Board Endorsed October 2015
Specialist Mathematics
integrating International Baccalaureate &
Australian Curriculum
T
Type 2
Written under the
Mathematics Course
Framework 2013
Accredited from 2016 – 2020
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Board Endorsed October 2015
Student Capabilities
All courses of study for the ACT Year 12 Certificate should enable students to develop essential
capabilities for twenty-first century learners. These ‘capabilities’ comprise an integrated and
interconnected set of knowledge, skills, behaviours and dispositions that students develop and use
in their learning across the curriculum.
The capabilities include:
 Literacy
 Numeracy
 Information and communication technology (ICT) capability
 Critical and creative thinking
 Personal and social capability
 Ethical behaviour
 Intercultural understanding.
Courses of study for the ACT Year 12 Certificate should be both relevant to the lives of students and
incorporate the contemporary issues they face. Hence, courses address the following three
priorities:
 Aboriginal and Torres Strait Islander histories and cultures
 Asia and Australia’s engagement with Asia
 Sustainability.
Elaboration of these student capabilities and priorities is available on the ACARA website at:
www.australiancurriculum.edu.au.
This course has been written for those schools that offer the International Baccalaureate in addition
to the ACT Year 12 Certificate. For this reason, implementation of this course should encompass the
IB learner profile.
Figure 1 - IBO, (2015), Learner Profile [ONLINE]. Available at: http://www.ibo.org [Accessed 29/07/2015].
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Board Endorsed October 2015
Course Adoption Form for Accredited Courses
B S S S
AUSTRALIAN CAPITAL TERRITORY
College:
Course Title: Specialist Mathematics
(integrating International Baccalaureate & integrating
Australian Curriculum)
Classification: T
Framework: Mathematics
Course Area:
Course Code:
Dates of Course Accreditation:
From
to
2016
2020
Identify units to be adopted by ticking the check boxes
Adopt

Unit Title
Unit 1: Specialist Mathematics
Value
(1.0/0.5)
Length
1.0
S

Unit 1a: Specialist Mathematics
0.5
Q

Unit 1b: Specialist Mathematics
0.5
Q
1.0
S

Unit 2: Specialist Mathematics

Unit 2a: Specialist Mathematics
0.5
Q

Unit 2b: Specialist Mathematics
0.5
Q
1.0
S

Unit 3: Specialist Mathematics

Unit 3a: Specialist Mathematics
0.5
Q

Unit 3b: Specialist Mathematics
0.5
Q
1.0
S

Unit 4 : Specialist Mathematics

Unit 4a: Specialist Mathematics
0.5
Q

Unit 4b: Specialist Mathematics
0.5
Q
1.0
S

Unit 5: Specialist Mathematics

Unit 5a: Specialist Mathematics
0.5
Q

Unit 5b: Specialist Mathematics
0.5
Q
1.0
S

Unit 6: Specialist Mathematics

Unit 6a: Specialist Mathematics
0.5
Q

Unit 6b: Specialist Mathematics
0.5
Q
1.0
S

Unit 7: Specialist Mathematics

Unit 7a: Specialist Mathematics
0.5
Q

Unit 7b: Specialist Mathematics
0.5
Q
1.0
S

Unit 8 : Specialist Mathematics
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Board Endorsed October 2015

Unit 8a: Specialist Mathematics
0.5
Q

Unit 8b: Specialist Mathematics
0.5
Q
Adoption The course and units named above are consistent with the philosophy and goals of the
college and the adopting college has the human and physical resources to implement the course.
Principal:
/
/20
BSSS Office Use
Entered into database: /
College Board Chair:
/20
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/
/20
Board Endorsed October 2015
Table of Contents
Course Adoption Form for Accredited Courses........................................................................................3
Course Name
……………………………………………………………………………………..6
Course Classification
……………………………………………………………………………………..6
Course Framework
……………………………………………………………………………………..6
Course Developers
……………………………………………………………………………………..6
Evaluation of Previous Course
……………………………………………………………………………………..6
Course Length and Composition
……………………………………………………………………………………..6
Implementation Guidelines
……………………………………………………………………………………..7
Subject Rationale
……………………………………………………………………………………8
Goals
……………………………………………………………………………………8
Content
……………………………………………………………………………………9
Teaching and Learning Strategies
……………………………………………………………………………………11
Assessment
……………………………………………………………………………………12
Achievement Standards
……………………………………………………………………………………14
Student Capabilities
……………………………………………………………………………………16
Representation of Cross-curriculum Prioritie ..………………………………………………………………………………..17
International Baccalaureate Learner Profile ……………………………………………………………………………………18
Moderation
……………………………………………………………………………………19
Resources
……………………………………………………………………………………20
Proposed Evaluation Procedures
……………………………………………………………………………………21
Unit 1: Specialist Mathematics IB T
Value: 1.0 …………………………………………………………………..22
Unit 2: Specialist Mathematics IB T
Value: 1.0 …………………………………………………………………..26
Unit 3: Specialist Mathematics IB T
Value: 1.0 …………………………………………………………………..30
Unit 4: Specialist Mathematics T
Value: 1.0 …………………………………………………………………..33
Unit 5: Specialist Mathematics T
Value: 1.0 …………………………………………………………………..36
Unit 6: Specialist Mathematics T
Value: 1.0 …………………………………………………………………..39
Unit 7: Specialist Mathematics T
Value: 1.0 …………………………………………………………………..42
Unit 8: Specialist Mathematics T
Value: 1.0
…………………………………………………………………..459
Appendix B – IB DP and AC mapping document ....................................................................................51
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Board Endorsed October 2015
Course Name
Specialist Mathematics (integrating the International Baccalaureate and Australian Curriculum)
Course Classification
T
Course Framework
Mathematics
Course Developers
Name
College
Taneile Harris
Narrabundah College
Nancy Lee
Canberra Girls’ Grammar School
Alice Wann
Melba Copland Secondary School
Jacob Woolley
Canberra College
Evaluation of Previous Course
This Specialist Mathematics course integrates the International Baccalaureate and the Australian
Curriculum.
Course Length and Composition
The following standard units will be the usual mode of delivery. Half standard units 0.5 (‘a’ and ‘b’)
are available.
Unit Titles
Unit Value
Unit 1: Specialist Mathematics
1.0
Unit 2: Specialist Mathematics
1.0
Unit 3: Specialist Mathematics
1.0
Unit 4: Specialist Mathematics
1.0
Unit 5: Specialist Mathematics
1.0
Unit 6: Specialist Mathematics
1.0
Unit 7: Specialist Mathematics
1.0
Unit 8: Specialist Mathematics
1.0
Available course pattern
A standard 1.0 value unit is delivered over at least 55 hours and can be as long as 63 hours. To be
awarded a course, students must complete at least the minimum number of hours and units over
the whole minor or major – both requirements must be met. The number of units may vary
according to the school timetable.
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Board Endorsed October 2015
Course
Number of standard units to meet course requirements
Minor
Minimum of 2 units
Major
Minimum of 3.5 units
Major Minor
Minimum of 5.5 units
Double Major
Minimum of 7 units
Implementation Guidelines
Suggested Implementation Patterns
Implementation Pattern
Units
Semester 1, Year 11
Unit 1: Specialist Mathematics IB/AC
Semester 2 , Year 11
Unit 5: Specialist Mathematics IB/AC
Unit 2: Specialist Mathematics IB/AC
Unit 6: Specialist Mathematics IB/AC
Semester 1, Year 12
Unit 3: Specialist Mathematics IB/AC
Unit 7: Specialist Mathematics IB/AC
Semester 2, Year 12
Unit 4: Specialist Mathematics IB/AC
Unit 8: Specialist Mathematics IB/AC
Note: Units 5 - 8 may be studied in any order.
Prerequisites for the course or units within the course
It is recommended that the sequential units 1-4 are studied concurrently with units 5-8.
Students may change from the Mathematical Methods course integrating Australian Curriculum to
the Specialist Mathematics course integrating AC/IB course at the discretion of the executive
teacher, taking into account there should be no duplication of content.
Students wishing to change to the Mathematical Methods course from the Specialist Mathematics
integrating Australian Curriculum and International Baccalaureate course may opt to change at the
completion of Units 1a, 1b or 2a only. Students may include these units into a Mathematical
Methods T course integrating Australian Curriculum. Students may not change to the Mathematical
Methods course if they have completed study in Units 2b, 3, or 4 due to duplication of content.
Compulsory units
A major in Specialist Mathematics course integrating IB and AC must include Units 1, 2 and 3.
Duplication of Content Rules
Students cannot be given credit towards the requirements for a Year 12 Certificate for a unit that
significantly duplicates content in a unit studied in another course. The responsibility for preventing
undesirable overlap of content studied by a student rests with the principal and the teacher
delivering the course. Substantial overlap of content is not permitted and students will only be given
credit for covering the content once.
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Board Endorsed October 2015
Subject Rationale
Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it
has evolved in highly sophisticated and elegant ways to become the language now used to describe
much of the modern world. Statistics is concerned with collecting, analysing, modelling and
interpreting data in order to investigate and understand real world phenomena and solve problems
in context. Together, mathematics and statistics provide a framework for thinking and a means of
communication that is powerful, logical, concise and precise.
Because both mathematics and statistics are widely applicable as models of the world around us,
there is ample opportunity for problem solving throughout Specialist Mathematics integrating AC/IB.
There is also a sound logical basis to this subject, and in mastering the subject students will develop
logical reasoning skills to a high level.
Specialist Mathematics course integrating AC/IB provides opportunities to develop rigorous
mathematical arguments and proofs, and to use mathematical and statistical models more
extensively. Topics are developed systematically and lay the foundations for future studies in
quantitative subjects in a coherent and structured fashion. Students of Specialist Mathematics course
integrating AC/IB will be able to appreciate the true nature of mathematics, its beauty and its
functionality.
Specialist Mathematics course integrating AC/IB is designed for students with a strong interest in
mathematics, including those intending to study mathematics, statistics, all sciences and associated
fields, economics or engineering at university.
For all content areas of Specialist Mathematics integrating AC/IB, the proficiency strands of the F–10
curriculum are still applicable and should be inherent in students’ learning of the subject. These
strands are Understanding, Fluency, Problem solving and Reasoning and they are both essential and
mutually reinforcing. In Specialist Mathematics integrating AC/IB, the formal explanation of
reasoning through mathematical proof takes on an important role and the ability to present the
solution of any problem in a logical and clear manner is of paramount importance. Theory of
knowledge identifies ways of knowing and logic and proof are the basis on which all mathematics is
built. The relationship between theory of knowledge and mathematics is a strong and important one,
and students’ attention should be drawn to questions relating to theory of knowledge and
mathematics and they should be encouraged to raise questions themselves. Mathematics has
provided important knowledge about the world, and the use of mathematics in science and
technology has been one of the driving forces for scientific advances. The ability to transfer skills
learned to solve one class of problems, for example integration, to solve another class of problems,
such as those in biology, kinematics or statistics, is a vital part of mathematics learning in this subject.
Goals
Specialist Mathematics integrating AC/IB aims to develop students’:
 understanding of concepts and techniques drawn from functions, algebra, trigonometry,
calculus, discrete maths, vectors, logic and proofs, probability and statistics
 ability to solve applied problems using concepts and techniques drawn from functions,
algebra, trigonometry, calculus, discrete maths, vectors, logic and proofs, probability and
statistics
 reasoning in mathematical and statistical contexts and interpretation of mathematical and
statistical information including ascertaining the reasonableness of solutions to problems
 capacity to communicate in a concise and systematic manner using appropriate mathematical
and statistical language
 capacity to choose and use technology appropriately and efficiently.
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Board Endorsed October 2015
Student Group
Links to Foundation to Year 10
For all content areas of Specialist Mathematics IB/AC, the proficiency strands of the F–10 curriculum
are still very much applicable and should be inherent in students’ learning of the subject. The strands
of Understanding, Fluency, Problem solving and Reasoning are essential and mutually reinforcing. For
all content areas, practice allows students to achieve fluency in skills, such as finding the scalar
product of two vectors, or finding the area of a region contained between curves. Achieving fluency
in skills such as these allows students to concentrate on more complex aspects of problem solving. In
Specialist Mathematics IB/AC, the formal explanation of reasoning through mathematical proof takes
an important role, and the ability to present the solution of any problem in a logical and clear
manner is of paramount significance. The ability to transfer skills learned to solve one class of
problems, such as integration, to solve another class of problems, such as those in biology,
kinematics or statistics, is a vital part of mathematics learning in this subject.
In order to study Specialist Mathematics IB/AC, it is desirable that students complete topics from
10A. The knowledge and skills from the following content descriptions from 10A are highly
recommended as preparation for Specialist Mathematics IB/AC:

Establish the sine, cosine and area rules for any triangle, and solve related problems

Use the unit circle to define trigonometric functions, and graph them with and without the
use of digital technologies

Investigate the concept of a polynomial, and apply the factor and remainder theorems to
solve problems.
Content
Specialist Mathematics course integrating AC/IB provides opportunities to develop rigorous
mathematical arguments and proofs, and to use mathematical models extensively. Specialist
Mathematics course integrating AC/IB also extends understanding and knowledge of probability and
statistics and introduces the topics of vectors, complex numbers and matrices. Specialist
Mathematics course integrating AC/IB can be studied as a minor, major, major/minor or double
major.
Specialist Mathematics is structured over eight units. Additional IB content will be covered in
tutorials.
Unit 1
Unit 2
Unit 3
Unit 4

functions and
graphs


the Logarithm
function

counting and
probability

trigonometric
functions
arithmetic and
geometric sequences
and series


integrals

basic Statistics
exponential functions



matrices
differential calculus
differential
Equations
Unit 5
Unit 6
Unit 7

vectors
Choice of 2 from:

statistics

complex numbers

logic and Proof

statistics Extension

Unit 8
Choice of 2 from:

logic and Proof
abstract Algebra

abstract Algebra

discrete Maths

discrete Maths

further Calculus

further Calculus

conics and
Independent Project
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Board Endorsed October 2015
Units
Unit 1 begins with a review of linear and quadratic relationships. This is followed by a study of
inverse proportion and polynomials including investigating the behaviour and features of a variety of
graphs. The concept of a relation (in particular circles, ellipses and “sideways parabola”) leads into
functions, which looks at the concept, notation, transformations, composition and other properties
of functions. The study of the trigonometric functions begins with a consideration of the unit circle
using degrees and radians and the trigonometry of triangles and its application. The graphs of the
trigonometric functions, including their reciprocals are examined and their applications in a wide
range of settings are explored. Trigonometric equations are solved both graphically and algebraically
and a variety of trigonometric identities are proved and applied.
In Unit 2, exponential functions are introduced and their properties and graphs examined. Arithmetic
and geometric sequences and their applications are introduced and their recursive definitions
applied. Rates and average rates of change are introduced, and this is followed by the key concept of
the derivative as an ‘instantaneous rate of change’. These concepts are reinforced numerically (by
calculating difference quotients), geometrically (as slopes of chords and tangents), and algebraically.
Differentiation rules as well as the second derivative are explored. This calculus topic also includes
derivatives of polynomial, trigonometric and exponential functions, using applications of the
derivative and second derivative to sketch curves, calculate slopes and equations of tangents,
determine instantaneous velocities, and solve optimisation problems.
In Unit 3, the logarithmic function and its derivative are studied. The study of calculus continues by
introducing integration, both as a process that reverses differentiation and as a way of calculating
areas. The fundamental theorem of calculus as a link between differentiation and integration is
emphasised. The integrals of polynomials, exponentials, trigonometric functions and rational
functions are established and used. Integration techniques of substitution, partial fractions and
integration by parts are explored. Integration is used to find volumes of solids of revolution about
either axis. Differential Equations start with an investigation of implicit differentiation and then a
study of first order differential equations, including slope fields and their applications.
In Unit 4, counting and probability investigates the language of sets and events, reviews the
fundamentals of probability before looking at conditional probability and independence. The
notations of combinations and permutations are explored and they are used in a variety of
applications including to solve counting problems and applied to binomial expansions. In Basic
Statistics, discrete random variables are introduced. Binomial distributions and associated problems
are used to solve practical problems. The appropriate use and graphs of normal distributions are
investigated, as well as calculating probabilities and quantiles. Matrices are introduced through a
study of matric arithmetic. Matrices are then used to study transformations in the plane.
In Unit 5, vectors are introduced by using them to represent directed line segments. The algebra of
vectors is then explored in both 2D and 3D. Both the geometric and algebraic definition of cross and
dot product are explored. Vectors and Cartesian equations are used to investigate the intersection of
lines and planes. Complex numbers begins with defining i and performing complex number
arithmetic. Then the complex plane is examined and determining linear factors of polynomials allows
roots of equations to be found. Complex arithmetic using polar form is studied including De Moivre’s
Theorem and Euler’s form.
In Unit 6, two of the following topics are chosen for study: Logic and Proof; Abstract Algebra;
Discrete Maths; and Further Calculus. Logic and Proof investigates the nature of proof and logical
statements. Proof patterns are investigated as well as using proof in Euclidean Geometry. Abstract
Algebra investigates sets including subsets and operations as well as relations, binary operations and
cayley tables. Groups including subgroups, cyclical groups, homomorphisms and isomorphisms are
studied. Discrete Maths investigates modular arithmetic, graph theory, algorithms on graphs and
recurrence relations, as well as linear Diophantine equations. Further Calculus looks at infinite
series, limits of sequences and functions, extension of differential equations and the Maclaurin and
Taylor series.
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Board Endorsed October 2015
In Unit 7, the unit begins with a study of Special Distributions. Following this interval estimates for
proportions are investigated. This includes looking at random sampling, sample proportions,
confidence intervals for sample proportions and t-distribution. Then unbiased estimators and
confidence intervals are looked at, including the central limit theorem. Hypothesis testing is studied,
including significance levels, critical regions and type I and II errors. Bivariate distributions are then
investigated, including covariance and correlations, as well as linear regression.
In Unit 8, two of the following topics are chosen for study: Logic and Proof; Abstract Algebra;
Discrete Maths; Further Calculus; and Conics and Independent Project. Logic and Proof investigates
the nature of proof and logical statements. Proof patterns are investigated as well as using proof in
Euclidean Geometry. Abstract Algebra investigates sets including subsets and operations as well as
relations, binary operations and cayley tables. Groups including subgroups, cyclical groups,
homomorphisms and isomorphisms are studied. Discrete Maths investigates modular arithmetic,
graph theory, algorithms on graphs and recurrence relations, as well as linear Diophantine equations.
Further Calculus looks at infinite series, limits of sequences and functions, extension of differential
equations and the Maclaurin and Taylor series. Conics looks at conics sections, their behaviour and
applications and is an extension of some of their earlier work in functions. The independent project
allows students to engage in mathematical research to enable them to showcase their understanding
of a topic chosen by them. It is envisioned that the final product will be a presentation and/or a
paper.
Teaching and Learning Strategies
Course developers are encouraged to outline teaching strategies that are grounded in the BSSS’
Learning Principles and encompass quality teaching. Pedagogical techniques and assessment tasks
should promote intellectual quality, establish a rich learning environment and generate relevant
connections between learning and life experiences.
Teaching strategies that are particularly relevant and effective in Mathematics include, but are not
limited to the following techniques.
Review prior learning
 brainstorming, individual, pair and group work
 student reflection of relevant concepts and skills
 diagnostic tests
Introduce new material
 link topic to prior mathematical knowledge, practical applications exposure to quality visual
imagery/materials through a variety of media
 experimentation and manipulation of concrete materials
 investigation through the use of technology
 motivate study through the intrinsic beauty of the topic and relevance to future life
experiences
 narrative and historical contexts
Provide demonstration, guided practice and application
 teacher demonstration, modelling and peer tutoring
 teacher scaffolding to facilitate analysis of concepts
 engagement of industry professionals, including guest speakers, demonstrators and mentors
 simulated real life and work scenarios
 online materials
 opportunities to develop modelling or problem solving skills in practical contexts
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Board Endorsed October 2015
Promote independent practice and application
 research strategies and time management
 problem solving strategies
 mentoring and peer tutoring
 practice and reinforcement of learning by way of revision, worksheets, tests and
demonstrations
 encourages responsibility for their own learning
 regular and meaningful feedback
 discussions, debates and student presentations
 longer-term activities such as investigative, research and project tasks
 development of student prepared summaries to be used in supervised assessment tasks
(reducing the need to memorise formulas and procedures). This allows equity of access,
especially for students whose first language is not English
Link to next task or skill area
 links with the broader Mathematics curriculum
Assessment
The identification of assessment task types, together with examples of tasks, provides a common and
agreed basis for the collection of evidence of student achievement. This collection of evidence
enables a comparison of achievement within and across colleges, through moderation processes.
This enables valid, fair and equitable reporting of student achievement on the Year 12 Certificate.
The identification of assessment criteria and assessment tasks types and weightings provide a
common and agreed basis for the collection of evidence of student achievement.
Assessment Criteria (the dimensions of quality that teachers look for in evaluating student work)
provide a common and agreed basis for judgement of performance against unit and course goals,
within and across colleges. Over a course, teachers must use all of these criteria to assess students’
performance, but are not required to use all criteria on each task. Assessment criteria are to be used
holistically on a given task and in determining the unit grade.
Assessment Tasks elicit responses that demonstrate the degree to which students have achieved the
goals of a unit based on the assessment criteria. The Common Curriculum Elements (CCE) is a guide
to developing assessment tasks that promote a range of thinking skills (see Appendix A). It is highly
desirable that assessment tasks engage students in demonstrating higher order thinking.
Rubrics use the assessment criteria relevant for a particular task and can be used to assess a
continuum that indicates levels of student performance against each criterion.
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Board Endorsed October 2015
General Assessment Criteria
Technology, its selection and appropriate use, is an integral part of all the following criteria. Students
will be assessed on the degree to which they demonstrate:
 Knowledge – knowledge of mathematical facts, techniques and formulae presented in the unit
