mathematics syllabus for 2013 wace

Mathematics: Accredited March 2008 (updated June 2012) For teaching 2013, examined in 2013

MATHEMATICS


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IMPORTANT INFORMATION Syllabus review

Once a course syllabus has been accredited by the School Curriculum and Standards Authority, the implementation of that syllabus will be monitored by the Course Advisory Committee. This committee can advise the Board of the Authority about any need for syllabus review. Syllabus change deemed to be minor requires schools to be notified of the change at least six months before implementation. Major syllabus change requires schools to be notified 18 months before implementation. Formal processes of syllabus review and requisite reaccreditation will apply.

Other sources of information

The Western Australian Certificate of Education (WACE) Manual contains essential information on assessment, moderation and examinations that need to be read in conjunction with this course.

The School Curriculum and Standards Authority website www.scsa.wa.edu.au and extranet provides support materials including sample programs, course outlines, assessment outlines, assessment tasks with marking keys, past WACE examinations with marking keys, grade descriptions with annotated student work samples and standards guides.

WACE providers

Throughout this document the term ‘school’ is intended to include both schools and other WACE providers.

Currency

This document may be subject to minor updates. Users who download and print copies of this document are responsible for checking for updates. Advice about any changes made to the document is provided through the Authority communication processes.

Copyright

© School Curriculum and Standards Authority, 2007. This document—apart from any third party copyright material contained in it—may be freely copied or communicated for non-commercial purposes by educational institutions, provided that it is not changed in any way and that the School Curriculum and Standards Authority is acknowledged as the copyright owner. Copying or communication for any other purpose can be done only within the terms of the Copyright Act or by permission of the School Curriculum and Standards Authority. Copying or communication of any third party

2008/16128[v16]


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Rationale There are strong, enduring reasons for the prominence of mathematics in the school curriculum. According to one leading mathematics educator these reasons are: ‘To teach basic skills; to help children learn to think logically; to prepare students for productive life and work; and to develop quantitatively literate citizens.’ – Lynn Arthur Steen

Adapted from: Steen, L. A. (1999). On mathematical reasoning [quotation]. Retrieved September, 2007 from http://www.math.wisc.edu/~wilson/Courses/Math903/SteenQuestions.htm.

Others have commented on the true artistic nature of mathematics:

‘Mathematics, rightly viewed, possesses not only truth, but supreme beauty… [it is] sublimely pure, and capable of a stern perfection such as only the greatest art can show.’ – Bertrand Russell.

Adapted from: Russell, Bertrand. (1919). On mathematical beauty [quotation]. Retrieved September, 2007 from Wikipedia website http://en.wikipedia.org/wiki/Bertrand_Russell. Licenced under the Creative Commons Attribution-ShareAlike 3.0 licence. The Mathematics course has been created with these sentiments in mind. It offers senior secondary students the opportunity to advance their mathematical skills, to build and use mathematical models, to solve problems, to learn how to reason logically, and to gain an appreciation of the elegance, beauty and creative nature of mathematics. Mathematics during schooling has traditionally been viewed as the study of number, algebra and geometry, and chance and data ideas. This Mathematics course has a greater emphasis on pattern recognition, recursion, mathematical reasoning, modelling, and the use of technology, in keeping with recent trends in mathematics education, and in response to the growing impact of computers and technology. Students develop fluency in a suite of standard mathematical outcomes in number, algebra, space, measurement, chance and data, including the thoughtful and selective use of appropriate technology. They develop fluency with mathematical methods to deal with applications in today’s world, and also come to appreciate changes in the role and practice of mathematics over time in a range of contexts. Students who choose the Mathematics course will already be familiar with the importance of mathematics in their daily lives. In the course, they learn how mathematics is used to describe and model a vast array of scientific and social phenomena. They develop a richer understanding of the role of mathematical techniques and applications in modelling real problems in a range

of contexts. They engage in posing and solving problems within mathematics itself, and thus appreciate mathematics as a creative endeavour. This gives students the ability to solve mathematical problems in a wide variety of contexts, thereby helping them to gain an appreciation of the wide applicability of mathematics. Students are encouraged to investigate patterns and relationships, draw inferences, make and test conjectures, and convince others of their findings using mathematical reasoning. In this manner they experience firsthand the creative and dynamic aspects of mathematics, and they have the opportunity to improve their reasoning skills and their ability to think logically. This course allows students to appreciate mathematics, as well as helping them to develop the necessary understanding and skills to prepare them for productive working lives. The Mathematics course has been designed to cater for the full range of student abilities and their mathematics achievement at the beginning of their senior years of schooling. The units are written as a sequential development of mathematical concepts, understandings and skills. They are grouped in four stages. Preliminary units provide opportunities for practical and well supported learning to help students develop skills. Stage 1 units emphasise practical uses of mathematics for daily life and the workplace. Stage 2 and Stage 3 units extend the mathematical development in all areas, providing preparation for daily life, the workplace and further studies. People who are mathematically able can contribute greatly towards dealing with many difficult issues facing the world today; problems such as health, environmental sustainability, climate change, and social injustice. We need to understand these problems thoroughly before we can expect to solve them, and this is where mathematics and mathematical modelling are so important.

Course outcomes The Mathematics course is designed to facilitate the achievement of three outcomes. Outcome 1: Number and algebra Students use mathematical language and processes to apply concepts of number and algebra to develop mathematical models, solve practical problems and explain and justify relationships. In achieving this outcome, students: • decide how to represent information, solve

problems and investigate issues; • use number, algebra and calculus concepts and

skills to work mathematically; and • interpret, evaluate and justify numerical and

algebraic results.


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Outcome 2: Space and measurement Students use mathematical language and processes to apply the concepts of space and measurement to develop mathematical models, solve practical problems and explain and justify relationships. In achieving this outcome, students: • decide how to represent information, solve

problems and investigate situations; • use spatial and measurement concepts and

skills to work mathematically; and • interpret, evaluate and justify spatial and

measurement results. Outcome 3: Chance and data Students conduct chance experiments, represent outcomes, quantify chance and interpret chance; and collect, organise, represent, summarise, interpret and report data. In achieving this outcome, students: • conduct chance experiments, quantify and

interpret chance; and • represent, interpret and report data.

Course content The course content is the focus of the learning program. The course content comprises concepts and relationships for: • Number and algebra • Space and measurement • Chance and data. Tools and procedures and the Practice of Mathematics are embedded in the unit content.

Concepts and relationships Number and algebra Numerical and algebraical concepts are of key importance in almost all mathematical activity. Quantification processes allow relationships to be represented numerically, whilst general relationships, including those that involve distinctive patterns of change, are expressed algebraically. For numbers to be used effectively, an understanding of the numbers involved is required, whether they be fractions, whole numbers or real numbers. Also, an understanding of the meaning, use and connections between arithmetic operations and the ability to use and interpret mental, written and technology-based calculations efficiently are required. Functions can be used to model situations, using equations and inequalities. The abstract representations are powerful as they can be manipulated independently of the original contexts from which they were derived. There are many different methods for solving equations and context

is taken into account in choosing the most appropriate equation in a given situation. Using formulas and solving equations allow for a thorough understanding of the settings to which they are connected. Space and measurement Measurement concepts and relationships enable sense to be made of the natural world. The concepts used in this course are length, area, volume, mass, capacity, time, ratio and rate. Direct measurements are made and measurements are calculated indirectly using arithmetic, algebra and calculus. In addition, the space concepts of shape, location, transformation and network are explored.

Chance and data The chance and data component focuses on mathematical concepts and techniques that underpin activities that shape our lives, such as market research and opinion polling, quality control, and assessing claims in the media. A sound understanding of how data are obtained, organised, represented, modelled and interpreted is of key significance. Other connections between our lives and the mathematics of chance and data involve distinctive mathematical thinking and working. These involve systematic enumeration, deterministic and probabilistic thinking, and inference to predict outcomes on the basis of limited data. The production, handling and interpretation of data and the cultivation of mathematical ways of thinking and working are the foci of the study of chance and data.

Tools and procedures Forms and representations The use of standard mathematical tools and procedures is necessary when a problem situation is recognised as being of a certain mathematical form or in a form that can be transformed to facilitate analysis. The ability to make decisions about whether to present exact or approximate numerical results, the extent to which symbolic results should be simplified, and whether to present reasoning as a specific case or as a general solution is necessary. Mathematical results need to be interpreted, judged for reasonableness and presented in context with appropriate units of measure.

Standard procedures need to be recalled by students in this course, whilst fluency is expected for frequently used processes. Appropriate methods are expected to be chosen from an array of symbolic, numerical, graphical or technology-based algorithms.


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Algorithms Computations involving number, data, algebra and calculus need to be performed with facility, reliability and accuracy. Suitable algorithms must be chosen from a collection of symbolic, numerical, graphical or technology-based algorithms. Decisions are needed regarding whether results ought to be numerical or symbolic, and the level of precision or generality required. Tools and procedures are chosen to be consistent with these decisions. Technology Technology of various kinds (spreadsheets, calculators, computer algebra systems, dedicated and dynamic mathematics software, interactive whiteboards and the internet) can support students to investigate, generate, create and explore mathematical ideas. Once selected for use, such technology should be used deliberately, carefully, and frequently. Decisions about the appropriate presentation of results must be considered. These decisions help to influence the optimal use of technologies. The internet is an increasingly important resource that allows students to access mathematically significant information and visually rich dynamic demonstrations of many ideas in this course.

Practice of mathematics Working mathematically The working mathematically outcomes for this course are embedded within the content of the units and in the outcome progressions. In particular, the processes of planning tasks, checking assumptions, selecting appropriate techniques, tools and skills, interpreting results and checking them for reasonableness, and linking results to contexts can be found. Mathematics is recognised as useful because it can be used to model real situations, but care is needed to ensure that chosen models and methods effectively represent the relevant aspects of the reality under investigation. Appreciating mathematics An awareness of the nature of mathematics; how it is created, used and communicated underpins this course. An appreciation of mathematics develops through doing and applying mathematics. The processes include observing, representing, conjecturing, justifying and using methods of formal proof. Mathematics is often an intuitive and creative process. Conjectures, initially tentative and error-prone, require rigorous justification. Mathematical ideas and their appearance and application in our culture, and in historical settings, assist in understanding the relationship between mathematics and contemporary living.

Communicating mathematics Communication skills are central to the development of informed numeracy. The skills that are needed include listening, reading and watching a range of sources of information about mathematics; talking and writing about mathematics to a range of audiences; and interpreting mathematical terms, notations and explanations.

Course units The cognitive difficulty of the content increases with units and stages. The pitch of the content for each stage is notional and there will be overlap between stages. Preliminary Stage units provide opportunities for practical and well supported learning to help students develop skills required for them to be successful upon leaving school or in the transition to Stage 1 units. Stage 1 units provide a practical and applied focus. Stage 2 units provide opportunities for applied learning but there is a greater focus on abstract mathematics. Stage 3 units provide opportunities to extend academic knowledge and understandings in challenging learning contexts.

Unit PAMAT In this unit, students use whole numbers for purposes to meet their daily needs, including money matters. They respond to terms about comparative measurement and the passing of time, follow simple directions and recognise familiar shapes. They engage in counting and sorting familiar objects or events.

Unit PBMAT In this unit, students develop understanding of counting, addition and subtraction of numbers. They study applications of number in everyday situations involving money and measurement. They distinguish length and other attributes of objects, compare attributes of different objects, identify shapes and read time. They describe position and movement, including on informal maps. Students recognise and describe chance in familiar activities and collect and analyse categorical and measurement data. They calculate using mental strategies, written methods and calculators.


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Unit 1AMAT In this unit, students develop understanding of multiplication and division. They use whole numbers and the four operations for practical purposes, including financial matters useful to them personally and in employment. Students measure length and mass of objects and calculate perimeters. They interpret timetables. They explore three-dimensional shapes and use informal maps. Students recognise and describe chance in familiar activities and produce data using probability devices. They collect and describe categorical and time series data. They calculate using mental strategies, written methods and calculators.

Unit 1BMAT In this unit, students use decimals, fractions and percentages for practical purposes. They apply mathematics for personal budgeting, banking and shopping. They estimate and measure length and mass of objects using a variety of instruments, and derive and use methods for calculating perimeter and basic areas. They translate, reflect and rotate shapes in design. Students use repeated measurement to collect data relevant to them, display data in tables and graphs and interpret the displays. They calculate using mental strategies, written methods and calculators.

Unit 1CMAT In this unit, students use decimals, fractions, percentages and ratios for practical purposes. They apply mathematics to financial matters in the workplace. They write and use algebraic rules for number patterns. They measure volume and other attributes of objects, and derive and use formulas for area and volume. They read and draw maps with scales, describe and draw shapes in three dimensions. Students describe likelihood for chance events, and design and test simple probability devices. They collect time series data relevant to them, display data in tables and graphs and interpret the displays. They calculate using mental strategies, written methods and calculators.

Unit 1DMAT In this unit, students use integers, decimals, fractions, percentages and ratios for practical purposes. They apply mathematics in making financial decisions. They write word sentences algebraically and solve simple algebraic equations. They calculate area and perimeters of circles and use the Pythagoras’ theorem for calculating the length of the sides of right triangles. They describe the effects of reflecting, rotating and translating shapes in design, and enlarge, reduce and distort figures. They interpret detailed maps. Students collect measurement data from fair samples, display data in tables and graphs, calculate averages and describe spread of data, and compare datasets. They use mental strategies, written methods, calculators and computer technologies where appropriate.

Unit 1EMAT In this unit, students use positive and negative numbers and numbers with powers for practical purposes. They calculate interest and repayments for loans. They draw graphs to represent real situations, and use them to describe how quantities are related. They use trigonometry to calculate measurements in right triangles, and calculate volume and surface area of shapes. They analyse networks. Students simulate everyday chance events, calculate probabilities and predict using probabilities. They collect bivariate data relevant to them, display the data in tables and graphs, and describe trends. They use mental strategies, written methods, calculators and computer technologies where appropriate.

Unit 2AMAT In this unit, students apply ratios and direct proportion in practical situations. They calculate profit, loss, discount and commission in financial contexts. They study introductory algebra and linear relationships in numeric, algebraic and graphical forms. They use Pythagoras’ theorem for the sides of triangles and analyse the reflection, rotation and translation of shapes in design. Students collect data from fair samples, and represent and interpret the data. They use mental and written methods and technologies where appropriate.

Unit 2BMAT In this unit, students study and apply exponential relationships. They develop skills for solving equations algebraically and graphically, and investigate and generalise number patterns. They use coordinate geometry in two dimensions. They use formulas directly and inversely for calculations involving three-dimensional shapes. They apply trigonometry in right triangles. They represent information using network diagrams. Students simulate everyday chance events, calculate and interpret probabilities, and collect and analyse bivariate and time series data. They use mental and written methods and technologies where appropriate.

Unit 2CMAT In this unit, students calculate interest and repayments in order to make decisions about savings and loans, and they interpret information on financial statements that are part of everyday living. They study and apply quadratic relationships. They extend their knowledge of coordinate geometry, and represent information in networks and interpret network diagrams. Students calculate and interpret probabilities for events with more than one chance component. They analyse datasets, determine trends in data and use trend lines for prediction. They use mental and written methods and technologies where appropriate.


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Unit 2DMAT In this unit, students study functions and their graphs. They formulate recursion rules and apply recursion in practical situations. They explore patterns, making conjectures and testing them. They use trigonometry for the solution of right and acute triangles. Students simulate chance events on technologies, and calculate and interpret probabilities for chance events that occur in two- or three- stages. They plan random samples, collect, and analyse data from them, and infer results for populations. They use mental and written methods and technologies where appropriate.

Unit 3AMAT In this unit, students explore and analyse the properties of functions and their graphs. They develop and use algebraic skills for solving equations. They apply recursion in practical situations, including for finance. They use trigonometry for the solution of triangles. Students use counting principles to calculate probabilities and analyse normally distributed data. They plan sampling methods, analyse data from samples and infer results for populations. They use mental and written methods and technologies where appropriate.

Unit 3BMAT In this unit, students study differential calculus of polynomial functions and use calculus in optimisation problems. They develop algebraic skills for solving systems of linear equations. They analyse and construct project networks. They reason deductively in algebra and geometry. Students analyse bivariate data, and argue to support or contest conclusions about data. They use mental and written methods and technologies where appropriate.

Unit 3CMAT In this unit, students develop their knowledge of calculus concepts and their algebraic, graphing and calculus skills, and apply these in mathematical modelling. They use counting techniques and probability laws, and calculate and interpret probabilities for the binomial, uniform and normal random variables. They use mental and written methods and technologies where appropriate.

Unit 3DMAT In this unit, students extend and apply their understanding of differential and integral calculus. They solve systems of equations in three variables and linear programming problems. They verify and develop deductive proofs in algebra and geometry. Students model data with probability functions and analyse data from samples. They justify decisions and critically assess claims about data. They use mental and written methods and technologies where appropriate.

Time and completion requirements The notional hours for each unit are 55 class contact hours. Units can be delivered typically in a semester or in a designated time period up to a year depending on the needs of the students. Pairs of units can also be delivered concurrently over a one year period. Schools are encouraged to be flexible in their timetabling in order to meet the needs of all of their students. Refer to the WACE Manual for more information about unit and course completion.

Resources Teacher support materials are available on the School Curriculum and Standards Authority website extranet and can be found at www.scsa.wa.edu.au

Vocational Education and Training information Vocational Education and Training (VET) is nationally recognised training that provides people with occupational knowledge and skills and credit towards, or attainment of, a vocational education and training qualification under the Australian Qualifications Framework (AQF). When considering VET delivery in WACE courses it is necessary to: • refer to the WACE Manual, Section 5: Vocational

Education and Training, and • contact education sector/systems representatives

for information on operational issues concerning VET delivery options in schools.

Australian Quality Training Framework (AQTF) AQTF is the quality system that underpins the national vocational education and training sector and outlines the regulatory arrangements in states and territories. It provides the basis for a nationally consistent, high-quality VET system. The AQTF Essential Conditions and Standards for Registered Training Organisations outline a set of auditable standards that must be met and maintained for registration as a training provider in Australia.


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VET integrated delivery VET integrated within a WACE course involves students undertaking one or more VET units of competency concurrently with a WACE course unit. No unit equivalence is given for units of competency attained in this way. VET integrated can be delivered by schools providing they meet AQTF requirements. Schools need to become a Registered Training Organisation (RTO) or work in a partnership arrangement with an RTO to deliver training within the scope for which they are registered. If a school operates in partnership with an RTO, it will be the responsibility of the RTO to assure the quality of the training delivery and assessment. Units of competency from selected training package qualifications have been taken into account during the development of this course. Teachers delivering Stage 1 or Stage 2 units of Mathematics may be able to contextualise aspects of the unit to assist students who are enrolled in VET units of competency elsewhere in their WACE program to gain a greater understanding and appreciation of workplace numeracy. Schools seeking to link delivery of this course with units of competency must read the training package rules for the relevant units of competency and associated qualifications on the Training.gov.au website: www.training.gov.au. This should be done in consultation with the RTO they are in partnership with for certification of the competencies in order to establish suitability of units intended for integration with this course.


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Assessment The WACE Manual contains essential information on principles, policies and procedures for school-based assessment and WACE examinations that needs to be read in conjunction with this document.

School-based assessment The table below provides details of the assessment types for this course and the weighting range for each assessment type. Teachers are required to use the assessment table to develop their own assessment outline for each unit (or pair of units) of the course. This outline includes a range of assessment tasks and indicates the weighting for each task and each assessment type. It also indicates the content and course outcomes each task covers. If a pair of units is assessed using a combined assessment outline, the assessment requirements must still be met for each unit.

