s perspective, including consideration of ratesof pay, penalty rates, overtime, holiday loading, deductions for taxation

• make calculations related to paying wages from an employer’s perspective,including consideration of superannuation, tax, hours worked, total wage bill fora number of employees, payroll tax

• make calculations involving the cost of running a hospitality establishment, egamount of food to be ordered, equipment needed, staffing rosters, costs of waterand energy usage, rental or leasing arrangements, insurance, other overheads

• calculate and compare occupancy rates of different establishments, eg hotels,motels, guesthouses

• investigate and compare tariffs (room rates, discounts, packages and seasonalvariations)

• interpret and draw plans of a small section in the establishment

• interpret prepared displays of statistics on tourism and hospitality, eg graphs ofnumbers of tourists, hotel occupancy rates, rate of employment in the industry.

188





B) HospitalityStudents could:

◊ plan a social event, eg class/school picnic, school dance, excursion, schoolbarbecue, a cake stall, and:

– calculate the cost of food to be purchased for the expected number attending

– calculate the cost of entertainment, venue hire and other overheads

– work out the amount to be charged per head to cover costs or make a profit

– plan the organisation and setting up of the event in a flow chart or otherorganised way

– draw up the anticipated budget for the event and compare to the actual costof the event

◊ visit a restaurant to investigate the methods used to record orders, plan meals toarrive together and calculate bills for tables, including working out how mucheach person should pay and calculating change

◊ investigate cruise liners, considering catering/provisioning, cabin sizes, berthingcosts, and planning cruise routes

◊ simulate reservations and bookings

◊ invite a hotel manager to visit the class to provide information about staffing,rosters, planning for increased occupancy, holiday packages etc

◊ collect reports on tourism and hospitality from newspaper, write a shortdescription of their content and interpret any graphs included

◊ design a section of a hotel, camping ground or caravan park and:

– investigate the area required for each room/site

– plan the number and location of rooms/sites

– plan the number and location of toilets/showers

– consider the location of other facilities: TV/recreation room, laundry,swimming pool etc

– draw a sketch map of the building, ground or park

– present a report on the design to the class.

189





190


Space

1) Drawing figuresand making models


Number

1) Number skills

Through the theme, Tourism and hospitality, students shoulddevelop the ability to:





iii) read, interpret and make simple maps and plans usingdistance, direction, coordinates and scales

iv) use and draw network diagrams to represent the order ofevents and paths between locations.





v) mentally perform simple numerical calculations with wholenumbers, money and common fractions (eg 35 + 15, 84 ÷ 4,

23 × 5, + , of 39)



viii) mentally give conversions for common fractions, decimals

and percentages (eg 0.5 = = 50%)



12

13

12

12




191


Number (continued)

2) Ratio and rate

Measurement

3) Estimating

5) Area

8) Time

Chance and data


Through the theme, Tourism and hospitality, students shoulddevelop the ability to:







iii) calculate the area of quadrilaterals (parallelogram,trapezium, rhombus): Parallelogram: Area = base ×perpendicular height, A = bhTrapezium: Area = half the sum of the parallel sides ×


Rhombus: Area = half the product of the diagonals,A = xy

iv) calculate the area of a circle given the diameter or theradius: Area of a circle = π × radius squared; A = πr 2

v) find the area of composite figures which can be divided upinto rectangles, triangles and/or circles.

i) place a series of events in order



iv) use timelines and a range of types of timetables to solveproblems

vi) plan an event which must satisfy a set of time constraints.


12

a + b2


192



Option 3: Geometrical Patterns

Symmetry, Tessellations, Fractals

ConsiderationsIn this topic, students’ interest in art should be encouraged, along with theirrecognition and appreciation of geometrical patterns.

Students should appreciate tessellations as tiling patterns where there are no gaps.They will have had some experience of tessellations in the core of this course inthe theme Building design. This option provides students with further experience inthe area of tessellations and also introduces them to elementary fractals.

Regular tessellations use only one key shape and students should investigate howmany tessellations of regular polygons exist.