 Application – appropriate selection and application of mathematical skills in mathematical
modelling and problem solving
 Reasoning – ability to use reasoning to support solutions and conclusions (in T courses only)
 Communication – interpretation and communication of mathematical ideas in a form
appropriate for a given use or audience.
Guide to Assessment for T Course
Task Type
Weighting for 1.0 and 0.5 units
Tests:
For example:
-
multiple choice
-
short answer
-
extended questions
40-75%
Non-Test Tasks (in-class):
For example:
-
validation activities
-
modelling
-
investigations
-
problem solving
-
journals
-
portfolios
-
presentations
-
practical activities
0-60%
25-60%
Take Home Tasks:
For example:
-
modelling
-
investigations
-
portfolios
-
practical activities
0-30%
Additional Assessment Advice for T Courses
 For a standard 1.0 unit, a minimum of three and a maximum of five assessment items.
 For a half-standard 0.5 unit, minimum of two and a maximum of three assessment items.
 Assessment items should not be a compilation of small discrete tasks (e.g. mini tests) as they
distract from assessing depth of knowledge and skills.
 Each unit (standard 1.0 or half standard 0.5) should include at least two different types of
tasks.
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Board Endorsed October 2015
 It is recommended that, in standard 1.0 units, no assessment item should carry a weighting of
greater than 45% of the unit assessment.
 Where possible, for tasks completed in unsupervised circumstances, validation of the students’
work should be undertaken.
 It is recommended that students undertake a take home task. It may be worth 0% and lead
into a non-zero weighted in-class validation.
 It is desirable that students studying at T level investigate Mathematics beyond the classroom
and this should be reflected in the task type.
Achievement Standards
Achievement standards in the form of unit grades provide a guide for teacher judgement of students’
achievement, based on the assessment criteria, over a unit of work. Grades are organised on an A-E
basis.
Grades are awarded on the proviso that the assessment requirements have been met. When
allocating grades, teachers will consider the degree to which students demonstrate their ability to
complete and submit tasks within a specified time frame.
The following descriptors are consistent with the system grade descriptors, which describe generic
standards of student achievement across all courses.
Teachers may wish to consult the ACARA achievement standards for Specialist Mathematics.
14
Unit Grades for T Courses
Technology, its selection and appropriate use, is an integral part of all the following descriptors.
Knowledge
Application
Reasoning
Communication
A student
who achieves
the grade A
Demonstrates very high level of
proficiency in the use of facts,
techniques and formulae.
Selects, extends and applies
appropriate modelling and
problem solving techniques.
Uses mathematical reasoning to
develop logical arguments in
support of conclusions, results
and/or decisions; justifies
procedures.
Is consistently accurate and
appropriate in presentation of
mathematical ideas in different
contexts.
A student
who achieves
the grade B
Demonstrates high level of
proficiency in the use of facts,
techniques and formulae.
Selects and applies appropriate
modelling and problem solving
techniques.
Uses mathematical reasoning to
develop logical arguments in
support of conclusions, results
and/or decisions.
Is generally accurate and
appropriate in presentation of
mathematical ideas in different
contexts.
A student
who achieves
the grade C
Demonstrates some proficiency
in the use of facts, techniques
and formulae studied.
With direction, applies a model.
Solves most problems.
Uses some mathematical
reasoning to develop logical
arguments.
Presents mathematical ideas in
different contexts.
A student
who achieves
the grade D
Demonstrates limited use of the
facts, techniques and formulae
studied.
Solves some problems
independently.
Uses some mathematical
reasoning to develop simple
logical arguments.
Presents some mathematical
ideas.
A student
who achieves
the grade E
Demonstrates very limited use
of the facts, techniques and
formulae studied.
Solves some problems with
guidance.
Uses limited reasoning to justify
conclusions.
Presents some mathematical
ideas with guidance.
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Board Endorsed October 2015
Student Capabilities
Literacy in Mathematics
In the senior years these literacy skills and strategies enable students to express, interpret, and
communicate complex mathematical information, ideas and processes. Mathematics provides a specific
and rich context for students to develop their ability to read, write, visualise and talk about complex
situations involving a range of mathematical ideas. Students can apply and further develop their literacy
skills and strategies by shifting between verbal, graphic, numerical and symbolic forms of representing
problems in order to formulate, understand and solve problems and communicate results. This process of
translation across different systems of representation is essential for complex mathematical reasoning and
expression. Students learn to communicate their findings in different ways, using multiple systems of
representation and data displays to illustrate the relationships they have observed or constructed.
Numeracy in Mathematics
The students who undertake this subject will continue to develop their numeracy skills at a more
sophisticated level than in Years F to 10. This subject contains financial applications of Mathematics that
will assist students to become literate consumers of investments, loans and superannuation products. It
also contains statistics topics that will equip students for the ever-increasing demands of the information
age. Students will also learn about the probability of certain events occurring and will therefore be well
equipped to make informed decisions.
ICT in Mathematics
In the senior years students use ICT both to develop theoretical mathematical understanding and to apply
mathematical knowledge to a range of problems. They use software aligned with areas of work and society
with which they may be involved such as for statistical analysis, algorithm generation, data representation
and manipulation, and complex calculation. They use digital tools to make connections between
mathematical theory, practice and application; for example, to use data, to address problems, and to
operate systems in authentic situations.
Critical and creative thinking in Mathematics
Students compare predictions with observations when evaluating a theory. They check the extent to which
their theory-based predictions match observations. They assess whether, if observations and predictions
don't match, it is due to a flaw in the theory itself or the method of applying the theory to make predictions
– or both. They revise, or reapply their theory more skilfully, recognising the importance of self-correction
in the building of useful and accurate theories and making accurate predictions.
Personal and social capability in Mathematics
In the senior years students develop personal and social competence in Mathematics through setting and
monitoring personal and academic goals, taking initiative, building adaptability, communication, teamwork
and decision-making.
The elements of personal and social competence relevant to Mathematics mainly include the application of
mathematical skills for their decision-making, life-long learning, citizenship and self-management. In
addition, students will work collaboratively in teams and independently as part of their mathematical
explorations and investigations.
Ethical understanding in Mathematics
In the senior years students develop ethical understanding in Mathematics through decision-making
connected with ethical dilemmas that arise when engaged in mathematical calculation and the
dissemination of results and the social responsibility associated with teamwork and attribution of input.
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Board Endorsed October 2015
The areas relevant to Mathematics include issues associated with ethical decision-making as students work
collaboratively in teams and independently as part of their mathematical explorations and investigations.
Acknowledging errors rather than denying findings and/or evidence involves resilience and an examination
of ethical behaviour. Students develop increasingly advanced communication, research, and presentation
skills to express viewpoints.
Intercultural understanding in Mathematics
Students understand Mathematics as a socially constructed body of knowledge that uses universal symbols
but has its origin in many cultures. Students understand that some languages make it easier to acquire
mathematical knowledge than others. Students also understand that there are many culturally diverse
forms of mathematical knowledge, including diverse relationships to number and that diverse cultural
spatial abilities and understandings are shaped by a person’s environment and language.
Representation of Cross-curriculum Priorities
The senior secondary Mathematics curriculum values the histories, cultures, traditions and languages of
Aboriginal and Torres Strait Islander Peoples past and ongoing contributions to contemporary Australian
society and culture. Through the study of mathematics within relevant contexts, opportunities will allow for
the development of students’ understanding and appreciation of the diversity of Aboriginal and Torres
Strait Islander Peoples histories and cultures.
There are strong social, cultural and economic reasons for Australian students to engage with the countries
of Asia and with the past and ongoing contributions made by the peoples of Asia in Australia. It is through
the study of mathematics in an Asian context that students engage with Australia’s place in the region.
Through analysis of relevant data, students are provided with opportunities to further develop an
understanding of the diverse nature of Asia’s environments and traditional and contemporary cultures.
Each of the senior Mathematics subjects provides the opportunity for the development of informed and
reasoned points of view, discussion of issues, research and problem solving. Therefore, teachers are
encouraged to select contexts for discussion connected with sustainability. Through analysis of data,
students have the opportunity to research and discuss sustainability and learn the importance of respecting
and valuing a wide range of world perspectives.
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Board Endorsed October 2015
International Baccalaureate Learner Profile
The aim of all IB programmes is to develop internationally minded people who, recognising their common
humanity and shared guardianship of the planet, help to create a better and more peaceful world.
IB learners strive to be:
Inquirers
Knowledgeable
Thinkers
Communicators
Principled
Open-minded
Caring
Risk-takers
Balanced
Reflective
They develop their natural curiosity. They acquire the skills necessary
to conduct inquiry and research and show independence in learning.
They actively enjoy learning and this love of learning will be sustained
throughout their lives.
They explore concepts, ideas and issues that have local and global
significance. In so doing, they acquire in-depth knowledge and
develop understanding across a broad and balanced range of
disciplines.
They exercise initiative in applying thinking skills critically and
creatively to recognize and approach complex problems, and make
reasoned, ethical decisions.
They understand and express ideas and information confidently and
creatively in more than one language and in a variety of modes of
communication. They work effectively and willingly in collaboration
with others.
They act with integrity and honesty, with a strong sense of fairness,
justice and respect for the dignity of the individual, groups and
communities. They take responsibility for their own actions and the
consequences that accompany them.
They understand and appreciate their own cultures and personal
histories, and are open to the perspectives, values and traditions of
other individuals and communities. They are accustomed to seeking
and evaluating a range of points of view, and are willing to grow from
