AC Mathematical Methods T 14-20 Updated
AC Mathematical Methods T 14-20 Updated
AC Mathematical Methods T 14-20 Updated
Mathematical Methods
integrating Australian Curriculum
Table of Contents
The ACT Senior Secondary System .........................................................................................1
ACT Senior Secondary Certificate .........................................................................................2
Learning Principles .........................................................................................3
General Capabilities .........................................................................................4
Cross-Curriculum Priorities .........................................................................................6
Rationale .........................................................................................7
Goals .........................................................................................8
Unit Titles .........................................................................................8
Organisation of Content .........................................................................................9
Assessment .......................................................................................11
Achievement Standards .......................................................................................13
Unit 1: Mathematical Methods Value: 1.0.......................................................................15
Unit 2: Mathematical Methods Value: 1.0.......................................................................19
Unit 3: Mathematical Methods Value: 1.0.......................................................................22
Unit 4: Mathematical Methods Value: 1.0.......................................................................26
Unit 5: Mathematical Methods Value: 1.0.......................................................................29
Appendix A – Implementation Guidelines .......................................................................................30
Appendix B – Course Developers .......................................................................................33
Appendix C – Common Curriculum Elements .......................................................................................34
Appendix D – Glossary of Verbs .......................................................................................35
Appendix E – Glossary for ACT Senior Secondary Curriculum...............................................................36
Appendix F – Glossary for Mathematical Methods...............................................................................37
Appendix G – Course Adoption Form .......................................................................................49
Board Endorsed November 2014 (Amended October 2016)
ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
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Underpinning beliefs
All students are able to learn.
Learning is a partnership between students and teachers.
Teachers are responsible for advancing student learning.
Learning Principles
1. Learning builds on existing knowledge, understandings and skills.
(Prior knowledge)
2. When learning is organised around major concepts, principles and significant real world issues,
within and across disciplines, it helps students make connections and build knowledge structures.
(Deep knowledge and connectedness)
3. Learning is facilitated when students actively monitor their own learning and consciously develop
ways of organising and applying knowledge within and across contexts.
(Metacognition)
4. Learners’ sense of self and motivation to learn affects learning.
(Self-concept)
5. Learning needs to take place in a context of high expectations.
(High expectations)
6. Learners learn in different ways and at different rates.
(Individual differences)
7. Different cultural environments, including the use of language, shape learners’ understandings
and the way they learn.
(Socio-cultural effects)
8. Learning is a social and collaborative function as well as an individual one.
(Collaborative learning)
9. Learning is strengthened when learning outcomes and criteria for judging learning are made
explicit and when students receive frequent feedback on their progress.
(Explicit expectations and feedback)
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General Capabilities
All courses of study for the ACT Senior Secondary 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)
critical and creative thinking
personal and social
ethical behaviour
intercultural understanding
Courses of study for the ACT Senior Secondary Certificate should be both relevant to the lives of
students and incorporate the contemporary issues they face. Hence, courses address the following
three priorities. These priorities are:
Aboriginal and Torres Strait Islander histories and cultures
Asia and Australia’s engagement with Asia
Sustainability
Elaboration of these General Capabilities and priorities is available on the ACARA website at
www.australiancurriculum.edu.au.
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.
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Cross-Curriculum Priorities
Aboriginal and Torres Strait Islander Histories and Cultures
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.
Asia and Australia’s Engagement with Asia
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.
Sustainability
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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Mathematical Methods T
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.
The major themes of Mathematical Methods are calculus and statistics. They include as necessary
prerequisites studies of algebra, functions and their graphs, and probability. They are developed
systematically, with increasing levels of sophistication and complexity. Calculus is essential for
developing an understanding of the physical world because many of the laws of science are
relationships involving rates of change. Statistics is used to describe and analyse phenomena
involving uncertainty and variation. For these reasons this subject provides a foundation for further
studies in disciplines in which mathematics and statistics have important roles. It is also
advantageous for further studies in the health and social sciences. In summary, the subject
Mathematical Methods is designed for students whose future pathways may involve mathematics
and statistics and their applications in a range of disciplines at the tertiary level.