Templates, cardboard cut outs, pattern shapes and the like are useful in showingthe repetition of shapes.

This topic has obvious links with congruence and transformational geometry. Suchlinks should be discussed.

Other areas of geometry can be investigated in depth if time and interest allows —eg the four-colour map problem, envelopes, conic sections, platonic solids,moebius strips, topology, special curves (catenary, spiral, parabola, cardioid etc).

193




Content

i) SymmetryLearning experiences should provide students with the opportunity to:

• identify the incidence of symmetry in nature

• identify objects as symmetrical about a plane, a line or a point.

ii) TessellationsLearning experiences should provide students with the opportunity to:

• recognise and use the transformations: reflection, translation and rotation

• identify the transformations used to create different tessellations

• create their own tessellations including those based on irregular figures

• identify the regular polygons that will tessellate

• construct a variety of tessellations of common geometrical shapes usingappropriate tools

• develop a ‘rule’ that could be used to check if a regular polygon will tessellate

• construct semi-regular tessellations involving two different geometrical shapes.

194





i) SymmetryStudents could:

◊ investigate the incidence of symmetry in nature, eg leaves, starfish, butterfliesetc

◊ identify the letters of the alphabet which, when written in capitals, have an axisof symmetry or point symmetry

◊ identify 2D figures with one, two, three, four and more axes of symmetry

◊ answer questions like: Which of the following set of objects have planes ofsymmetry — a pair of scissors, a pair of pliers, bowling ball, golf club, spoon?

◊ find words that read the same in a mirror

◊ complete irregular shapes so that they have an axis of symmetry, as below.

ii) TessellationsStudents could:

◊ collect frieze patterns and describe the transformations that could be used tocreate the patterns from a single component, eg:

◊ describe the transformations which could be used for a tessellation of commonand unusual shapes, eg:

◊ go on a walk in the locality, considering, sketching and analysing the tilingpatterns they meet along the way

(continued)

195


��



Content

iii) FractalsLearning experiences should provide students with the opportunity to:

• use construction methods to produce fractals based on simple shapes such astriangles and squares

• find the scale factor of reduction for fractals.

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ii) Tessellations (continued)

Students could:

◊ investigate which polygons will tessellate either by drawing or using shape tilesand describe the transformation needed to produce the tessellation

◊ using regular polygons of up to eight sides, and templates, decide which regularpolygons tessellate

◊ explain why some regular polygons tessellate and others do not◊ describe some of the methods Escher used to draw tessellations and produce

Escher-type tessellations◊ draw tessellations using computer software such as LOGO or drawing packages

and repetitions of the tessellating shape. ◊ draw a semi-regular tessellation using regular shapes such as octagons and

squares, or irregular shapes, eg:

iii) FractalsStudents could:

◊ draw fractals for several stages; for example, the Sierpinski triangle, a fractaltree, 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

◊ 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

◊ find the scale factor of reduction for fractals such as the Sierpinski triangle,fractal tree and carpet

◊ use the software LOGO to generate fractals.

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198



Option 4: Trigonometry

Right-angled triangles and trigonometry,Applications of trigonometry

ConsiderationsTrigonometry can be introduced in a problem-solving context, where studentsdevelop the awareness that in right-angled triangles the ratios of sides for aparticular angle are constant. It is important to emphasise the real-life applicationsof trigonometry in building construction, surveying and navigation etc. Furtherexperience with right-angled triangle trigonometry is included in Option 7, Furthermeasurement.

Students must have a scientific calculator with which they can find trigonometricratios of angles and angles given the trigonometric ratios. Care should be takenwith calculators which do not have algebraic logic, so that students are competentwith the necessary order of operations with their calculators, but can write theworking correctly. For example, the calculator might require the order to be ‘30cos’, but students need to write cos 30°.

Trigonometry reinforces ratio, enlarged triangles, Pythagoras’ theorem, scaledrawing and has wide application to problems in a number of areas.