In developing an assessment outline and teaching program the following guidelines should be taken into account. • All assessment tasks should take into account

the teaching, learning and assessment principles outlined in the WACE Manual.

• There is flexibility for teachers to design school-based assessment tasks to meet the learning needs of students.

• The assessment table outlines the forms of student response required for this course.

• Student work submitted to demonstrate achievement should only be accepted if the teacher can attest that, to the best of her/his knowledge, all uncited work is the student’s own.

• Evidence collected for each unit must include assessment tasks conducted under test conditions together with other forms of assessment tasks.

Assessment table

Weightings for types Type of assessment

Unit PA

Unit PB and

Stage 1 Stage 2 Stage 3

100% 40–50% 65–75% 75–85%

Response In this type of assessment, students apply their mathematical understanding and skills to analyse, interpret and respond to questions and situations. The assessment type provides for the assessment of conceptual understandings, knowledge of mathematical facts and terminology, problem-solving skills and the use of algorithms. Questions in this type of assessment can range from those that are routine and familiar to students through to non-routine, unfamiliar questions. The questions may be closed and, so, target particular methods and results, or they may be open-ended and allow for choice in the methods and a variety of results. Open-ended questions typically call for high level reasoning. Evidence gathering tools may include assignments, tests, examinations, observation check lists and quizzes. Written assessments may be done under timed conditions.

Best suited to the collection of evidence of student achievement of all course outcomes.

nil 50–60% 25–35% 15–25%

Investigation In this type of assessment, students plan, research, conduct and communicate the findings of an investigation. The assessment type provides for the assessment of mathematical-inquiry skills, problem-solving and modelling skills and course-specific knowledge and skills. Students may investigate mathematical patterns, making and testing conjectures and generalising mathematical relationships. They may select, apply and adapt models and procedures to solve complex problems in contexts and, then, justify their results to themselves and others. They may identify social issues, collect and analyse relevant data in order to reach conclusions and make recommendations. They may develop, over an extended period of time, a theme or project related to the practical application of mathematics. Evidence gathering tools may include diagrams and tables used to organise thoughts and processes, written investigation reports, journals, project reports, posters, oral and multimedia presentations, extended pieces of work, observation checklists and interviews.



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Grades Schools report student achievement in a completed unit at Stage 1, 2 or 3 in terms of grades. The following grades are used: Grade Interpretation A Excellent achievement B High achievement C Satisfactory achievement D Limited achievement E Inadequate achievement Schools report student achievement in Preliminary Stage units as either completed or not completed. Each grade is based on the student’s overall performance for the unit as judged by reference to a set of pre-determined standards. These standards are defined by grade descriptions and annotated work samples. The grade descriptions for this course are provided in Appendix 1. They can also be accessed, together with annotated work samples, through the Guide to Grades link on the course page of the Authority website at www.scsa.wa.edu.au Refer to the WACE Manual for further information regarding grades.

WACE Examinations In their final year, students who are studying at least one Stage 2 pair of units (e.g. 2A/2B) or one Stage 3 pair of units (e.g. 3A/3B) are required to sit an examination in this course, unless they are exempt. WACE examinations are not held for Stage 1 units and/or Preliminary Stage units. Any student may enrol to sit a Stage 2 or Stage 3 examination as a private candidate. There will be four external examinations for the Mathematics course: • Units 2A/2B • Units 2C/2D • Units 3A/3B • Units 3C/3D These examinations will be scheduled at the same time and reflect the last pair of units completed within this course. Each examination will consist of two sections; a calculator-free section and a calculator-assumed section. Each examination assesses the specific content described in the syllabus for the pair of units studied.

Time allowed These examinations will require three hours in total, including approximately 15 minutes changeover period. Details of the WACE examinations in this course are prescribed in the WACE examination design briefs (pages 55–63). Refer to the WACE Manual for further information regarding WACE examinations.

Standards Guides Standards for this course are exemplified in Standards Guides. They include examination questions, annotated candidate responses at the ‘excellent’ and ‘satisfactory’ achievement bands, statistics for each question and comments from examiners. The guides are published on the Authority’s web site at www.scsa.wa.edu.au and are accessed under Examination materials. An extranet log-in is required to view the guides.


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UNIT PAMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students use whole numbers for purposes to meet their daily needs, including money matters. They respond to terms about comparative measurement and the passing of time, follow simple directions and recognise familiar shapes. They engage in counting and sort familiar objects or events.

Suggested learning contexts The unit content will be introduced and applied in a variety of contexts that are accessible to students. Suggested contexts may include: • numbers in relation to everyday living e.g. on

buses, letter boxes, lockers, numerical labels and for shopping (Number)

• measurement in relation to social activities e.g. faster or slower movements, longer or shorter time for tasks (Space and measurement)

• spatial relationships associated with personal health and safety e.g. waiting and moving as requested, following sequenced movements (Space and measurement)

• spatial relationships associated with therapy activities e.g. indicating, moving, squeezing requested shapes (Space and measurement)

• chance associated with familiar events e.g. a bus or train being early or late, an expected baby is a girl (Chance and data)

• data from students e.g. eye and hair colour, preferred TV shows (Chance and data)

• data in relation to life skills or simulated workplace activities e.g. sorting, counting or putting away cutlery, clothing and packaged food (Number, Chance and data).

Unit content This unit includes the content areas: • number • space and measurement • chance and data. This unit includes knowledge, understandings and skills to the degree of complexity described below. Students will be provided with opportunities to: • engage in mathematical activities • carry through tasks • seek assistance to solve problems • communicate results.

The number formats for the unit are counting numbers, whole numbers and ordinal numbers.

1. Number 1.1 Number 1.1.1 use and match relevant numbers 1.1.2 count using one-to-one correspondence

between numbers and objects, for small numbers

1.1.3 represent small numbers with objects 1.1.4 use counting numbers in everyday activities 1.1.5 use ordinal numbers in everyday activities. 1.2 Estimation and calculation 1.2.1 participate in calculation activities 1.2.2 use objects on templates to represent

and/or solve problems 1.2.3 use trial and error strategies. 1.3 Equivalence, equations and inequalities 1.3.1 use terms ‘none’, ‘nothing’, ‘haven’t got any’

or zero. 1.4 Finance 1.4.1 exchange money for goods or services and

receive change.

2. Space and measurement 2.1 Time 2.1.1 use relevant time vocabulary 2.1.2 recognise calendars, and analogue and/or

digital clocks and their uses 2.1.3 sequence familiar daily and weekly events 2.1.4 follow timetables and work routines. 2.2 Length, area, mass, volume and capacity

and angle 2.2.1 compare objects directly using length 2.2.2 heft to compare masses of objects 2.2.3 use everyday comparative language

associated with length and mass 2.2.4 experiment with measuring equipment. 2.3 Location 2.3.1 locate significant areas in buildings and the

environment e.g. toilets, bus stops 2.3.2 place objects in their appropriate locations 2.3.3 respond to, and use directional terms e.g.

under 2.3.4 follow simple spatial directions in games

and practical situations. 2.4 Shape 2.4.1 match like shapes 2.4.2 link objects to names of shapes e.g. wheels

are circles 2.4.3 create shapes with different materials. 2.5 Transformations 2.5.1 orient objects to fit containers 2.5.2 rearrange objects to suit purposes.


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3. Chance and data 3.1 Collect and organise data 3.1.1 identify attributes of self, others and objects 3.1.2 use trial and error to complete tasks 3.1.3 classify self and familiar things. 3.2 Represent data 3.2.1 identify groups of objects or people. 3.3 Interpret data 3.3.1 recognise themselves or objects as part of

particular groups.

Assessment The type of assessment in the table below is consistent with the teaching and learning strategies considered to be the most supportive of student achievement of the outcomes in this unit of the Mathematics course. The table provides details of the assessment type, examples of different ways it can be applied and the weighting.

WeightingUnit PA Type of assessment

100%

Response

Students apply their mathematical understanding and skills to respond to questions and situations.

Questions in this type of assessment can range from those that are routine and familiar to students through to non-routine, unfamiliar questions. The questions will be closed and, so, target particular methods and results.

Evidence gathering tools may include observation check lists and quizzes.

Suited to the collection of evidence of student achievement of all course outcomes.


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UNIT PBMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students develop understanding of counting, addition and subtraction of numbers. They study applications of number in everyday situations involving money and measurement. They distinguish length and other attributes of objects, compare attributes of different objects, identify shapes and read time. They describe position and movement, including on informal maps. Students recognise and describe chance in familiar activities and collect and analyse categorical and measurement data. They calculate using mental strategies, written methods and calculators.

Suggested learning contexts The unit content will be introduced and applied in a variety of contexts that are accessible to students. Suggested contexts may include: • number from students’ experience e.g. scoring

in sport, bag limits for fishing, stock counts, number of stock in farming, individual’s school attendance over a month (Number)

• finance in relation to everyday personal spending e.g. on food, public transport fares, movie tickets, petrol and parking meter tickets (Number)

• measurement related to time e.g. watches and calendars (Space and measurement)

• space in relation to student environment e.g. position, maps of familiar areas (Space and measurement)

• shapes in the environment e.g. international road signs (Space and measurement)

• transformations associated with patterns e.g. in tiling, logos, tattoos, nature (Space and measurement)

• chance associated with familiar events e.g. a particular team will win a competition, some students in the class will play sport after school, a die shows a six (Chance and data)

• categorical data from students e.g. after school activities, preferred snacks (Number, Chance and data)

• categorical data from the school environment e.g. visible shapes, mode of transport to school (Number, Space and measurement, Chance and data)

• measurement data from students e.g. head circumference, length of long jump (Number, Space and measurement, Chance and data)

• measuring instruments, calculations, and mathematical representation that are relevant to students in vocational programs (Number, Space and measurement, Chance and data).

Unit content This unit builds on the content covered by the previous unit. This unit includes the content areas: • number • space and measurement • chance and data. This unit includes knowledge, understandings and skills to the degree of complexity described below. Students will be provided with opportunities to: • choose and use mathematical methods to carry

through tasks • check and correct answers • communicate reasoning and results. The number formats for the unit are counting numbers, whole numbers and ordinal numbers.

1. Number 1.1 Number 1.1.1 read, write, say and use counting numbers

into the hundreds 1.1.2 represent numbers with drawings 1.1.3 compare collection sizes 1.1.4 order whole numbers 1.1.5 use ordinal numbers to indicate place in a

sequence, such as third place. 1.2 Estimation and calculation 1.2.1 use mental imagery and students’ own

mental counting strategies to add and subtract small whole numbers

1.2.2 use materials to illustrate addition and subtraction

1.2.3 use counting to add and subtract small numbers

1.2.4 use addition facts to 10 + 10 in mental computation

1.2.5 derive subtraction from addition facts 1.2.6 use counting and repeated addition to

achieve multiplication 1.2.7 use counting and repeated subtraction to

achieve division 1.2.8 add and subtract whole numbers on a

calculator 1.2.9 partition quantities into two or four equal

parts and name the parts halves and quarters.

1.3 Equivalence, equations and inequalities 1.3.1 partition small whole numbers and recognise

equivalences e.g. 2 + 7 is the same as 3 + 6 1.3.2 write statements of equality using small

whole numbers and +, – and = 1.3.3 solve ‘missing number’ equations that

involve addition and subtraction.


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1.4 Finance 1.4.1 Use money:

– making purchases – recognising the denominations of notes

and coins – deciding the amount to tender – counting money to tender – counting change – comparing the purchase value of similar

items. 1.4.2 order spending preferences 1.4.3 saving and spending with banking

institutions: – using savings accounts for depositing and

withdrawing money – using debit cards – using ATMs, EFTPOS.

1.4.4 add and subtract whole number amounts of money.

2. Space and measurement

2.1 Time 2.1.1 estimate time using natural or artificial

phenomena such as time for people to eat lunch

2.1.2 read and use time on analogue and digital watches and clocks

2.1.3 place familiar events in order of occurrence. 2.2 Length, area, mass, volume and

capacity, angle 2.2.1 compare capacities by pouring 2.2.2 use everyday comparative language

associated with capacity e.g. greater than 2.2.3 use everyday language associated with

approximation of measures e.g. nearly as high as

2.2.4 measure by counting informal uniform units (including hand span, paper clips, jars and marbles) and whole number metric units of length, capacity and mass.

2.3 Location 2.3.1 respond to, and use the language of

position, orientation and movement e.g. in front of, left, near

2.3.2 place key features and objects on maps of familiar areas e.g. the school, showing relative position e.g. of one object between two others.

2.4 Shape 2.4.1 name and draw everyday 2D geometric

shapes e.g. circle, diamond, square, capturing the essence of the shape

2.4.2 identify everyday 3D shapes e.g. cone, sphere

2.4.3 use everyday words to describe shapes e.g. flat, straight, curved, round, side, corner

2.4.4 identify spatial features of figures and objects e.g. number of sides and vertices

2.4.5 discriminate between like and unlike shapes giving geometric reasoning

2.4.6 recognise shapes in the environment that resemble geometric shapes

2.4.7 link shapes of objects to their function e.g. wheels are round so they will roll.

2.5 Transformations 2.5.1 move, match and position shapes to build

structures 2.5.2 fit shapes together to form larger shapes 2.5.3 generate simple patterns by sliding, turning

and flipping shapes 2.5.4 describe transformations using turn, flip and

slide.

3. Chance and data 3.1 Describe chance 3.1.1 list outcomes for familiar chance events 3.1.2 distinguish likely, unlikely and impossible

outcomes. 3.2 Interpret chance 3.2.1 classify familiar events as likely, unlikely or

impossible 3.2.2 recognise that repetitions of chance actions

are likely to produce different results 3.2.3 list outcomes of familiar chance events that

satisfy given criteria. 3.3 Collect and organise data 3.3.1 collect:

– objects – categorical data through simple survey – measurement data (record using informal

uniform units or whole number metric units)

3.3.2 record data with materials and pictures in organised lists with structure provided, and in provided one-way tables

3.3.3 record data with tallies in provided tables 3.3.4 sort objects and pictures into groups using

familiar, agreed-to criteria and draw pictures of objects in groups.

3.4 Represent data 3.4.1 construct block graphs and pictographs with

a one-to-one or many-to-one correspondence between data and symbols

3.4.2 construct column graphs for measurement data, so that each column represents a single measurement

3.4.3 include titles on graphs and labels on axes 3.4.4 count the number of objects, pictures or

data in categories and name the count frequency.

3.5 Interpret data 3.5.1 read information from lists and one-way

tables, block graphs, pictographs and column and bar graphs using labelled calibrations on scales

3.5.2 link information from lists, tables and graphs to real contexts


15

3.5.3 compare individual objects with other individual objects such as measurements, colour and group size for grouped objects

3.5.4 compare individual data with other individual data such as frequency of data in categories

3.5.5 report on collected data, describing how graphs display the data and describing what data show.

Assessment The two types of assessment in the table below are consistent with the teaching and learning strategies considered to be the most supportive of student achievement of the outcomes in the Mathematics course. The table provides details of the assessment type, examples of different ways that these assessment types can be applied and the weighting range for each assessment type.

WeightingUnit PB Type of assessment

40–50%

Response

In this type of assessment, students apply their mathematical understanding and skills to analyse, interpret and respond to questions and situations. The assessment type provides for the assessment of conceptual understandings, knowledge of mathematical facts and terminology, problem-solving skills and the use of algorithms.

Questions in this type of assessment can range from those that are routine and familiar to students through to non-routine, unfamiliar questions. The questions may be closed and, so, target particular methods and results, or they may be open-ended and allow for choice in the methods and a variety of results.

Evidence gathering tools may include assignments, tests, observation check lists and quizzes. Written assessments may be done under timed conditions.


50–60%

Investigation

In this type of assessment, students plan, research, conduct and communicate the findings of an investigation. The assessment type provides for the assessment of mathematical-inquiry skills, problem-solving and modelling skills and course-specific knowledge and skills.

Students may identify social issues, collect and analyse relevant data in order to reach conclusions and make recommendations. They may develop, over an extended period of time, a theme or project related to the practical application of mathematics.

Evidence gathering tools may include diagrams and tables used to organise thoughts and processes, journals, project reports, posters, oral and multimedia presentations, observation checklists and interviews.



16

UNIT 1AMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students develop understanding of multiplication and division. They use whole numbers and the four operations for practical purposes, including financial matters useful to them personally and in employment. Students measure lengths and masses of objects and calculate perimeters. They interpret timetables that they are likely to use. They explore three-dimensional shapes and use informal maps. Students recognise and describe chance in familiar activities and produce data using probability devices. They collect and describe categorical and time series data. They calculate using mental strategies, written methods and calculators.

Suggested learning contexts The unit content will be introduced and applied in a variety of contexts that are accessible to students. Suggested contexts may include: • number in relation to budgeting e.g. running a

car, planning a fundraising activity, budgeting income and living expenses (Number and algebra)

• finance in relation to personal budgeting and banking and reading financial statements (Number and algebra)

• time in relation to bus, train and school timetables, estimates of people’s ages (Space and measurement)

• shapes and their uses in design, sculpture, packing and navigating (Space and measurement)

• location using informal maps e.g. of the school and school locality, the bush (Space and measurement)

• transformations e.g. in corroborees, Latin and square dancing, the LOGO software program and computer games (Space and measurement)

• chance associated with spinners, dice, coins, tacks, colour and count of sweets in packets (Number and algebra, Chance and data)

• chance associated with familiar events e.g. an expected baby is a boy, winning first prize in a lottery (Chance and data)

• categorical data from students e.g. favourite CDs and sport and corresponding data from an overseas class, first names that are common (Number and algebra, Chance and data)

• measurement data from students e.g. time per day spent watching TV to the nearest half hour, time spent getting to school to the nearest quarter of an hour (Number and algebra, Space and measurement, Chance and data)

• time series data e.g. shadow length over time, volume of water collected from a dripping tap over time, plant and animal growth (Number and algebra, Space and measurement, Chance and data)

• measuring instruments, calculations, and mathematical representations that are relevant to students in vocational programs (Number and algebra, Space and measurement, Chance and data).

Unit content This unit includes the content areas: • number and algebra • space and measurement • chance and data. This unit includes knowledge, understandings and skills to the degree of complexity described below. Students will be provided with opportunities to: • carry through tasks

identify information choose and use mathematical methods choose methods of processing—mental,

written, with a calculator. • interpret solutions

check and correct answers link answers to contexts.

• communicate methods, reasoning and results. The number formats for the unit are whole numbers, decimals to two decimal places, common fractions.

1. Number and algebra (19 hours) 1.1 Number 1.1.1 read, write, say and use:

– whole numbers into the thousands – common fractions including halves,

quarters, thirds – decimals to two decimal places for money

and measurement. 1.1.2 recognise place value for whole numbers. 1.2 Estimation and calculation 1.2.1 recognise addition, subtraction,

multiplication and division as distinct processes

1.2.2 use multiplication facts to 10 × 10 in mental computation

1.2.3 use addition facts to calculate subtraction results

1.2.4 use multiplication facts to calculate division results

1.2.5 partition quantities into two or more equal parts, naming the parts with unit fractions

1.2.6 count with fractions including ¼, ½, ¾ 1.2.7 order and record on a number line unit

fractions and related fractions including ¼, ½, ¾, 1, 1¼


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1.2.8 describe fractions of things such as the fraction of a pie that is eaten and the fraction left over

1.2.9 divide whole numbers by whole numbers, giving the answer as a whole number and fraction or whole number and remainder

1.2.10 add and subtract fractions with the same denominator, using written computation

1.2.11 describe the practical meaning of decimals for money and measurements

1.2.12 add and subtract decimals for money and measurement using written computation

1.2.13 use a calculator to operate on whole numbers and decimals (one or more operations where order is determined by contexts)

1.2.14 estimate sums and products by rounding to single digit numbers or multiples of 10.

