MATHEMATICS: Specialist
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
IMPORTANT INFORMATION
Syllabus review
Once a course syllabus has been accredited by the Curriculum Council, the implementation of that syllabus will be monitored by the
syllabus committee. This committee can advise council 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 other
procedures that need to be read in conjunction with this course.
The Curriculum Council will support teachers in delivering the course by providing resources and professional development online.
The council website www.curriculum.wa.edu.au provides support materials including sample programs, assessment outlines,
assessment tasks, with marking keys, sample examinations with marking keys and grade descriptions with annotated student work
samples.
Training package support materials are developed by Registered Training Organisations (RTOs), government bodies and industry
training advisory bodies to support the implementation of industry training packages. Approved support materials are listed at
www.ntis.gov.au
WACE providers
Throughout this course booklet the term ‘school’ is intended to include both schools and other WACE providers.
Currency statement
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 Curriculum Council communication processes.
Copyright
© Curriculum Council, 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 Curriculum Council 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 Curriculum Council.
Copying or communication of any third party copyright material contained in this document can be done only within the terms of the Copyright Act or by permission of the copyright
owners.
2008/16171[v10]
2
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
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
Others have commented on the 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.
This Mathematics: Specialist 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, but this 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.
This course provides a solid foundation for the
many students who will continue their study of
mathematics beyond the compulsory years of
schooling. Students will already be familiar with the
importance of mathematics in their daily lives. In
this 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 also 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 investigate patterns and relationships,
draw inferences, make and test conjectures, and
convince others of their findings using mathematical
reasoning and proof. In this manner they
experience first-hand the creative and dynamic
aspects of mathematics, and they 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.
It should be emphasised that 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 is
so important.
Students studying Mathematics: Specialist will be
strongly advantaged by also studying Mathematics.
This course provides students with the opportunity
to further their achievement of specific overarching
learning outcomes from the Curriculum Framework
together with the development of the core-shared
values.
Course outcomes
The Mathematics: Specialist course is designed to
facilitate the achievement of two outcomes. These
outcomes are based on the Mathematics learning
area outcomes in the Curriculum Framework.
Outcomes are statements of what students should
know, understand, value and be able to do as a
result of the syllabus content taught.
Outcome 1: Functional relationships
Students use mathematical language and
processes to apply the concepts of function,
measurement and change to develop mathematical
models, solve practical problems, and explain and
justify relationships.
In achieving this outcome, students:
 understand
mathematical
concepts,
relationships and processes, by recognising
and making decisions about what mathematics
to use to represent information, solve problems
and investigate situations involving variation,
numerically, symbolically and graphically;
 use functional and numerical relationships, and
associated differentiation and integration; and
 apply concepts of function and change to
interpret findings, judge whether results are
realistic, ensure degrees of accuracy, evaluate
solutions
and
mathematically
justify
relationships
through
explanation
and
generalisation.
Outcome 2: Spatial relationships
Students use mathematical language and
processes to apply the concepts of space,
measurement and change to develop mathematical
models, solve practical problems, and explain and
justify relationships.
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
3
In achieving this outcome, students:
 understand
mathematical
concepts,
relationships and processes, recognising and
making decisions about what mathematics to
use to represent information, solve problems
and investigate situations involving spatial
relationships
and
change,
numerically,
symbolically and graphically;
 use vector and trigonometric relationships, and
associated differentiation and integration; and
use matrices to describe transformations; and
 apply concepts of spatial relationships and
change to interpret findings, judge whether
results are realistic, ensure degrees of
accuracy,
evaluate
solutions
and
mathematically justify relationships through
explanation and generalisation.
Outcome progressions
Each of the outcomes is described as a learning
progression across three broad levels (see
Appendix 1). In teaching a particular course unit,
teachers can use the outcome progressions along
with the unit content and contexts to:

plan appropriate lessons and activities for their
students, and

develop specific assessment tasks and
marking keys.
Course content
The course content needs to be the focus of the
learning program. It enables students to maximise
their achievement of both the overarching learning
outcomes from the Curriculum Framework and of
the course outcomes.
The course content is divided into three content
areas:
 concepts and relationships
 tools and procedures
 the practice of mathematics.
Concepts and relationships
Vectors
Vectors are mathematical tools used to describe
various types of physical quantities that cannot be
measured by a single number. They arise naturally
in two- and three-dimensional geometry, and as
descriptions of numerous aspects of the physical
world around us.
Trigonometry
Extending simple trigonometric ratios leads to the
concepts and power of trigonometric functions.
Their periodic properties as functions defined on the
entire real line make them ideally suited for
modelling periodic phenomena. They also have
useful applications within the general framework of
calculus, and interesting and powerful connections
with complex numbers.
4
Exponentials/logarithms
Exponential functions have an important role in the
study of growth and decay, the primary reason for
which is the close relationship between an
exponential function and its derivative. The
theoretical properties of exponential functions, and
their logarithmic inverses, are studied, and their
application as solutions of differential equations
describing population growth and decay are
explored in depth.
Functions
Functions and their graphs are powerful ways of
representing and describing relationships between
numerical quantities. They are also the starting
points for the calculus, which has fundamentally
changed the sciences, engineering and business
and our understanding of the world around us.
Several new types of functions are studied,
including exponentials and logarithms, and
trigonometric functions. By building up a supply of
functions with distinctive properties, good choices
can be made for the purposes of mathematical
modelling. In addition, a number of generic
properties of functions are investigated, including
an informal treatment of limits, rates of change, and
the effect of linear scale changes on their graphs.
Mathematical reasoning
Mathematical theories are developed using some
fundamental assumptions and by establishing
properties and relationships. Generating a proof is
an art and allows a creative development from
generalising patterns and making conjectures to
exploring forms of existing proof.
Complex numbers
Complex numbers draw together, in an elegant and
illuminating manner, key elements of number,
algebra,
equation
solving,
geometry
and
trigonometry. In doing so, it provides a satisfying
conclusion to several key elements of the school
mathematics curriculum.
Polar coordinates
Polar coordinates provide an alternative way of
specifying position in a plane, using the concepts of
distance and direction. They have strong links with
the vector concepts of magnitude and direction, and
they form the basis of polar or exponential
representations of complex numbers. These
alternatives enhance our understanding of key
elements of the course, and provide additional
problem-solving techniques. The use of technology
in this topic is commonsense and can contribute to
a new, and importantly, a different understanding,
with its direct appeal for visual learners.
Matrices
Matrices are introduced as arrays of coefficients of
linear equations. They have an algebra worthy of
study which provides a powerful way of handling
large systems of equations in many unknowns
when modelling the growth and decay of interacting
populations. The interpretation of matrices, and
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
their products, as transformations of various types
have applications in transformation geometry and in
trigonometry. The use of technology in this topic is
commonsense and can contribute to a new, and
importantly, a different understanding.
Tools and procedures
Forms and representations
The use of standard mathematical tools and
procedures is expected when a problem situation is
recognised as being of a particular mathematical
form, or is one which can be transformed to
facilitate analysis. The ability to make decisions is
important. Such decisions may concern whether a
numerical result should be presented exactly or
approximately; or about the extent to which a
symbolic result needs to be simplified or presented
as a specific case or as a general solution.
Mathematical results need to be interpreted and
presented in context, and in the appropriate units of
measure, of the original problem situation or task.
Sensible and sound judgements about the
reasonableness and accuracy of one’s own results
are
expected.
Standard
conventions
and
procedures need to be readily recalled, whilst
fluency is expected for frequently-used processes.
Algorithms
Computations involving number, 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 a suitable level of
precision or generality decided upon in either case;
tools and procedures are chosen to be consistent
with these decisions. Judgements are needed
regarding the reasonableness of results.
Technology
Technology of various kinds (spreadsheets,
graphics calculators, computer algebra systems,
dedicated and dynamic mathematics software,
interactive whiteboards and the internet) can
support the investigation, generation, creation and
exploration of mathematical ideas. When selected
for use, such technology should be used carefully
and frequently. Decisions about the appropriate
presentation of results are made. These decisions
affect the technology chosen and help to influence
its optimal use. The internet is an increasingly
important resource in accessing mathematically
significant information and visually-rich, dynamic
demonstrations of many ideas in this course.
The practice of mathematics
Working mathematically
The working mathematically content for this course
is embedded within the explanations in the outcome
progressions. These processes involve selecting
techniques, tools or skills appropriately and
understanding the consequences of using them.
Mathematics is recognised as useful because it can
be used to model real situations, but care is needed
to ensure that chosen models faithfully and
effectively represent the relevant aspects of the
context under investigation.
Appreciating mathematics
An understanding of the nature and power of
mathematics; how it is created, used and
communicated is a feature of this course. Doing
and applying mathematics involves observing,
representing, conjecturing and, in several
instances, justifying using various methods of
formal proof. Mathematics at this level is often an
intuitive and creative process. Conjectures, initially
tentative and error-prone, require rigorous
justification.
Communicating mathematics
The communication skills needed for the
mathematics in this course extend to interpreting
and writing concise mathematical notations and
multiple representations from a variety of
mathematical systems. For example, this is vividly
exemplified by the variety of representations of the
circular motion of a point through 90o anticlockwise
in two dimensions being appropriately represented,
depending on the mathematical context, either by:
 a basic triangular figure, diagrammatically
showing the point’s movement (relative to the
coordinate axes)
 a simple Cartesian coordinate relationship
x, y    y, x 