This option involves trigonometry in right-angled triangles only. Students shouldbe familiar with the appropriate terminology of trigonometry, and should becomecompetent with finding the hypotenuse, adjacent and opposite for angles in right-angled triangles in a variety of orientations.

Problems involving bearings, angles of elevation and depression should be keptfairly straightforward at this level and diagrams would usually be given to students,although they should be able to interpret the diagrams. Bearings should be themain compass points (north, north-east, east etc) or three-figure bearings fromnorth. Angles need not be in minutes, but should be expressed to the nearestdegree or to a decimal place.

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Content

i) Right-angled triangles and trigonometryLearning experiences should provide students with the opportunity to:

• identify the hypotenuse in right-angled triangles

• identify the sides which are adjacent or opposite a given angle in right-angledtriangles

• demonstrate understanding that the ratio of matching sides in right triangles (suchas opposite to adjacent) is constant for equal angles

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

• use calculators to find trigonometric ratios of acute angles

• use calculators to find angles given trigonometric ratios (angles to be measured indegrees or decimal degrees)

• use sine, cosine and tangent ratios to find the unknown sides (including thehypotenuse) in right-angled triangles

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

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i) Right-angled triangles and trigonometryStudents could:

◊ investigate the ratios of the sides of similarright-angled triangles, eg explore the pair oftriangles in the diagram, paying particularattention to the ratios of matching sides

◊ identify the hypotenuse in right-angled triangles, and the sides which areopposite or adjacent marked angles, eg for the triangles below

◊ for angles in right-angled triangles, find the value of the sine, cosine and tangentratios

◊ use calculators to find trigonometric ratios, eg cos 25°, tan 72.57°, sin 63°etc

◊ relate the tangent ratio to slope, eg find the angle of slope of a ski jump wherethe vertical rise is 6 m and the horizontal run is 9 m

◊ answer questions relating to right-angled triangles like; given cosA = 0.5 find thesize of angleA, given tan B = 1.8, find the size of angle B to the nearest degree

◊ find out everything they can about a right–angled triangle like the one in thediagram.

201


62° 62°

Ax

Ba

43°

35m



Content

ii) Applications of trigonometryLearning experiences should provide students with the opportunity to:

• identify right-angled triangles in diagrams and use trigonometry to find sides orangles

• use trigonometry to solve simple problems involving right-angled triangles

• find and use directions as bearings measured from north

• solve simple problems which involve bearings

• identify angles of elevation and depression in diagrams involving right-angledtriangles

• solve problems involving angles of elevation and depression.

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ii) Applications of trigonometry Students could:

◊ decide which trigonometric ratio is needed for a particular problem

◊ solve problems like: an escalator at anairport slopes at an angle of 30° and is20 m long. Through what height would aperson be lifted when travelling on theescalator?

◊ match a series of right-angled triangles to a set of written problems

◊ write a problem to go with a particular diagram and see if another studentagrees that the problem is correct for the diagram

◊ 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° and solvingword problems, given the diagrams, to find lengths of unknown sides in theright-angled triangles that result

◊ answer questions by interpreting thediagram and solving the resultingtrigonometric ratio, eg ‘an aircraft leavesSydney and flies 400 km on a bearing of135°. How far south of Sydney is theplane at this time?’

◊ use a clinometer to read angles of elevation and depression, draw relateddiagrams with guidance, and find the heights of buildings, trees etc in theirschool environment

◊ answer questions like: a ski slope falls120 m over a 420 m run. What is theangle of depression from the top of theslope?

203


20m

30°

135°S400km

420m120m


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Option 5: Further Geometry

Angles, Quadrilaterals, Polygons, Solving geometrical problems

ConsiderationsGroup learning methods, writing assignments and projects should be an integralpart of the format in which geometry is taught. Students should talk about theiranswers, realise the need for reasons and begin to learn to give a reason for theiranswers, at least informally.