1.3 Equivalence, equations and inequalities 1.3.1 write statements of equality using whole

numbers, operations and the ‘equals’ symbol

1.3.2 solve ‘missing number’ problems involving any of the four operations

1.3.3 recognise addition is commutative and subtraction is not.

1.4 Finance 1.4.1 calculate with money and the four

arithmetic operations using a calculator 1.4.2 make simple budgets for personal use 1.4.3 calculate earnings:

– weekly/fortnightly pay from hourly pay – weekly pay from annual salary.

1.4.4 use mathematics in personal banking: – understand the concept of receiving

interest on savings without calculating interest

– understand the concept of paying interest on loans such as on money borrowed for purchases, credit card debit balances, cash advances, personal loans.

1.4.5 interpret financial statements such as pay slips, debit/credit card statements, bank statements, telephone accounts, rent statements, invoices, retail store statements, income tax statements and loan repayment statements

1.4.6 complete simple financial forms.

2. Space and measurement (18 hours) 2.1 Time 2.1.1 estimate, measure, record, order and

compare time using standard units 2.1.2 read and use stopwatches and calendars 2.1.3 interpret everyday timetables and

programs.

2.2 Length, area, mass, volume and capacity, angle

2.2.1 estimate, measure, order and compare length, mass and capacity using metric units in whole units

2.2.2 calculate perimeter of shapes and generate shapes with specific perimeters (without formulas).

2.3 Location 2.3.1 use directional language associated with

turns and compass bearings to describe a route e.g. right angle, quarter turn, left turn, north, west

2.3.2 find key features on maps and interpret relative position and proximity from maps

2.3.3 place key features on maps of known locations, attending to relative position and proximity.

2.4 Shape 2.4.1 name geometric shapes and their features

(circle, ellipse, triangle, rectangle, square, cube, cone, cylinder, sphere, rectangular prism, pyramid; and vertex, side, edge, face, base)

2.4.2 identify essential attributes of named shapes

2.4.3 draw 2D geometric shapes given the name 2.4.4 copy plans/pictures made of geometric

shapes e.g. bird’s-eye-view of a table setting

2.4.5 match 3D objects and drawings of them, attending to shape and placement of parts

2.4.6 describe and draw cross-sections of simple 3D objects e.g. a cube of cheese, a pineapple

2.4.7 link the structure of objects to their flexibility, stability and ease of storage.

2.5 Transformations 2.5.1 describe symmetry of figures and

arrangements in students’ own words 2.5.2 produce tiling patterns by systematically

translating, rotating and reflecting a given shape

2.5.3 describe transformations using rotation, reflection and translation.

3. Chance and data (18 hours) 3.1 Conduct chance experiments 3.1.1 generate data using simple probability

devices. 3.2 Quantify chance 3.2.1 list outcomes for familiar chance events,

describe them as being more or less likely than each other or equally likely, and provide reasoning from personal experience

3.2.2 list all possible outcomes for one-stage experiments before experimentation and use them to decide chance, such as most likely, equally likely


18

3.2.3 check likelihood predictions against experimental data.

3.3 Interpret chance 3.3.1 classify familiar events as certain, likely,

unlikely or impossible and classify them as equally likely or not

3.3.2 order chance events from least likely to most likely, providing reasoning from personal experience or reasoning based on data

3.3.3 interpret chance in real contexts 3.3.4 recognise that repetitions of chance events

are likely to produce different results 3.3.5 describe how to influence the chance of an

event happening. 3.4 Collect and organise data 3.4.1 formulate research questions including

about topics that require data collection beyond the classroom

3.4.2 collect categorical data by simple survey 3.4.3 collect measurement data, including at

equal time intervals 3.4.4 predict what data will show 3.4.5 record data with tallies in one- and two-way

tables that are provided 3.4.6 record time series data in provided tables 3.4.7 classify and sort data according to agreed

criteria, and reword classifications to clarify what is included and excluded.

3.5 Represent data 3.5.1 calculate frequency 3.5.2 construct one-way frequency tables for

number of objects in group and data recorded with pictures and tallies

3.5.3 assign data to two-way tables and to Venn diagrams with two overlapping categories, and replace data with frequencies

3.5.4 construct block graphs and pictographs with a one-to-one and many-to-one correspondence between data and symbols

3.5.5 construct column graphs for measurement data, so that each column represents a single measurement

3.5.6 construct column and horizontal bar graphs showing frequencies for different categories, using frequency scales labelled with consecutive whole numbers

3.5.7 construct column and horizontal bar graphs for time series data, so that times are treated as categories and each column/bar represents a measurement

3.5.8 include titles on graphs and labels on axes. 3.6 Interpret data 3.6.1 read information from tables with tallies,

one- and two-way frequency tables, Venn diagrams with two categories, block graphs and pictographs, and bar and column graphs, using labelled calibrations on scales

3.6.2 compare individual data with other individual data such as frequency of data in categories

3.6.3 compare individual data using group characteristics (rank data in order of magnitude, and select lowest, highest and middle scores)

3.6.4 report on collected data: – stating research questions – describing and explaining what data and

graphs show – commenting on results in relation to

predicted results.


19


Weighting Stage 1 Type of assessment

40–50%

Response


Questions in this type of assessment can range from those that are routine and familiar to students through to non-routine, unfamiliar questions. The questions may be closed and, so, target particular methods and results, or they may be open-ended and allow for choice in the methods and a variety of results. Open-ended questions typically call for high level reasoning.

Evidence gathering tools may include assignments, tests, examinations, observation check lists and quizzes. Written assessments may be done under timed conditions.


50–60%

Investigation


Students may investigate mathematical patterns, making and testing conjectures and generalising mathematical relationships. They may select, apply and adapt models and procedures to solve complex problems in contexts and, then, justify their results to themselves and others. They may identify social issues, collect and analyse relevant data in order to reach conclusions and make recommendations. They may develop, over an extended period of time, a theme or project related to the practical application of mathematics.

Evidence gathering tools may include diagrams and tables used to organise thoughts and processes, written investigation reports, journals, project reports, posters, oral and multimedia presentations, self or peer evaluations, observation checklists and interviews.



20

UNIT 1BMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students use decimals, fractions and percentages for practical purposes. They apply mathematics for personal budgeting, banking and shopping. They estimate and measure length and mass of objects using a variety of instruments, and derive and use methods for calculating perimeter and basic areas. They translate, reflect and rotate shapes in design. Students use repeated measurement to collect data relevant to them, display data in tables and graphs and interpret the displays. They calculate using mental strategies, written methods and calculators.

Suggested learning contexts The unit content will be introduced and applied in a variety of contexts that are accessible to students. Suggested contexts may include: • number in relation to students’ experience e.g.

personal finance, measurement in sport (Number and algebra)

• financial budgeting for personal use e.g. phone account, keeping a horse or other animals (Number and algebra)

• methods of scoring in events and of making comparisons between competitors e.g. diving, football, rowing, boxing, darts, dancing (Number and algebra)

• measuring e.g. with thermometers, stopwatches, long measuring tapes, graduated jugs (Space and measurement)

• transformations and their use in patterns e.g. in dress fabric, tiling, wallpaper, wrapping paper, lace, wrought iron, art (Space and measurement)

• data from estimation, with and without a ruler being visible e.g. arm length of a student, height of teacher, length of pipe for garden reticulation, length and area of paving (Space and measurement, Chance and data)

• measurement data (repeated measurement of single object) e.g. arm length of a student (Space and measurement, Chance and data)

• categorical data from students (two attributes for Venn diagrams) e.g. students’ preferences for one, none or both of two foods, students’ enrolment in one, none or both of two school subjects (Chance and data)

• census data for column graphs e.g. population by age, population by state (Chance and data)

• misleading graphs in advertising (Chance and data)

• averages in the media e.g. number of children in families, rainfall (Number and algebra, Chance and data)


Unit content This unit builds on the content covered by the previous unit. This unit includes the content areas: • number and algebra • space and measurement • chance and data.

This unit includes knowledge, understandings and skills to the degree of complexity described below. Students will be provided with opportunities to:

• carry through tasks identify and organise information choose and use mathematical methods choose methods of processing—mental,

written, with a calculator record working.

• interpret solutions check answers fit specifications link solutions to contexts and reach

conclusions. • communicate methods, reasoning and results.

The number formats for the unit are whole numbers, decimals, common fractions, common percentages.

1. Number and algebra (19 hours) 1.1 Number 1.1.1 read, write, say and use:

– whole numbers into the millions – decimals to three decimal places – common fractions—denominators 2, 3, 4,

5, 6, 8 and 10 – common percentages including 10%,

20%, 25%. 1.1.2 recognise place value for whole numbers

and decimals. 1.2 Estimation and calculation 1.2.1 use addition facts to 10 + 10 and

multiplication facts to 10 × 10 in mental computation, including for inverse calculation of subtraction and division

1.2.2 estimate sums, differences, products and quotients by rounding to single digit numbers or multiples of 10


21

1.2.3 calculate fractions, decimals and percentages in contexts such as the fraction/decimal/percentage of the class owning a hand-held computer game

1.2.4 write simple fractional equivalences e.g.

63

21 =

1.2.5 order and record on the number line fractions that are easily visualised e.g.

32 and 4

3 , or that have a common

numerator e.g. 43 , 5

3

1.2.6 add and subtract decimals with written methods

1.2.7 order and record on the number line decimals with the same number of decimal places

1.2.8 use a calculator to add, subtract, multiply and divide decimals in contexts: – use brackets and the calculator memory

for interim results (not relying on rule of order in a calculator)

– interpret whole number and remainder results arising from division

– interpret negative signs in results – round results to suit contexts.

1.2.9 convert between common fractions, decimals and percentages, mentally and with a calculator

1.2.10 use a calculator to calculate fractions, decimals and percentages of quantities.

1.3 Equivalence, equations and inequalities 1.3.1 write and verify numerical statements of

equality and inequality, using the four arithmetic operations and the symbols =, ≠, < ,>

1.3.2 write number statements that recognise the commutative and associative laws for addition and multiplication

1.3.3 recognise that the commutative and associative laws do not apply to subtraction and division

1.3.4 write word equations symbolically with blocks for missing numbers e.g. 2 × + 3 = 11, and determine missing numbers by guess, check-and-improve and working-backwards methods.

1.4 Finance 1.4.1 budget for personal use:

– identify income and expenditure – make budget estimates – follow budgets for a period of time.

1.4.2 use mathematics in personal banking: – understand the concept of interest on

savings and loans, such as on term deposits, store cards, credit card debit balances, cash advances and personal loans

– calculate simple interest, repayments and balance amounts.

1.4.3 use mathematics in shopping: – calculate discount and price after discount – interpret advertising specials and

discounts – calculate unit cost given the price of

several items e.g. $3.50 for 5 items.

2. Space and measurement (20 hours) 2.1 Time 2.1.1 estimate, measure, order and compare time

using: – standard units – 12- and 24-hour time.

2.1.2 record time on time sheets to the nearest quarter of an hour and calculate hours worked

2.1.3 interpret everyday timetables and programs with 12- and 24-hour time.

2.2 Length, area, mass, volume and

capacity, angle 2.2.1 estimate, measure, order and compare

length and mass using metric units 2.2.2 read labelled and unlabelled calibrations on

whole number scales on a variety of instruments

2.2.3 use metric prefixes e.g. kilo and abbreviations e.g. kg

2.2.4 measure and calculate perimeters of polygons and other shapes

2.2.5 generate shapes with specific perimeters 2.2.6 estimate and measure areas of polygons

and other shapes by counting squares 2.2.7 demonstrate and use: area of a

rectangle = length × breadth 2.2.8 compare angles directly and order them by

size 2.2.9 measure, draw and estimate angles in

degrees 2.2.10 classify angles as acute, right, obtuse,

straight and reflex. 2.3 Shape 2.3.1 name polygons (types of triangles and

quadrilaterals, pentagon, hexagon, octagon)

2.3.2 draw polygons that meet criteria for angles, sides and vertices

2.3.3 identify 2D shapes that tessellate and draw tessellations

2.3.4 modify simple shapes that tessellate to form complicated shapes that tessellate and use them to form patterns

2.3.5 use geometric language to describe 2D figures so others can draw them e.g. logos, patterns with reflection and translation.


22

2.4 Transformations 2.4.1 recognise and describe the translation,

reflection and rotation of figures 2.4.2 produce patterns by systematically

translating, rotating or reflecting a given shape

2.4.3 produce figures that meet transformation criteria e.g. a shape with order-6 rotational symmetry.

3. Chance and data (16 hours) 3.1 Collect and organise data 3.1.1 plan the collection of data from repeated

measurement of a quantity 3.1.2 plan recording sheets, organising data in

tables 3.1.3 record data and check and edit the record 3.1.4 group data involving whole numbers into

equal sized class intervals, with guidance. 3.2 Represent data 3.2.1 construct one- and two-way frequency

tables 3.2.2 construct frequency Venn diagrams for two

events 3.2.3 construct pictographs with a many-to-one

correspondence between data and symbols 3.2.4 construct dot frequency plots 3.2.5 construct column and horizontal bar graphs

showing frequencies for whole number data grouped in intervals, treating intervals as categories, and using scales calibrated with whole numbers but not all calibrations are labelled

3.2.6 calculate mean, median, and mode for listed data and ungrouped frequency data

3.2.7 choose averages to suit the contexts of data

3.2.8 describe spread in terms of lowest and highest scores and range

3.2.9 calculate proportions of data in simple fractional and percentage forms.

3.3 Interpret data 3.3.1 read information from frequency graphs,

using labelled and unlabelled calibrations on scales

3.3.2 discern if graphs of univariate data are misleading and explain why

3.3.3 compare individual data using group characteristics (rank data in order of magnitude, and select lowest, highest and middle scores)

3.3.4 compare proportions of data satisfying criteria

3.3.5 compare datasets using mean/median and spread indicated by lowest and highest scores and range

3.3.6 calculate sum of scores from the mean and numbers of data in categories from simple proportions

3.3.7 link spread to accuracy in repeated measurement and natural variation in phenomena, and link mean, median, mode to true measurement and typical values

3.3.8 report on collected data, including describing the collection and explaining what data and graphs show.


23



40–50%

Response





50–60%

Investigation






24

UNIT 1CMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students use decimals, fractions, percentages and ratios for practical purposes. They apply mathematics to financial matters in the workplace. They write and use algebraic rules for number patterns. They measure volume and other attributes of objects, and derive and use formulas for area and volume. They read and draw maps with scales, describe and draw shapes in three dimensions. Students describe likelihood for chance events, and design and test simple probability devices. They collect time series data relevant to them, display data in tables and graphs and interpret the displays. They calculate using mental strategies, written methods and calculators.

Suggested learning contexts The unit content will be introduced and applied in a variety of contexts that are accessible to students. Suggested contexts may include: • graphs for personal records over time e.g.

hours of part-time work by week (Number and algebra, Chance and data)

• shapes in architecture and industry e.g. wheat silos, warehouses, cooling towers, stockpiles of iron ore (Space and measurement)

• measurement in practice e.g. garden design, area of paving and garden beds, fencing, stage design, reticulation, in agriculture and horticulture (Space and measurement)

• probability devices e.g. spinners, dice, cards, coloured or numbered counters in a container (Chance and data)

• likelihood e.g. in relation to winning a raffle, moves in a board game, a sports team winning, rain the next day, being taller than the next person you meet (Chance and data)

• time series data e.g. World and Olympic sports records, plant growth, money raised in a fundraising effort, volume of water collected from a dripping tap, populations for Australian States and the nation over time (Space and measurement, Chance and data)


Unit content This unit builds on the content covered by the previous units. This unit includes the content areas: • number and algebra • space and measurement • chance and data.

This unit includes knowledge, understandings and skills to the degree of complexity described below. Students will be provided with opportunities to: • plan and carry through tasks

identify and organise information develop systematic approaches choose and use mathematical methods choose methods of processing—mental,

written, with a calculator. • interpret solutions:

check answers fit specifications link solutions to contexts and reach

conclusions generalise results.

• communicate methods, reasoning and results. The number formats for the unit are whole numbers, decimals, common fractions, common percentages, simple ratios, square and cubic numbers written with powers.

1. Number and algebra (20 hours) 1.1 Estimation and calculation 1.1.1 use the rule of order with +, –, ×, ÷ and

brackets when calculating with whole numbers and decimals

1.1.2 round to the nearest whole number and one decimal place

1.1.3 distinguish factors from multiples and write numbers as products of factors

1.1.4 use mental/written computation with fractions to: – calculate fractions of whole numbers

e.g. ¾ of 12 – add and subtract fractions with the same

denominator – divide whole numbers by unit fractions

e.g. the number of quarters in 10 oranges. 1.1.5 calculate simple decimal quantities with

mental/written computation e.g. 0.5 of 3 kg 1.1.6 estimate common percentages of quantities

with mental/written computations e.g. 20% of $16

1.1.7 use a calculator to calculate fractions, decimals and percentages of quantities and to express one quantity as a percentage of another quantity e.g. the percentage of students in a class who have a part time job

1.1.8 use diagrams to show simple ratios e.g. 1 to 4


25

1.1.9 calculate ratios in contexts e.g. the ratio of sugar to butter in recipes

1.1.10 partition quantities according to simple ratios e.g. share $12 in the ratio 1 to 2

1.1.11 describe simple ratios with percentages and fractions

1.1.12 calculate with unit rates in contexts including hourly rate of pay, kilometres per hour.

1.2 Functions and graphs 1.2.1 recognise everyday variables that change

with time and describe the nature of the changes

1.2.2 describe how quantities appear to be related in tables of ordered pairs

1.2.3 locate and plot points in the first quadrant 1.2.4 use points on a graph to compare the

quantities they represent 1.2.5 read and interpret line graphs of situations

including travel graphs 1.2.6 describe and interpret changes in the

steepness of line graphs 1.2.7 choose graphs to match given situations. 1.3 Patterns 1.3.1 create tables to show position numbers and

elements in a number pattern 1.3.2 describe in words:

– one-step recursive rules for patterns with a given starting number

– two-step rules that link each element in a number pattern to position.

1.3.3 write algebraic rules that relate each element of a pattern to its position in the pattern e.g. 12 +×= nb

1.3.4 support or refute conjectured rules for number patterns by testing cases

1.3.5 use rules to continue number patterns and generate terms from rules.

1.4 Finance 1.4.1 use mathematics in the workplace:

– calculate weekly/fortnightly pay from hourly rate

– calculate pay for different conditions including retainer and commission, salary, base pay and overtime

– compare the advantages and disadvantages of casual, part-time, full-time, temporary and permanent conditions

– calculate employer superannuation contributions

– complete personal tax returns – calculate income tax from tax tables.

2. Space and measurement (19 hours)


2.1.1 derive and use area formulas for squares, rectangles, triangles, parallelograms

2.1.2 measure the volume of rectangular prisms by counting cubes

2.1.3 demonstrate and use volume formulas for cubes and right rectangular prisms (volume of prism = area of base × perpendicular height)

2.1.4 use ratios to describe concentrations and to interpret information on food labels

2.1.5 convert between units of length. 2.2 Location 2.2.1 use whole number scales on maps e.g.