a transformation matrix
 x
0  1 
1 0  acting on  y 


a complex number operation (multiplication of
x  iy by i to give the result
a pair of differential equations x'   y, y'  x
where xt   cos t, yt   sin t
each describing the same motion.
Course units
All units are at Stage 3 and provide opportunities to
extend
knowledge
and understandings in
challenging academic learning contexts. The
content is notionally pitched at levels 6 to 8.
Unit 3AMAS
The focus for this unit is on representation and
students use a variety of forms. A strong distinction
is drawn between exact and approximate results
and their practical applications in particular contexts
when solving problems. Students use mathematical
models to understand situations defined in terms of
change. Mathematical reasoning is introduced and
used to establish laws and investigate functions.
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
5
Unit 3BMAS
Students explore new ways of expressing and
analysing change, including limiting behaviour and
continuity. Students establish and use properties to
develop deductive proofs. By building strong
algebraic skills to support mathematical arguments,
supplemented by the use of appropriate technology,
students investigate more complex models to solve
practical problems.
Unit 3CMAS
The focus for this unit is the abstract development
of a range of sophisticated relationships. Spatial
contexts are extended from two dimensions to three
dimensions. This unit develops abstraction as an
increasingly powerful way of expressing and
analysing change and introduces exhaustion and
contradiction as methods of proof to be explored.
Unit 3DMAS
The focus for this unit is on the use of differential
and integral calculus to understand a range of
phenomena. By increasing familiarity with
transformation and the use of matrices, students
can extend their theoretical understanding of growth
and decay models. This unit introduces
mathematical induction to complete the suite of
proof processes developed in mathematical
reasoning to a satisfactory, pre-tertiary level.
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.
A unit is completed when all assessment
requirements for that unit have been met. Only
completed units will be recorded on a student's
statement of results.
Refer to the WACE Manual for details about unit
completion and course completion.
6
Vocational Education
Training information
Vocational Education Training (VET) is nationally
recognised training that provides practical work skills
and credit towards, or attainment of, a vocational
education and training qualification.
When considering VET delivery in courses it is
necessary to:
 refer to the WACE Manual, Section 5: Vocational
Education 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 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.
VET delivery
VET can be delivered by schools providing they
meet Australian Quality Training Framework (AQTF)
requirements. Schools need to become a Registered
Training Organisation (RTO) or work in partnership
(auspicing 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. Qualifications identified in this course
must be on the scope of registration of the RTO
delivering or auspicing training.
Units of competency from related training package
qualifications have been considered during the
development of this course but no units of
competency have been suggested for integration.
Resources
Teacher support materials are available on the
Curriculum Council website extranet and can be
found at: http://www.curriculum.wa.edu.au/
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
Assessment
Refer to the WACE Manual for policy and principles
for
both
school-based
assessment
and
examinations.
School-based 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:
Specialist course. The table provides details of the
assessment types, including examples of different
ways that they can be applied and the weighting
range for each assessment type.
Teachers are to use the assessment table to
develop their own assessment outlines.
An assessment outline needs to be developed for
each class group enrolled in each unit of the course.
This outline includes a range of assessment tasks
that cover all assessment types and course
outcomes with specific weightings. If units are
delivered concurrently, assessment requirements
must still be met for each unit.
In developing assessment outlines and teaching
programs the following guidelines should be taken
into account.

All tasks should take into account teaching,
learning and assessment principles from the
Curriculum Framework.

There is flexibility within the assessment
framework for teachers to design school-based
assessment tasks to meet the learning needs
of students.

Student responses may be communicated in
any appropriate form e.g. written, oral,
graphical, multimedia or various combinations
of these.