Aspects of angles and triangles have already been completed in the core of thiscourse, in Geometrical facts, properties and relationships. This option continues theconsideration of angles, including those on parallel lines. The core work ongeometry concentrated on students being able to recognise and name angles, whilethis option requires students to use angle properties. This option also extendsstudents’ knowledge of quadrilaterals and polygons. Some aspects of solving fairlysimple geometrical problems numerically has also been included. It is intendedthat students would use an experimental approach, often using measurement toconfirm properties and relationships.

Mathematical templates are additional tools that are very useful in helping to drawgeometrical figures and graphs of functions. Geometrical software could also beuseful here to help students establish and confirm relationships within shapes.

Extensive links can be made with artistic design, technical drawing and thebuilding industry.

An emphasis on the development and use of appropriate language is important.Language activities where students need to describe geometrical shapes to anotherstudent, or set up a flowchart so that working through a series of questions relatingto a shape would lead to the name of the shape, are valuable in aiding languagedevelopment. Students should be encouraged to give a reason for their result tosimple numerical questions which require only one step in reasoning. Suchjustifications or reasons could be given orally or written as a single word or phrase.Students are not expected to give formal proofs of results.

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Content

i) AnglesLearning experiences should provide students with the opportunity to:

• identify angles that are congruent, complementary and supplementary

• identify and name alternate, corresponding and cointerior angles

• recognise that for parallel lines, alternate and corresponding angles are equal,and cointerior angles are supplementary

• apply the angle relationships above to solve problems, giving a reason for theirsolution.

ii) QuadrilateralsLearning experiences should provide students with the opportunity to:

• describe angle and side properties of quadrilaterals (square, rectangle,parallelogram, trapezium, rhombus)

• establish and use the following:

The angle sum of a quadrilateral is 360°.

• apply properties of quadrilaterals to solve numerical problems.

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i) AnglesStudents could:

◊ Answer questions like: identify pairs of congruent, complementary andsupplementary angles in the following diagrams:

◊ identify pairs of corresponding, alternate and cointerior angles in the followingdiagrams:

◊ draw pairs of parallel lines with the transversal cutting the lines at differentangles, find the size of the angles (by measurement) and draw conclusions aboutthe angle relationships that exist in parallel lines

◊ find the size of as many angles as possible inthe diagram.

ii) QuadrilateralsStudents could:

◊ match quadrilaterals to descriptions or names by measuring if necessary, egmatch the names parallelogram, rectangle, square, kite, trapezium, rhombuswith the following geometric shapes and give a description of each:

◊ by measuring, find the sum of the angles of a number of quadrilaterals, such asthose above

◊ draw any quadrilateral and tear off the corners, fitting them together to showthat the sum of the angles of a quadrilateral is 360°

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

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60°



Content

iii) PolygonsLearning experiences should provide students with the opportunity to:

• find the interior angle sum of a polygon

• determine and use the exterior angle sum of a polygon

• apply the above relationships to solve numerical problems involving polygons.

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ii) Quadrilaterals (continued)

Students could:

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

◊ 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, by drawing, the types of quadrilaterals formed if two intersectingdiagonalsa) cut each other in halfb) do not cut each other in halfc) are the same length and cut each other in halfd) meet at right angles and cut each other in half

◊ find the value of the unknown angles in quadrilaterals such as the ones shown,giving a reason.

iii) PolygonsStudents could:

◊ identify and name polygons, both regular and irregular◊ use a template to draw regular quadrilaterals, pentagons, hexagons and octagons

and count how many triangles (drawn from a vertex) the polygons can bedivided into

◊ draw regular quadrilaterals, pentagons, hexagons and octagons and count howmany triangles (drawn from an internal point) the polygons can be divided into

◊ draw various polygons (equilateral triangle, square, pentagon, hexagon,octagon …) and, using division into triangles, develop a rule that describes therelationship between the number of sides of a polygon and its angle sum(computer software such as LOGO or other commercially available geometricalsoftware can enhance this activity)

◊ investigate the sum of the exterior angles for various polygons and generalise arule for this sum (continued)

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a° x°y°120°

c° a°

65° 50°

f° g°

130°



Content

iv) Solving geometrical problemsLearning experiences should provide students with the opportunity to:

• solve simple geometrical problems numerically giving a reason for the solutions.