the experience.
They show empathy, compassion and respect towards the needs and
feelings of others. They have a personal commitment to service, and
act to make a positive difference to the lives of others and to the
environment.
They approach unfamiliar situations and uncertainty with courage and
forethought, and have the independence of spirit to explore new
roles, ideas and strategies. They are brave and articulate in defending
their beliefs.
They understand the importance of intellectual, physical and
emotional balance to achieve personal well-being for themselves and
others.
They give thoughtful consideration to their own learning and
experience. They are able to assess and understand their strengths
and limitations in order to support their learning and personal
development.
IB Learner Profile, “International Baccalaureate Mathematics HL Subject Guide”, August 2014, Available from: www.ibo.org. [29th July 2015].
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Board Endorsed October 2015
Moderation
Moderation is a system designed and implemented to:
 provide comparability in the system of school-based assessment
 form the basis for valid and reliable assessment in senior secondary schools
 involve the ACT Board of Senior Secondary Studies and colleges in cooperation and partnership
 maintain the quality of school-based assessment and the credibility, validity and acceptability of
Board certificates.
Moderation commences within individual colleges. Teachers develop assessment programs and
instruments, apply assessment criteria, and allocate Unit Grades, according to the relevant Course
Framework. Teachers within course teaching groups conduct consensus discussions to moderate marking
or grading of individual assessment instruments and unit grade decisions.
The Moderation Model
Moderation within the ACT encompasses structured, consensus-based peer review of Unit Grades for all
accredited courses, as well as statistical moderation of course scores, including small group procedures, for
T courses.
Moderation by Structured, Consensus-based Peer Review
In the review process, Unit Grades awarded by teachers on the basis of school assessment are moderated
by peer review against system wide assessment criteria and achievement standards. This is done by
matching student performance as demonstrated in portfolios of assessment tasks against the criteria and
standards. Advice is then given to colleges to assist teachers with, and/or reassure them on, their
judgments.
Preparation for Structured, Consensus-based Peer Review
Each year, teachers taking a Year 11 class are asked to retain originals or copies of student work completed
in Semester 2. Similarly, teachers taking a Year 12 class should retain originals or copies of student work
completed in Semester 1. Assessment and other documentation required by the Office of the Board of
Senior Secondary Studies should also be kept. Year 11 work from Semester 2 of the previous year is
presented for review at Moderation Day 1 in March, and Year 12 work from Semester 1 is presented for
review at Moderation Day 2 in August.
In the lead up to Moderation Day, a College Course Presentation (comprised of a document folder and a set
of student portfolios) is prepared for each A, M and T course/units offered by the school, and is sent in to
the Office of the Board of Senior Secondary Studies.
The College Course Presentation
The package of materials (College Course Presentation) presented by a college for review on moderation
days in each course area will comprise the following:
 a folder containing supporting documentation as requested by the Office of the Board through
memoranda to colleges
 a set of student portfolios containing marked and/or graded written and non-written assessment
responses and completed criteria and standards feedback forms. Evidence of all assessment
responses on which the unit grade decision has been made is to be included in the student review
portfolios.
Specific requirements for subject areas and types of evidence to be presented for each Moderation Day will
be outlined by the Board Secretariat through memoranda and Information Papers.
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Board Endorsed October 2015
Visual evidence for judgements made about practical performances
(also refer to BSSS Website Guidelines)
It is a requirement that schools’ judgements of standards to practical performances (A/T/M) be supported
by visual evidence (still photos or video).
The photographic evidence submitted must be drawn from practical skills performed as part of the
assessment process.
Teachers should consult the BSSS guidelines at
http://www.bsss.act.edu.au/grade_moderation/information_for_teachers when preparing photographic
evidence.
Resources
The following links provide up-to-date resources on Australian Curriculum and International Baccalaureate:
www.haesemathematics.com.au
www.oup.com.au
www.pearson.com.au
www.amsi.org.au/SAM.SeniorYears
www.nelsonsecondary.com.au
www.cambridge.com.au
www.macmillan.com.au
Glossary
It is recommended that teachers use glossaries from resources for the International Baccalaureate.
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Board Endorsed October 2015
Proposed Evaluation Procedures
Course evaluation will be a continuous process. Teachers will meet regularly to discuss the content of the
course and any requirements for modification of activities, teaching strategies and assessment instruments.
The current trends and innovations in the teaching of Specialist Mathematics course integrating AC/IB will
be considered as teachers attend workshops, seminars and participate in discussion groups with other
teachers such as on Moderation Day.
Teachers will monitor student performance and progress and student responses to various teaching,
learning and assessment strategies. Students and teachers will complete evaluation questionnaires at the
end of each unit. The results of these will be collated and reviewed from year to year. There will also be a
continuous monitoring of student numbers between Years 11 and 12.
Informal discussions between teachers and students, past students, parents and other teachers will
contribute to the evaluation of the course.
In the process of evaluation; students, teachers and others should, as appropriate, consider:
 Are the course and Course Framework still consistent?
 Were the goals achieved?
 Was the course content appropriate?
 Were the teaching strategies used successful?
 Was the assessment program appropriate?
 Have the needs of the students been met?
 Was the course relevant?
 How many students completed the course in each of the years of accreditation?
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Board Endorsed October 2015
Unit 1: Specialist Mathematics IB T
Value: 1.0
Unit 1a: Specialist Mathematics IB
Value: 0.5
Unit 1b: Specialist Mathematics IB
Value: 0.5
Prerequisites
Nil
Duplication of Content
Nil
Specific Unit Goals
By the end of this unit, students:
 understand the concepts and techniques used in functions and graphs and trigonometry
 solve problems in functions and graphs and trigonometry
 apply reasoning skills in functions and graphs and trigonometry
 interpret and evaluate mathematical and statistical information and ascertain the reasonableness of
solutions to problems
 communicate arguments and strategies when solving problems.
Content
Further elaboration on the content of this unit is available at:
http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/MathematicalMethods/Curriculum/SeniorSecondary
Topic 1: Functions and graphs
Lines and linear relationships
 determine the coordinates of the midpoint of two points
 examine examples of direct proportion and linearly related variables
 recognise features of the graph of 𝑦 = 𝑚𝑥 + 𝑐, including its linear nature, its intercepts and its slope
or gradient
 find the equation of a straight line given sufficient information; parallel and perpendicular lines
 solve linear equations.
Review of quadratic relationships:
 examine examples of quadratically related variables
 recognise features of the graphs of 𝑦 = 𝑥 2 , 𝑦 = 𝑎(𝑥 − 𝑏)2 + 𝑐, and 𝑦 = 𝑎(𝑥 − 𝑏)(𝑥 − 𝑐), including
their parabolic nature, turning points, axes of symmetry and intercepts
 solve quadratic equations using the quadratic formula and by completing the square
 find the equation of a quadratic given sufficient information
 find turning points and zeros of quadratics and understand the role of the discriminant
 classify the nature of the roots of a quadratic
 Sum and products of roots
 recognise features of the graph of the general quadratic 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
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Board Endorsed October 2015
Inverse proportion:
 examine examples of inverse proportion
1
𝑎
 recognise features of the graphs of 𝑦 = 𝑥 and 𝑦 = 𝑥−𝑏, including their hyperbolic shapes, and their
asymptotes
Powers and polynomials:
 recognise features of the graphs of 𝑦 = 𝑥 𝑛 for 𝑛 ∈ 𝑵, 𝑛 = −1 and 𝑛 = ½, including shape, and
behaviour as 𝑥 → ∞ and 𝑥 → −∞
 identify the coefficients and the degree of a polynomial
 expand quadratic and cubic polynomials from factors
 recognise features of the graphs of 𝑦 = 𝑥 3 , 𝑦 = 𝑎(𝑥 − 𝑏)3 + 𝑐 and 𝑦 = 𝑘(𝑥 − 𝑎)(𝑥 − 𝑏)(𝑥 − 𝑐),
including shape, intercepts and behaviour as 𝑥 → ∞ and 𝑥 → −∞
 factorise cubic polynomials in cases where a linear factor is easily obtained
 solve cubic equations using technology, and algebraically in cases where a linear factor is easily
obtained
Graphs of relations:
 recognise features of the graphs of 𝑥 2 + 𝑦 2 = 𝑟 2 and (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2 , including their
circular shapes, their centres and their radii
 recognise features of the graph of 𝑦 2 = 𝑥 including its parabolic shape and its axis of symmetry
Functions:
 understand the concept of a function as a mapping between sets, and as a rule or a formula that
defines one variable quantity in terms of another
 use function notation, domain and range (including implied), independent and dependent variables
 understand the concept of the graph of a function
 examine translations and the graphs of 𝑦 = 𝑓(𝑥) + 𝑎 and 𝑦 = 𝑓(𝑥 + 𝑏)
 examine dilations and the graphs of 𝑦 = 𝑐𝑓(𝑥) and 𝑦 = 𝑓(𝑘𝑥)
 recognise the distinction between functions and relations, and the vertical line test.
 determine when the composition of two functions is defined
 find the composition of two functions
 determine if a function is one-to-one
 consider inverses of one-to-one function
 determine odd and even functions
 examine the reflection property of the graph of a function and the graph of its inverse.
 graphing absolute value of functions
 graphing reciprocals of functions
 identify solutions of 𝑔(𝑥) ≥ 𝑓(𝑥)
Topic 2: Trigonometric functions
Cosine and sine rules:
 review sine, cosine and tangent as ratios of side lengths in right-angled triangles
 understand the unit circle definition of cos 𝜃, sin 𝜃 and tan 𝜃 and periodicity using degrees
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Board Endorsed October 2015
1
 establish and use the sine and cosine rules and the formula 𝐴𝑟𝑒𝑎 = 2 𝑏𝑐 sin 𝐴 for the area of a
triangle
Circular measure and radian measure:
 define and use radian measure and understand its relationship with degree measure
 calculate lengths of arcs, areas of sectors and segments in circles
Trigonometric functions:
 understand the unit circle definition of cos 𝜃, sin 𝜃 and tan 𝜃 and periodicity using radians
𝜋 𝜋
𝜋
 recognise the exact values of sin 𝜃, cos 𝜃 and tan 𝜃 at integer multiples of 6 , 4 and 3
𝜋
 use exact value to find and apply angles of any magnitude (including in the form sin ( 2 −
𝜋
𝜃) 𝑎𝑛𝑑 sin ( 2 + 𝜃) )
 recognise the graphs of 𝑦 = sin 𝑥, 𝑦 = cos 𝑥 , and 𝑦 = tan 𝑥 on extended domains
 examine amplitude changes and the graphs of 𝑦 = 𝑎 sin 𝑥 and 𝑦 = 𝑎 cos 𝑥
 examine period changes and the graphs of 𝑦 = sin 𝑏𝑥, 𝑦 = cos 𝑏𝑥, and 𝑦 = tan 𝑏𝑥
 examine phase changes and the graphs of 𝑦 = sin(𝑥 + 𝑐), 𝑦 = cos(𝑥 + 𝑐) and
𝜋
𝜋
𝑦 = tan (𝑥 + 𝑐) and the relationships sin (𝑥 + 2 ) = cos 𝑥 and cos (𝑥 − 2 ) = sin 𝑥
 identify contexts suitable for modelling by trigonometric functions and use them to solve practical
problems
 solve equations involving trigonometric functions using technology, and algebraically.
The basic trigonometric functions:

find all solutions of 𝑓(𝑎(𝑥 − 𝑏)) = 𝑐 where f is one of sin, cos or tan

graph functions with rules of the form 𝑦 = 𝑓(𝑎(𝑥 − 𝑏)) where 𝑓 is one of sin, cos, or tan.

Compound angles:

prove and apply the angle sum, difference and double angle identities.
The reciprocal trigonometric functions, secant, cosecant and cotangent:

define the reciprocal trigonometric functions, sketch their graphs, and graph simple
transformations of them.
Sketch the inverse functions of sin, cos and tan and identify their domains and ranges
Trigonometric identities:

prove and apply the Pythagorean identities

prove and apply the identities for products of sines and cosines expressed as sums and differences

convert sums a cos x + b sin x to R cos(x ± α) or R sin(x ± α) and apply these to sketch graphs,
solve equations of the form a cos x + b sin x = c and solve problems

prove and apply other trigonometric identities such as cos 3x = 4 cos3 x − 3 cos x .
Applications of trigonometric functions to model periodic phenomena:

model periodic motion using sine and cosine functions and understand the relevance of the period
and amplitude of these functions in the model.
Teaching and Learning Strategies
Refer to page 14.
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Board Endorsed October 2015
Assessment
Refer to Assessment Criteria, Task Types table on pages 15-16.
Suggested Unit Resources
Refer to page 23.
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Board Endorsed October 2015
Unit 2: Specialist Mathematics IB T
Value: 1.0
Unit 2a: Specialist Mathematics IB
Value: 0.5
Unit 2b: Specialist Mathematics IB
Value: 0.5
Prerequisites
Nil
Duplication of Content
Nil
Specific Unit Goals
By the end of this unit, students:
 understand the concepts and techniques used in sequences and series, exponential functions and
calculus
 solve problems in sequences and series, exponential functions and calculus
 apply reasoning skills in sequences and series, exponential functions and calculus
 interpret and evaluate mathematical and statistical information and ascertain the reasonableness of
solutions to problems
 communicate arguments and strategies when solving problems.
Content
Further elaboration on the content of this unit is available at:
http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/MathematicalMethods/Curriculum/SeniorSecondary
Topic 1 Arithmetic and geometric sequences and series
Arithmetic sequences:
 recognise and use the recursive definition of an arithmetic sequence: 𝑡𝑛+1 = 𝑡𝑛 + 𝑑
 use the formula 𝑡𝑛 = 𝑡1 + (𝑛 − 1)𝑑 for the general term of an arithmetic sequence and recognise its
linear nature
 use arithmetic sequences in contexts involving discrete linear growth or decay, such as simple
interest
 establish and use the formula for the sum of the first 𝑛 terms of an arithmetic sequence
Geometric sequences:
 recognise and use the recursive definition of a geometric sequence: 𝑡𝑛+1 = 𝑟𝑡𝑛
 use the formula 𝑡𝑛 = 𝑟 𝑛−1 𝑡1 for the general term of a geometric sequence and recognise its
exponential nature
 understand the limiting behaviour as 𝑛 → ∞ of the terms 𝑡𝑛 in a geometric sequence and its
dependence on the value of the common ratio 𝑟
 establish and use the formula 𝑆𝑛 = 𝑡1
𝑟 𝑛 −1
𝑟−1
for the sum of the first 𝑛 terms of a geometric sequence
 use geometric sequences in contexts involving geometric growth/decay, such as compound interest
 establish and use the formula 𝑆∞ =
𝑡1
1−𝑟
for the sum to infinity
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Board Endorsed October 2015
Topic 2: Exponential functions
Indices and the index laws:
 review indices (including fractional indices) and the index laws
 use radicals and convert to and from fractional indices
 understand and use scientific notation and significant figures
Exponential functions:
 establish and use the algebraic properties of exponential functions
 recognise the qualitative features of the graph of 𝑦 = 𝑎 𝑥 (𝑎 > 0) including asymptotes, and of its
translations (𝑦 = 𝑎 𝑥 + 𝑏 and 𝑦 = 𝑎 𝑥+𝑐 )
 understand the inverse behaviour of exponentials and logarithms
 use logarithms where necessary to solve exponentials
 identify contexts suitable for modelling by exponential functions and use them to solve practical
problems
 solve equations involving exponential functions using technology, and algebraically in simple cases
Topic 3: Differential calculus
Rates of change:
 interpret the difference quotient
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
as the average rate of change of a function 𝑓
 use the Leibniz notation 𝛿𝑥 and 𝛿𝑦 for changes or increments in the variables 𝑥 and 𝑦
 use the notation
𝛿𝑦
𝛿𝑥
 interpret the ratios
𝑦 = 𝑓(𝑥).
for the difference quotient
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
and
𝛿𝑦
𝛿𝑥
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
where 𝑦 = 𝑓(𝑥)
as the slope or gradient of a chord or secant of the graph of
The concept of the derivative:
 examine the behaviour of the difference quotient
to the concept of a limit
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
ℎ→0
𝑑𝑦
𝛿𝑦
for the derivative: 𝑑𝑥 = lim 𝛿𝑥
𝛿𝑥→0
as ℎ → 0 as an informal introduction
 define the derivative 𝑓 ′ (𝑥) as lim
 use the Leibniz notation
𝑑𝑦
and the correspondence 𝑑𝑥 = 𝑓 ′ (𝑥) where
𝑦 = 𝑓(𝑥)
 interpret the derivative as the instantaneous rate of change
 interpret the derivative as the slope or gradient of a tangent line of the graph of 𝑦 = 𝑓(𝑥).
Computation of derivatives:
 estimate numerically the value of a derivative, for simple power functions
 examine examples of variable rates of change of non-linear functions
𝑑
 establish the formula 𝑑𝑥 (𝑥 𝑛 ) = 𝑛𝑥 𝑛−1 for positive integers 𝑛 by expanding (𝑥 + ℎ)𝑛 or by
factorising (𝑥 + ℎ)𝑛 − 𝑥 𝑛 .
Properties of derivatives:
 understand the concept of the derivative as a function
 recognise and use linearity properties of the derivative
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Board Endorsed October 2015
 calculate derivatives of polynomials and other linear combinations of power functions
Exponential functions:
 estimate the limit of
𝑎 ℎ −1
as
ℎ
ℎ → 0 using technology, for various values of 𝑎 > 0
 recognise that 𝑒 is the unique number 𝑎 for which the above limit is 1
 establish and use the formula
𝑑
(𝑒 𝑥 )
𝑑𝑥
= 𝑒𝑥
 use exponential functions and their derivatives to solve practical problems.
Trigonometric functions:
𝑑
𝑑
 establish the formulas 𝑑𝑥 (sin 𝑥) = cos 𝑥, and 𝑑𝑥 (cos 𝑥) = − sin 𝑥 by numerical estimations of the
limits and informal proofs based on geometric constructions
 use trigonometric functions and their derivatives to solve practical problems.
Differentiation rules:
 differentiation of sum and multiples of functions
 understand and use the product and quotient rules
 understand the notion of composition of functions and use the chain rule for determining the
derivatives of composite functions
 apply the product, quotient and chain rule to differentiate functions such as 𝑥𝑒 𝑥 , tan 𝑥,
𝑒 − 𝑥 sin 𝑥 and 𝑓(𝑎𝑥 + 𝑏).
1
𝑥𝑛
, 𝑥 sin 𝑥,
 differentiate functions implicitly
Applications of derivatives:
 find instantaneous rates of change
 find the slope of a tangent and the equation of the tangent
 construct and interpret position-time graphs, with velocity as the slope of the tangent
 sketch curves associated with simple polynomials; find stationary points, and local and global
maxima and minima, points of inflexion, concavity and second derivative; and examine behaviour as
𝑥 → ∞ and 𝑥 → −∞
 solve optimisation problems arising in a variety of contexts involving simple polynomials on finite
interval domains.
The second derivative and applications of differentiation:
𝑑𝑦
 use the increments formula: 𝛿𝑦 ≅ 𝑑𝑥 × 𝛿𝑥 to estimate the change in the dependent variable 𝑦
resulting from changes in the independent variable 𝑥
 understand the concept of the second derivative as the rate of change of the first derivative function
 recognise acceleration as the second derivative of position with respect to time
 understand the concepts of concavity and points of inflection and their relationship with the second
derivative
 understand and use the second derivative test for finding local maxima and minima
 sketch the graph of a function using first and second derivatives to locate stationary points and
points of inflection
 solve optimisation problems from a wide variety of fields using first and second derivatives.
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Board Endorsed October 2015
Teaching and Learning Strategies
Refer to page 14.
Assessment
Refer to Assessment Criteria, Task Types table on pages 15-16.
Suggested Unit Resources
Refer to page 23.
29
Board Endorsed October 2015
Unit 3: Specialist Mathematics IB T
Value: 1.0
Unit 3a: Specialist Mathematics IB
Value: 0.5
Unit 3b: Specialist Mathematics IB
Value: 0.5
Students are expected to study the accredited semester 1.0 unit unless enrolled in a 0.5 unit due to late
entry or early exit in a semester.
Prerequisites
Nil
Duplication of Content
Nil
Specific Unit Goals
By the end of this unit, students:
 understand the concepts and techniques in logarithms, integrals and differential equations
 solve problems in logarithms, integrals and differential equations
 apply reasoning skills in logarithms, integrals and differential equations
 interpret and evaluate mathematical and statistical information and ascertain the reasonableness of
solutions to problems.
 communicate their arguments and strategies when solving problems.
Content
Further elaboration on the content of this unit is available at:
http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/MathematicalMethods/Curriculum/SeniorSecondary
Topic 1: The Logarithm Function
Logarithmic functions:
 define logarithms as indices: 𝑎 𝑥 = 𝑏 is equivalent to 𝑥 = log 𝑎 𝑏 i.e. 𝑎log𝑎 𝑏 = 𝑏
 establish and use the algebraic properties of logarithms
 recognise the inverse relationship between logarithms and exponentials: 𝑦 = 𝑎 𝑥 is equivalent to 𝑥 =
log 𝑎 𝑦
 interpret and use logarithmic scales such as decibels in acoustics, the Richter Scale for earthquake
magnitude, octaves in music, pH in chemistry
 solve equations involving indices using logarithms
 recognise the qualitative features of the graph of 𝑦 = log 𝑎 𝑥 (𝑎 > 1) including asymptotes, and of its
translations 𝑦 = log 𝑎 𝑥 + 𝑏 and 𝑦 = log 𝑎 (𝑥 + 𝑐)
 solve simple equations involving logarithmic functions algebraically and graphically
 identify contexts suitable for modelling by logarithmic functions and use them to solve practical
problems
Calculus of logarithmic functions:
 define the natural logarithm ln 𝑥 = log 𝑒 𝑥
 recognise and use the inverse relationship of the functions 𝑦 = 𝑒 𝑥 and 𝑦 = ln 𝑥
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Board Endorsed October 2015
 establish and use the formula
𝑑
(ln 𝑥)
𝑑𝑥
1
=𝑥
 use logarithmic functions and their derivatives to solve practical problems.
Topic 2: Integrals
Anti-differentiation:
 recognise anti-differentiation as the reverse of differentiation
 use the notation ∫ 𝑓(𝑥)𝑑𝑥 for anti-derivatives or indefinite integrals
 establish and use the formula ∫ 𝑥 𝑛 𝑑𝑥 =
1
𝑥 𝑛+1
𝑛+1
𝑥
+ 𝑐 for 𝑛 ≠ −1
 establish and use the formula ∫ 𝑒 𝑑𝑥 = 𝑒 + 𝑐
𝑥
 establish and use the formulas ∫ sin 𝑥 𝑑𝑥 = −cos 𝑥 + 𝑐 and ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝑐
1
𝑥
 establish and use the formula ∫ 𝑑𝑥 = ln 𝑥 + 𝑐, for 𝑥 > 0
 recognise and use linearity of anti-differentiation
 determine indefinite integrals of the form ∫ 𝑓(𝑎𝑥 + 𝑏)𝑑𝑥
 identify families of curves with the same derivative function
 determine 𝑓(𝑥), given 𝑓 ′ (𝑥)𝑎nd an initial condition 𝑓(𝑎) = 𝑏
 determine displacement given velocity in linear motion problems.
Definite integrals:
 examine the area problem, and use sums of the form ∑𝑖 𝑓(𝑥𝑖 ) 𝛿𝑥𝑖 to estimate the area under the
curve 𝑦 = 𝑓(𝑥)
𝑏
 interpret the definite integral ∫𝑎 𝑓(𝑥)𝑑𝑥 as area under the curve 𝑦 = 𝑓(𝑥) if 𝑓(𝑥) > 0
𝑏
 recognise the definite integral ∫𝑎 𝑓(𝑥)𝑑𝑥 as a limit of sums of the form ∑𝑖 𝑓(𝑥𝑖 ) 𝛿𝑥𝑖
𝑏
 interpret ∫𝑎 𝑓(𝑥)𝑑𝑥 as a sum of signed areas
 recognise and use the additivity and linearity of definite integrals.
Fundamental theorem:
𝑥
 understand the concept of the signed area function 𝐹(𝑥) = ∫𝑎 𝑓(𝑡)𝑑𝑡
𝑑
𝑥
 understand and use the theorem: 𝐹 ′ (𝑥) = 𝑑𝑥 (∫𝑎 𝑓(𝑡)𝑑𝑡) = 𝑓(𝑥), and illustrate its proof
geometrically
𝑏
 understand the formula ∫𝑎 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) and use it to calculate definite integrals.
Integration techniques:
 use substitution 𝑢 = 𝑔(𝑥)to integrate expressions of the form 𝑓(𝑔(𝑥))𝑔′ (𝑥)
 find and use the derivative of the inverse trigonometric functions: arcsine, arccosine and arctangent
 integrate expressions of the form
±1
√𝑎 2 −𝑥 2
and
𝑎
𝑎 2 +𝑥 2
 use partial fractions where necessary for integration in simple cases
 integrate by parts.
Applications of integral calculus:

calculate areas between curves determined by functions

determine volumes of solids of revolution about either axis
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Topic 3: Differential Equations
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑢
𝑑𝑢
𝑑𝑥

related rates as instances of the chain rule:

solve simple first-order differential equations of the form
=
×
𝑑𝑦
form 𝑑𝑥
𝑑𝑦
𝑑𝑥
= 𝑓(𝑥), differential equations of the
𝑑𝑦
= 𝑔(𝑦) and, in general, differential equations of the form 𝑑𝑥 = 𝑓(𝑥)𝑔(𝑦) using separation
of variables

examine slope (direction or gradient) fields of a first order differential equation

applications of differentials equations
Teaching and Learning Strategies
Refer to page 14.
Assessment
Refer to Assessment Criteria, Task Types table on pages 15-16.
Suggested Unit Resources
Refer to page 23.
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Board Endorsed October 2015
Unit 4: Specialist Mathematics T
Value: 1.0
Unit 4a: Specialist Mathematics IB
Value: 0.5
Unit 4b: Specialist Mathematics IB
Value: 0.5
Students are expected to study the accredited semester 1.0 unit unless enrolled in a 0.5 unit due to late
entry or early exit in a semester.
Prerequisites
Nil
Duplication of Content Rules
Nil
Specific Unit Goals
By the end of this unit, students:
 understand the concepts and techniques in probability, statistics and matrices
 solve problems in probability, statistics and matrices
 apply reasoning skills in probability, statistics and matrices
 interpret and evaluate mathematical and statistical information and ascertain the reasonableness of
solutions to problems
 communicate their arguments and strategies when solving problems.
Content
Further elaboration on the content of this unit is available at:
http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/MathematicalMethods/Curriculum/SeniorSecondary
Topic 1: Counting and probability
Language of events and sets:
 review the concepts and language of outcomes, sample spaces and events as sets of outcomes
̅ (or 𝐴′ ) for the complement of an event 𝐴, 𝐴 ∩
 use set language and notation for events, including A
𝐵 for the intersection of events 𝐴 and 𝐵, and 𝐴 ∪ 𝐵 for the union, and recognise mutually exclusive
events
 use everyday occurrences to illustrate set descriptions and representations of events, and set
operations.
Review of the fundamentals of probability:
 review probability as a measure of ‘the likelihood of occurrence’ of an event
 review the probability scale: 0 ≤ 𝑃(𝐴) ≤ 1 for each event 𝐴, with 𝑃(𝐴) = 0 if 𝐴 is an impossibility
and 𝑃(𝐴) = 1 if 𝐴 is a certainty
 review the rules: 𝑃′(𝐴) = 1 − 𝑃(𝐴) and 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
 use relative frequencies obtained from data as point estimates of probabilities.
Conditional probability and independence:
 understand the notion of a conditional probability and recognise and use language that indicates
conditionality
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Board Endorsed October 2015
 use the notation 𝑃(𝐴|𝐵) and the formula P(A  B) = P(A|B)P(B)
 understand the notion of independence of an event 𝐴 from an event 𝐵, as defined by 𝑃(𝐴|𝐵) =
𝑃(𝐴)
 establish and use the formula 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵) for independent events 𝐴 and 𝐵, and
recognise the symmetry of independence
 use of Bayes theorem for a maximum of three events
 use relative frequencies obtained from data as point estimates of conditional probabilities and as
indications of possible independence of events.
Combinations:
 understand the notion of a combination as an ordered set of 𝑟 objects taken from a set of 𝑛 distinct
objects
𝑛
𝑛
𝑛!
 use the notation ( ) and the formula ( ) = 𝑟!(𝑛−𝑟)! for the number of combinations of 𝑟 objects
𝑟
𝑟
taken from a set of 𝑛 distinct objects
 expand (𝑥 + 𝑦)𝑛 for small positive integers 𝑛
𝑛
 recognise the numbers ( ) as binomial coefficients, (as coefficients in the expansion of (𝑥 + 𝑦)𝑛 )
𝑟
 use Pascal’s triangle and its properties.
Permutations (ordered arrangements):

solve problems involving permutations

use the multiplication principle

use factorial notation

solve problems involving permutations and restrictions with or without repeated objects
Topic 2: Basic Statistics
Discrete and continuous random variables:

concept of discrete and continuous random variables and their probability distributions

definition and use of probability density functions

calculate mean, median, mode, variance and standard deviation of probability density function
Binomial distributions:
 understand the concepts of Bernoulli trials
 identify contexts suitable for modelling by binomial random variables
𝑛
 determine and use the probabilities 𝑃(𝑋 = 𝑟) = ( ) 𝑝𝑟 (1 − 𝑝)𝑛−𝑟 associated with the binomial
𝑟
distribution with parameters 𝑛 and p ; note the mean 𝑛𝑝 and variance 𝑛𝑝(1 − 𝑝) of a binomial
distribution
 use binomial distributions and associated probabilities to solve practical problems
Normal distributions:
 identify contexts such as naturally occurring variation that are suitable for modelling by normal
random variables
 recognise features of the graph of the probability density function of the normal distribution with
mean 𝜇 and standard deviation 𝜎 and the use of the standard normal distribution
 calculate probabilities and quantiles associated with a given normal distribution using technology,
and use these to solve practical problems
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Board Endorsed October 2015
Topic 3: Matrices
Matrix arithmetic:

understand the matrix definition and notation

define and use addition and subtraction of matrices, scalar multiplication, matrix multiplication,
multiplicative identity and inverse

calculate the determinant and inverse of 2 × 2 matrices and solve matrix equations of the form AX =
B, where A is a 2 × 2 matrix and X and B are column vectors.
Transformations in the plane:

translations and their representation as column vectors

define and use basic linear transformations: dilations of the form (x, y) ⟶ (λ1 x, λ2 y), rotations
about the origin and reflection in a line which passes through the origin, and the representations of
these transformations by 2  2 matrices

apply these transformations to points in the plane and geometric objects

define and use composition of linear transformations and the corresponding matrix products

define and use inverses of linear transformations and the relationship with the matrix inverse

examine the relationship between the determinant and the effect of a linear transformation on
area

establish geometric results by matrix multiplications; for example, show that the combined effect
of two reflections in lines through the origin is a rotation.
Teaching and Learning Strategies
Refer to page 14.
Assessment
Refer to Assessment Criteria, Task Types table on pages 15-16.
Suggested Unit Resources
Refer to page 23.
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Board Endorsed October 2015
Unit 5: Specialist Mathematics T
Value: 1.0
Unit 5a: Specialist Mathematics IB
Value: 0.5
Unit 5b: Specialist Mathematics IB
Value: 0.5
Prerequisites
Nil
Duplication of Content Rules
Nil
Specific Unit Goals
By the end of this unit, students:

understand the concepts and techniques in vectors and complex numbers

apply reasoning skills and solve problems in vectors and complex numbers

communicate their arguments and strategies when solving problems

construct proofs of results

interpret mathematical information and ascertain the reasonableness of their solutions to
problems.
Content
Topic 1 : Vectors in 2D and 3D
Representing vectors by directed line segments:

examine examples of vectors including displacement and velocity

define and use the magnitude and direction of a vector

represent a scalar multiple of a vector

use the triangle rule to find the sum and difference of two vectors.
Algebra of vectors:

use ordered pair notation and column vector notation to represent a vector

define and use unit vectors and the perpendicular unit vectors i , j and k

express a vector in component form using the unit vectors i , j and k

examine and use addition and subtraction of vectors in component form

define and use multiplication by a scalar of a vector in component form

define and use scalar (dot) product

apply the scalar product to vectors expressed in component form

examine properties of parallel and perpendicular vectors and determine if two vectors are parallel
or perpendicular

solve problems involving displacement and velocity involving the above concepts.

prove geometric results in the plane and construct simple proofs in three-dimensions.
Vector and Cartesian equations:

introduce Cartesian coordinates for three-dimensional space, including plotting points
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Board Endorsed October 2015

use vector equations of curves in two or three dimensions involving a parameter, and determine a
‘corresponding’ Cartesian equation in the two-dimensional case

determine a vector equation of a straight line and straight-line segment, given the position of two
points, or equivalent information, in both two and three dimensions

points of intersection of lines in space

examine the position of two particles each described as a vector function of time, and determine if
their paths cross or if the particles meet

use the cross product to determine a vector normal to a given plane

geometric interpretation of magnitude of cross product

determine vector and Cartesian equations of a plane and of regions in a plane.