Student work submitted to demonstrate
achievement of outcomes 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 should include
tasks conducted under test conditions.
Assessment table
Weightings for types
Stage 3
Type of assessment
Response
Students apply their understanding and skills in mathematics to analyse, interpret and respond to questions
and situations. This assessment type provides for the assessment of knowledge, conceptual understandings
and the use of algorithms.
75–85%
Written assessments, which may be done under timed conditions, require students to demonstrate use of
terminology, knowledge of factual information, understanding of concepts, use of algorithms and problemsolving skills.
Questions in this type of assessment can range from those of a routine nature to students, assessing lower
level concepts, through to open-ended questions that require responses at the highest level of achievement.
Evidence-gathering tools may include tests and examinations.
Best suited to the collection of evidence of student achievement of both outcomes.
Investigation
Students plan, research, conduct and communicate the findings of an investigation. They may investigate
problems to identify the underlying mathematics, or select, adapt and apply models and procedures to solve
problems. This assessment type provides for the assessment of general inquiry skills, course specific
knowledge and skills, and modelling skills.
Students may be given an investigation of a practical or theoretical situation involving mathematical concepts
and relationships, for which they need to generalise, construct proofs and make conjectures.
15–25%
Students may also be given unfamiliar situations for which they need to solve a problem, choosing and using
mathematical models with adaptations where necessary, comparing their solutions with the situations
concerned and then presenting their findings. Presentation of findings may be in written, oral or in multimedia
form, using appropriate conventions.
Evidence-gathering tools may include: lists, tables or diagrams used to organise thoughts and processes,
journals or reports, observation checklists, self or peer evaluation, posters or computer-based presentations,
projects, extended pieces of work, interviews or multimedia presentations.
Best suited to the collection of evidence of student achievement of both outcomes.
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
7
Grades
Schools assign grades following the completion of
the course unit. The following grades may be used:
Grade
A
B
C
D
E
Interpretation
Excellent achievement
High achievement
Satisfactory achievement
Limited achievement
Inadequate achievement
Time allowed
These examinations will require three hours in total,
including approximately 15 minutes changeover
period
Details of the examinations in this course are
prescribed in the examination design briefs (pages
17–21).
Preliminary Stage units are not graded.
Achievement in these units is reported as either
Completed or Not Completed.
Each grade is based on the student’s overall
performance for the course unit as judged by
reference to a set of pre-determined standards.
These standards are defined by grade descriptions.
Grade descriptions:
 describe the range of performances and
achievement characteristics of grades A, B, C, D
and E in a given stage of a course
 can be used at all stages of planning,
assessment and implementation of courses, but
are particularly important as a final point of
reference in assigning grades
 are subject to continuing review by the Council.
The grade descriptions for this course can be
accessed
on
the
course
page
at
http://www.curriculum.wa.edu.au/
Examination details
There are separate examinations for Stage 2 pairs
of units and Stage 3 pairs of units.
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) will sit an examination in
this course, unless they are exempt.
There will be two external examinations for the
Mathematics: Specialist course
 Units 3A/3B
 Units 3C/3D
Each examination will assess the specific content,
knowledge and skills described in the syllabus for
the pair of units studied.
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.
8
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
UNIT 3AMAS
2. Trigonometry (11 hours)
Using trigonometry to find distances and angles (in
degrees and radians) in geometric figures in two
and three dimensions is the focus of this section.
Unit description
Radians are introduced because of their direct
association with arc length.
2.1 establish the relationship between radian
measure and degree measure of angles and
convert from one measure to the other
2.2 determine arc lengths in circles, exactly and
approximately
2.3 establish and use the formula for the area of
The focus for this unit is on representation and
students use a variety of forms. A strong distinction
is drawn between exact and approximate results
and their practical applications in particular contexts
when solving problems. Students use mathematical
models to understand situations defined in terms of
change. Mathematical reasoning is introduced and
used to establish laws and investigate functions.
triangles area ΔABC =
2.4
Unit learning contexts
Within
the
broad
area
of
mathematical
relationships, teachers may choose a variety of
contexts appropriate for the age group, interests
and locality of the students.
2.5
2.6
Unit content
2.7
This unit includes the content areas:
 concepts and relationships
 tools and procedures
 the practice of mathematics
to the degree of complexity described below:
1. Vectors (11 hours)
This section is an introduction to vector terminology,
representation and methods based on coordinate
geometry and trigonometry.
1.1 distinguish between vector and scalar
quantities
1.2 represent a directed line segment in the
plane with magnitude and direction using
1.3
1.4
1.5
1.6
vector displacement notation AB or a
develop the concept of equality of vectors,
opposite vectors, unit vectors and the zero
vector
represent a vector as an ordered pair (a,b)
represent vectors in the form ai + bj, where i
and j are the standard unit vectors
establish
and
use
the
formula
a, b 
a 2  b2
for
the
magnitude
1.7
1.8
define the position vector OP , from the
origin, of a point P in the Cartesian plane
use the parallelogram law or triangle law of
vector addition and the triangle inequality
determine areas of sectors and segments in
circles using exact and approximate values
as appropriate
establish and use the sine and cosine rules
to find distances and angles in triangles in
two- and three-dimensional situations,
including obtuse triangles and those triangles
with two solutions (the ambiguous case)
use the triangle inequality for the lengths of
the sides of a triangle
solve practical problems including angles of
elevation
and
depression,
surveying,
bearings and navigation distances along
circles of constant latitude or constant
longitude on the surface of the Earth.
3. Exponentials and logarithms (13 hours)
This section reviews, consolidates and extends the
concepts of exponential and logarithmic functions,
their graphs, the index and logarithmic laws and
their application to solving simple equations as
preparation for the calculus of exponential and
logarithmic functions in Unit 3BMAS.
3.1 develop and use the index laws for positive
bases and rational exponents
3.2 establish and use the properties of
exponential functions y  Ca x a  0 and
3.3
draw their graphs
develop the inverse relationship between
logarithmic and exponential functions:
x  log a y and y  a x
3.4
or
modulus of a vector as its length in the plane
1
absin C
2
3.5
3.6
investigate and use the properties of the
logarithmic functions y  log x for a > 0, and
a
draw their graphs
use the laws of logarithms
solve practical growth and decay problems
using exponential and logarithmic functions.
ab  a  b
1.9
multiply a vector by a scalar and subdivide
line segments internally
1.10 represent relative displacement and relative
velocity as vectors.
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
9
4. Functions (13 hours)
The study of calculus begins with the basic concepts
of functions which are explored in detail. The
approach is informal and intuitive and the underlying
ideas are illustrated wherever possible by graphs
and sketches to provide understanding without
overwhelming with technical detail.
4.1 develop the concept of function composition
and obtain expressions for the composites of
simple functions
4.2 identify the domain and range of simple
functions and their composites
4.3 investigate the inverse of a function as a
reflection in y = x
4.4 investigate relationships between domains
and ranges of functions and their inverses
4.5 solve, algebraically and geometrically, simple
equations and inequalities involving absolute
values of linear functions
4.6 investigate the effects of varying a, b, c and d
on the graph of y  af(b(x  c))  d where
f  x  is an exponential, logarithmic, power,
reciprocal or absolute value function.
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:
Specialist 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.
Weighting
Stage 3
Response
Students apply their understanding and skills in
mathematics to analyse, interpret and respond to
questions and situations. This assessment type
provides for the assessment of knowledge,
conceptual understandings and the use of
algorithms.
75–85%
Written assessments, which may be done under
timed conditions, require students to demonstrate
use of terminology, knowledge of factual
information, understanding of concepts, use of
algorithms and problem-solving skills.
Questions in this type of assessment can range
from those of a routine nature to students,
assessing lower level concepts, through to openended questions that require responses at the
highest level of achievement.
5. Mathematical reasoning (3 hours)
Mathematical reasoning is an explicit focus
throughout the MAS units. Here, we examine
conjectures from number patterns and establish
laws and properties needed elsewhere in this unit.
5.1 identify and generalise number patterns for
powers,
exponential,
and
inverse
relationships
5.2 establish the laws of logarithms
5.3 investigate properties of the absolute value
function (real)—analytically and graphically.
Evidence-gathering tools may include tests and
examinations.
Best suited to the collection of evidence of student
achievement of both outcomes.
Investigation
Students plan, research, conduct and communicate
the findings of an investigation. They may
investigate problems to identify the underlying
mathematics, or select, adapt and apply models
and procedures to solve problems. This
assessment type provides for the assessment of
general inquiry skills, course specific knowledge
and skills, and modelling skills.
6. Complex numbers (0 hours)
Content for complex numbers is specified only for
units 3B–3DMAS.
7. Polar coordinates (2 hours)
Polar coordinates are a means of specifying
position in the plane by magnitude and direction.
7.1 develop the concept of polar coordinates (r,)
in the plane, where r  0
7.2 use the relationship between Cartesian and
polar coordinates in the plane to convert from
one system to the other.
Type of assessment
Students may be given an investigation of a
practical or theoretical situation involving
mathematical concepts and relationships, for which
they need to generalise, construct proofs and make
conjectures.
15–25%
Students may also be given unfamiliar situations for
which they need to solve a problem, choosing and
using mathematical models with adaptations where
necessary, comparing their solutions with the
situations concerned and then presenting their
findings. Presentation of findings may be in written,
oral or in multimedia form, using appropriate
conventions.
Evidence-gathering tools may include: lists, tables
or diagrams used to organise thoughts and
processes, journals or reports, observation
checklists, self- or peer evaluation, posters or
computer-based presentations, projects, extended
pieces of work, interviews or multimedia
presentations.
Best suited to the collection of evidence of student
achievement of both outcomes.
10
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
UNIT 3BMAS
Unit description
Students explore new ways of expressing and
analysing change, including limiting behaviour and
continuity. Students establish and use properties to
develop deductive proofs. By building strong
algebraic skills to support mathematical arguments,
supplemented by the use of appropriate technology
students investigate more complex models to solve
practical problems.
2. Trigonometry (10 hours)
A thorough understanding of the trigonometric
functions is an important foundation for the
successful study of mathematics at higher levels. In
this unit special emphasis is given to periodicity,
amplitude and phase, which are illustrated by
graphs wherever possible.
2.1 develop the concept of sine, cosine and
tangent as functions, and establish and use
the following properties:
Pythagorean: sin 2 x  cos 2 x  1
cos x   cos x
parity:
sin  x    sin x
tan  x    tan x
complementarity:
sin x  cos2  x 
Unit learning contexts
cos x  sin 2  x 
Within the broad area of mathematical change,
teachers may choose a variety of contexts
appropriate for the age group, interests and locality
of the students.
periodicity:
sin x  2   sin x
cosx  2   cos x
tan x     tan x
Unit content
This unit includes the content areas:
 concepts and relationships
 tools and procedures
 the practice of mathematics
to the degree of complexity described below:
1. Vectors (16 hours)
This section extends vector methods to include
vector representations of lines and the dot product.
1.1 develop the concept of the dot product of
vectors in a plane, using projections, and the
formula a  b  a1b1  a 2 b2 , and establish the
phase:
2.2
2.3
2.4
formula a  b  a b cos where a  a1 , a 2 
1.2
1.3
1.4
and b  b1 , b2 
calculate the angle between two vectors and
identify parallelism and perpendicularity
establish and use the vector equation of a
line in the plane in its various forms:
one point and the slope: r  r1  λl
two points: r  r1   r2  r1 
normal: r  n  c
establish and use the vector form of the
equation of a circle in the plane:
1.6
solve practical problems using vector
equations of lines and the dot product
including tangency and shortest distance
problems
solve practical problems using vectors
including the study of bearings, forces and
navigation problems involving apparent and
true velocities.
investigate the transformations of sine,
cosine and tangent functions such as
y = a sin b(x + c) + d and identify the effects
of the constants a, b, c and d on amplitude,
period, phase, and the locations of zeros and
turning points (see Unit 3AMAS 4.6)
use appropriate technology to investigate and
represent diagrammatically the roles of a, b, c
and d in the linear scale changes studied
in 2.2 above
use the addition and double angle formulas
for sine, cosine and tangent:
sin      sin  cos  cos sin 
cos     cos cos  sin  sin 
tan   tan 
1  tan  tan 
sin 2  2 sin  cos
cos 2  cos2   sin 2 
tan     
 2 cos2   1
 1  2 sin 2 
2 tan 
tan 2 
1  tan 2 
r d  ρ
1.5
sin x  cosx  2 , cos x  sin x  2 
2.5
solve trigonometric equations of the form
sin(ax) = k, cos(ax) = k and tan(ax) = k
for a given finite domain.
3. Exponentials and logarithms (8 hours)
In this section, limit concepts are introduced via an
important limit associated with the number e. This
is followed by the study of the natural exponential
and logarithm functions and differentiation of these
functions.
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
11
3.1
investigate the limiting behaviour
r n as n  , (r fixed and r  1)
3.2
investigate the limiting behaviour
a