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iv) Polygons (continued)

Students could:

◊ use their knowledge of regular polygons to explain why regular hexagonstessellate but some other regular polygons may not; and determine whichregular 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°.

vi) Solving geometrical problems Students could:

◊ answer questions such as:

a) find the size of all the angles in the diagrams below

b) find the value of all the angles in the diagram below and give reasons. Whattype of triangle is ABC?

Sample answer, with perhaps a verbal description:

Triangle ABC has 2 angles equal, so it must have

2 sides equal. It is isosceles.

c) Find the value of a and b in the diagram below, giving reasons.

Sample answer:

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25°

114°B

A

D66°E

C 114°B 66° 66°

A

48°vertically opposite

angles in a triangle add to 180°

D66°E adds to 180°

C

135° a°

b°

135°

angles add to 180°angles are alternate and lines parallel

45°

135°


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Option 6: Further Number

Directed number, Index notation, Scientific notation,Applying the index laws

ConsiderationsStudents need to be aware that some representations of numbers are needed inparticular contexts (eg scientific notation for very large or very small numbers ordirected numbers for numbers less than zero). Through this option, studentsdevelop confidence in using different representations of numbers and movetowards an intuitive understanding of the size of such numbers.

The need for directed numbers has already been established through working withtemperature in the core of this course (Mathematics involving food and Mathematics ofour environment ). In this option students further their skills for operating withdirected numbers. They should become competent with directed numbers usingmental and written computation, as well as with the help of their calculator.

The need for scientific notation should be identified from our need to work inmeasurement with very large numbers (distances between planets etc) and verysmall numbers (eg nanotechnology). Students should be able to use a scientificcalculator for calculations with indices and scientific notation. The need for correctinterpretation of calculator displays in scientific notation should be stressed.

Indices in this option are confined to those relating to numbers, algebra withindices is beyond the scope of this course. Some time would need to be spent onnegative indices, so that students recognise that a negative index relates toreciprocals. Usually calculations with indices are restricted to integral indices,however there is a small section on indices for square roots. These would usuallybe the only fractional indices considered at this level.

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Content

i) Directed numberLearning experiences should provide students with the opportunity to:

• recognise directed number as having both direction and magnitude

• order directed numbers on a number line

• perform the four operations (addition, subtraction, multiplication and division)with directed numbers.

ii) Index notationLearning experiences should provide students with the opportunity to:

• write xn in expanded form where x > 0, n > 0

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

• use patterns of indices or other approaches to establish the meaning of the zeroindex and negative indices and to demonstrate the reasonableness of thedefinitions:

x0 = 1 and x –m = , where x > 0, m > 0

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

• use indices expressed in expanded form to establish the index laws:

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

• use a calculator in a variety of ways to verify the index laws (integral indices).

1xm

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i) Directed numberStudents could:

◊ describe examples of directed numbers used in society, eg temperature

◊ place directed numbers on a number line

◊ give true/false answers for inequalities such as:

–3 < –5, 6 > –2, – (–2) < 2, 0 > –12, –1000 > –2000

◊ perform operations with directed numbers, eg

3 – 5, –4 + 7, –5 – 16, 5 × (–3), (–4) × (–12), 16 ÷ (–8), 23 – 5 × 6, 6 – (–5),2 × (–5) – 3 × 4

◊ relate directed number to real life situations and solve problems, eg thetemperature at Charlotte’s Pass was –4°C and dropped 5° overnight. What wasthe minimum temperature overnight?

ii) Index notationStudents could:

◊ write numbers such as the following in expanded form:23, (–4)2, 105, 71, 3 × 24,

◊ use patterns with indices such as 34, 33, 32, 31, 30 to illustrate the zero index

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

◊ explain why 32 × 34 ≠ 96

◊ 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

◊ decide whether statements like 2–1 > 1, 1 < 3–1 are true or false

◊ write expressions like 23 × 24, 56 ÷ 54 and (26)3 in expanded form and henceexpress the result in index form

◊ use a calculator to evaluate questions involving indices like 47 × 43, 34 × 3–2,86 ÷ 89 , (43)4, comparing each result to 410, 32, 8–1, 412 respectively.