For all content areas of Mathematical Methods, the proficiency strands of the F-10 curriculum are
still applicable and should be inherent in students’ learning of this subject. These strands are
Understanding, Fluency, Problem solving and Reasoning, and they are both essential and mutually
reinforcing. For all content areas, practice allows students to achieve fluency in skills, such as
calculating derivatives and integrals, or solving quadratic equations, and frees up working memory
for more complex aspects of problem solving. The ability to transfer skills to solve problems based
on a wide range of applications is a vital part of mathematics in this subject. Because both calculus
and statistics are widely applicable as models of the world around us, there is ample opportunity for
problem solving throughout this subject.
Mathematical Methods is structured over four units. The topics in Unit 1 build on students’
mathematical experience. The topics ‘Functions and graphs’, ‘Trigonometric functions’ and ‘Counting
and probability’ all follow on from topics in the F-10 curriculum from the strands, Number and
Algebra, Measurement and Geometry and Statistics and Probability. In Mathematical Methods there
is a progression of content and applications in all areas. For example, in Unit 2 differential calculus is
introduced, and then further developed in Unit 3 where integral calculus is introduced. Discrete
probability distributions are introduced in Unit 3, and then continuous probability distributions and
an introduction to statistical inference conclude Unit 4.
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Goals
Mathematical Methods aims to develop students’:
understanding of concepts and techniques drawn from algebra, the study of functions,
calculus, probability and statistics
ability to solve applied problems using concepts and techniques drawn from algebra,
functions, calculus, 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.
Student Group
Links to Foundation to Year 10
In Mathematical Methods, there is a strong emphasis on mutually reinforcing proficiencies in
Understanding, Fluency, Problem solving and Reasoning. Students gain fluency in a variety of
mathematical and statistical skills, including algebraic manipulations, constructing and interpreting
graphs, calculating derivatives and integrals, applying probabilistic models, estimating probabilities
and parameters from data, and using appropriate technologies. Achieving fluency in skills such as
these allows students to concentrate on more complex aspects of problem solving. In order to study
Mathematical Methods, it is desirable that students complete topics from 10A. The knowledge and
skills from the following content descriptions from 10A are highly recommended for the study of
Mathematical Methods:
define rational and irrational numbers, and perform operations with surds and fractional
indices
factorise monic and non-monic quadratic expressions, and solve a wide range of quadratic
equations derived from a variety of contexts
calculate and interpret the mean and standard deviation of data, and use these to compare
datasets.
Unit Titles
Unit 1: Mathematical Methods
Unit 2: Mathematical Methods
Unit 3: Mathematical Methods
Unit 4: Mathematical Methods
Unit 5: Mathematical Methods
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Organisation of Content
Mathematical Methods focuses on the development of the use of calculus and statistical analysis.
The study of calculus in Mathematical Methods provides a basis for an understanding of the physical
world involving rates of change, and includes the use of functions, their derivatives and integrals, in
modelling physical processes. The study of statistics in Mathematical Methods develops the ability to
describe and analyse phenomena involving uncertainty and variation.
Mathematical Methods is organised into four units. The topics broaden students’ mathematical
experience and provide different scenarios for incorporating mathematical arguments and problem
solving. The units provide a blending of algebraic and geometric thinking. In this subject there is a
progression of content, applications, level of sophistication and abstraction. The probability and
statistics topics lead to an introduction to statistical inference.
Unit 1 Unit 2 Unit 3 Unit 4
Mathematical Functions and Exponential Further The
Methods graphs functions differentiation logarithmic
Trigonometric Arithmetic and and function
functions geometric applications Continuous
Counting and sequences and Integrals random
probability series Discrete variables and
Introduction to random the normal
differential variables distribution
calculus Interval
estimates for
proportions
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Assessment
The identification of criteria within the achievement standards 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 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 B). It is
highly desirable that assessment tasks engage students in demonstrating higher order thinking.