1  
n

3.3
3.4
3.5
of
n
6.5
6.6
add, subtract, multiply and divide complex
numbers in Cartesian form
develop the concept of conjugates of
complex numbers.
as n   , (a fixed)
define e as the limit of  1  1 

n
n
as n  
investigate growth and decay problems of the
form y = a.ekx
differentiate exponential and logarithmic
functions including ef(x) and ln[f(x)].
4. Functions (12 hours)
An intuitive approach using the zoom facility of
calculators, numerical experimentation as well as
algebraic manipulation is still appropriate in this
section.
4.1 investigate the continuity and limiting
behaviour of functions
4.2 define the derivative of functions from first
principles and apply to familiar functions (not
trigonometric)
4.3 investigate the differentiability of functions
using limits
4.4 draw and interpret graphs of gradient
functions
4.5 investigate piecewise-defined functions and
continuity (including absolute value, the sign
function sgn [x], and the greatest integer
function int [x])
4.6 apply the chain rule with appropriate notation
to differentiate composite functions
4.7 use the product and quotient rules to
differentiate polynomial, exponential (base e)
and natural logarithmic functions
4.8 develop the concept of the integral of a
function as a limiting sum.
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:
Specialist 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.
Weighting
Stage 3
Response
Students apply their understanding and skills in
mathematics to analyse, interpret and respond to
questions and situations. This assessment type
provides for the assessment of knowledge,
conceptual understandings and the use of
algorithms.
75–85%
12
Written assessments, which may be done under
timed conditions, require students to demonstrate
use of terminology, knowledge of factual
information, understanding of concepts, use of
algorithms and problem-solving skills.
Questions in this type of assessment can range
from those of a routine nature to students,
assessing lower level concepts, through to openended questions that require responses at the
highest level of achievement.
Evidence-gathering tools may include tests and
examinations.
Best suited to the collection of evidence of student
achievement of both outcomes.
Investigation
Students plan, research, conduct and communicate
the findings of an investigation. They may
investigate problems to identify the underlying
mathematics, or select, adapt and apply models and
procedures to solve problems. This assessment
type provides for the assessment of general inquiry
skills, course specific knowledge and skills, and
modelling skills.
5 Mathematical reasoning (4 hours)
5.1 make conjectures regarding limiting patterns
5.2 establish the addition and double angle
formulas for sine, cosine and tangent
5.3 develop the chain rule for differentiating
composite functions
5.4 prove simple trigonometric identities by
deduction, using the properties listed in 2.1
and 2.4.
6. Complex numbers (5 hours)
Complex numbers have applications in many
branches of science and engineering. The study of
complex numbers enriches and unifies studies in
algebra, geometry, trigonometry and calculus.
6.1 define the number i as a solution of x2 = −1
6.2 investigate complex solutions of quadratic
equations
6.3 represent geometrically a complex number z
as a point in the complex plane
6.4 represent the Cartesian form of z as the sum
of its real and imaginary parts: z = a + bi,
where i2 = −1
Type of assessment
Students may be given an investigation of a
practical or theoretical situation involving
mathematical concepts and relationships, for which
they need to generalise, construct proofs and make
conjectures.
15–25%
Students may also be given unfamiliar situations for
which they need to solve a problem, choosing and
using mathematical models with adaptations where
necessary, comparing their solutions with the
situations concerned and then presenting their
findings. Presentation of findings may be in written,
oral or in multimedia form, using appropriate
conventions.
Evidence-gathering tools may include: lists, tables
or diagrams used to organise thoughts and
processes, journals or reports, observation
checklists, self- or peer evaluation, posters or
computer-based presentations, projects, extended
pieces of work, interviews or multimedia
presentations.
Best suited to the collection of evidence of student
achievement of both outcomes.
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
UNIT 3CMAS
Unit description
The focus for this unit is on the abstract
development of a range of sophisticated
relationships. Spatial contexts are extended from
two dimensions to three. This unit develops
abstraction as an increasingly powerful way of
expressing and analysing change. It introduces
exhaustion and contradiction as methods of proof to
be explored.
Unit learning contexts
Within
the
broad
area
of
mathematical
generalisation, teachers may choose a variety of
contexts appropriate for the age group, interests
and locality of the students.
Unit content
This unit includes the content areas:
 concepts and relationships
 tools and procedures
 the practice of mathematics
to the degree of complexity described below:
1 Vectors (12 hours)
The geometry of three-dimensional space is
conceptually more difficult than the geometry of the
plane. However, the transition from two to three
dimensions is facilitated by the vector approach.
In this section the similarities, rather than the
differences, between two- and three-dimensional
geometry are emphasised. The applications studied
here should have an emphasis on real life
situations.
1.1 review vector properties in 2D and extend
into 3D, namely: represent vectors in space
in Cartesian form as ordered triples (a,b,c)
1.2 develop the concept of displacement vectors
in space, including equality of vectors,
opposite vectors and the zero vector
1.3 establish
and
use
the
formula
a, b, c 
1.4
1.5
1.6
1.7
1.8
a 2  b 2  c 2 for the length of a
vector in space
represent the vector (a,b,c) in the form
ai + bj + ck, where i, j and k are the standard
unit vectors
develop the concept of the position vector of
a point in space
add vectors in space using the parallelogram
rule and addition of components
multiply vectors by scalars and extend this to
subdividing line segments internally
develop the concept of the dot product of
vectors in a plane, using projections, and the
a  b  a1b1  a2b2  a3b3
formula
and
establish the formula a  b  a b cos where
a  a1 , a 2 , a3  and b  b1 , b2 , b3  and  is
the angle between the vectors
1.9 calculate the angle between vectors and
identify parallelism and perpendicularity
1.10 establish and use the vector equation of a
plane in space in the form
r n c
or
r = a + λb + μc
together with its Cartesian and parametric
forms
1.11 establish and use the vector equation of a
line in space in the form r  r1  l together
with its parametric equivalent
1.12 solve practical problems in three-dimensional
geometry using vector concepts and
formulas, and graphical methods where
appropriate.
2. Trigonometry (6 hours)
Finding derivatives and integrals of elementary
functions, which partly commenced in units 3BMAT
and 3BMAS, is extended here to include a study of
the trigonometric functions. The starting point is the
basic trigonometric approximation sin x  x for
small x.
2.1
establish the limit sin x  1 as
x  0 using
x
2.2
inequalities, graphically and numerically
establish the limit 1  cos x  0 as x  0
x
2.3
2.4
determine the derivative of sin x, from first
principles
differentiate and integrate the sine, cosine
and tangent functions.
3. Exponentials and logarithms (7 hours)
This section reviews differentiation of exponential
and logarithmic functions and uses a more formal
approach to the integration and differentiation of
logarithmic and exponential functions examined.
3.1 review the inverse relationship between
exponentials and logarithms
3.2 investigate the logarithmic properties of the
x1
function 1 dt , define this as the natural
t
logarithm ln x and review its basic properties
3.3 use the change of base formula to convert
logarithms from one base to another
3.4
integrate functions of the form
kf x 
and
f x 
kf x e f  x  using the change of variable (or
substitution technique) either by observation,
or provided.
4. Functions (16 hours)
This section combines, reinforces and extends the
calculus techniques learnt so far and assumes
knowledge from Calculus in Units 3BMAT and
3CMAT. Appropriate applications are also studied.
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
13
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
determine the derivative of polynomial