16

12

19

103

5

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Content

iii) Scientific notation Learning experiences should provide students with the opportunity to:

• 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.

iv) Applying the index lawsLearning experiences should provide students with the opportunity to:

• use the following results for positive and negative integral indices

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

• apply these generalisations to numerical and algebraic expressions

• use index laws to define fractional indices for square and cube roots

(eg for x positive, since (√x)2 = x, √x = x )

• use a calculator in a variety of ways to verify operations with numbers in indexform (fractional indices for square and cube roots)

• solve numerical problems involving indices.

12

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iii) Scientific notationStudents could:

◊ explain the difference between 2 × 104 and 24

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

◊ recognise and interpret calculator displays in scientific notation

◊ 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


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 takes light toreach Earth from the sun. Compare this value with that for other stars.

iv) Applying the index lawsStudents could:

◊ find pairs of terms that can be multiplied to give 27, 70, 34, …

◊ find pairs of terms that can be divided to give terms like 3 × 52

◊ simplify expressions like:

23 × 24, 27 ÷ 23, 25 × 2–4, 35 × 3–5, (32)4

◊ write, √5,√7 as expressions with fractional indices

◊ find the value of 4 × 4

◊ use a calculator to compare the values of (√5)2 to (5 )2

and 5

◊ consider historical puzzles such as the grain of rice problem: if one grain of riceis placed on the first square of the chess board, 2 on the second, 4 on the thirdetc, how many grains will be on the last square? How many grains altogether?

12

12

12

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Option 7: Further Measurement

Surveying, Navigation, Navigation on land

ConsiderationsThis option involves surveying and navigation, as practical applications ofmeasurement.

Students should have some practical experience of the surveying methods whichthey will encounter in this option. Surveying requires students to use scaledrawings and to work with bearings. This may need to be reviewed if studentsneed more experience and time to develop these skills. Students should have anunderstanding of right-angled triangle trigonometry before studying the methods ofsurveying.

The work on navigation should be centred on navigational charts. Students shouldbe encouraged to use the language of navigation (eg parallels of latitude, meridiansof longitude, great circles, poles, tropics, principal compass points, nautical milesand knots) 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 and the distortions could be discussed.

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 members of the class orgroup who belong to Scouts, Guides or orienteering clubs, or who are experiencedin bushwalking and reading maps. Students could also investigate surveyingequipment such as the Global Positioning System, which gives very accuratereadings for latitude and longitude.

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Content

i) SurveyingLearning experiences should provide students with the opportunity to:

• choose and use the most appropriate method of measuring horizontal lengths

• draw a perpendicular to a fixed line from a point outside the line

• 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.

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i) SurveyingStudents could:

◊ measure the length of a section of the school playground using differentmethods, eg pacing, measuring tape, trundle wheel, deciding which methodwould be the most accurate

◊ use a set square, or straight edge and compasses to draw a perpendicular to aline from a point outside the line

◊ using a directional compass, determine the bearings of a number of objects inthe school playground and calculate the horizontal angles between the objects

◊ use the offset or radial methods to make a scale drawing of an irregular shapedfield which has been marked out, and calculate the perimeter and area of thefield

◊ use field book entries and the offset method to mark out a shape on the ground


a) use the field book entries to make ascale drawing and find the perimeterand area of the shape

b) this sketch is the result of a radialsurvey. Find the perimeter of the fieldby using a scale drawing.