1 cm represents 10 km 2.2.2 use map coordinates and whole number

scales to interpret maps and plans 2.2.3 make maps and plans with whole number

scales, showing key features. 2.3 Shape 2.3.1 draw prisms, pyramids, cylinders and cones

so that they are recognisable without necessarily being precise

2.3.2 match 3D objects with drawings of them e.g. front, back and side views or 3D views

2.3.3 draw side, front, back and top views of 3D shapes e.g. prisms, pyramids, cylinders and cones

2.3.4 select 3D objects that meet geometric criteria e.g. curved and straight edges, number of faces and vertices

2.3.5 make models of simple 3D objects e.g. a TV, attending to shape.

3. Chance and data (16 hours) 3.1 Conduct chance experiments 3.1.1 plan and make simple probability devices

for particular orders of probability and test the devices.

3.2 Quantify chance 3.2.1 systematically list or display on diagrams

outcomes for one-stage experiments and use them to decide chance (most likely, least likely, equally likely etc.) and justify choices

3.2.2 describe likelihood with simple ratios, fractions and percentages

3.2.3 place chance expressions (‘impossible’, ‘poor chance’, ‘even chance’ etc.) on a scale from 0 to 1.

3.3 Interpret chance 3.3.1 use likelihood values to predict number of

outcomes that are likely to satisfy provided criteria in n trials

3.3.2 recognise that likely events may not happen and very unlikely events are possible

3.3.3 reason about the number of trials needed for reliable conclusions about the likelihood of a result, basing reasoning on experimental data


26

3.3.4 order outcomes from least likely to most likely, using simple ratios, fractions and percentages

3.3.5 explain numerical values for likelihood in terms of contexts

3.3.6 identify real world events with given chance (0, ¼,½, ¾ and 1)

3.3.7 identify factors that could affect the chance of an event happening.

3.4 Collect and organise data 3.4.1 frame and revise questions in context from

general questions to specific ones that can be tested with time series data

3.4.2 plan how to collect measurement data at regular time intervals

3.4.3 plan recording sheets linking the design to the question and choose measurement units

3.4.4 record data, attending to accuracy. 3.5 Represent data 3.5.1 plot points and construct line graphs for

time series data using scales where not all calibrations are labelled

3.5.2 describe trend as increasing or decreasing, for time series data with a one-to-one correspondence between time and the dependent variable.

3.6 Interpret data 3.6.1 read information from time series graphs,

using labelled and unlabelled calibrations on scales

3.6.2 discern if graphs of time series data are misleading and explain why

3.6.3 describe trend in terms of time and other variables

3.6.4 report on collected data, including assessing how to improve data collection.


WeightingStage 1 Type of assessment

40–50%

Response





50–60%

Investigation






27

UNIT 1DMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students use integers, decimals, fractions, percentages and ratios for practical purposes. They apply mathematics in making financial decisions. They write word sentences algebraically and solve simple equations. They calculate area and perimeters of circles and use Pythagoras’ theorem for the sides of triangles. They describe the effects of reflecting, rotating and translating shapes in design, and enlarge, reduce and distort figures. They interpret detailed maps. Students collect measurement data from fair samples, display data in tables and graphs, calculate averages and describe spread of data, and compare datasets. They use mental strategies, written methods, calculators and computer technologies where appropriate.

Suggested learning contexts The unit content will be introduced and applied in a variety of contexts that are accessible to students. Suggested contexts may include: • finance in relation to record keeping with

spreadsheets e.g. budgets for independent living—rent, accommodation, food, transport; review and adjust budgets (Number and algebra)

• financial plans for personal use e.g. mobile phones, internet charges, rewards programs that exchange points for goods/services (Number and algebra)

• ratio and proportion e.g. planning for a morning tea, adjusting recipes for group size (Number and algebra)

• number contexts that may include aspects of mathematics e.g. planning a school ball, birthday party (18/21), parents’ wedding anniversary, special birthdays (Number and algebra)

• transformations in relation to design e.g. fabric, lace, wrought-iron work, Wingdings fonts (Space and measurement)

• scale drawing e.g. garden design, farm sheds, sheep yards, stage design, selecting furniture for rooms, tourist routes (Space and measurement)

• time in relation to schedules e.g. sports carnival events (Space and measurement)

• surveys of e.g. boys and girls, or year group opinions on school issues (Chance and data)

• data for comparison e.g. predicted and actual test marks (Chance and data)

• census data for column graphs e.g. income by sex and age for each state and the nation (Chance and data)

• averages in real life e.g. median house prices, mean monthly rainfall, mode number of people attending popular events such as AFL finals, mean of circumference/diameter for round objects (Number and algebra, Space and measurement, Chance and data)

• statistics in the media e.g. about exports, sport, estimated death toll in a disaster (Chance and data)


Unit content This unit builds on the content covered by the previous units. This unit includes the content areas: • number and algebra • space and measurement • chance and data. This unit includes knowledge, understandings and skills to the degree of complexity described below. Students will be provided with opportunities to: • plan and carry through tasks





• communicate methods, reasoning and results. The number formats for the unit are decimals, fractions, percentages, square numbers, square roots, positive and negative numbers, simple ratios.

1. Number and algebra (19 hours) 1.1 Estimation and calculation 1.1.1 use rule of order with +, –, ×, ÷, squares,

square roots and brackets when calculating with whole numbers and decimals

1.1.2 round decimals to the nearest multiple of ten, unit or tenth

1.1.3 estimate sums, difference, products and quotients by rounding


28

1.1.4 use mental/written computation: – calculate simple percentages of quantities – add and subtract integers, decimals and

common fractions e.g. 43

61 +

– multiply integers, decimals, common fractions by single digit whole numbers

– divide integers and decimals by single digit counting numbers.

1.1.5 convert between common fractions, decimals and percentages of quantities

1.1.6 order fractional, decimal and percentage quantities that are close e.g. ⅓ and 30% of $100

1.1.7 convert between fractions, decimals and percentages with a calculator

1.1.8 use calculators and spreadsheets to add, subtract, multiply and divide numbers, adhering to and utilising conventions of the technologies

1.1.9 recognise that multiplying and dividing by fractions and decimals can increase or decrease the original amounts

1.1.10 order and record on the number line: – decimals with different numbers of

decimal places – positive and negative whole and decimal

numbers. 1.2 Equivalence, equations and inequalities 1.2.1 generate number equivalent statements

that recognise the associative, commutative and distributive properties e.g.

2504625425625 =+×=×+× )( 1.2.2 recognise that letters stand for variable

numbers in algebra 1.2.3 use algebraic conventions, such as

kk 22 =× and 2kkk =× 1.2.4 substitute into formulas directly to evaluate

quantities such as area A given r for the

formula, 2rA π=

1.2.5 write word sentences and constraints symbolically including sentences like ‘a taxi charge is $4.80 flag fall and $1.35 per kilometre’

1.2.6 solve one- and two-step equations with a single algebraic term using the guess, check and improve, work-backwards and

balance methods e.g. 20002 =x , 1232 =+x

1.2.7 substitute to validate solutions of equations 1.2.8 state truth sets for word statements and

graph them on the number line e.g. numbers between 4 and 10

1.2.9 generate pairs of numbers that satisfy equations e.g. 36=yx .

1.3 Finance 1.3.1 construct spreadsheets and use them to

make financial decisions, including budgeting for personal use, budgeting for an activity

1.3.2 read and interpret information from financial statements, such as pay slips, debit/credit card statements, bank statements, telephone accounts, rent statements, invoices, retail store statements and income tax statements

1.3.3 compare financial statements, such as mobile phone plans, rewards programs, internet plans, interest-free terms.

2. Space and measurement

(18 hours) 2.1 Time 2.1.1 calculate elapsed time including the

duration of events 2.1.2 interpret complex timetables and schedules

such as tide charts 2.1.3 integrate information to schedule events in

which time is a variable 2.1.4 calculate and use everyday rates including

speed in kilometres per hour (conversions not included).

2.2 Length, area, mass, volume and

capacity, angle 2.2.1 read between calibrations on scales when

measuring 2.2.2 relate the diameter of a circle to its

circumference 2.2.3 estimate area of a circle by counting

squares 2.2.4 use decimal approximations for π 2.2.5 use formulas to calculate:

– circumference of a circle – area of a circle.

2.2.6 use Pythagoras’ theorem to calculate the sides of right triangles.

2.3 Location 2.3.1 describe scales with ratios 2.3.2 interpret maps using map coordinates, ratio

scales, compass directions and bearings. 2.4 Transformations 2.4.1 use a grid to enlarge, reduce or distort a 2D

figure by whole number and unit fraction scales

2.4.2 describe the properties of transformations e.g. corresponding points on the image and object are the same distance from a line of reflection

2.4.3 describe the position and orientation of 3D objects after translation, reflection and rotation e.g. in the context of moving furniture

2.4.4 use geometric language to describe transformed figures.


29

3. Chance and data (18 hours) 3.1 Collect and organise data 3.1.1 frame short sets of survey questions in

context 3.1.2 choose a ‘fair’ sample for a survey (not a

formal random sample) 3.1.3 plan recording sheets for survey data 3.1.4 collect and record data, and check and edit

the record 3.1.5 group measurement data in tables with

provided equal sized class intervals.

3.2 Represent data 3.2.1 construct one- and two-way frequency

tables 3.2.2 construct column graphs showing

frequency and compound (i.e. clustered) column graphs for two sets of data

3.2.3 construct frequency histograms for ungrouped data and data grouped in equal sized class intervals

3.2.4 calculate mean, median and mode for ungrouped frequency data

3.2.5 calculate relative frequency, and proportions of data in fractional, decimal and percentage forms

3.2.6 describe spread of datasets informally (data are spread out, tightly packed)

3.2.7 describe spread using range and lowest and highest scores.

3.3 Interpret data 3.3.1 read information from tables, circle graphs

(pie charts with simple percentages, and frequency graphs, reading between calibrations on scales)

3.3.2 discern advantages and disadvantages of using frequency graphs rather than tables to display data

3.3.3 discern the relative advantages of the various ‘averages’ (mean, median and mode)

3.3.4 compare datasets using mean, median, lowest and highest scores and range

3.3.5 use words that acknowledge uncertainty when comparing data sets such as ‘scores for … tend to be more spread than scores for …’

3.3.6 calculate numbers of data in categories from relative frequency and proportions

3.3.7 report on collected data, including sampling methods and justification for them.



40–50%

Response





50–60%

Investigation






30

UNIT 1EMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students use positive and negative numbers and numbers with powers for practical purposes. They calculate interest and repayments for loans. They draw graphs to represent real situations, and use them to describe how quantities are related. They use trigonometry to calculate measurements in right triangles, and calculate volume and surface area of shapes. They analyse networks. Students simulate everyday chance events, calculate probabilities and predict using probabilities. They collect bivariate data relevant to them, display the data in tables and graphs, and describe trends. They use mental strategies, written methods, calculators and computer technologies where appropriate.

Suggested learning contexts The unit content will be introduced and applied in a variety of contexts that are accessible to students. Suggested contexts may include: • number in relation making financial decisions

e.g. about large purchases (Number and algebra)

• finance in relation to: − purchasing with borrowed money e.g. trail

bike, car, computer, TV, property, stock for farms

− contracts e.g. commercial advertisements, interest-free period contracts

− tables of rates e.g. income tax (Number and algebra).

• algebra in relation to sequences of shapes and numbers; length, area and volume when doubled, tripled etc. (Number and algebra)

• networks e.g. main road system, delivery and courier rounds (Space and measurement)

• surface area/volume ratios and heat transfer e.g. in babies, animals, freezer storage (Space and measurement)

• simulation of two-choice situations where the population proportions are known e.g. having /not-having a disease, being/not-being a member of a particular racial group, being male or female (Number and algebra, Chance and data)

• bivariate data e.g. pulse rate before and after exercise, hand span and foot length for individuals, left-hand and right-hand reaction time under stimulus for individuals, predicted and actual daily temperatures, predicted and actual test marks (Chance and data)


Unit content This unit builds on the content covered by the previous units. This unit includes the content areas: • number and algebra • space and measurement • chance and data. This unit includes knowledge, understandings and skills to the degree of complexity described below. Students will be provided with opportunities to: • plan and carry through tasks:





• communicate methods, reasoning and results. The number formats for the unit are percentages, ratios, positive and negative numbers, numbers expressed with positive integer powers, square roots, cube roots.

1. Number and algebra (23 hours) 1.1 Estimation and calculation 1.1.1 use common percentage, decimal and

fractional equivalents 1.1.2 round whole and decimal numbers to given

degrees of accuracy and to suit contexts 1.1.3 calculate with powers using written

methods and calculators 1.1.4 calculate with square and cube roots using

calculators 1.1.5 calculate with ratios and rates in contexts

such as the number of litres of petrol to travel 450 km if consumption is 7.8 L/100 km.

1.2 Functions and graphs 1.2.1 locate and plot points in the four quadrants

of the Cartesian plane, joining points if appropriate for problem contexts

1.2.2 sketch graphs to represent function relationships in contexts including travel graphs


31

1.2.3 describe how quantities appear to be related using graphs of points and continuous graphs, including travel graphs

1.2.4 distinguish dependent and independent variables and assign independent variables to the horizontal axis on graphs.

1.3 Patterns 1.3.1 describe symbolically:

– one- and two-step rules that link each element in a number pattern to position

– one- and two-step recursive rules for patterns with a given starting number, using arithmetic operations and powers.

1.3.2 check and revise rules to ensure correct meaning, taking into account order of operations

1.3.3 follow one- and two-stage rules to extend sequences and predict results.

1.4 Finance 1.4.1 make decisions about loans and

investments: – calculate simple interest – calculate interest compounded yearly,

monthly and daily, using recursion with technology

– calculate repayments and balance amounts for loans

– compare loans and investments with simple and compound interest.

1.4.2 calculate depreciation and inflation 1.4.3 determine ‘best buys’ using ratio and

proportion.



2.1.1 use formulas to calculate surface area and volume of prisms, pyramids, cones, cylinders, spheres and hemispheres

2.1.2 use inversely the formulas for: – areas of squares, rectangles, circles – volumes of rectangular right prisms,

cylinders, cones, spheres and hemispheres

2.1.3 convert between metric units e.g. kilometres to metres

2.1.4 convert between derived metric units and across units e.g. cubic metres to cubic centimetres, square metres to hectares

2.1.5 use angle measure in degrees 2.1.6 use trigonometric ratios (sine, cosine,

tangent) to calculate sides and angles (degree measure) of right triangles

2.1.7 use direct proportion to estimate where direct measurement is not possible e.g. estimate the height of a tree.

2.2 Networks 2.2.1 represent information as networks 2.2.2 investigate the traversability of networks,

intuitively and with algorithms 2.2.3 develop and use systematic methods for

the shortest path between vertices of simple networks e.g. two-directional flow.

3. Chance and data (17 hours) 3.1 Conduct chance experiments 3.1.1 conduct simulations to model real world

events with outcomes that are not equally likely.

3.2 Quantify chance 3.2.1 use long run relative frequency to estimate

probabilities 3.2.2 list sample spaces for one-stage events

with repetition to reflect likelihood of outcomes

3.2.3 calculate simple probabilities using sample spaces and the number of favourable outcomes divided by the total number of outcomes

3.2.4 use fractions, decimals and percentages to describe probability and move freely between them

3.2.5 use the facts that probabilities sum to 1 and range from 0 to 1 to check probabilities.

3.3 Interpret chance 3.3.1 predict the results for repetition of

simulations with the same number of trials 3.3.2 use probabilities to predict proportions and

number of outcomes that are likely to satisfy provided criteria in n trials

3.3.3 recognise predictions are not always realised

3.3.4 recognise the law of large numbers (that outcomes for successive trials follow no describable pattern but relative frequency of outcomes is predictable for a large number of trials)

3.3.5 order outcomes from least likely to most likely, using fractional, decimal and percentage probabilities

3.3.6 explain probability statements in common usage

3.3.7 identify factors that could compromise the simulation of real world events

3.3.8 use chance terminology when describing events (‘probability of’, ‘complement of’).

3.4 Collect and organise data 3.4.1 plan the collection of bivariate data to

investigate situations specified by the teacher

3.4.2 predict what data will show 3.4.3 plan recording sheets involving tables 3.4.4 collect and record data and check and edit

the record.


32

3.5 Represent data 3.5.1 construct scatterplots for bivariate data,

plotting between calibrations on scales 3.5.2 describe trend as increasing or decreasing,

for bivariate data 3.5.3 sketch notional increasing and decreasing

trends (not from points). 3.6 Interpret data 3.6.1 read information from scatterplots, reading

between calibrations on scales 3.6.2 use sketches of trend in interpretation of

data in contexts 3.6.3 report on collected data, including

assessing how to improve data collection and handling.



40–50%

Response





50–60%

Investigation






33

UNIT 2AMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students apply ratios and direct proportion in practical situations. They calculate profit, loss, discount and commission in financial contexts. They study introductory algebra and linear relationships in numeric, algebraic and graphical forms. They use Pythagoras’ theorem for the sides of triangles and analyse the reflection, rotation and translation of shapes in design. Students collect data from fair samples, and represent and interpret the data. They use mental and written methods and technologies where appropriate. The unit content will be introduced and applied in a variety of contexts that are accessible to students.


This unit includes knowledge, understandings and skills to the degree of complexity described below and comprises the examinable content of the course.

Students will be provided with opportunities to: • plan and carry through tasks:





• communicate methods, reasoning and results. They use mental and written methods and technologies where appropriate. The number formats for the unit are decimals, fractions, percentages, positive and negative numbers, numbers expressed with positive integer powers, square roots, cube roots, simple ratios and rates.

1. Number and algebra (28 hours) 1.1 Estimation and calculation 1.1.1 use the connections between the four

arithmetic operations 1.1.2 apply the rule of order when calculating 1.1.3 use mental and written methods to

calculate and estimate with integers, decimals, fractions between 0 and 1, common percentages of whole numbers and of decimals to two decimal places

1.1.4 round and truncate as part of estimation and calculation

1.1.5 use multiplication and division in situations involving ratios

1.1.6 use calculators to calculate with integers, decimals, fractions and percentages, powers, square roots and cube roots.

1.2 Functions and graphs 1.2.1 locate, plot and interpret points in the four

quadrants of the Cartesian plane; reason whether or not to join the points

1.2.2 recognise that continuous lines and curves on a graph consist of points

1.2.3 sketch and interpret graphs that represent relationships in contexts such as water filling a vase

1.2.4 identify linear relationships in the form

cmxy +=

– formulate linear rules for tables of values – determine tables of values for linear rules,

recognise many values are possible besides the ones chosen

– use tables of values to graph linear rules – read points from a line graph and

recognise they satisfy the rule for the line – read gradient and vertical intercept of line

graphs and link gradient to difference pattern in tables

– formulate linear rules from graphs – graph lines from rules using gradient and

vertical intercept. 1.2.5 use the function facilities of calculators and

computer spreadsheets. 1.3 Equivalence, equations and inequalities 1.3.1 recognise that letters stand for variable

numbers in algebra, including when translating between word and algebraic statements

1.3.2 use algebraic conventions such as

kk 22 =× , and 2kkk =× 1.3.3 collect like terms in algebraic expressions 1.3.4 use the distributive property:

– to expand expressions of the form )dcx)(bax( ±±

– to factorise linear expressions such as 48 −k .

1.3.5 solve linear equations graphically and algebraically, expanding and gathering like terms where appropriate


34

1.3.6 solve equations arising from application of Pythagoras’ theorem

1.3.7 solve direct proportion problems 1.3.8 relate the ideas of proportion and direct

variation to linear functions. 1.4 Finance 1.4.1 calculate profit, loss, discount and

commission 1.4.2 determine ‘best buys’, use comparison,

ratio and proportion.