functions from first principles
use the product, quotient and chain rules to
differentiate functions including exponential,
logarithmic and trigonometric functions
find the area under and between curves
determine the equation(s) of the tangent(s) to
a function
differentiate functions defined implicitly
solve practical problems involving parametric
and
differential
equations
(variables
separable)
integrate combinations of functions using
antiderivatives
integrate functions using the change of
variable or substitution technique (either by
observation, or provided).
5. Mathematical reasoning (5 hours)
This section affords students the opportunity to
study specific methods of proof and to develop an
understanding of some famous proofs.
5.1 make conjectures and generalisations about
properties of natural and figurate numbers
and recurrence relations
5.2 find
counter
examples
to
disprove
mathematical statements
5.3 distinguish between axioms and theorems
5.4 develop geometric proofs by deduction using
vector methods.
5.5 prove harder trigonometric identities by
deduction, using the properties in Unit
3BMAS points 2.1 and 2.4
5.6 explore proof by exhaustion
5.7 explore proof by contradiction including
Euclid’s proof of ‘infinitely many primes’.
7. Polar coordinates (2 hours)
Polar coordinates are a means of specifying
position in the plane by magnitude and direction.
These will recur in this unit within the vectors and
the complex numbers topics.
7.1 find the distance between points whose
position is expressed in polar form
7.2 draw and interpret polar graphs (including
inequalities) of r = constant, θ = constant and
r =k θ.
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:
Specialist 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.
Weighting
Stage 3
Response
Students apply their understanding and skills in
mathematics to analyse, interpret and respond to
questions and situations. This assessment type
provides for the assessment of knowledge, conceptual
understandings and the use of algorithms.
75–85%
6.4
Evidence-gathering tools may include tests and
examinations.
Best suited to the collection of evidence of student
achievement of both outcomes.
Investigation
Students plan, research, conduct and communicate
the findings of an investigation. They may investigate
problems to identify the underlying mathematics, or
select, adapt and apply models and procedures to
solve problems. This assessment type provides for the
assessment of general inquiry skills, course specific
knowledge and skills, and modelling skills.
determine the conjugate z of a complex
number z, expressed in Cartesian or polar
form, and locate it in the complex plane
establish algebraically and geometrically, the
conjugation properties:
Students may be given an investigation of a practical
or theoretical situation involving mathematical
concepts and relationships, for which they need to
generalise, construct proofs and make conjectures.
2
z z  z ; z1  z 2  z1  z 2 ; z1 z 2  z1 z 2
6.5
6.6
z
establish
z
2
as the reciprocal of a non-zero
complex number, z
describe regions in the complex plane and
Argand diagrams defined by means of simple
systems of equalities and inequalities.
Written assessments, which may be done under timed
conditions, require students to demonstrate use of
terminology, knowledge of factual information,
understanding of concepts, use of algorithms and
problem-solving skills.
Questions in this type of assessment can range from
those of a routine nature to students, assessing lower
level concepts, through to open-ended questions that
require responses at the highest level of achievement.
6. Complex numbers (7 hours)
This section extends the introduction to complex
numbers and their representation in Cartesian form
in Unit 3BMAS to include polar representation and
Argand diagrams.
6.1
express a complex number z in polar form:
z = r cis , where r = z and  = arg z
6.2
multiply and divide complex numbers
expressed in polar form
6.3
Type of assessment
15–25%
Students may also be given unfamiliar situations for
which they need to solve a problem, choosing and
using mathematical models with adaptations where
necessary, comparing their solutions with the
situations concerned and then presenting their
findings. Presentation of findings may be in written,
oral or in multimedia form, using appropriate
conventions.
Evidence-gathering tools may include: lists, tables or
diagrams used to organise thoughts and processes,
journals or reports, observation checklists, self- or peer
evaluation, posters or computer-based presentations,
projects, extended pieces of work, interviews or
multimedia presentations.
Best suited to the collection of evidence of student
achievement of both outcomes.
14
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
UNIT 3DMAS
1.8
1.9
Unit description
The focus for this unit is on the use of differential
and integral calculus to understand a range of
phenomena. By increasing familiarity with
transformation and the use of matrices, students
can extend their theoretical understanding of growth
and decay models. This unit introduces
mathematical induction to complete the suite of
proof processes developed in mathematical
reasoning, to a satisfactory, pre-tertiary level.
establish and apply the relationship between
the determinant and areas of shapes before
and after transformation
solve practical problems involving the use of
Leslie matrices and other examples of
transition matrices.
2. Trigonometry (4 hours)
This section requires the application of the
techniques of calculus to solve simple harmonic
motion problems.
2.1 investigate
the
differential
equation
d2y
 k2y  0
dt 2
and
its
solutions
y (t )  C cos kt  D sin kt
 A coskt  1 
Unit learning contexts
Within the broad area of mathematical modelling,
teachers may choose a variety of contexts
appropriate for the age group, interests and locality
of the students.
Unit content
This unit includes the content areas:
 concepts and relationships
 tools and procedures
 the practice of mathematics
to the degree of complexity described below:
1. Matrices (14 hours)
In this section, matrices are studied in their own
right and the basic properties of matrix algebra are
examined. The usefulness of matrices and matrix
algebra is illustrated by the study of linear
transformations in the plane, the solution of linear
equations and in the solving of practical problems
by the use of transition matrices.
1.1 add, subtract and multiply matrices (including
multiply by a scalar)
1.2 examine the algebraic properties of matrix
addition
and
multiplication,
including
commutativity for addition and not for
multiplication
1.3 examine the properties of special matrices:
identity, unit, singular, diagonal, row and
column matrices
1.4 calculate the determinant and inverse of a
2 × 2 matrix and recognise a singular 2 × 2
matrix
1.5 solve systems of up to five simultaneous
linear equations with no more than five
unknowns, using matrix algebra
1.6 examine the geometric properties of 2 × 2
matrices as linear transformations in the plane
including general rotations and reflections,
and dilations and shears parallel to the
coordinate axes
1.7 use matrix multiplication to determine the
combined effect of two linear transformations
in the plane
 A sinkt   2 
as models of simple harmonic motion.
3. Exponentials and logarithms (7 hours)
This section consolidates earlier work with
exponential and logarithmic functions and continues
with the solution of practical problems.
3.1 integrate linear combinations of powers and
exponentials
3.2 solve practical problems involving models of
dP
 kP
growth and decay of the form
dt
3.3 solve practical problems involving logarithmic
scales.
4. Function (10 hours)
This section combines, reinforces and extends the
calculus techniques and applications studied
previously; assuming knowledge from Calculus in
Unit 3DMAT.
4.1 investigate graphical, geometric and algebraic
properties of absolute value functions (in the
complex and Cartesian planes)
4.2 integrate functions and composite integrands
involving power, polynomial, exponential,
logarithmic and trigonometric functions
studied, using the change of variable (or
substitution technique) either by observation,
or as provided
4.3 solve related rates problems
4.4 solve practical problems by applying calculus
techniques to problems from various
branches of the sciences including rectilinear
motion, optimisation and marginal cost.
5. Mathematical reasoning (5 hours)
5.1 investigate
a
variety
of
traditional
mathematical
conjectures
including
Goldbach’s conjecture about two primes and
the twin prime conjecture
5.2 explore proof by induction including