Intersections of a line with a plane; two planes ; three planes

Angle between a line and a plane and the angle between two planes.
Topic 2: Complex Numbers
Complex numbers

define the imaginary number i as a root of the equation x 2 = −1

use complex numbers in the form a + bi where a and b are the real and imaginary parts

determine and use complex conjugates

perform complex-number arithmetic: addition, subtraction, multiplication and division.
The complex plane:

consider complex numbers as points in a plane with real and imaginary parts as Cartesian
coordinates

examine addition of complex numbers as vector addition in the complex plane

understand and use location of complex conjugates in the complex plane.
Roots of equations:

use the general solution of real quadratic equations

determine complex conjugate solutions of real quadratic equations

determine linear factors of real quadratic polynomials.
Complex arithmetic using polar form:

use the modulus |z|of a complex number z and the argument Arg (z) of a non-zero complex
number z and prove basic identities involving modulus and argument

convert between Cartesian and polar form

define and use multiplication, division, and powers of complex numbers in polar form and the
geometric interpretation of these

prove and use De Moivre’s theorem for integral powers.

Complex numbers in Euler’s form

Ability to convert between forms is expected.
The complex plane (the Argand plane):

examine and use addition of complex numbers as vector addition in the complex plane
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Board Endorsed October 2015
Roots of complex numbers

determine and examine the 𝓃𝔱h roots of unity and their location on the unit circle

determine and examine the 𝓃𝔱hroots of complex numbers and their location in the complex plane.
Factorisation of polynomials:

prove and apply the factor theorem and the remainder theorem for polynomials

consider conjugate roots for polynomials with real coefficients

solve simple polynomial equations.
Teaching and Learning Strategies
Refer to page 14.
Assessment
Refer to Assessment Criteria, Task Types table on pages 15-16.
Suggested Unit Resources
Refer to page 23.
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Board Endorsed October 2015
Unit 6: Specialist Mathematics T
Value: 1.0
Unit 6a: Specialist Mathematics IB
Value: 0.5
Unit 6b: Specialist Mathematics IB
Value: 0.5
Prerequisites
Nil
Duplication of Content Rules
Nil
Specific Unit Goals
By the end of this unit, students:

understand the concepts and techniques in logic or abstract algebra or discrete mathematics or
calculus

apply reasoning skills and solve problems in logic or abstract algebra or discrete mathematics or
calculus

communicate their arguments and strategies when solving problems

construct proofs of results

interpret mathematical information and ascertain the reasonableness of their solutions to
problems.
Content – Choose two of the following four topics for Unit 6:
Topic 1 : Logic and Proof
The nature of proof:

truth tables

use implication, converse, equivalence, negation, contrapositive

argument, validity, syllogisms

venn diagrams

use proof by contradiction

use the symbols for implication (), equivalence (), and equality (=)

use the quantifiers ‘for all’ and ‘there exists’
 use examples and counter-examples.
Proof Patterns:
 proof by counter-example
 contradiction
 deduction
 mathematical induction
Proof in the context of Euclidean Geometry
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Board Endorsed October 2015
Topic 2: Abstract Algebra
Sets:

finite and infinite sets. Subsets and Operations on sets

De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).
 ordered pairs: the Cartesian product of two sets.
Relations:

relations: equivalence relations; equivalence classes

functions: injections; surjections; bijections

composition of functions and inverse functions

binary Operations

operation tables (cayley tables)

binary operations: associative, distributive and commutative properties

the identity element e
 the inverse of an element
Groups:

the definition and examples of a group

permutations under composition of permutations

subgroups

cyclical groups, including the proof that cyclical groups are Abelian

homomorphism and Isomorphism
Topic 3: Discrete Maths
Methods of proof:
 strong Induction and Pigeon Hole principle
Divisibility and prime numbers:
 divisibility, gcd, lcm, primes and Euclidean Algorithm
Linear Diophantine Equations
Modular arithmetic:

modular Arithmetic, linear congruence

representation of integers in different bases
 Fermat’s Little Theorem
Graph theory:

graph definitions and terminology

simple graphs, connected graphs, complete graphs, bipartite graphs, planar graphs, trees weighted
graphs, subgraphs

Eulerian trails and circuits
 Hamiltonian paths and cycles
Algorithms on graphs:

Kruskal’s Dijkstra’s algorthims

Chinese postman problems

travelling salesman problems
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Board Endorsed October 2015
 nearest-neighbour algorithm
Recurrence relations:

solution to linear homogeneous recurrence relations with constant coefficients

modelling with recurrence relations
Topic 4: Further Calculus
Infinite series:

infinite sequences

convergence of infinite series

convergence tests: comparison test, limit comparison test, ratio test, integral test
 P series, alternating series (abs and conditionally), power series
Limits of sequences and functions:

continuity and differentiability of a function at a point

Riemann sums (upper, lower)

fundamental Theorem of Calculus
 L’Hopital’s rule
Differential Equations:

Euler’s method

homogeneous differential equations using y = vx
 solutions using integrating factor
Maclaurin and Taylor series:

Rolle’s Theorem, Mean value theorem,

Taylor polynomials, Lagrange form of the error term

Maclaurin series for e, sinx, cosx, ln (1+x)

use of substitution, products, integration and differentiation to obtain other series

Taylor series from differential equations.
Teaching and Learning Strategies
Refer to page 14.
Assessment
Refer to Assessment Criteria, Task Types table on pages 15-16.
Suggested Unit Resources
Refer to page 23.
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Board Endorsed October 2015
Unit 7: Specialist Mathematics T
Value: 1.0
Unit 7a: Specialist Mathematics IB
Value: 0.5
Unit 7b: Specialist Mathematics IB
Value: 0.5
Prerequisites
Nil
Duplication of Content Rules
Nil
Specific Unit Goals
By the end of this unit, students:

understand the concepts and techniques in applications of statistical inference

apply reasoning skills and solve problems in applications of statistical inference

communicate their arguments and strategies when solving problems

construct proofs of results

interpret mathematical and statistical information and ascertain the reasonableness of their
solutions to problems.
Content
Topic 1: Statistics

cumulative distribution functions for discrete and continuous distributions

poisson, Geometric, negative binomial distributions

probability generating functions

linear transformations of single random variable; mean, variance, expectation
Random sampling:
 understand the concept of a random sample
 discuss sources of bias in samples, and procedures to ensure randomness
 use graphical displays of simulated data to investigate the variability of random samples from various
types of distributions, including uniform, normal and Bernoulli.
Samples:
 central Limit Theorem
 understand the concept of the sample proportion 𝑝̂ as a random variable whose value varies
between samples, and the formulas for the mean 𝑝 and standard deviation √(𝑝(1 − 𝑝)/𝑛 of the
sample proportion 𝑝̂
 unbiased estimators and estimates for mean and variance.
 confidence intervals for the mean or a normal population.
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Board Endorsed October 2015
 examine the approximate normality of the distribution of 𝑝̂ for large samples
 simulate repeated random sampling, for a variety of values of p and a range of sample sizes, to
illustrate the distribution of 𝑝̂ and the approximate standard normality of
𝑝̂ −𝑝
√(𝑝̂(1−𝑝̂)/𝑛
where the
closeness of the approximation depends on both n and p.
Confidence intervals for proportions:
 the concept of an interval estimate for a parameter associated with a random variable
 use the approximate confidence interval (𝑝̂ − 𝑧√(𝑝̂ (1 − 𝑝̂ )/𝑛, 𝑝̂ + 𝑧√(𝑝̂ (1 − 𝑝̂ )/𝑛), as an interval
estimate for 𝑝, where 𝑧 is the appropriate quantile for the standard normal distribution
 define the approximate margin of error 𝐸 = 𝑧√(𝑝̂ (1 − 𝑝̂ )/𝑛 and understand the trade-off between
margin of error and level of confidence
 use simulation to illustrate variations in confidence intervals between samples and to show that most
but not all confidence intervals contain 𝑝.
Students’ t-distribution
Topic 2: Statistics Extension
Hypothesis Testing:
 null and alternative hypotheses
 significance level
 critical regions
 type 1 and 11 errors
 testing hypothesis for mean of a normal population
 hypothesis testing in binomial, Poisson, using z-test and t-tests, binomial proportion, difference
between means of two normal populations.