2.1 Measurement 2.1.1 use Pythagoras’ theorem to calculate the

length of sides of right triangles 2.1.2 use direct proportion to estimate where

direct measurement is not possible. 2.2 Transformations 2.2.1 identify translations, reflections and

rotations of figures in two dimensions 2.2.2 produce patterns which exhibit symmetries,

rotations, reflections and translations 2.2.3 use geometric conventions in drawing and

geometric language to describe figures and patterns.

3. Chance and data (18 hours) 3.1 Collect and organise data 3.1.1 plan the collection of measurement data

and fair (unbiased) samples to investigate situations specified by the teacher

3.1.2 plan recording sheets 3.1.3 collect and record data, and check and edit

the record 3.1.4 group data in tables with provided equal

sized class intervals. 3.2 Represent data 3.2.1 construct one- and two-way frequency

tables and dot frequency plots 3.2.2 construct frequency histograms for

ungrouped data and data grouped in equal sized class intervals

3.2.3 sketch the notional shape of frequency graphs (not from points)

3.2.4 calculate mean, median and mode for ungrouped frequency data

3.2.5 calculate mean for grouped data and median and modal classes

3.2.6 calculate relative frequency, and proportions of data in fractional, decimal and percentage forms

3.2.7 describe spread of datasets informally, using terms such as data are spread out, tightly packed, clusters, gaps, more/less dense regions, outliers

3.2.8 describe spread using range and lowest and highest scores.

3.3 Interpret data 3.3.1 read information from tables, circle graphs

(pie charts) with percentages, and frequency graphs, reading between calibrations on scales

3.3.2 discern advantages/disadvantages of using frequency graphs rather than tables to display data

3.3.3 discern the relative advantages and disadvantages of the various ‘averages’ (mean, median, mode)

3.3.4 compare datasets by describing spread of graphed data, and using mean, median, lowest and highest scores and range

3.3.5 use mathematical words that acknowledge uncertainty when comparing data sets such as ‘scores for … tend to be more spread than scores for …’

3.3.6 calculate numbers of data in categories from relative frequency and proportions

3.3.7 report on collected data (to include justifying sampling methods and explaining what graphs and summary values show) .


35



65–75%

Response





25–35%

Investigation






36

UNIT 2BMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students study and apply exponential relationships. They develop skills for solving equations algebraically and graphically, and investigate and generalise number patterns. They use coordinate geometry in two dimensions. They use formulas directly and inversely for calculations involving three-dimensional shapes. They apply trigonometry in right triangles. They represent information using network diagrams. Students simulate everyday chance events, calculate and interpret probabilities, and collect and analyse bivariate and time series data. They use mental and written methods and technologies where appropriate. The unit content will be introduced and applied in a variety of contexts that are accessible to students.

Unit content This unit builds on the content covered by the previous units. This unit includes the content areas: • number and algebra • space and measurement • chance and data. This unit includes knowledge, understandings and skills to the degree of complexity described below and comprises the examinable content of the course. Students will be provided with opportunities to: • plan and carry through tasks:

identify and organise information develop systematic approaches partition problems into sub-problems identify simpler, related problems choose and use mathematical methods choose methods of processing—mental,




• communicate methods, reasoning and results. They use mental and written methods and technologies where appropriate. The number formats for the unit are positive and negative numbers, square roots, cube roots and numbers expressed with integer powers.

1. Number and algebra (21 hours) 1.1 Estimation and calculation 1.1.1 use mental and written methods to

calculate and estimate with integers, decimals, fractions between 0 and 1, common percentages of whole numbers and of decimals to two decimal places

1.1.2 use calculators to calculate with integers, decimals, fractions and percentages, powers, square roots, cube roots and trigonometric ratios

1.1.3 convert numbers to and from scientific notation.

1.2 Functions and graphs

1.2.1 graph exponential relationships, xby = for

1>b , 0≥x 1.2.2 recognise linear and exponential functions

from equations, tables and graphs 1.2.3 interpret linear and exponential functions for

practical situations. 1.3 Equivalence, equations and inequalities 1.3.1 without a calculator solve equations with

one algebraic term such as 1912 2 =+x ,

243 3 =x 1.3.2 solve linear equations and simple linear

inequalities algebraically 1.3.3 set up simultaneous linear equations, using

the forms cmxy += and cbyax =+ , and

solve the equations algebraically (unique solution only)

1.3.4 solve simultaneous linear equations graphically

1.3.5 solve exponential equations graphically and algebraically (simple cases with no leading

coefficients e.g. 642 =x ). 1.4 Patterns 1.4.1 state recursive word rules for number

sequences, identifying starting numbers 1.4.2 describe number sequences recursively

using algebraic notation such as 31 +=+ nn TT , 41 =T

1.4.3 use recursive rules to continue sequences, including rules that involve simple percentages

1.4.4 investigate numbers: – identify patterns – conjecture generalisations – test conjectures with further cases – provide explanations that support or refute

conjectures – use mathematical language to explain

patterns.


37


2.1 Measurement 2.1.1 use surface area and volume formulas

directly and inversely for – cubes, right prisms and pyramids – cylinders, cones and spheres (decimal

answers only). 2.1.2 use sine, cosine and tangent ratios to

calculate sides and angles (degree measure) of right triangles (two-dimensional contexts only; exact trigonometric ratios involving surds are not required).

2.2 Coordinate geometry 2.2.1 determine:

– distance between two points – the gradient of a line joining two points.

2.2.2 determine the equation of a line given: – a point on the line and the gradient – two points on the line.

2.3 Networks 2.3.1 represent information using network

diagrams and interpret the diagrams (basic networks only, project networks not included)

2.3.2 investigate the traversability of networks 2.3.3 develop systematic methods to determine

the shortest path between two vertices of a network.

3. Chance and data (17 hours) 3.1 Conduct chance experiments 3.1.1 simulate everyday chance events with

outcomes that are not equally likely. 3.2 Quantify chance 3.2.1 use long run relative frequency to estimate

probabilities 3.2.2 list sample spaces for one-stage events,

with repetition to reflect possible outcomes 3.2.3 calculate simple probabilities, using sample

spaces and the number of favourable outcomes divided by the total number of outcomes, for one-stage events


3.2.5 use the fact that probabilities sum to 1 to calculate probabilities for complementary events



simulations with the same number of trials 3.3.2 use probabilities to predict proportions and


3.3.3 recognise predictions are not always realised

3.3.4 recognise the law of large numbers (that individual outcomes of chance events are unpredictable but the relative frequency stabilises as the number of trials becomes large)

3.3.5 order outcomes from least likely to most likely, using fractional, decimal and percentage probabilities

3.3.6 explain probability statements in common usage

3.3.7 identify factors that could compromise a simulation of everyday chance events


3.4 Collect and organise data 3.4.1 plan the collection of bivariate or time series

data to investigate situations specified by the teacher

3.4.2 predict what data will show 3.4.3 collect and record data, and check and edit

the record. 3.5 Represent data 3.5.1 plot points and construct line graphs for

time series data, plotting between calibrations on scales, if necessary

3.5.2 construct scatterplots for bivariate data, plotting between calibrations on scales, if necessary

3.5.3 describe trend as increasing or decreasing, for bivariate and time series data

3.5.4 sketch notional increasing and decreasing trends (not from points)

3.5.5 fit trend lines ‘by eye’ over plotted points. 3.6 Interpret data 3.6.1 read information from scatterplots and plots

of time series data, reading between calibrations on scales

3.6.2 predict using interpolation and extrapolation and trend lines fitted by eye, recognising the dangers of extrapolation

3.6.3 report on collected data (to include assessing how to improve data collection and handling).


38



65–75%

Response





25–35%

Investigation






39

UNIT 2CMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students calculate interest and repayments in order to make decisions about savings and loans, and they interpret information on financial statements that are part of everyday living. They study and apply quadratic relationships. They extend their knowledge of coordinate geometry, and represent information in networks and interpret network diagrams. Students calculate and interpret probabilities for events with more than one chance component. They analyse datasets, determine trends in data and use trend lines for prediction. They use mental and written methods and technologies where appropriate. The unit content will be introduced and applied in a variety of contexts that are accessible to students.


This unit includes knowledge, understandings and skills to the degree of complexity described below and comprises the examinable content of the course. Students will be provided with opportunities to: • plan and carry through tasks:

identify and organise information develop systematic approaches partition problems into sub-problems identify simpler, related problems choose and use mathematical methods choose methods of processing—mental,




• communicate methods, reasoning and results. They use mental and written methods and technologies where appropriate. The number formats for the unit are positive and negative numbers, square roots, cube roots, recurring decimals and numbers expressed with integer powers.

1. Number and algebra (24 hours) 1.1 Estimation and calculation 1.1.1 use calculators and written methods

efficiently 1.1.2 round numbers to a given number of

significant figures 1.1.3 round, truncate and decide on appropriate

accuracy as part of calculation and estimation

1.1.4 recognise the effects of errors due to truncating and rounding

1.1.5 convert numbers to, and from scientific notation.

1.2 Functions and graphs 1.2.1 recognise properties of linear functions:

– the meaning of m and c in cmxy +=

– families of lines from their equations. 1.2.2 sketch quadratic functions in the following

forms: )cx)(bx(ay −−=

cbxay +−= 2)(

cbxaxy ++= 2 , where a, b and c are

integers 1.2.3 identify families of quadratic functions from

their equations 1.2.4 identify features of parabolas:

– intercepts – lines of symmetry – turning points – concavity.

1.2.5 interpret parabolas: – relationships between variables – turning points and optimization.

1.2.6 use function notation. 1.3 Equivalence, equations and inequalities 1.3.1 factorise differences of two squares such

as: 22 ba − , 222 bxa − and readily factorised

quadratic expressions of the form

cbxx ++2 with and without a calculator 1.3.2 solve quadratic equations:

– algebraically, if in factored form or readily factorised form

– graphically. 1.4 Finance 1.4.1 calculate compound interest recursively

with technology 1.4.2 calculate repayments and amount owing for

loans 1.4.3 interpret and compare loans and

investments with simple and compound interest

1.4.4 make decisions about loans and investments

1.4.5 calculate inflation and depreciation


40

1.4.6 interpret financial information including tax tables, commercial advertisements, and credit card rates, charges and credit limits.

2. Space and measurement (10 hours) 2.1 Coordinate geometry 2.1.1 determine the gradient and equations of

parallel and perpendicular lines 2.1.2 apply distance and gradient relations to

solve problems in the Cartesian plane. 2.2 Networks 2.2.1 interpret information represented in network

diagrams (basic networks only, project networks not included)

2.2.2 develop systematic methods to determine the shortest path between two vertices of a network

2.2.3 determine minimal spanning trees for networks using network diagrams and Prim’s algorithm

2.2.4 determine the maximal flow for networks with one source and one sink.

3. Chance and data (21 hours) 3.1 Quantify chance 3.1.1 use Venn diagrams to represent sample

spaces for two events and to illustrate set concepts (subset, intersection, union, complement)

3.1.2 use two-way tables to represent sample spaces for two events

3.1.3 use Venn diagrams and two-way tables to calculate simple probabilities for compound events (event A or B, event A and B, event A given event B, complement of A)


3.1.5 use set and probability notation n(U), n(A),

n(A') or n( A ), n(A ∪ B), n(A ∩ B), n(A|B), Ø

and P(A), P( A ), P(A ∩ B), … 3.2 Interpret chance 3.2.1 use probabilities to predict proportions and



3.3 Collect and organise data 3.3.1 plan how to group data in equal sized class

intervals, taking into account the range of measurements

3.3.2 assign data to the intervals. 3.4 Represent data 3.4.1 construct frequency histograms for

ungrouped and grouped data

3.4.2 calculate mean using n

xx = and

=

f

fxx notation, and median and mode

for ungrouped frequency data 3.4.3 calculate weighted mean, mean for grouped

data, and modal and median classes 3.4.4 describe spread between data displayed in

frequency tables and graphs using terms such as gaps, clusters, more dense/less dense regions, outliers

3.4.5 calculate range for ungrouped and grouped data

3.4.6 calculate relative frequency and proportions of data in fractional, decimal and percentage forms and use them to describe spread

3.4.7 plot time series and bivariate data, fit trend lines ‘by eye’ and calculate their equations.

3.5 Interpret data 3.5.1 read information from frequency tables,

nested and layered tables, frequency graphs and time series graphs and scatterplots

3.5.2 discern the suitability of mean, median and mode for indicating central location

3.5.3 calculate numbers of data in categories from relative frequencies and proportions

3.5.4 identify independent and dependent variables for experimental and time series data

3.5.5 predict using interpolation and extrapolation and trend line graphs and equations, recognising the risks of extrapolation

3.5.6 explain why predicted and actual results are likely to differ.


41



65–75%

Response





25–35%

Investigation






42

UNIT 2DMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students study functions and their graphs. They formulate recursion rules and apply recursion in practical situations. They explore patterns, making conjectures and testing them. They use trigonometry for the solution of right and acute triangles. Students simulate chance events on technologies, and calculate and interpret probabilities for chance events that occur in two- or three- stages. They plan random samples, collect, and analyse data from them, and infer results for populations. They use mental and written methods and technologies where appropriate. The unit content will be introduced and applied in a variety of contexts that are accessible to students.

Unit content This unit builds on the content covered by the previous units. This unit includes the content areas:

• number and algebra • space and measurement • chance and data.

This unit includes knowledge, understandings and skills to the degree of complexity described below and comprises the examinable content of the course. Students will be provided with opportunities to:

• plan and carry through tasks identify and organise information choose and use mathematical methods choose methods of processing—mental,




• communicate methods, reasoning and results. They use mental and written methods and technologies where appropriate. The number formats for the unit are positive and negative numbers, recurring decimals, square roots, cube roots and numbers expressed with integer powers.

1. Number and algebra (27 hours)

1.1 Estimation and calculation 1.1.1 use index laws to multiply and divide

numbers with integer powers. 1.2 Functions and graphs 1.2.1 sketch graphs of:

xby = , 0>b nxy = , for n = 2, 3, -1

1.2.2 recognise functions of the forms xby = ,

0>b and nxy = for n = 2, 3, -1 from

tables and graphs 1.2.3 describe the effects of varying a , b and c

on the graph of c)bx(afy +−=

where 2)( xxf = , 3)( xxf = or xkxf =)(

(vary up to two parameters in any one example)

1.2.4 distinguish linear, quadratic, cubic, exponential and reciprocal functions from equations and graphs

1.2.5 sketch the cubic functions: ))()(( dxcxbxay −−−=

2))(( cxbxay −−= 3)( bxay −=

1.2.6 use technology to graph

dcxbxaxy +++= 23 .

1.3 Equivalence, equations and inequalities 1.3.1 rearrange and simplify algebraic

expressions into forms useful for computation

1.3.2 estimate the solutions for cab x = using substitution, where a , b and c are constants

1.3.3 solve quadratic, cubic and exponential equations graphically

1.3.4 solve simultaneous equations graphically. 1.4 Patterns 1.4.1 link arithmetic sequences to linear

functions, and geometric sequences to exponential functions

1.4.2 determine recursive rules for terms of arithmetic, geometric and Fibonacci sequences and write the rules with recursive notation such as 31 +=+ nn TT , 41 =T

1.4.3 test generalisations by systematically checking cases and searching for counter examples

1.4.4 investigate real world applications of arithmetic, geometric and Fibonacci sequences.


43

2. Space and measurement (10 hours) 2.1 Measurement 2.1.1 use sine, cosine and tangent ratios to

calculate sides and angles (degree measure) of right triangles (two-dimensional contexts only; exact trigonometric ratios involving surds are not required)

2.1.2 use the formula area ΔABC = Cabsin21

2.1.3 use sine and cosine rules to determine sides and angles of acute triangles (two-dimensional contexts only).

3. Chance and data (18 hours) 3.1 Conduct chance experiments 3.1.1 plan and conduct simulations using

technology-based random number generators.

3.2 Quantify chance 3.2.1 use long run relative frequency to estimate

probabilities 3.2.2 use lists, tables and tree diagrams to

determine sample spaces for one-, two- and three-stage events

3.2.3 use sample spaces to calculate simple probabilities and probabilities for compound events (event A or B, event A and B, event A given event B, complement of A)

3.2.4 use the relationship P(A) + P( A ) = 1 to calculate probabilities for complementary events



simulations with different numbers of trials 3.3.2 recognise that a first-stage result in a two-

stage experiment may or may not affect a second stage result

3.3.3 estimate population size using the capture-recapture technique.

3.4 Collect and organise data 3.4.1 identify problems/situations that require

comparison of data, formulate research questions, and revise the questions to reduce ambiguity

3.4.2 plan what data to collect (primary data by observation, experiment or survey; or secondary data, from published materials or databases)

3.4.3 check the credibility of secondary data 3.4.4 plan random sampling and sample size that

will allow reliable conclusions 3.4.5 predict what data will show 3.4.6 plan how to record data to facilitate analysis

including units of measurement and possible grouping

3.4.7 collect and record data, and check and edit the record.

3.5 Represent data 3.5.1 produce tables and graphs and summary

statistics to support analysis 3.5.2 determine the standard deviation for

grouped and ungrouped data using the inbuilt facility on a calculator.

3.6 Interpret data 3.6.1 discern viability of range and standard

deviation for ranking datasets in order of spread

3.6.2 interpret spread summaries in terms of their mathematical definitions

3.6.3 compare datasets using mean and standard deviation, and noting features of tabulated or graphed data

3.6.4 use words that acknowledge uncertainty when comparing data sets such as ‘scores for … tend to be more spread than scores for …’

3.6.5 infer results for populations from samples, recognising possible chance variation between them

3.6.6 report on collected data (to include commenting on external factors i.e. hidden variables that might have affected data and recognising possible chance variation in samples).


44



65–75%

Response





25–35%

Investigation






45

UNIT 3AMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students explore and analyse the properties of functions and their graphs. They develop and use algebraic skills for solving equations. They apply recursion in practical situations, including for finance. They use trigonometry for the solution of triangles. Students use counting principles to calculate probabilities and analyse normally distributed data. They plan sampling methods, analyse data from samples and infer results for populations. They use mental and written methods and technologies where appropriate. The unit content will be introduced and applied in a variety of contexts that are accessible to students.


identify and organise information choose and use mathematical methods choose methods of processing—mental,


check answers fit specifications link solutions to contexts generalise results.

• argue to support or contest mathematical conclusions

• communicate methods, reasoning and results. They use mental and written methods and technologies where appropriate. The number formats for the unit are positive and negative numbers, recurring decimals, square roots, cube roots and numbers expressed with rational powers.

1. Number and algebra (27 hours) 1.1 Estimation and calculation 1.1.1 use mental strategies for estimation in

context 1.1.2 evaluate the absolute value of rational

numbers 1.1.3 use calculators efficiently 1.1.4 round numbers to a given number of

significant figures 1.1.5 round, truncate and choose appropriate

accuracy as part of calculation and estimation

1.1.6 recognise the effects of rounding and truncating on the accuracy of results

1.1.7 use the laws of indices to simplify numerical and algebraic expressions and to solve equations.

1.2 Functions and graphs 1.2.1 sketch graphs of:

xby = , 0>b , eb ≠ , nxy = , for n = 2, 3, ½, ⅓, -1

1.2.2 describe the effects of varying a , b , c and d on the graph of

dcxbafy +−= )]([ where: nxxf =)( , for n = 2, 3, ½,⅓, -1 xkxf =)( and determine the equation from

their graphs (vary up to two parameters in any one example)

1.2.3 identify domain and range of functions 1.2.4 distinguish linear, quadratic, cubic,

exponential and reciprocal functions in algebraic and graphical forms

1.2.5 describe the graphs of functions qualitatively (calculations not required) considering: – intercepts – lines of symmetry – turning points – asymptotes – concavity – points of inflection.