de Moivre’s theorem: z cis 
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
n  z n cis n .
15
6. Complex numbers (12 hours)
This section is the culmination of the study of
complex numbers at the senior school level and
links algebraic, trigonometric and geometric ideas
studied in previous units.
6.1 establish properties of sums, products,
division and exponentiation (including
combinations of these) of complex numbers
and their conjugates (using real and
‘imaginary’ components)
6.2 use
de Moivre’s
theorem:
 z cis  n  z n cis n
6.3
6.4
to
establish
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:
Specialist 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.
Weighting
Stage 3
trigonometric relationships
find and locate in the complex plane, solutions
Response
Students apply their understanding and skills in
mathematics to analyse, interpret and respond to
questions and situations. This assessment type
provides for the assessment of knowledge,
conceptual understandings and the use of
algorithms.
of z n  C
establish the exponential properties of
cis  cos  i sin  and use Euler’s formula
e i  cos  i sin 
to
cis    , cis 0 and cis  n  .
Type of assessment
investigate
75–85%
Written assessments, which may be done under
timed conditions, require students to demonstrate
use of terminology, knowledge of factual
information, understanding of concepts, use of
algorithms and problem-solving skills.
Questions in this type of assessment can range
from those of a routine nature to students,
assessing lower level concepts, through to openended questions that require responses at the
highest level of achievement.
Evidence-gathering tools may include tests and
examinations.
Best suited to the collection of evidence of student
achievement of both outcomes.
Investigation
Students plan, research, conduct and communicate
the findings of an investigation. They may
investigate problems to identify the underlying
mathematics, or select, adapt and apply models
and procedures to solve problems. This
assessment type provides for the assessment of
general inquiry skills, course specific knowledge
and skills, and modelling skills.
Students may be given an investigation of a
practical or theoretical situation involving
mathematical concepts and relationships, for which
they need to generalise, construct proofs and make
conjectures.
15–25%
Students may also be given unfamiliar situations for
which they need to solve a problem, choosing and
using mathematical models with adaptations where
necessary, comparing their solutions with the
situations concerned and then presenting their
findings. Presentation of findings may be in written,
oral or in multimedia form, using appropriate
conventions.
Evidence-gathering tools may include: lists, tables
or diagrams used to organise thoughts and
processes, journals or reports, observation
checklists, self- or peer evaluation, posters or
computer-based presentations, projects, extended
pieces of work, interviews or multimedia
presentations.
Best suited to the collection of evidence of student
achievement of both outcomes.
16
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
Examination details
Stage 3
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
17
Mathematics: Specialist
Examination design brief
Stage 3—3A/3B
This examination consists of two sections.
Section One: Calculator-free
Time allowed
Reading time before commencing work:
Working time for section:
5 minutes
50 minutes
Permissible items
Standard items:
pens, pencils, pencil sharpener, eraser, correction fluid, ruler, highlighters
Special items:
nil
Changeover period – no student work:
approximately 10 minutes
Section Two: Calculator-assumed
Time allowed
Reading time before commencing work:
Working time for section:
10 minutes
100 minutes
Permissible items
Standard items:
pens, pencils, pencil sharpener, eraser, correction fluid, ruler, highlighters
Special items:
drawing instruments, templates, notes on two unfolded sheets of A4 paper, and up to
three calculators satisfying the conditions set by the Curriculum Council for this
examination
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.
Content area
Percentage of exam
Vectors
20–25%
Trigonometry
15–20%
Exponentials and logarithms
15–20%
Functions
20–25%
Mathematical reasoning
5–10%
Complex numbers and Polar coordinates
5–10%
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.
18
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
Section
Section One
Calculator-free
40 marks
5–10 questions
Working time: 50 minutes
Supporting information
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
80 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.
Working time: 100 minutes
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.
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
19
Mathematics: Specialist
Examination design brief
Stage 3—3C/3D
This examination consists of two sections.
Section One: Calculator-free
Time allowed
Reading time before commencing work:
Working time for section:
5 minutes
50 minutes
Permissible items
Standard items:
pens, pencils, pencil sharpener, eraser, correction fluid, ruler, highlighters
Special items:
nil
Changeover period – no student work:
approximately 10 minutes
Section Two: Calculator-assumed
Time allowed
Reading time before commencing work:
Working time for section:
10 minutes
100 minutes
Permissible items
Standard items:
pens, pencils, pencil sharpener, eraser, correction fluid, ruler, highlighters
Special items:
drawing instruments, templates, notes on two unfolded sheets of A4 paper, and up to
three calculators satisfying the conditions set by the Curriculum Council for this
examination
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.
Content area
Percentage of exam
Matrices
10–15%
Vectors
10–15%
Trigonometry
5–10%
Exponentials and logarithms
10–15%
Functions
20–25%
Mathematical reasoning
5–10%
Complex numbers and Polar coordinates
15–20%
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.
20
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
Section
Section One
Calculator-free
40 marks
5–10 questions
Working time: 50 minutes
Supporting information
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
80 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.
Working time: 100 minutes
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.
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
21
22
Mathematics: Specialist: Accredited March 2008 (updated June 2010)
For teaching 2011, examined in 2011
Appendix 1: Outcome progressions
Mathematics: Specialist: Accredited March 2008 (updated June 2010) Appendix 1
For teaching 2011, examined in 2011
Outcome progressions
Outcome 1: Functional relationships
Students use mathematical language and processes to apply the concepts of function, measurement and change to develop mathematical models, solve
practical problems, and explain and justify relationships.
Level 6
Level 7
Level 8
Students investigate, understand and apply inequalities,
functions and relationships to model practical situations in
numeric, symbolic and graphical forms; understand and use
indices, rates of change and the basics of complex numbers;
and interpret findings in context.
Students investigate, understand and apply linear
inequalities, exponential, piecewise and periodic functions
and relationships to model practical situations in numeric,
symbolic and graphical forms and the transformation effect
of constants in an expression; understand and use complex
numbers in polar and coordinate forms, matrix arithmetic and
the relationship between indices and logarithms; use
differentiation and integration to model change, relating it to
graphical features; evaluate limits through an iterative
approach; and interpret findings in context.
Students investigate, understand and apply techniques of
differential and integral calculus to model and analyse
situations; understand the relationships between different
processes and representations and know when each is
appropriate to use; evaluate solutions; and provide
mathematically rigorous justifications of relationships; and
interpret findings in context.
 understand mathematical
concepts, relationships and
processes by recognising and
making decisions about what
mathematics to use to
represent information, solve
problems and investigate
situations involving variation,
numerically, symbolically and
graphically.
 investigate occurrence of complex numbers; investigate
 investigate representations of complex numbers;