Rubrics are constructed for individual tasks, informing 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.
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 and understanding 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.
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Achievement Standards
Years 11 and 12 achievement standards are written for A/T courses. A single achievement standard
is written for M courses.
A Year 12 student in any unit is assessed using the Year 12 achievement standards. A Year 11 student
in any unit is assessed using the Year 11 achievement standards. Year 12 achievement standards
reflect higher expectations of student achievement compared to the Year 11 achievement
standards. Years 11 and 12 achievement standards are differentiated by cognitive demand, the
number of dimensions and the depth of inquiry.
An achievement standard cannot be used as a rubric for an individual assessment task. Assessment
is the responsibility of the college. Student tasks may be assessed using rubrics or marking schemes
devised by the college. A teacher may use the achievement standards to inform development of
rubrics. The verbs used in achievement standards may be reflected in the rubric. In the context of
combined Years 11 and 12 classes, it is best practice to have a distinct rubric for Years 11 and 12.
These rubrics should be available for students prior to completion of an assessment task so that
success criteria are clear.
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level of proficiency in the proficiency in the use of proficiency in the use of of mathematical facts, use of mathematical facts,
use of mathematical facts, mathematical facts, mathematical facts, techniques and formulae techniques and formulae
techniques and formulae techniques and formulae techniques and formulae studied studied
studied
selects, extends and applies selects and applies with direction, applies a solves some mathematical solves some mathematical
Application
appropriate mathematical appropriate mathematical mathematical model. problems independently problems with guidance
modelling and problem modelling and problem solves most problems
solving techniques solving techniques
uses mathematical uses mathematical uses some mathematical uses some mathematical uses limited reasoning to
reasoning to develop reasoning to develop reasoning to develop reasoning to develop justify conclusions
Reasoning
is consistently accurate and is generally accurate and presents mathematical presents some presents some
Communication
appropriate in presentation appropriate in presentation ideas in different contexts mathematical ideas mathematical ideas with
of mathematical ideas in of mathematical ideas in guidance
different contexts different contexts
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Content Descriptions
Further elaboration on the content of this unit is available at:
http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/Mathematical-
Methods/Curriculum/SeniorSecondary
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Inverse proportion:
examine examples of inverse proportion (MMT12)
1 a
recognise features of the graphs of y= and y= , including their hyperbolic shapes,
x x−b
and their asymptotes. (MMT13)
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 →−∞ (MMT14)
identify the coefficients and the degree of a polynomial (MMT15)
expand quadratic and cubic polynomials from factors (MMT16)
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 →−∞ (MMT17)
factorise cubic polynomials in cases where a linear factor is easily obtained (MMT18)
solve cubic equations using technology, and algebraically in cases where a linear factor is
easily obtained. (MMT19)
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 (MMT20)
recognise features of the graph of y 2=x including its parabolic shape and its axis of
symmetry. (MMT21)
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 (MMT22)
use function notation, domain and range, independent and dependent variables (MMT123)
understand the concept of the graph of a function (MMT24)
examine translations and the graphs of y=f ( x )+ a and y=f ( x +b) (MMT25)
examine dilations and the graphs of y=cf ( x ) and y=f ( kx ) (MMT26)
recognise the distinction between functions and relations, and the vertical line test. (MMT27)
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Trigonometric functions:
understand the unit circle definition of cos θ , sinθ and tanθ and periodicity using radians
(MMT34)
π π