1.2.6 use function notation.

1.3 Equations and inequalities 1.3.1 rearrange algebraic expressions into forms

useful for computation, including factorising 222 bxa − and cbxx ++2

1.3.2 solve algebraically and graphically: – quadratic equations in factored form – cubic equations in factored form

– exponential equations cab kx = , 0>b (logarithms not required)

– simple power equations cxn = , n = 2, 3, ½, ⅓, -1.

1.3.3 solve simultaneous equations graphically, including linear and quadratic equations


46

1.3.4 describe how one quantity varies with another by inspecting the formula that relates them, including quantities that are inversely proportional

1.3.5 solve inverse proportion problems 1.3.6 relate the ideas of inverse proportion and

reciprocal functions.

1.4 Patterns 1.4.1 use recursion to determine terms and sums

for sequences including arithmetic and geometric sequences

1.4.2 use recursion to study growth and decay. 1.5 Finance 1.5.1 use, construct and interpret spreadsheets

for making financial decisions 1.5.2 judge adequacy of spreadsheets and make

refinements if necessary 1.5.3 calculate loans with reducible interest,

including determining the number of years for the balance to fall to a specified amount

1.5.4 calculate annuities using a spreadsheet 1.5.5 interpret and make decisions about loan

and repayment amounts with reducible interest.


2.1 Rate 2.1.1 convert between rate units such as

kilometres per hour and metres per second 2.1.2 interpret function of time relationships

)(tfy = including distance and

displacement relationships 2.1.3 sketch and interpret graphs for )(tfy =

relationships 2.1.4 recognise that rate of change is constant for

linear relationships. 2.2 Measurement 2.2.1 use the unit circle to identify sine and

cosine ratios for acute and obtuse angles (degree measure only)

2.2.2 use the formula area ΔABC = Cabsin21

2.2.3 use the sine and cosine rules to determine sides and angles of triangles (two-dimensional contexts only).

3. Chance and data (20 hours) 3.1 Quantify chance 3.1.1 use lists, tree diagrams and two-way tables

to determine sample spaces for two- and three-stage events

3.1.2 use Venn diagrams to represent sample spaces for two events and to illustrate subset, intersection, union and complement

3.1.3 use sample spaces to calculate simple probabilities and probabilities for compound events (event A or B, event A and B, event A given event B, complement of A)

3.1.4 use addition and multiplication principles for counting, and use the counts to calculate probabilities

3.1.5 use the relationship P(A) + P( A ) = 1 to calculate probabilities for complementary events

3.1.6 use set and probability notation such as n(U), n(A), n(A') or n( A ), n(A ∪ B), n(A ∩ B), n(A|B), Ø and P(A), P(A') or

P( A ), P(A ∪ B), P(A' ∩ B) 3.1.7 calculate probabilities for normal

distributions with known mean μ and standard deviation σ

3.1.8 use the 68%, 95%, 99.7% rule for data one, two and three standard deviations from the mean

3.1.9 use probability notation for normal random variables such as P(X < x).

3.2 Interpret chance 3.2.1 use probabilities to predict proportions and


3.2.2 estimate population size using the capture/recapture technique

3.2.3 calculate quantiles for normally distributed data with known mean and standard deviation

3.2.4 use number of standard deviations from the mean (standard scores) to describe deviations from the mean in normally distributed data sets.

3.3 Collect and organise data 3.3.1 plan sampling methods (systematic,

random, stratified, self-selection, convenience) and justify choosing a sample instead of a census.

3.4 Represent data 3.4.1 construct frequency histograms for grouped

and ungrouped data 3.4.2 construct boxplots for ungrouped data,

outliers not distinguished 3.4.3 calculate mean, median and mode for

ungrouped frequency data and recognise that averages indicate location of frequency distributions

3.4.4 calculate weighted mean, mean for grouped data, and median and modal classes

3.4.5 describe spread between data displayed in frequency tables and graphs using terms such as gaps, clusters, more dense/less dense regions, outliers, symmetry and skewness

3.4.6 calculate cumulative frequency, quartiles and interquartile range for ungrouped data and use them to describe spread


47

3.4.7 determine the standard deviation for grouped and ungrouped data using the inbuilt facility on a calculator

3.4.8 identify extreme and unexpected values 3.4.9 calculate outliers (values more than

1.5 × interquartile range beyond the upper and lower quartiles).

3.5 Interpret data 3.5.1 discern connections between frequency

histograms and boxplots, including the shape of histograms for provided boxplots

3.5.2 discern the advantages/disadvantages of using frequency histograms and boxplots to display data

3.5.3 discern effects of different equal sized class intervals on histograms

3.5.4 discern viability of interquartile range, range and standard deviation for ranking datasets in order of spread

3.5.5 interpret spread summaries in terms of their mathematical definitions

3.5.6 reason to include or exclude outliers 3.5.7 discern effects on summary statistics of

cropping data (including outliers) 3.5.8 compare datasets, combining interpretation

of mean, standard deviation, and skewness or symmetry about the mean

3.5.9 compare datasets, combining interpretation of median, interquartile range and skewness or symmetry about the median

3.5.10 compare scores from two or more sets of data using number of standard deviations from the mean (standard scores)

3.5.11 infer results for populations from samples, recognising possible chance variation between them

3.5.12 show how data can be manipulated to serve different purposes.



75–85%

Response





15–25%

Investigation






48

UNIT 3BMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students study differential calculus of polynomial functions and use calculus in optimisation problems. They develop algebraic skills for solving systems of linear equations. They analyse and construct project networks. They reason deductively in algebra and geometry. Students analyse bivariate data, and argue to support or contest conclusions about data. They use mental and written methods and technologies where appropriate. The unit content will be introduced and applied in a variety of contexts that are accessible to students.


choose and use mathematical models and methods

choose methods of processing—written, with a calculator.

• interpret solutions: check answers fit specifications link solutions to contexts generalise results.

• argue to support or contest mathematical conclusions


1. Number and algebra (31 hours) 1.1 Functions and graphs 1.1.1 apply polynomial, exponential and power

functions to practical situations including optimisation and use numerical and graphical techniques

1.1.2 interpret graphs: – domain and range – intercepts and points – slope at a point – local and global maxima and minima.

1.2 Equations and inequalities 1.2.1 formulate and solve one-variable equations

and inequalities (absolute value terms not included)

1.2.2 formulate systems of linear equations in two variables from word descriptions

1.2.3 solve systems of linear equations in two variables algebraically

1.2.4 graph two variable linear inequalities. 1.3 Calculus 1.3.1 understand the calculus of polynomial

functions: – average rate of change – derivative as instantaneous rate of

change and slope of a curve at a point – limit (informally).

1.3.2 differentiate nxy = , n a whole number

1.3.3 use the sum and product rules to differentiate polynomials

1.3.4 use differentiation to determine tangent lines at a point for polynomial functions

1.3.5 use differentiation to sketch polynomial functions (points of inflection not required)

1.3.6 use differentiation to solve optimisation problems with polynomial functions

1.3.7 use notations for the derivative: y ′ , 'f ,

)x(f ′ , dx

dy,

dx

df and )(xf

dx

d

1.4 Patterns 1.4.1 make conjectures about numbers such as

‘the sum of two odd numbers is even’ 1.4.2 search for counter examples to conjectures

in order to disprove them 1.4.3 construct simple deductive proofs using

algebra such as ‘prove that the sum of two odd numbers is even’

1.4.4 follow algebraic deductive arguments and ascertain their validity.


49

2. Space and measurement (8 hours) 2.1 Networks 2.1.1 analyse project networks 2.1.2 construct project networks 2.1.3 determine critical paths and minimum

completion times for projects with fixed activity times.

2.2 Reason geometrically 2.2.1 distinguish general geometric arguments

from those based on specific cases 2.2.2 follow geometric deductive arguments and

ascertain their validity.

3. Chance and data (16 hours) 3.1 Represent data 3.1.1 describe association (positive, negative,

weak, strong or none) 3.1.2 determine Pearson’s correlation coefficient

r using a calculator 3.1.3 describe properties of regression lines

(least-squares relationship and passing through )y,x( )

3.1.4 calculate and graph regression models for data with linear trends

3.1.5 calculate residuals for linear models and construct residual plots

3.1.6 calculate moving averages, regression lines for moving averages, and seasonal adjustments for periodic time series data.

3.2 Interpret data 3.2.1 place expressions of association (weak,

strong etc.) on a scale from –1 to 1 3.2.2 recognise correlation does not imply

causality 3.2.3 discern ‘goodness of fit’ for regression lines,

using visual inspection of scatterplots, residual plots and correlation coefficient

3.2.4 consider regression lines: – to include or crop outliers – effects on the lines of cropping outliers

and other data – whether intercepts are valid – variables that explain data above and

below the lines – alternative models that might fit data

better than a line including quadratic, exponential.

3.2.5 predict from regression lines, recognising the risks of extrapolation, and assess reliability

3.2.6 explain why regression lines are used for prediction, rather than data points and why predicted and actual results are likely to differ

3.2.7 recognise that regression lines for samples and populations may differ due to chance variation

3.2.8 predict from regression lines, making seasonal adjustments for periodic data.



75–85%

Response





15–25%

Investigation






50

UNIT 3CMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students develop their knowledge of calculus concepts and their algebraic, graphing and calculus skills, and apply these in mathematical modelling. They use counting techniques and probability laws, and calculate and interpret probabilities for the binomial, uniform and normal random variables. They use mental and written methods and technologies where appropriate. The unit content will be introduced and applied in a variety of contexts that are accessible to students.


This unit includes knowledge, understandings and skills to the degree of complexity described below and comprises the examinable content of the course.

Students will be provided with opportunities to: • plan and carry through tasks, choosing and

using mathematical models and methods • interpret solutions:

▪ link solutions to contexts ▪ generalise results.

• critically assess mathematical reasoning and conclusions


1. Number and algebra (25 hours)

1.1 Estimation and calculation 1.1.1 choose levels of accuracy to suit contexts

and distinguish between exact values, approximations and estimates

1.1.2 manipulate numerical and algebraic expressions to facilitate calculation.

1.2 Functions and graphs

1.2.1 investigate the limiting behaviour n

n

a

+1

as ∞→n , ( a fixed)

1.2.2 define e as the limit of n

n

+ 11 as ∞→n

1.2.3 sketch the graph of xey =

1.2.4 describe the effects of varying a, b, c and d on the graph of d)]cx(b[afy +−=

where xe)x(f =

1.2.5 form composite functions ))x(g(f or

)(xgf where f and g are linear,

quadratic or exponential functions or take

the form nx , =n ½ or –1 1.2.6 determine the domain and range of

composite functions. 1.3 Equations and inequalities 1.3.1 simplify sums and differences, products and

quotients of algebraic fractions with constants, linear expressions or quadratic expressions in the numerator or denominator.

1.4 Calculus

1.4.1 differentiate nxy = , n rational

1.4.2 use the sum, product and quotient rules to differentiate polynomials and other simple algebraic combinations

1.4.3 introduce the derivative of kxe by considering the slope of the graph of

kxey =

1.4.4 differentiate composite functions using the chain rule with Leibniz notation

dx

dy

dy

dz

dx

dz ×=

1.4.5 obtain second derivatives of polynomials and other simple algebraic combinations

1.4.6 solve optimisation problems using calculus 1.4.7 sketch polynomial functions using calculus

(turning points and points of inflection identified and classified with sign and derivative tests)

1.4.8 use differentiation to determine tangent lines

1.4.9 treat the integral as the limiting sum of the signed area of rectangles

1.4.10 recognise that the integral of a sum is the sum of the integrals

1.4.11 recognise the link between the integral as a signed area and antidifferentiation

1.4.12 use )()()(' afbfdxxfb

a−= to calculate

definite integrals

1.4.13 integrate nax , 1−≠n ,

1.4.14 integrate kxe


51

1.4.15 integrate )('))(( xfxfa n , 1−≠n for a

polynomial function f .

2. Space and measurement (10 hours) 2.1 Rate 2.1.1 distinguish between average rate of change

and instantaneous rate of change of a function at any point

2.1.2 use exponential functions kxAey = as

solutions of the differential equation to solve simple problems involving growth and decay

2.1.3 use instantaneous rate of change and the derivative to approximate marginal rate of change and use the increments formula

xdx

dyy δδ ≈ .

2.2 Measurement 2.2.1 calculate area under, and between, curves

defined by polynomial functions 2.2.2 use calculus to optimise quantities relating

to measurement in space.

3. Chance and data (20 hours) 3.1 Quantify chance 3.1.1 use combinations and arrangements for

counting and calculating probabilities

3.1.2 use rn C and

r

n notation

3.1.3 use the laws P(A ∪ B) = P(A) + P(B) – P(A ∩ B), P(A ∩ B) = P(A) × P(B|A) to calculate probabilities (to include cases where A and B are independent, mutually exclusive, complementary)

3.1.4 use P(B|A) = P(A ∩ B)/P(A) to calculate conditional probabilities

3.1.5 list sample spaces and calculate probabilities for discrete random variables associated with one- and two-stage events 3.1.6 calculate the probability of x successes in n independent trials 3.1.7 calculate probabilities for compound events and probability of at least x successes in n trials 3.1.8 use the concepts of continuous random

variable and probability density function to calculate probabilities for uniform and normal distributions

3.1.9 use probability notation for random variables such as P(X = x), P(X < x) .

3.2 Interpret chance 3.2.1 classify everyday events as

complementary, mutually exclusive, independent and not independent

3.2.2 prove events are independent or not independent using the laws of probability

3.2.3 classify variables as discrete or continuous and justify the choice

3.2.4 use probabilities associated with discrete, binomial, uniform and normal distributions to calculate proportions, quantiles and mean values

3.2.5 calculate mean np and standard deviation

)1( pnp − for binomial distributions.



75–85%

Response





15–25%

Investigation





kyy =′


52

UNIT 3DMAT

Unit description The unit description provides the focus for teaching the specific unit content. In this unit, students extend and apply their understanding of differential and integral calculus. They solve systems of equations in three variables and linear programming problems. They verify and develop deductive proofs in algebra and geometry. Students model data with probability functions and analyse data from samples. They justify decisions and critically assess claims about data. They use mental and written methods and technologies where appropriate. The unit content will be introduced and applied in a variety of contexts that are accessible to students.

Unit content This unit builds on the content covered by the previous units. This unit includes the content areas: • number and algebra • space and measurement • chance and data. This unit includes knowledge, understandings and skills to the degree of complexity described below and comprises the examinable content of the course. Students will be provided with opportunities to: • plan and carry through tasks, choosing and

using mathematical models and methods • interpret solutions, considering limitations of

models, exclusions and assumptions • critically assess mathematical reasoning and

conclusions • communicate methods, reasoning and results. They use mental and written methods and technologies where appropriate. The number formats for the unit are real number forms that facilitate problem solving including surds and scientific notation.

1. Number and algebra (20 hours) 1.1 Equations and inequalities 1.1.1 solve equations and inequalities that

involve algebraic fractions with constant terms and linear or quadratic expressions in the numerator and denominator (absolute value expressions not included)

1.1.2 solve two-variable linear programming problems, including sensitivity analysis

1.1.3 solve systems of equations in three variables systematically by elimination.

1.2 Calculus 1.2.1 understand the relationships between the

critical points and graphs of )(xf , )(xf ′

and )(xf ′′

1.2.2 develop and use the concept of a function

defined as an integral = x

adttfxF )()(

1.2.3 examine and use the two parts of the Fundamental Theorem of Calculus:

=x

axfdttf

dx

d )()(

and )()()(' afbfdxxfb

a−=

1.3 Patterns 1.3.1 make and test conjectures 1.3.2 disprove conjectures with counter examples 1.3.3 construct algebraic deductive proofs.


2.1 Rate 2.1.1 evaluate total change from given rates of

change 2.1.2 determine and use time-related derivatives

for motion in a straight line—velocity, speed and acceleration

2.1.3 model rectilinear motion with the differential equations ( )tfx =' and )(tgx =′′ solving

for x 2.1.4 determine rate of change of derived

attributes with respect to time including rate of change of surface area and volume of a sphere given the rate of change of the radius

2.1.5 solve related rates problems (functions of time only).

2.2 Measurement 2.2.1 calculate volumes of solids of revolution

around the x- and y-axes using dxy2π

and dyx2π

2.2.2 derive mensuration formulas. 2.3 Reason geometrically 2.3.1 distinguish general geometric arguments

from those based on specific cases 2.3.2 follow and ascertain the validity of

geometric arguments 2.3.3 construct deductive proofs involving:

– isosceles, right, similar and congruent triangles

– angles in circles – tangents to circles.


53

3. Chance and data (17 hours) 3.1 Quantify chance 3.1.1 select and use appropriate methods to

calculate probabilities (sample spaces, addition and multiplication principles and combinations for counting, probability laws, and binomial, uniform and normal probability distributions)

3.1.2 use long run relative frequency to estimate probabilities.

3.2 Interpret chance 3.2.1 model experimental data with discrete,

binomial, uniform and normal probability distributions

3.2.2 identify the limitations of models for predicting real behaviour.

3.3 Represent data 3.3.1 investigate the behaviour of the sample mean using simulation and other sampling techniques for samples of different sizes 3.3.2 recognise that:

– for any population the sample mean ( )X is

a random variable – for any population with mean μ and

standard deviation σ the distribution of

X over all samples of size n has the following characteristics:

o the mean of the sample means ( )X is

the population mean

o the standard deviation of X is n

σ

– the distribution of the sample means approaches normality as the sample size increases (Central Limit Theorem).

3.4 Interpret data 3.4.1 use statistics from a random sample to

determine a 90%, 95% or 99% confidence

interval x z x zn n

σ σμ− < < + for the

population mean μ , where:

– x is the sample mean – z is the appropriate cut-off point from the

standard normal distribution, and where – the population standard deviation σ is

either known or estimated by the sample standard deviation.

3.4.2 recognise that the level of confidence is the probability that the population mean μ is in

the confidence interval and that x differs

from μ by less than zn

σ

3.4.3 recognise the relationship between the width of a confidence interval, the confidence level, the sample size and the population standard deviation

3.4.4 determine the sample size necessary to obtain a confidence interval of a given width and a given level of confidence (90%, 95% or 99%).



75–85%

Response





15–25%

Investigation






54


55

Examination details Stage 2 and Stage 3


56

Mathematics Examination design brief

Stage 2—2A/2B This examination consists of two sections. Section One: Calculator-free Time allowed Reading time before commencing work: 5 minutes Working time for section: 50 minutes Permissible items Standard items: pens (blue/black preferred), pencils (including coloured), sharpener, correction tape/fluid,

eraser, ruler, highlighters Special items: nil Changeover period – no candidate work: approximately 15 minutes Section Two: Calculator-assumed Time allowed Reading time before commencing work: 10 minutes Working time for section: 100 minutes Permissible items Standard items: pens (blue/black preferred), pencils (including coloured), sharpener, correction tape/fluid,

eraser, ruler, highlighters Special items: drawing instruments, templates, notes on two unfolded sheets of A4 paper, and up to three

calculators approved for use in the WACE examinations Additional information It is assumed that candidates sitting this examination have a calculator with graphics capabilities for Section Two. The examination assesses the syllabus content areas using the following percentage ranges. These apply to the whole examination rather than individual sections.