investigate relationships between functions, their inverses
and their transformations; investigate the behaviour of
functions using calculus techniques; use understandings
of number, functions and calculus concepts to recognise
their application in practical situations and formulate
algebraic representations autonomously; recognise when
changing the form or representation of an algebraic or
numeric expression will facilitate solution of a problem,
including the use of matrix representations; make
decisions about processes based on context and
accuracy; and adjust graphics calculator display(s) based
on understanding of functions to ensure complete
visualisation and accuracy of information.
 use functional and numerical
relationships, and associated
differentiation and integration.
 use graphical methods to determine feasible regions for

work flexibly with complex numbers in a variety of
representations to solve problems; prove (by induction)
de Moivre’s theorem and use it to prove trigonometric
relationships; establish and use matrix equations to solve
systems of simultaneous linear equations; use logarithms
and exponentials to solve problems, including growth and
decay; understand and use relationships between
inverse functions; use first principles to find derivatives of
functions using limits; apply calculus concepts and
processes flexibly and autonomously to solve practical
applications, including the use of related rates, and use
analytic processes to determine limits of functions.
 apply concepts of function and
change to interpret findings,
judge whether results are
realistic, ensure degrees of
accuracy, evaluate solutions
and mathematically justify
relationships through
explanation and
generalisation.
 apply linear, quadratic and basic exponential concepts to