221


410m

440m

365m311°

237°123°

046° N

270m

24

17

B463322120A

21



Content

ii) NavigationLearning experiences should provide students with the opportunity to:

• recognise that Earth is approximately a sphere and identify the important parts ofa sphere

• explain the use of parallels of latitude and meridians of longitude to determineand record the position of a point on Earth’s surface

• appreciate that a chart is a flat representation of the curved surface of Earth

• 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.

iii) Navigation on landLearning experiences should provide students with the opportunity to:

• orient a topographical map by compass

• read a topographical map

• orient a map by physical features

• use a compass proficiently

• navigate a simple course

• use nature’s compass (sun or stars) to find the four main compass points.

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ii) NavigationStudents could:

◊ from a wire model of the earth, identify the important parts of the sphere suchas the centre, radius, diameter, great circles and small circles

◊ relate the great circles and small circles on a sphere to the parallels of latitudeand meridians of longitude

◊ investigate the history of navigation and the use of various map projections

◊ convert distances in nautical miles to kilometres and speeds in knots to km/hand vice versa


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 variationis 8° E

c) find cities or towns which have the same longitude as Tamworth

d) 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 farapart are these cities?

f) find the city that is 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.

iii) Navigation on landStudents could:

◊ name and describe the main parts of a compass

◊ explain what a compass bearing is, and set and follow 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

(continued)

223




Content

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iii) Navigation on land (continued)

Students could:

◊ obtain a local street map and describe how to travel from the school to the postoffice, giving grid references and compass bearings

◊ make a simple map of a small area

◊ have excursions to parkland where semi-permanent courses are set up

◊ trace out the following course at a park:

1) walk 20 m north

2) 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 thisposition

5) draw a representation of this course on paper

◊ locate the direction south using the Southern Cross.

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226



Option 8: Further Algebra

Algebraic skills, Equations, Graphs of straight lines

ConsiderationsStudents have covered an introductory topic on Algebra in the core of this course.They have had experience with algebra from generalising number patterns, andwith translation of oral and written statements into algebraic statements and viceversa. They have dealt with simple manipulation of algebraic expressions (notinvolving the use of negative numbers) and with relatively straightforwardequations and formulae.

Some further consideration of algebraic patterns before commencing this optionwould provide students with an opportunity to improve their understanding of thepronumeral. An ability to perform operations with negative numbers and integersis important in this option, and experience in this area can be gained through thestudy of the option Further number.

Further algebra provides students with an opportunity to further develop theiralgebraic skills by considering simplification of expressions, further solutions tolinear equations and consideration of the straight line. Students have practisedsome graphing techniques in the topic Introductory algebra — in this option studentsstudy linear graphs and their equations in depth, and develop an appreciation ofthe significance of m and b in the equation y = mx + b. An investigative approachthrough graphics calculators or graphing software would be useful to supportstudents’ development of understanding of graphs in this option.

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Content

i) Algebraic skillsLearning experiences should provide students with the opportunity to:

• represent equivalent algebraic expressions using concrete materials or diagrams

• recognise equivalent algebraic expressions

• recognise and use simple algebraic conventions involving the four operations

• recognise like algebraic terms and collect like terms to simplify expressions,including those that contain more than one variable and involve operations withnegative numbers

• remove grouping symbols and simplify the resulting expression

• identify common factors and use them to factorise algebraic expressions

• substitute into algebraic expressions and evaluate the result.

228





i) Algebraic skillsStudents could:

◊ verify that 9 × (3 + 7)= 9 × 3 + 9 × 7= 90 and use this to remove the brackets in

9 × (a + 7) or 9(a + 7)

◊ using three cups containing an equal but unknown number of counters andadditional counters, set up representations of 3(n + 4) and 3n + 12 and showthat these expressions are equivalent

◊ draw a diagram using rectangles and an array of dots to show equivalences suchas 4(n + 2) = 4n + 8, eg

is the same as

◊ distinguish between 3(a + b) and 3a + b and explain the difference◊ simplify expressions like:

2x2 – 5x2

9ab + 3ba 2x2 + 5y – 3x2

2x × 3x 12a ÷ 3a3a × 2a × –4a 4x (3x + 2) – (x – 1)

4a2 ÷ a –

◊ answer questions like:a) the factors of 6 are 1, 2, 3 and 6. What are the factors of 6x2?