Content area Percentage of exam

Number and algebra 40–50%

Space and measurement 20–25%

Chance and data 30–35% The candidate is required to demonstrate knowledge of mathematical facts, conceptual understandings, use of algorithms, use and knowledge of notation and terminology, and problem-solving skills. Questions could require the candidate to investigate mathematical patterns, make and test conjectures, generalise and prove mathematical relationships. Questions could require the candidate to apply concepts and relationships to unfamiliar problem-solving situations, choose and use mathematical models with adaptations, compare solutions and present conclusions. A variety of question types that require both open and closed responses could be included. Instructions to candidates indicate that for any question or part question worth more than two marks, valid working or justification is required to receive full marks. A Formula Sheet is provided.


57

Section Supporting information

Section One Calculator-free

331/3% of the total examination

50 marks

5–10 questions

Working time: 50 minutes

Questions examine content and procedures that can reasonably be expected to be completed without the use of a calculator i.e. without undue emphasis on algebraic manipulations or time-consuming calculations. The candidate could be required to provide answers that include calculations, tables, graphs, interpretation of data, descriptions and conclusions. Stimulus material could include diagrams, tables, graphs, drawings, print text and data gathered from the media that are organised around scenarios or concepts relevant to these units.

Section Two

Calculator-assumed


100 marks

8–13 questions


Questions examine content and procedures for which the use of a calculator is assumed. The candidate could be required to provide answers that include calculations, tables, graphs, interpretation of data, descriptions and conclusions. Stimulus material could include diagrams, tables, graphs, drawings, print text and data gathered from the media that are organised around scenarios or concepts relevant to these units.


58


Stage 2—2C/2D This examination consists of two sections. Section One: Calculator-free Time allowed Reading time before commencing work: 5 minutes Working time for section: 50 minutes Permissible items Standard items: pens (blue/black preferred), pencils (including coloured), sharpener, correction tape/fluid,



calculators approved for use in the WACE examinations Additional information It is assumed that candidates sitting this examination have a calculator with CAS capabilities for Section Two. The examination assesses the syllabus content areas using the following percentage ranges. These apply to the whole examination rather than individual sections.




Chance and data 35–40%

The candidate is required to demonstrate knowledge of mathematical facts, conceptual understandings, use of algorithms, use and knowledge of notation and terminology, and problem-solving skills. Questions could require the candidate to investigate mathematical patterns, make and test conjectures, generalise and prove mathematical relationships. Questions could require the candidate to apply concepts and relationships to unfamiliar problem-solving situations, choose and use mathematical models with adaptations, compare solutions and present conclusions. A variety of question types that require both open and closed responses could be included. Instructions to candidates indicate that for any question or part question worth more than two marks, valid working or justification is required to receive full marks. A Formula Sheet is provided.


59




50 marks

5–10 questions



Section Two

Calculator-assumed


100 marks

8–13 questions




60


Stage 3—3A/3B This examination consists of two sections. Section One: Calculator-free Time allowed Reading time before commencing work: 5 minutes Working time for section: 50 minutes Permissible items Standard items: pens (blue/black preferred), pencils (including coloured), sharpener, correction tape/fluid,










61




50 marks

5–10 questions



Section Two

Calculator-assumed


100 marks

8–13 questions




62


Stage 3—3C/3D This examination consists of two sections. Section One: Calculator-free Time allowed Reading time before commencing work: 5 minutes Working time for section: 50 minutes Permissible items Standard items: pens (blue/black preferred), pencils (including coloured), sharpener, correction tape/fluid,










63




50 marks

5–10 questions



Section Two

Calculator-assumed


100 marks

8–13 questions




64

Mathematics: Accredited March 2008 (updated June 2012)—Appendix 1 For teaching 2013, examined in 2013

Appendix 1: Grade descriptions

Grade descriptions Mathematics 1A Stage 1


A Identifies and organises relevant information. Identifies and organises relevant information that is dense and scattered, e.g. reading an unfamiliar table to identify the necessary data. Chooses effective models and methods and carries the methods through correctly. Carries extended responses through, e.g. giving explicit directions for finding a route from A to B on a street map. Interprets mathematics in unpractised ways, e.g. estimating the likelihood or probability of a common or real life event. Obeys mathematical conventions and attends to accuracy. Uses the equals sign correctly and calculates money problems which are set out horizontally, e.g. $21.00 + $100.00 +… = $... Links mathematical results to data and contexts to reach reasonable conclusions. Rounds to give whole number answers by considering the context, e.g. converts 26 deaths per year ≈ 2 per month. Recognises specified conditions in extended responses, e.g. picks the optimum result from an annual table of temperatures which documents the best time to travel. Communicates mathematical reasoning, results and conclusions. Shows main steps in reasoning to explain the solution to a problem, e.g. describes a route with simple terms such as left, right, “go as far as…” Justifies working by thoroughly stating methods or properties that have been applied, e.g. details the stages and times involved in a journey.

B Identifies and organises relevant information. Identifies and organises relevant information that is dense and scattered, e.g. reading an unfamiliar map to identify necessary locations or routes. Chooses effective models and methods and carries the methods through correctly. Carries extended responses through on most occasions, e.g. adds and subtracts sums of money involving many steps. Translates between representations in unpractised ways including using yearly results to estimate monthly answers. Carries familiar deductive reasoning through to answer open questions regarding timetables, e.g. “When should Joe leave to catch the bus?” Interprets data in unpractised ways, e.g. to compare the likelihood or probability of two possible events using data in table form. Obeys mathematical conventions and attends to accuracy. Uses spatial terms, e.g. left, right, up, down, and geometrical conventions for diagrams as well as for giving directions. Links mathematical results to data and contexts to reach reasonable conclusions. Attends to units in extended responses and rounds to a specified degree when required. Communicates mathematical reasoning, results and conclusions. Justifies working by stating methods or properties that have been applied, e.g. explains the conventions of a street map using one-way arrows.

C Identifies and organises relevant information. Identifies and organises relevant information from information that is grouped together and is narrow in scope, e.g. identifies the nearest hospital to a given railway station on a map. Chooses effective models and methods and carries the methods through correctly. Calculates specific cases of known generalisations, e.g. reading a timetable to calculate the time needed for a journey. Makes commonsense connections by reading a map to locate places or landmarks. Interprets mathematics in practised ways, e.g. adds or subtracts two amounts of money. Obeys mathematical conventions and attends to accuracy. Uses the equals sign correctly when carrying a single thread of reasoning through. Correctly enters numbers into a calculator and copies them from a calculator to solve short answer money problems. Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions in short responses, e.g. identifies the finish time given: “The activity starts now and runs for 10 minutes.” Attends to units in short responses. Communicates mathematical reasoning, results and conclusions. Justifies answers with a simple or routine statement, e.g. “Travel to B because it is closer.”

Grade descriptionsMathematics 1A Stage 1


D Identifies and organises relevant information. Identifies and organises relevant information that is grouped together and is narrow in scope, e.g. railway stations on a map. Interprets single relative terms, e.g. hottest month, from an annual table of temperatures. Chooses effective models and methods and carries the methods through correctly. Answers structured questions which require short responses that include adding times and using schedules. Carries a single thread of reasoning through to reading a street map. Translates between representations and compares distances on a scale map. Applies mathematics in practised ways, e.g. adds or subtracts two amounts of money. Obeys mathematical conventions and attends to accuracy. Applies some basic conventions for diagrams, graphs and maps and recognises symbols on a street map, e.g. one-way streets and hospitals. Attends to accuracy when using a calculator to do single-step calculations. Links mathematical results to data and contexts to reach reasonable conclusions. Attends to units in short responses in an inconsistent manner, e.g. dollar and cents answers with money problems. Communicates mathematical reasoning, results and conclusions. Shows working, including expressions entered into a calculator, to do single-step money calculations. Make appropriate use of basic symbols, e.g. %, $, am, pm, =.

E Does not meet the requirements of a D grade.

Grade descriptions Mathematics 1B/1C Stage 1


A Identifies and organises relevant information. Identifies and organises information that is dense and scattered, e.g. discriminates between terms used when dealing with wages and tax; incorporates results from previous calculations when working on an extended word problem; or extracts the correct information from a table of values in an extended word problem and interprets conjectures. Chooses effective models and methods and carries the methods through correctly. Carries deductive reasoning through and makes calculations in the correct order when following an extended word problem, e.g. calculates the simple interest of a loan as defined by the narrative in the word problem. Makes counterintuitive connections to extended examples involving order of operations, where operations of the same order are next to each other, e.g. 147 x 3 ÷ 7. Determines the effects of changed conditions, e.g. makes a reasonable conjecture on what might happen as a result of changes to a data set. Obeys mathematical conventions and attends to accuracy. Applies conventions with consistent accuracy to extended problems involving order of operations. Uses the equals sign correctly and sets out successive steps in finance calculations. Links mathematical results to data and contexts to reach reasonable conclusions. Attends to units in extended responses and demonstrates consistency with units from different data sources. Recognises specified conditions in extended responses, e.g. labelling diagrams using appropriate notation and symbols. Makes and specifies appropriate conjectures. Communicates mathematical reasoning, results and conclusions. Shows the main steps in reasoning to provide clear evidence of which calculations were used, e.g. balancing a budget. Justifies by stating methods or properties that have been applied in an extended problem. Draws conclusions using information from more than one data source.

B Identifies and organises relevant information. Identifies and organises relevant dense and scattered information when interpreting extended word problems, e.g. recognises the correct terminology associated with loans; recognises the appropriate dimensions in extended word problems in space and measurement; or draws information about the likelihood of an event from a word problem. Chooses effective models and methods and carries the methods through correctly. Solves unstructured problems and identifies the appropriate stages of an extended word problem, e.g. problems involving interest, loans and repayments. Generalises mathematical patterns using words, e.g. squaring the term. Translates between representations in unpractised ways and draws to scale the hidden faces of a 3-D prism, given a sketch of one face and information about the missing dimensions. Calculates and applies rates to be used, e.g. grams per square metre of fertiliser for a garden, from the narrative of a word problem. Obeys mathematical conventions and attends to accuracy. Applies conventions for diagrams, graphs and maps, e.g. drawing 3-D shapes and labelling the sides and vertices properly. Rounds to suit contexts and specified accuracies in extended responses, e.g. rounds to a whole number when the answer requires the number of people. Links mathematical results to data and contexts to reach reasonable conclusions. Attends to units in extended responses and demonstrates consistency with dollars and cents in finance questions. Makes an appropriate conjecture using data in table form. Communicates mathematical reasoning, results and conclusions. Shows most of the main steps in reasoning to provide clear evidence of which calculations were used in each step, e.g. evaluating loan repayments. Justifies with a simple or routine statement and an appropriate numerical example when dealing with number patterns.

Grade descriptionsMathematics 1B/1C Stage 1


C Identifies and organises relevant information. Identifies and organises relevant information that is grouped together and is relatively narrow in scope, e.g. substitutes the correct values into simple percentage equations, completes profit and loss calculations or reads values from a scale diagram. Chooses effective models and methods and carries the methods through correctly. Answers structured questions that require short responses, e.g. calculates a percentage of an amount or number. Applies mathematics in practised ways, e.g. calculates simple interest. Calculates specific cases of generalisations, e.g. a set of square numbers. Makes commonsense connections when comparing the unit price of articles to determine the better buy. Uses a calculator in practised ways to make calculations, e.g. the mean of a column of values. Obeys mathematical conventions and attends to accuracy. Applies basic conventions for diagrams, graphs and maps. Uses the equals sign appropriately when setting out short responses. Applies the Rule of Order of Operations for a given context in short response problems involving finance. Rounds to suit contexts and specified accuracies in short responses, e.g. answers for finance problems are expressed in dollars and cents with dollar symbols or, where relevant, indicating cents only, e.g. $0.53 = 53 cents.Links mathematical results to data and contexts to reach reasonable conclusions. Attends to units appropriately in short responses, e.g. using the dollar and cents symbols, using hours and minutes appropriately when dealing with time intervals. Recognises specified conditions in short responses, e.g. compares costs to select the cheapest option. Communicates mathematical reasoning, results and conclusions. Shows working, including expressions entered into a calculator, for problems that involve two-step calculations. Justifies with a simple or routine statement, e.g. “Choose Item B because A costs more per litre than B.”

D Identifies and organises relevant information. Identifies and organises most of the relevant information that is grouped together and is relatively narrow in scope, e.g. reading a correct value from a data table. Chooses effective models and methods and carries the methods through correctly. Answers structured questions that require short calculations where an example is supplied. Calculates specific cases of generalisations, e.g. calculating the area of a rectangle. Makes commonsense connections to calculate and apply everyday rates using cost per unit in dollars and the number purchased to calculate the total cost.Obeys mathematical conventions and attends to accuracy. Applies basic conventions for diagrams, graphs and maps, e.g. uses broken lines to indicate hidden edges when drawing 3-D diagrams. Rounds to suit most contexts and specified accuracies in short responses, e.g. rounding dollar answers to two decimal places. Links mathematical results to data and contexts to reach reasonable conclusions. Attends to units in short responses that involve money, e.g. using the dollar symbol ($) appropriately.Communicates mathematical reasoning, results and conclusions. Shows some working, including expressions entered into a calculator, which involve single-step calculations. Uses basic symbols, e.g. %, $, pm or am.


Grade descriptions Mathematics 1D/1E Stage 1


A Identifies and organises relevant information. Identifies and organises relevant dense and scattered information in tasks that contain a series of complex processes, e.g. completes a balance sheet by calculating debits and credits, recognises the role of nodes and paths when considering a traversable network. Chooses effective models and methods and carries the methods through correctly. Carries extended responses through, e.g. breaking a complex geometrical figure into its component parts to calculate the total area. Uses deductive reasoning to show how the dimensions of one part of a diagram can be used to help in another part. Applies mathematics in unpractised ways to deal with unfamiliar diagrams, e.g. considers the geometric properties of its parts. Obeys mathematical conventions and attends to accuracy. Uses the equals sign correctly and introduces new operations into a calculation. Rounds to suit contexts and specified accuracies in extended responses when given the required degree of accuracy, e.g. converting 3.456 hours to hours and minutes. Rounds off appropriately after multiplying by π in circle measurement problems. Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions in extended responses, e.g. identifies a section of a diagram as a fraction of a circle. Communicates mathematical reasoning, results and conclusions. Shows main steps in reasoning, e.g. makes a systematic list to find the shortest path on a network diagram. Justifies working by stating methods or properties that have been applied to explain the mathematics used in an open investigation, e.g. estimating the volume of water lost from a leaky tap over a period of time by making use of volume, time, sampling and averages.

B Identifies and organises relevant information. Identifies and organises relevant dense and scattered information in tasks that contain a series of simple processes, e.g. finding the interest rate per month given a yearly (per annum) rate and then making the number of iterations needed to calculate the interest compounded monthly. Chooses effective models and methods and carries the methods through correctly. Solves unstructured problems using familiar sub-problems, e.g. uses Pythagoras’ theorem to find a length and then uses this result in a further calculation. Carries an extended response through, involving familiar processes. Makes counterintuitive connections, e.g. recognises that debits get added to the total of a credit card balance. Generalises mathematical structures, e.g. extends a tessellation for a given number of steps. Translates between representations in unpractised ways, e.g. using a geometric pattern of dots in a given diagram to generate a numerical pattern or sequence. Determines the effects of changed conditions and extracts correct information, e.g. from a modified network diagram.Obeys mathematical conventions and attends to accuracy. Applies conventions for diagrams, graphs and maps, e.g. reading the appropriate distances on a network diagram in order to calculate the minimum time taken for a journey from A to B. Attends to units in extended responses and rounds to a specified degree when required. Checks results and makes adjustments where necessary. Links mathematical results to data and contexts to reach reasonable conclusions. Links and processes more than one piece of information which may be scattered. Uses units consistently throughout an extended perimeter, area or volume problem. Recognises that a compound figure is made up of many identical figures, e.g. rectangles or semicircles.Communicates mathematical reasoning, results and conclusions. Shows main steps in reasoning to explain a calculation with a simple formula e.g.

x

hoursxinlostvolumehourperlossofrate = .

Makes a simple statement based on calculated results. Justifies working with a simple or routine statement, e.g. explains a sampling method and justifies its use.

Grade descriptionsMathematics 1D/1E Stage 1


C Identifies and organises relevant information. Identifies and organises relevant information that is grouped together and is narrow in scope, e.g. uses a diagram to count terms in a number pattern or identifying simple number patterns in a table of consecutive terms. Interprets single relative terms and chooses the lowest value from a set of scores or chooses the smaller shape within a diagram. Chooses effective models and methods and carries the methods through correctly. Answers structured questions requiring short responses, e.g. working through consecutive terms of a table of values to form a generalisation. Calculates specific cases of generalisations including using Pythagoras’ Theorem to calculate the length of a side in a right triangle. Makes commonsense connections to interpret data in a graph to make everyday conjectures, e.g. the student with the highest score is the happiest. Applies mathematics in practised ways, e.g. calculates the volume of a rectangular prism; calculates the weekly total = daily rate × 7; and interprets the figures in a spreadsheet for loan repayments. Obeys mathematical conventions and attends to accuracy. Applies basic conventions for diagrams, graphs and maps, e.g. indicating pathways on a network diagram; marking the start and end points and showing direction using arrows; and measuring true bearings from the north. Applies the rule of order of operations to solve word problems, including rates and costs. Checks for accuracy when entering numbers into a calculator in multi-step calculations. Rounds to suit contexts and specified accuracies in short responses, including rounding off a square root appropriately in

a measurement problem, e.g. 7 2.645751311 2.65 or 2.646= ≈ ≈ . Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions in short responses and refers to the independent and dependent variables when referring to a scatter diagram. Attends to units in short responses as part of the answer.Communicates mathematical reasoning, results and conclusions. Shows working, including expressions entered into a calculator, e.g. when completing a sequence of compound interest calculations. Justifies results with a simple or routine statement to provide evidence when considering results of surveys.

D Identifies and organises relevant information. Interprets, with some inaccuracies, single relative terms, e.g. initial amount or how long, in a spreadsheet for a loan repayment. Chooses effective models and methods and carries the methods through correctly. Answers simple, structured questions that require short responses, e.g. using a set of diagrams to build a number pattern or converting single units from hours to minutes. Translates between representations when connecting the number of dots in a diagram to tables of values. Makes commonsense connections to calculate and apply everyday rates, e.g. the daily water consumption in a household. Obeys mathematical conventions and attends to accuracy. Applies basic conventions for diagrams, graphs and maps for simple problems, e.g. labels the angles measured in a bearings diagram. Enters numbers correctly into a calculator with single-step calculations.Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions, e.g. applying Pythagoras’ Theorem for given right triangles. In short responses, includes units in at least one part of the working. Communicates mathematical reasoning, results and conclusions. Shows working for single-step money calculations including expressions entered into a calculator. Converts units, e.g. litres (L) to cubic metres (m3). Includes appropriate symbols as part of the answer, e.g. %, $, =.


Grade descriptionsMathematics 2A/2B Stage 2


A Identifies and organises relevant information. Identifies and organises relevant information that is dense and scattered, e.g. interpreting word problems to set up algebraic equations; developing an extended table of values to look for number patterns; and drawing geometrical diagrams from descriptive passages. Chooses effective models and methods and carries the methods through correctly. Solves extended unstructured problems, e.g. adding the correct information to a given geometry diagram; choosing the correct information from a given network diagram; and recognising gradient and intercept as part of linear function modelling. Carries extended responses through, e.g. developing a diagram and using the result to solve related problems or using extended tables to find patterns both in the rows and columns. Generalises mathematical structures and solves linear equations with fractions in them. Makes counterintuitive connections by solving linear equations with multiple negatives in them. Determines the effects of changed conditions, e.g. recognising the effects of change of gradient or intercept on a linear graph or extracting correct information from a modified network diagram. Obeys mathematical conventions and attends to accuracy. Decides at which point to round in an extended response and determines the degree of accuracy based upon the context or units used. Defines the appropriate variables in word problems. Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions and attends to units in extended responses, e.g. taking distance travelled in kilometres and time taken in minutes and converting speed to metres per second. Communicates mathematical reasoning, results and conclusions. Shows the main steps in reasoning using appropriate mathematical language and terms of the related problem. Justifies reasoning in unpractised ways by stating properties that have been applied and meeting basic proof requirements, e.g. describing a line sufficiently to define it.