apply complex numbers and linear, quadratic, polynomial,
exponential, reciprocal, piecewise and periodic functions
to interpret answers in context, addressing accuracy and
appropriateness of answers; make connections between
different representations; generalise relationships and
provide mathematical justification through formal
deductive and inductive processes; intuitively detect
errors in calculations and processes from the final
answer; and use alternative methods to check solutions.
Students:
ways that two quantities vary with each other through
numeric, symbolic, and graphical approaches with
prompting; use understandings of exponential
relationships of the form y=a.bx to recognise when they
apply in practical situations; make decisions about
required accuracy of answers and appropriate units based
on given information; and adjust graphics calculator
display(s) to ensure complete visualisation and accuracy
of information.
simultaneous linear inequalities; use understanding of the
forms of equations to establish requirements for non-real
number solutions; use slope as a measure of rate of
change of one variable with respect to another; work
flexibly with rates expressed in familiar units; use
exponential functions of the form y = a.bx to solve
problems; use the inverse relationship y = ax and
x = logay; and use the index laws to work flexibly with
algebraic expressions.
interpret findings in context, addressing accuracy in light of
provided information and appropriateness of answers;
make connections between different representations;
generalise relationships based on common features and
structures of patterns; and use alternative methods when
prompted to check solutions.
investigate properties of piecewise and periodic functions
through a variety of approaches; use understandings of
complex numbers and linear, quadratic, exponential (of
the form y = a.ebx ) piecewise and periodic relationships
to recognise when they apply in practical situations; use
understanding of calculus concepts to recognise their
application in practical problems; make decisions about
required accuracy of answers and appropriate units based
on given information and processes followed, including
using exact values; and reflectively adjust calculator
display(s) to ensure complete visualisation and accuracy
of information.
 use complex numbers in coordinate and polar forms,
convert between them, and represent them graphically;
use understanding of index laws to establish and use laws
of logarithms to solve problems; perform operations on
matrices such as addition, multiplication and inversion;
use piecewise and periodic functions numerically,
analytically and graphically, showing understanding of
transformation effects of constants in the function;
differentiate and integrate using power, exponential and
logarithmic functions; use understanding of calculus to
determine local and global features of graphs; establish
limits through iterative processes; formulate expressions
for instantaneous rates of change and calculate values in
familiar and unfamiliar rates; and determine equations of
tangents to curves.
 apply understanding of basic complex numbers, indices,
logarithms, matrices, and piecewise and periodic
functions to interpret findings in context, addressing
accuracy in light of provided information and processes
followed, and appropriateness of answers; make
connections between different representations; generalise
relationships and explain why generalisations are true;
and use alternative methods to check solutions.
Mathematics: Specialist: Accredited March 2008 (updated June 2010) Appendix 1
For teaching 2011, examined in 2011
Outcome progressions
Outcome 2: Spatial relationships
Students use mathematical language and processes to apply the concepts of space, measurement and change to develop mathematical models, solve
practical problems, and explain and justify relationships.
Level 6
Level 7
Level 8
Students investigate, understand and apply trigonometric
and algebraic concepts to model and solve practical
problems in two-dimensional space; and understand and
use graphical transformation and polar coordinates.
Students investigate, understand and apply trigonometric
and algebraic concepts to model, and solve practical
problems in three-dimensional space; combine spatial
and measurement techniques; and use two-dimensional
vectors and matrices.
Students investigate and apply their understanding of
vectors and matrices to draw flexibly upon and make
connections between results about location and
transformation when solving analytical and practical
problems; and justify the results.
understand mathematical
concepts, relationships and
processes, recognising and
making decisions about
what mathematics to use to
represent information, solve
problems and investigate
situations involving spatial
relationships and change,
numerically, symbolically
and graphically.

investigate situations in two-dimensional space involving
trigonometric processes; and make decisions about
required accuracy of answers and units based on given
information.

investigate situations in dimensional space involving
right and non-right triangles; and understand the
possibility of error and make decisions about exact
solutions and precision.


choose polar or rectangular coordinate representations
appropriate to the use and context.

use the determinant to describe the effect on a figure
undergoing transformation.
use matrices to transform points and curves on the
coordinate plane; and use matrix multiplication and
matrix inverse to describe the effect on a figure
undergoing transformation. Investigate Leslie and other
transition matrices in context. Investigate algebraic
expressions and rearrange to facilitate computation,
including matrix methods; formulate algebraic
expressions from practical situations; break down threedimensional related rates problems into two-dimensional
steps.

use vector and trigonometric
relationships, and
associated differentiation
and integration; and use
matrices to describe
transformations.

work flexibly with radian measure and polar coordinates;
use Pythagoras’ theorem, right triangle trigonometry,
establish and use sine and cosine rules to solve right
and non-right triangle problems; deduce and use
½absinC to find the area of a triangle; find areas of
sectors and segments; solve problems involving arc
length, latitude and longitude; sketch two-dimensional
problems to model situations; and graph transformations
from written descriptions in four quadrants based on
translation, rotation, reflection and/or dilation.

use basic vectors in two-dimensional space to determine
resultant, unit, magnitude, direction and midpoint in
practical situations; represent circles with vectors; use
right and non-right triangle trigonometry to solve
problems in dimensional space; use two-dimensional
sketches to model problems; differentiate and integrate
trigonometric functions and apply concepts; use
matrices to transform points and curves on the
coordinate plane; and use the determinant and matrix
multiplication to describe the effect on a figure
undergoing transformation.

work flexibly with vectors; use vector properties to
describe the position of moving objects; use vectors to
find the closest position and collision point of two moving
objects; produce simple deductive proofs using vectors;
combine understandings of space, measurement and
algebra to solve problems using calculus; make
sketches to represent related rates problems to facilitate
solution; determine limits of trigonometric functions; and
transform shapes or describe transformations in four
quadrants from functional information or matrices,
including shears.

apply concepts of spatial
relationships and change to
interpret findings, judge
whether results are realistic,
ensure degrees of accuracy,
evaluate solutions and
mathematically justify
relationships through
explanation and
generalisation.

interpret findings in context and accuracy of solutions in
light of correct processes followed and accuracy of given
information; generalise relationships between concepts;
make connections between visual and algebraic
representations; use alternative methods to check
solutions; make generalisations from common features
based on understanding of relationships or structure of
patterns and testing additional cases; and recognise use
of counter example to disprove assertion.

use calculators efficiently for accounting for the effects
of rounding/truncating on accuracy; extend
generalisation from several trials to an algorithm or rule,
or verification of provided proof by use of examples and
by following process; and use alternative methods to
check solutions.

choose to work with exact answers where appropriate;
prove by deductive or inductive processes, identifying
errors in logic, and distinguishing general arguments
from those based on specific cases; and intuitively
detect errors in calculations and processes from final
solutions and use alternative methods to check
solutions.
Students:

Mathematics: Specialist: Accredited March 2008 (updated June 2010) Appendix 1
For teaching 2011, examined in 2011