b) choose the terms that have common factors: 9a4, 6a, 3b, 6, 7ab, c) Factorise and simplify the following expressions

3x + 15 a2 – 5a4ab + 14b –12x + 16xy

◊ make substitutions into algebraic expressions and evaluate the result, eg if x = 3and y = –4, evaluate:a) x + y b) 2x – 3y c) x2 + y2 d) 3xy e) f)

5x3 + y

12x + 2y4

12x – 8xy4x

16x + 88

4a3

8a4

3a 2

a

8ab2

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Content

ii) EquationsLearning experiences should provide students with the opportunity to:

• use algebraic expressions to write linear equations from a written description in areal-life context

• decide whether a suggested value could be a solution to a linear equation bysubstitution

• use a variety of analytical methods to solve a range of linear equations, includingequations that involve brackets and fractions (numerical denominators)

• compare and contrast different methods of solutions to the one equation

• check solutions, ensuring that the result makes sense in context

• solve problems involving equations.

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ii) EquationsStudents could:

◊ for the tables of values below, find equations that connect s and t

◊ set up equations from real situations such as taxi fares, postage and telephonerates

◊ decide on an equation to match a series of pairs of numbers, eg (0,–2), (1,0),(2,2), (3,4), (4,6)

◊ use the method of guess, check and improve to solve equations

◊ for the relationship c = 3a + 5, find a when c = 10; if y = 2x + 5, find xwhen y =11

◊ 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 linear equations of the type:

3a + 7 = 22 2(x + 5) = 9 = 4 + 5 = 7

◊ solve word problems that result in equations like:

2x – 3 = x + 7 or 3(x + 4) = 5(x – 3)

◊ compare different methods of answering questions like: the solution to3(x + 2) = x + 4 is x = –1. Change one term or sign in the initial equation sothat the answer will be x = 4


a) the area of a triangle is 42 cm2 and the base length is 12 cm. What is theheight of the triangle?

b) a student earns $5 for the first hour of babysitting and $4 for each hour afterthat. Write an equation to represent this information. If he earned $29 for ababysitting job, how many hours did he work for?

x3

x – 13

231


s 0 1 2 3 4

t 1 2 3 4 5

s 1 2 3 4 5 6

t 3 5 7 9 11 13



Content

iii) Graphs of straight linesLearning experiences should provide students with the opportunity to:

• set up a table of values for the relationship y = mx + b

• graph equations of the form y = mx + b

• describe the significance of m and b for the graph of y = mx + b

• define m as the gradient of the line y = mx + b where the gradient is

• use the graph of the straight line to find its gradient

• from the graph of a straight line determine its equation in the form y = mx + b

• solve linear simultaneous equations by finding the point of intersection of theirgraphs.

riserun

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iii) Graphs of straight linesStudents could:

◊ construct tables of values and use coordinates to graph straight lines, eg y = x,y = 2x – 1, y = 3 – x, y = + 2, y = 5, x = – 4

◊ describe what ‘gradient’ means

◊ answer questions like: When will a line have a negative gradient?

◊ for a variety of lines such as those above describe how the gradient changes fordifferent coefficients of x

◊ 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

◊ draw a line by using just the y intercept and gradient

◊ draw four lines with a gradient of 2 and write down the equations

◊ find the gradient for lines where the scales on the axes are different.


a) if x + 2y = 9, what could x and y be?

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?

◊ match graphs with particular linear relationships by inspecting the constants,choose from a set of given graphs, graphs to match y = x, y = 2x, y = 2x + 3,y = x + 3, y = 3 – x

◊ consider questions like: graph y = 2x + 1 and y = x + 3. Find the point ofintersection of the graphs, and check your answer by substitution

◊ use graphics calculators or graphing software packages to plot pairs of lines andread off the point of intersections.

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Mathematics - ARC · pedagogy for mathematics since the previous Mathematics 9–10 Syllabus, written in the early 1980s. The approach of this Mathematics 9–10 Syllabus reflects