recognise the exact values of sin θ , cos θ and tan θ at integer multiples of ∧¿ (MMT35)
6 4
recognise the graphs of y=sin x , y=cos x , and y =tan x on extended domains (MMT36)
examine amplitude changes and the graphs of y=a sin x and y=a cos x (MMT37)
examine period changes and the graphs of y=sin bx , y=cos bx , and y=tan bx (MMT38)
examine phase changes and the graphs of y=sin( x+ c) , y=cos(x +c ) and (MMT39)
y=tan (x +c )and the relationships sin x+ ( π2 )=cos x and cos ( x− π2 )=sin x (MMT40)
prove and apply the angle sum and difference identities (MMT41)
identify contexts suitable for modelling by trigonometric functions and use them to solve
practical problems (MMT42)
solve equations involving trigonometric functions using technology, and algebraically in simple
cases. (MMT43)
recognise the numbers (nr ) as binomial coefficients, (as coefficients in the expansion of
( x + y )n) (MMT47)
use Pascal’s triangle and its properties. (MMT48)
Language of events and sets:
review the concepts and language of outcomes, sample spaces and events as sets of outcomes
(MMT49)
use set language and notation for events, including Á (or A' ¿ for the complement of an event
A , A ∩ B for the intersection of events A and B, and A ∪B for the union, and recognise
mutually exclusive events (MMT50)
use everyday occurrences to illustrate set descriptions and representations of events, and set
operations. (MMT51)
Review of the fundamentals of probability:
review probability as a measure of ‘the likelihood of occurrence’ of an event (MMT52)
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 (MMT53)
review the rules: P( A ')=1−P( A) and P ( A ∪ B ) =P ( A )+ P ( B )−P ( A ∩B ) (MMT54)
use relative frequencies obtained from data as point estimates of probabilities. (MMT55)
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establish and use the formula P ( A ∩ B )=P ( A ) P(B) for independent events A and B, and
recognise the symmetry of independence (MMT59)
use relative frequencies obtained from data as point estimates of conditional probabilities and
as indications of possible independence of events. (MMT60)
Assessment
Refer to pages 11-13.
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Content Descriptions
Further elaboration on the content of this unit is available at:
http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/Mathematical-
Methods/Curriculum/SeniorSecondary
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Geometric sequences:
recognise and use the recursive definition of a geometric sequence: t n+1=r t n (MMT12)
n−1
use the formula t n=r t 1 for the general term of a geometric sequence and recognise its
exponential nature (MMT13)
understand the limiting behaviour as n → ∞ of the terms t n in a geometric sequence and its
dependence on the value of the common ratio r (MMT14)
r n −1
establish and use the formula Sn=t 1 for the sum of the first n terms of a geometric
r−1
sequence (MMT15)
use geometric sequences in contexts involving geometric growth or decay, such as compound
interest. (MMT16)
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interpret the derivative as the slope or gradient of a tangent line of the graph of y=f (x ).
(MMT25)
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Computation of derivatives:
estimate numerically the value of a derivative, for simple power functions (MMT26)
examine examples of variable rates of change of non-linear functions (MMT27)
d n
establish the formula ( x )=n x n−1 for positive integers n by expanding ( x +h)n or by
dx
factorising ( x +h) −x n. (MMT28)
n
Properties of derivatives:
understand the concept of the derivative as a function (MMT29)
recognise and use linearity properties of the derivative (MMT30)
calculate derivatives of polynomials and other linear combinations of power functions.
(MMT31)
Applications of derivatives:
find instantaneous rates of change (MMT32)
find the slope of a tangent and the equation of the tangent (MMT33)
construct and interpret position-time graphs, with velocity as the slope of the tangent (MMT34)
sketch curves associated with simple polynomials; find stationary points, and local and global
maxima and minima; and examine behaviour as x → ∞ and x →−∞ (MMT35)
solve optimisation problems arising in a variety of contexts involving simple polynomials on
finite interval domains. (MMT36)
Anti-derivatives:
calculate anti-derivatives of polynomial functions and apply to solving simple problems
involving motion in a straight line. (MMT37)
Assessment
Refer to pages 11-13.