 2 for uniform and given ratio

 2 for binomial, poisson, normal

 2 for independence
Covariance and correlation:

covariance and product moment correlation coefficient

the use of the product moment correlation coefficient as a measure of association between two
variables

the use of the sample product moment correlation coefficient in applications.
 scatter diagrams and correlation coefficient
Linear regression:

linear regression using least squares estimates

use of regression for prediction
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Board Endorsed October 2015
Teaching and Learning Strategies
Refer to page 14.
Assessment
Refer to Assessment Criteria, Task Types table on pages 15-16.
Suggested Unit Resources
Refer to page 23.
44
Board Endorsed October 2015
Unit 8: Specialist Mathematics T
Value: 1.0
Unit 8a: Specialist Mathematics IB
Value: 0.5
Unit 8b: Specialist Mathematics IB
Value: 0.5
Prerequisites
Nil
Duplication of Content Rules
Nil
Specific Unit Goals
By the end of this unit, students:

understand the concepts and techniques in logic or abstract algebra or discrete mathematics or
calculus or conics

apply reasoning skills and solve problems in logic or abstract algebra or discrete mathematics or
calculus or conics

communicate their arguments and strategies when solving problems

construct proofs of results

interpret mathematical information and ascertain the reasonableness of their solutions to
problems.
Content – Choose two of the following five topics for Unit 8
Topic 1 : Logic and Proof
The nature of proof:

truth tables

use implication, converse, equivalence, negation, contrapositive

argument, validity, syllogisms

venn diagrams

use proof by contradiction

use the symbols for implication (), equivalence (), and equality (=)

use the quantifiers ‘for all’ and ‘there exists’
 use examples and counter-examples.
Proof Patterns:
 proof by counter-example
 contradiction
 deduction
 mathematical induction
Proof in the context of Euclidean Geometry
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Board Endorsed October 2015
Topic 2: Abstract Algebra
Sets:

finite and infinite sets. Subsets and Operations on sets

De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).
 ordered pairs: the Cartesian product of two sets.
Relations:

relations: equivalence relations; equivalence classes

functions: injections; surjections; bijections

composition of functions and inverse functions

binary Operations

operation tables (cayley tables)

binary operations: associative, distributive and commutative properties

the identity element e
 the inverse of an element
Groups:

the definition and examples of a group

permutations under composition of permutations

subgroups

homomorphism and Isomorphism
Topic 3: Discrete Maths
Methods of proof:
 strong Induction and Pigeon Hole principle
Divisibility and prime numbers:
 divisibility, gcd, lcm, primes and Euclidean Algorithm
Linear Diophantine Equations
Modular arithmetic:

modular Arithmetic, linear congruence

representation of integers in different bases
 Fermat’s Little Theorem
Graph theory:

graph definitions and terminology

simple graphs, connected graphs, complete graphs, bipartite graphs, planar graphs, trees weighted
graphs, subgraphs

Eulerian trails and circuits
 Hamiltonian paths and cycles
Algorithms on graphs:

Kruskal’s Dijkstra’s algorthims

Chinese postman problems

travelling salesman problems

nearest-neighbour algorithm
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Recurrence relations:

solution to linear homogeneous recurrence relations with constant coefficients

modelling with recurrence relations
Topic 4: Further Calculus
Infinite series:

infinite sequences

convergence of infinite series

convergence tests: comparison test, limit comparison test, ratio test, integral test
 P series, alternating series (abs and conditionally), power series
Limits of sequences and functions:

continuity and differentiability of a function at a point

Riemann sums (upper, lower)

fundamental Theorem of Calculus
 L’Hopital’s rule
Improper integrals :
 first order differential equations, slope fields
Differential Equations:

Euler’s method

homogeneous differential equations using y = vx
 solutions using integrating factor
Maclaurian and Taylor series:

Rolle’s Theorem, Mean value theorem,

Taylor polynomials, Lagrange form of the error term

Maclaurian series for e, sinx, cosx, ln (1+x)

use of substitution, products, integration and differentiation to obtain other series

Taylor series from differential equations.
Topic 5: Conics and Independent Project
Conics sections:
 geometric sections of a right circular cone to obtain circle, parabola, ellipse, hyperbola
 Locus definitions, including terms – focus, directrix, eccentricity
 features and properties of circle, ellipse, parabola and hyperbola
 Cartesian, polar and parametric equations of circle, ellipse, parabola and hyperbola
 chords, tangents and normals
 applications of conics
Independent Project:

students engage in mathematical research to enable them to showcase their understanding of a
topic chosen by them. It is envisioned that the final product will be a presentation and/or a paper
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Teaching and Learning Strategies
Refer to page 14.
Assessment
Refer to Assessment Criteria, Task Types table on pages 15-16.
Suggested Unit Resources
Refer to page 23.
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Appendix A – Common Curriculum Elements
Common curriculum elements assist in the development of high quality assessment tasks by encouraging
breadth and depth and discrimination in levels of achievement.
Organisers
Elements
Examples
create, compose
and apply
analyse,
synthesise and
evaluate
organise,
sequence and
explain
identify,
summarise and
plan
apply
ideas and procedures in unfamiliar situations, content and processes
in non-routine settings
compose
oral, written and multimodal texts, music, visual images, responses
to complex topics, new outcomes
represent
images, symbols or signs
create
creative thinking to identify areas for change, growth and innovation,
recognise opportunities, experiment to achieve innovative solutions,
construct objects, imagine alternatives
manipulate
images, text, data, points of view
justify
arguments, points of view, phenomena, choices
hypothesise
statement/theory that can be tested by data
extrapolate
trends, cause/effect, impact of a decision
predict
data, trends, inferences
evaluate
text, images, points of view, solutions, phenomenon, graphics
test
validity of assumptions, ideas, procedures, strategies
argue
trends, cause/effect, strengths and weaknesses
reflect
on strengths and weaknesses
synthesise
data and knowledge, points of view from several sources
analyse
text, images, graphs, data, points of view
examine
data, visual images, arguments, points of view
investigate
issues, problems
sequence
text, data, relationships, arguments, patterns
visualise
trends, futures, patterns, cause and effect
compare/contrast
data, visual images, arguments, points of view
discuss
issues, data, relationships, choices/options
interpret
symbols, text, images, graphs
explain
explicit/implicit assumptions, bias, themes/arguments, cause/effect,
strengths/weaknesses
translate
data, visual images, arguments, points of view
assess
probabilities, choices/options
select
main points, words, ideas in text
reproduce
information, data, words, images, graphics
respond
data, visual images, arguments, points of view
relate
events, processes, situations
demonstrate
probabilities, choices/options
describe
data, visual images, arguments, points of view
plan
strategies, ideas in text, arguments
classify
information, data, words, images
identify
spatial relationships, patterns, interrelationships
summarise
main points, words, ideas in text, review, draft and edit
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Appendix A – Common Curriculum Elements
Glossary of Verbs
Verbs
Definition
Analyse
Consider in detail for the purpose of finding meaning or relationships, and identifying patterns,
similarities and differences
Apply
Use, utilise or employ in a particular situation
Argue
Give reasons for or against something
Assess
Make a Judgement about the value of
Classify
Arrange into named categories in order to sort, group or identify
Compare
Estimate, measure or note how things are similar or dissimilar
Compose
The activity that occurs when students produce written, spoken, or visual texts
Contrast
Compare in such a way as to emphasise differences
Create
Bring into existence, to originate
Demonstrate
Give a practical exhibition an explanation
Describe
Give an account of characteristics or features
Discuss
Talk or write about a topic, taking into account different issues or ideas
Evaluate
Examine and judge the merit or significance of something
Examine
Determine the nature or condition of
Explain
Provide additional information that demonstrates understanding of reasoning and /or
application
Extrapolate
Infer from what is known
Hypothesise
Put forward a supposition or conjecture to account for certain facts and used as a basis for
further investigation by which it may be proved or disproved
Identify
Recognise and name
Interpret
Draw meaning from
Investigate
Plan, inquire into and draw conclusions about
Justify
Show how argument or conclusion is right or reasonable
Manipulate
Adapt or change
Plan
Strategies, develop a series of steps, processes
Predict
Suggest what might happen in the future or as a consequence of something
Reflect
The thought process by which students develop an understanding and appreciation of their
own learning. This process draws on both cognitive and affective experience
Relate
Tell or report about happenings, events or circumstances
Represent
Use words, images, symbols or signs to convey meaning
Reproduce
Copy or make close imitation
Respond
React to a person or text
Select
Choose in preference to another or others
Sequence
Arrange in order
Summarise
Give a brief statement of the main points
Synthesise
Combine elements (information/ideas/components) into a coherent whole
Test
Examine qualities or abilities
Translate
Express in another language or form, or in simpler terms
Visualise
The ability to decode, interpret, create, question, challenge and evaluate texts that
communicate with visual images as well as, or rather than, words
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Appendix B – Course Mappings
Specialist Mathematics IB/AC Mapping to IB HL AC Courses
Please note: While many of the topic names in the Specialist Mathematics IB/AC course are similar to those
used in the AC courses, the IB/AC course topics often include further content not offered in the AC courses.
Unit
International
Baccalaureate HL
Australian Curriculum ACT
Specialist Mathematics (SM) or
Mathematical Methods (MM)
Course
Unit 1
 Functions and graphs
 Trigonometric functions
 IB HL Topic 2
 IB HL Topic 3
 MM Unit 1 - Functions and
graphs
AND
SM Unit 3 – Functions and
sketching graphics
 MM Unit 1 - Trigonometric
Functions
AND
SM Unit 2 – Trigonometry
Unit 2
 Sequences and Series
 Exponential functions
 Differential Calculus
 IB HL Topic 1: 1.1
 IB HL Topic 1: 1.2
 IB HL Topic 6: 6.1-6.3
 MM Unit 2 - Arithmetic and
geometric sequences and
series
 MM Unit 2 – Exponential
functions
 MM Unit 2 - Introduction to
differential calculus
AND
MM Unit 3 - Further
differentiation and applications
 IB HL Topic 1: 1.2
 IB HL Topic 6: 6.4-6.7
 IB HL Topic 9: some of
9.5
 MM Unit 4 - The
logarithmic
function
 MM Unit 3 – Integrals
Unit 3
 The logarithm function
 Integrals
 Differential Equations
AND
SM Unit 4 – Integration and
applications of integrations
 SM Unit 4 – Rates of change and
differential equations
Unit 4
 Counting and probability
 Basic statistics
 Matrices
 IB HL Topic 5: 5.2, 5.3,
5.4 AND IB HL Topic 1:
1.3
 IB HL Topic 5: 5.5, 5.6,
5.7
 IB HL Topic 1: 1.9
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 MM Unit 1 – Counting
and probability
AND
SM Unit 1 –
Combinatorics
 MM Unit 3 – Discrete
random variables
 SM Unit 2 – Matrices
Board Endorsed October 2015
Unit 5
 Vectors
 Complex numbers
 IB HL Topic 4
 IB HL Topic 1: 1.5 – 1.8
 SM Unit 1 – Vectors in the
plane
AND
SM Unit 3 – Vectors in 3D
 SM Unit 2 – Real and
complex numbers AND
SM Unit 3 – Complex
Numbers
Unit 6 and 8
 Logic and proof
 Abstract Algebra
 Discrete Maths
 Further Calculus
 Conics and independent
project
It is a requirement that
theory of knowledge be
integrated in all IB courses.
The logic and proof and
conics and independent
project topics meet this
requirement.





SM Unit 1 – Geometry
N/A
N/A
N/A
N/A
Abstract Algebra – IB HL
Topic 8
Discrete Maths – IB HL
Topic 10
Further Calculus – IB HL
Topic 9
Unit 7
 Statistics
 Statistics Extension
IB HL Topic 5: 5.1, 5.6
AND
IB HL Topic 7
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 SM Unit 4 – Statistical
Inference