B Identifies and organises relevant information. Identifies and organises relevant information that is dense and scattered, e.g. accurately labelling 2-D diagrams or network diagrams with part information included; completing tables of values to look for patterns; and identifying the correct information from a given geometry diagram. Chooses effective models and methods and carries the methods through correctly. Recognises the correct function model and reads the correct values from a graph. Carries through and solves multi-step equations; extracts correct information from a network diagram; extracts correct information from an extended table of values. Generalises obvious mathematical structures, e.g. using interpolation/extrapolation appropriately in graphing, finding obvious number patterns in tables of values.Obeys mathematical conventions and attends to accuracy. Applies conventions for diagrams and graphs including accurately labelling angles and sides in an extended 2-D diagram. Rounds to specified accuracies, e.g. converts a fraction of an hour to minutes and seconds. Checks results and makes adjustments where necessary. Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions and attends to units in extended word problems, e.g. “Ben is twice as old as Mary” implies units are in years. Links and processes more than one piece of information which may be scattered. Communicates mathematical reasoning, results and conclusions. Shows the main steps in reasoning, e.g. when solving trig-ratio problems from set diagrams. Justifies conclusions with a simple or routine statement which links to results. Uses routine methods in labelling networks to show results and shows main steps in a proof requirement, e.g. using the Pythagoras’ Theorem to identify a Pythagorean triad.



C Identifies and organises relevant information. Identifies and organises relevant information that is grouped together or is relatively narrow in scope, e.g. makes direct substitution of values into linear equations; calculates successive terms using a recursive formula; recognises the trend in a scatter graph; and selects the correct sides when using trig-ratios on a diagram that is supplied. Chooses effective models and methods and carries the methods through correctly. Answers structured questions that require short responses, e.g. solves two-step linear equations; completes missing values in a table of a given function or graph. Applies mathematics in practised ways, e.g. chooses the correct trig-ratio for a supplied diagram or to solve a right-triangle problem; interprets trend lines in practised ways. Carries a single thread of reasoning through, e.g. uses the correct order of operation in short response equations; applies the correct rules to a network diagram. Calculates specific cases of generalisations, e.g. substitutes values into a given formulas such as trig-ratios, and evaluates the unknown angle or side. Makes commonsense connections, e.g. works forwards or backwards in a number sequence to arrive at a term. Uses a calculator appropriately for calculations observing Order of Operation rules and showing these steps as part of the working. Obeys mathematical conventions and attends to accuracy. Applies the rule of Order of Operations to equations, e.g. Pythagoras’ Theorem. Applies conventions for diagrams and labels sides and angles in geometry; recognises nodes and paths in a network diagram; sets up graphs neatly and accurately when reading values from them. Rounds to suit contexts, e.g. shows dollar calculations to two decimal places; rounds distance to a sensible level with travel questions. Accurately plots and labels given points on a Cartesian plane in short responses. Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions and attends to units in short responses, e.g. expresses the answer using the units defined in the question; and links trig-ratios to a diagram on a Cartesian plane. Communicates mathematical reasoning, results and conclusions. Shows working, including intermediate steps and/or expressions entered into a calculator, when setting out short responses. Uses the Left hand side = Right hand side convention properly in short responses and appropriate subscript notation in solutions with sequence questions.

D Identifies and organises relevant information. Identifies and organises relevant information that is grouped together and narrow in scope, e.g. plotting points on a Cartesian plane; and making single value substitutions in short responses. Chooses effective models and methods and carries the methods through correctly. Answers familiarly structured questions that require short responses, e.g. drawing single geometric figures by connecting points on a Cartesian plane. Substitutes into familiar linear equations to evaluate the subject of the equation. Makes single-step, commonsense connections to recognise the use of the constant in a linear equation. Uses a calculator to correctly complete single-step calculations or parts of multi-step calculations. Obeys mathematical conventions and attends to accuracy. Plots graphs with a poor degree of accuracy, or labels diagrams and given points with little detail. Rounds to suit contexts in short answer questions, but only when asked. Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions and attends to units in short responses only in familiar and practised questions. Communicates mathematical reasoning, results and conclusions. Shows working but only in familiar and practised contexts.


Grade descriptionsMathematics 2C/2D Stage 2


A Identifies and organises relevant information. Identifies and organises relevant information from dense and scattered information that involves a series of complex processes, e.g. connecting results of earlier parts of an extended question to use in an overall solution; and identifying relevant information in extended network diagrams. Chooses effective models and methods and carries the methods through correctly. Generalises unpractised mathematical structures to identify and use algebraic techniques or strategies that can simplify the working of problems. Carries deductive reasoning and extended responses through when moving from descriptive text to graphical representation and then to algebraic equation. Makes counterintuitive connections, e.g. when determining the effects of changed conditions in a network and its effect on unused capacity. Obeys mathematical conventions and attends to accuracy. Uses introduced variables, e.g. the value of a network path to define a cost or a distance. Uses inequality signs correctly when solving word problems involving linear relations.Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions in extended responses and calculates which parts of a network diagram would give the optimum result for given constraints. Communicates mathematical reasoning, results and conclusions. Shows main steps in reasoning when meeting basic proof requirements, e.g. showing explicitly why a value does not meet particular criteria. Uses the properties of a scatter diagram to justify conclusions concerning the relationship of the variables.

B Identifies and organises relevant information. Identifies and organises relevant information that is dense and scattered including: interpreting multiple relative terms (lowest, more than, etc.); setting up relevant inequations with word problems; and defining an appropriate sample space from a description that leaves room for ambiguity. Chooses effective models and methods and carries the methods through correctly. Solves unstructured problems using familiar sub-problems, e.g. picking the correct linear regression model from a scatter diagram. Carries familiar deductive reasoning and extended responses through; identifies routes or paths on a network and checks if they meet the given constraints. Uses a calculator appropriately for calculation, statistics, algebra and graphing. Checks accuracy, e.g. of a conjecture in number pattern investigation. Looks for counter examples in deductive reasoning. Obeys mathematical conventions and attends to accuracy. Applies conventions for diagrams and graphs and draws a non-routine sample space appropriate to the needs of a problem. Links mathematical results to data and contexts to reach reasonable conclusions. Attends to units in extended responses and is consistent with units in a network diagram, e.g. an answer 1.253 is given as 125 since the units on the network are in hundreds. Communicates mathematical reasoning, results and conclusions. Shows main steps in reasoning and sets out proof using the Left hand side = Right hand side convention. Justifies with a simple or routine statement and draws conclusions using mathematical terminology, such as: “A negative gradient implies...”



C Identifies and organises relevant information. Identifies and organises relevant information that is grouped together and is relatively narrow in scope e.g. recognising the pattern of a simple sequence; reading values from a scatter graph; and identifying maximal flow in a network diagram. Chooses effective models and methods and carries the methods through correctly. Answers structured questions that require short responses including predicting the dependent variable by using a trend line on a scatter diagram. Applies mathematical methods in practised ways, e.g. determines the shortest path (cheapest route) between two vertices of a network. Calculates the value of the term Tn

of a sequence for a given value for n when the rule is supplied; applies the area formula, e.g. Area Δ ABC = 1

2 × 7 × 6 × sin 23°

Makes commonsense connections, e.g. the use of place value to maximise the value of a set of three given digits such as 543 > 534. Uses a calculator appropriately for calculations, statistics and straightforward algebra and graphing, and gives evidence that the correct values or parameters have been used with the correct expression to produce the answer. Obeys mathematical conventions and attends to accuracy. Applies basic conventions for diagrams and graphs, e.g. copying the values from a network diagram onto a list to find the maximum flow or labelling histograms appropriately with the scales accurately marked. Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions and attends to units in short responses, e.g. rounding 125.3 people to 125 people since whole numbers are used in most contexts when counting people. Communicates mathematical reasoning, results and conclusions. Justifies with a simple or routine statement. Shows working including expressions, writes down the correct formula or expression used in a calculator and indicates that the numbers were correctly substituted.

D Identifies and organises relevant information. Identifies and organises relevant information that is grouped together and is narrow in scope e.g. reading points on a Cartesian plane or calculating the sum of paths of a network. Chooses effective models and methods and carries the methods through correctly. Answers familiar, structured questions that require short responses, e.g. making straight substitution into

linear functions and simplifying answers. Applies simple Index Laws, e.g. 2 3 5a a a× = ; makes direct

substitution into a simple term or expression and calculates an answer. Makes commonsense connections when interpreting the sample space of a spinner. Gives some evidence that the correct values/parameters have been entered into the calculator and that the correct expression has been used to produce the answer. Obeys mathematical conventions and attends to accuracy. Lacks accuracy or detail with the scale or does not label key features with graphs. Rounds only when specifically asked to in short answer questions and may truncate the number instead. Links mathematical results to data and contexts to reach reasonable conclusions. Ignores the units in both working and answer form, e.g. may not round to whole number form in capture/recapture method when calculating the number of fish. Communicates mathematical reasoning, results and conclusions. Shows appropriate working only in familiar and practised contexts.


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A Identifies and organises relevant information. Identifies and organises relevant information that is dense and scattered, e.g. complex problems that contain a series of steps or processes such as identifying and stating the equations or inequations which adequately represent quantities that vary in a particular problem. Identifies and organises relevant information from key elements in ambiguous data, including identifying a cycle in time series data that does not follow the usual pattern. Chooses effective models and methods and carries the methods through correctly. Carries an extended response through, e.g. correctly connects the many stages in both tabular and graphical form when dealing with time series data. Carries deductive reasoning through when describing the effect on the linear model of removing a single data point from a residual plot. Translates between representations in unpractised ways, e.g. determines the seasonal adjustment values from a graph rather than a table. Obeys mathematical conventions and attends to accuracy. Uses inequality signs correctly and attends to open or closed intervals. Rounds, unprompted, to suit contexts and specified accuracies in extended responses. Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions in extended responses, such as to define intervals of the domain for which certain conditions apply, e.g. ( ) ( ) 1.0051 4k x c x for x> < ≤ . Determines limitations of a model, e.g.

excludes a section of data before fitting the rest of the data to a function. Communicates mathematical reasoning, results and conclusions. Shows main steps in reasoning, e.g. when graphing two functions to find their points of intersection and defining the solution set for the required conditions.

B Identifies and organises relevant information. Identifies and organises relevant information that is dense and scattered when dealing with time series data. Chooses effective models and methods and carries the methods through correctly. Solves unstructured problems graphically, including simultaneous equations, and applies appropriate boundary conditions according to a word problem. Carries deductive reasoning through to make appropriate conjectures concerning trend lines. Combines interpretation of multiple representations when comparing the table, graph and regression equation to make a conjecture concerning the slope or trend.Obeys mathematical conventions and attends to accuracy. Rounds to the correct number of decimal places to suit contexts and specified accuracies in extended responses. Links mathematical results to data and contexts to reach reasonable conclusions. Interprets multi-dimension quantities, e.g. interpreting a gradient of a ‘fertility rate’ versus ‘time’ graph. Determines limitations of a model and mentions point outliers when fitting a function to the data of a scatter plot. Communicates mathematical reasoning, results and conclusions. Justifies working by stating properties that have been applied, e.g. explaining, using the pattern on a graph, why a 12-point moving average was used to determine a trend line.



C Identifies and organises relevant information. Identifies and organises relevant information that is grouped together and is narrow in scope, e.g. using the residual plot of a given model to determine the suitability of the model to fit a set of data. Picks the most likely point to have been plotted in the wrong place using a residual plot.Chooses effective models and methods and carries the methods through correctly. Answers structured questions that require short responses, e.g. sketches a given trend line on the same axes as a time-series plot. Uses a regression line to predict a single value for the dependent variable. Makes commonsense connections to describe what the patterns within a set of data infer. Applies mathematical methods in practised ways, e.g. calculating the moving average; using a calculator to produce a mean or standard deviation from a set of data; using the midpoints of intervals in a grouped data set to calculate a mean or standard deviation; or using a calculator to get a regression line from a given data set. Obeys mathematical conventions and attends to accuracy. Applies basic conventions for diagrams, graphs and maps when labelling axes and significant points on a sketch. Defines introduced variables, e.g. the independent variable (year number) and dependent variable (fertility rate) when rewriting a linear regression line in terms of the two variables on the scatter graph, i.e. fertility rate = 0.042 × year number + 0.9773. Rounds, unprompted, to suit contexts and specified accuracies in short responses, e.g. providing solutions involving money to two decimal places. Links mathematical results to data and contexts to reach reasonable conclusions. Recognises specified conditions in short responses, e.g. gives the average to the nearest unit since the data set was recorded to the nearest unit. Refers to units, e.g. ‘minutes’, in short responses when giving the answer for mean or standard deviation. Communicates mathematical reasoning, results and conclusions. Shows working, including the expressions entered into a calculator, e.g. when using a regression line to predict a value or when calculating a moving average. Justifies answers with a simple or routine statement, e.g. stating: “the residuals are more random”.

D Identifies and organises relevant information. Identifies and organises relevant information that is grouped together and is narrow in scope such as describing the trend of a graph in terms, e.g. “is increasing” or “is decreasing”. Fails to recognise that a question has more than one answer and misses other solutions. Chooses effective models and methods and carries the methods through correctly. Carries a single thread of reasoning through, e.g. decides whether a set of residual plots are randomly placed relative to the zero line. Obeys mathematical conventions and attends to accuracy. Enters data correctly into a calculator, e.g. uses single variable data to calculate statistics such as mean and standard deviation. Links mathematical results to data and contexts to reach reasonable conclusions. Refers to units on most occasions for problems that are in context but not usually involving statistics, e.g. mean or standard deviation. Communicates mathematical reasoning, results and conclusions. Shows working, including expressions entered into a calculator, e.g. uses a regression line to predict a value or calculates a moving average. Uses basic symbols and notation, e.g. , , $x≤ as part of the answer.




A Identifies and organises relevant information. Identifies and organises relevant information that is dense and scattered for complex problems involving a series of steps or processes, e.g. correctly identifies a problem as a specific case, e.g. identifies a binomial model and defines the relevant expression or recognises a representation of Bernoulli trials. Incorporates information that is needed to define equations from text and diagrams, e.g. in problems involving maximum volume or volume of a solid of revolution; or a descriptive passage containing information for calculating probability values involving events which are noted to have equally likely outcomes. Chooses effective models and methods and carries the methods through correctly. Chooses an effective method and correctly carries the method through in an extended response, e.g. uses a given probability density function to calculate a probability value. Generalises and extends models from previous part/s of the question, e.g. chooses the correct equations and correctly applies the derivative to find the optimum result. Obeys mathematical conventions and attends to accuracy. Obeys conventions and attends to accuracy when calculating compound probability, e.g. using a tree diagram or the multiplication principle to calculate a probability. Obeys mathematical conventions when using piecewise functions to define a probability density function. Links mathematical results to data and contexts to reach reasonable conclusions. Uses the second derivative of a function to determine the nature of the concavity of the graph and hence to locate points of inflection. Makes links between displacement, velocity and acceleration of a particle to determine the distance travelled in a given time period. Communicates mathematical reasoning, results and conclusions. Sets out mathematical reasoning, results and conclusions for the Normal approximation. Defines a uniform probability density function using a piecewise function over a given domain.

B Identifies and organises relevant information. Identifies and organises relevant information that is dense and scattered for less complex problems, such as those involving only a few steps or processes, e.g. variables identified in a given diagram or for a related rates problem; probability values from a descriptive passage; the definite integral used to determine the volume of a solid of revolution. Identifies variables from the written text of a problem involving linear programming and uses this information to define the inequalities. Chooses effective models and methods and carries the methods through correctly. Chooses an effective method or variables then correctly carries the method through for problems that contains only a few steps, e.g. optimises the objective function in a linear programming problem; uses the incremental formula to obtain an expression for percentage change; and defines the parameters of the Normal approximation and carries the correct calculation through. Obeys mathematical conventions and attends to accuracy. Obeys conventions and attends to accuracy, e.g. when using an increments formula; drawing the graphs of inequalities accurately; using the second derivative to check for maximum or minimum; with units involving related rates; and with respect to units when accurately applying the chain rule. Links mathematical results to data and contexts to reach reasonable conclusions. Uses the graph of the derivative function to determine the nature of the turning points of the original function. Links inappropriate data to mathematical results that are wrong and states that the increments are too large. Links changes to the objective function to the constraints, defines the new objective function and determines the new solution. Communicates mathematical reasoning, results and conclusions. Sets out mathematical reasoning, results and conclusions when working with extended optimisation problems. Proves differential equations with multiple terms or calculating related rates.



C Identifies and organises relevant information. Identifies and organises relevant information that is grouped together and is narrow in scope, e.g. the fundamental theorem as the derivative of an integral; the relevant definite integral to determine the volume of a solid of revolution; the intersection of two graphs; the elements of a simple sample space and the range of possible values; information in a probability density function, recognising that the sum of probabilities equals one. Chooses effective models and methods and carries the methods through correctly. Chooses an effective integral by defining the area between the graphs to be rotated, and correctly carries through, finding the volume of solid of revolution. Chooses an effective method and correctly evaluates the

parameter k defined in the differential equation dPkP

dt= .Chooses and uses the correct parameters when

calculating a normal probability value Pr(X≤x) given and μ σ .

Obeys mathematical conventions and attends to accuracy. Applies basic conventions for diagrams and graphs. Applies rules and checks for accuracy when evaluating integrals; and applies the chain rule correctly. Obeys conventions and uses correct notation, e.g. differentiation and probability distributions. Checks for accuracy of calculations, including those where technology is used, e.g. to evaluate an integral. Links mathematical results to data and contexts to reach reasonable conclusions. Uses the graph of the derivative function to locate the turning points of the original function. Links the first derivative of a displacement function to velocity, including the appropriate units, and relates acceleration using the second derivative. Recognises specified conditions in short responses and rejects solutions to optimisation problems because they are outside the domain, e.g. t<0. Communicates mathematical reasoning, results and conclusions. Communicates mathematically when naming probability distributions. Uses the multiplication principle when calculating arrangements. Uses the second derivative of a function to determine the nature of the turning points of a function.

D Identifies and organises relevant information. Identifies the elements of a simple sample space. Identifies the range of possible scores in a probability experiment. Identifies and organises the relevant definite integral, from a simple diagram, to use for the volume of a solid of revolution. Finds the intersection of two graphs. Calculates a simple probability value using a probability density function, and recognises that sum of probabilities equals one. Chooses effective models and methods and carries the methods through correctly. Applies mathematical methods, e.g. differentiation and integration, in practised ways. Interprets selections

in practised ways and calculates each selection nrC accurately. Calculates the probability of a discrete

random variable using a table of values. Obeys mathematical conventions and attends to accuracy. Uses technology to evaluate an integral but gives only the answer. Enters data correctly into a calculator but tends to give numerical answers without working. Links mathematical results to data and contexts to reach reasonable conclusions. Attends to units in short responses and rounds to suit the context when required, e.g. number of batteries expressed as a whole number. Communicates mathematical reasoning, results and conclusions. Uses basic symbols and notation, e.g. ≤ , x, $ .


mathematics syllabus for 2013 wace

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