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Content Descriptions
Further elaboration on the content of this unit is available at:
http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/Mathematical-
Methods/Curriculum/SeniorSecondary
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Topic 2: Integrals
Anti-differentiation:
recognise anti-differentiation as the reverse of differentiation (MMT17)
use the notation ∫ f ( x ) dx for anti-derivatives or indefinite integrals (MMT18)
n 1 n+1
establish and use the formula ∫ x dx= x + c for n ≠−1 (MMT19)
n+1
x x
establish and use the formula ∫ e dx=e + c (MMT20)
establish and use the formulas ∫ sin x dx=−cos x +c and ∫ cos x dx=sin x +c (MMT21)
recognise and use linearity of anti-differentiation (MMT22)
determine indefinite integrals of the form ∫ f ( ax+ b ) dx (MMT23)
identify families of curves with the same derivative function (MMT24)
determine f ( x ) , given f ' ( x ) and an initial condition f ( a )=b (MMT25)
determine displacement given velocity in linear motion problems. (MMT26)
Definite integrals:
examine the area problem, and use sums of the form ∑ f ( x i ) δ xi to estimate the area
i
under the curve y=f (x ) (MMT27)
b
interpret the definite integral ∫ f ( x ) dx as area under the curve y=f ( x ) if f ( x ) >0 (MMT28)
a
b
recognise the definite integral ∫ f ( x ) dx as a limit of sums of the form ∑ f ( x i ) δ xi (MMT29)
i
a
b
interpret ∫ f ( x ) dx as a sum of signed areas (MMT30)
a
recognise and use the additivity and linearity of definite integrals. (MMT31)
Fundamental theorem:
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
x
understand the concept of the signed area function F ( x )=∫ f ( t ) dt (MMT32)
a
x
d
'
understand and use the theorem: F ( x )=
dx (∫ )
a
f ( t ) dt =f ( x ), and illustrate its proof
geometrically (MMT33)
b
understand the formula ∫ f ( x ) dx=F ( b )−F (a) and use it to calculate definite integrals.
a
(MMT34)
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
Applications of integration:
calculate the area under a curve (MMT35)
calculate total change by integrating instantaneous or marginal rate of change (MMT36)
calculate the area between curves in simple cases (MMT37)
determine positions given acceleration and initial values of position and velocity. (MMT38)
Bernoulli distributions:
use a Bernoulli random variable as a model for two-outcome situations (MMT46)
identify contexts suitable for modelling by Bernoulli random variables (MMT47)
recognise the mean pand variance p ( 1− p ) of the Bernoulli distribution with parameter p
(MMT48)
use Bernoulli random variables and associated probabilities to model data and solve practical
problems. (MMT49)
Binomial distributions:
understand the concepts of Bernoulli trials and the concept of a binomial random variable as
the number of ‘successes’ in n independent Bernoulli trials, with the same probability of
success p in each trial (MMT50)
identify contexts suitable for modelling by binomial random variables (MMT51)
distribution with parameters n and p ; note the mean np and variance np ( 1− p )of a binomial
distribution (MMT52)
use binomial distributions and associated probabilities to solve practical problems. (MMT53)
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Assessment
Refer to pages 11-13.
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
Content Descriptions
Further elaboration on the content of this unit is available at:
http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/Mathematical-
Methods/Curriculum/SeniorSecondary
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Assessment
Refer to pages 11-13.
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their responsibility to teach all content descriptions. It is mandatory that teachers address all
content descriptions and that students engage with all content descriptions.
Half standard 0.5 units
Half standard units appear on the course adoption form but are not explicitly documented in
courses. It is at the discretion of the college principal to split a standard 1.0 unit into two half
standard 0.5 units. Colleges are required to adopt the half standard 0.5 units. However, colleges are
not required to submit explicit documentation outlining their half standard 0.5 units to the BSSS.
Colleges must assess students using the half standard 0.5 assessment task weightings outlined in the
framework. It is the responsibility of the college principal to ensure that all content is delivered in
units approved by the Board.
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
Review is a subcategory of moderation, comprising the review of standards and the validation of
Unit Grades. In the review process, Unit Grades, determined for Year 11 and Year 12 student
assessment portfolios that have been assessed in schools by teachers under accredited courses, are
moderated by peer review against system wide criteria and standards. This is done by matching
student performance with the criteria and standards outlined in the unit grade descriptors as stated
in the Course Framework. 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 teaching a Year 11 class are asked to retain originals or copies of student work
completed in Semester 2. Similarly, teachers teaching 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.
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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, T and M course/units offered by the school
and is sent in to the Office of the Board of Senior Secondary Studies.
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Name College
Jacob Woolley Canberra College
Gary Pocock Canberra Institute of Technology
Marion McIntosh Melba Copland Secondary School
Wayne Semmens Melba Copland Secondary School
Jennifer Missen Merici College
Nicole Burg Narrabundah College
Rebecca Guinane Narrabundah College
Andrew Trost Narrabundah College
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
n n
The expansion ( x + y ) =x + ( n1) x n−1
y + ⋯+ n x n−r y r + ⋯+ y n is known as the binomial theorem.
()
r
n! n ×(n−1)× ⋯ ×(n−r +1)
The numbers (nr )= r ! ( n−r )!
=
r ×(r−1)×⋯ × 2× 1
are called binomial coefficients.
b 2 b2
The quadratic expression a x 2+ bx+ c can be rewritten as a x+
( 2a ) (
+ c−
4a )
. Rewriting it in this
Function
A function f is a rule that associates with each element x in a set S a unique element f (x) in a set
T . We write x ⟼ f ( x) to indicate the mapping of x to f (x). The set Sis called the domain of f and
the set T is called the codomain. The subset of T consisting of all the elements f ( x ) : x ∈ S is called
the range of f . If we write y=f (x ) we say that x is the independent variable and y is the
dependent variable.
Graph of a function
The graph of a function f is the set of all points ( x , y ) in Cartesian plane where x is in the domain of
f and y=f ( x )
Quadratic formula
−b ± √ b2−4 ac
If a x 2+ bx+ c=0 with a ≠ 0 , then x= . This formula for the roots is called the
2a
quadratic formula.
Vertical line test
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
A relation between two real variables x and y is a function and y=f (x ) for some function f , if and
only if each vertical line, i.e. each line parallel to the y−¿ axis, intersects the graph of the relation in
at most one point. This test to determine whether a relation is, in fact, a function is known as the
vertical line test.
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
Trigonometric functions
Circular measure is the measurement of angle size in radians.
Radian measure
The radian measure θ of an angle in a sector of a circle is defined by θ=l/ r , where r is the radius
and l is the arc length. Thus an angle whose degree measure is 180 has radian measure π .
Length of an arc
The length of an arc in a circle is given by l=rθ , where l is the arc length,r is the radius and θ is the
angle subtended at the centre, measured in radians. This is simply a rearrangement of the formula
defining the radian measure of an angle.
Sine rule and cosine rule
The lengths of the sides of a triangle are related to the sines of its angles by the equations
a b c
= =
sin A sin B sin C
This is known as the sine rule.
The lengths of the sides of a triangle are related to the cosine of one of its angles by the equation
The period of a function f ( x ) is the smallest positive number p with the property that f ( x + p )=f ( x )
for all x . The functions sin x and cos x both have period 2 π and tan x has period π
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
Pascal’s triangle is a triangular arrangement of binomial coefficients. The nth row consists of the
binomial coefficients (nr ) , for 0 ≤ r ≤ n, each interior entry is the sum of the two entries above it,
and sum of the entries in the nth row is 2n
Conditional probability
The probability that an event A occurs can change if it becomes known that another event B occurs.
The new probability is known as a conditional probability and is written as P ( A|B ) . If B has
occurred, the sample space is reduced by discarding all outcomes that are not in the event B . The
new sample space, called the reduced sample space, is B . The conditional probability of event A is
P( A ∩ B)
given by P( A∨B)= .
P(B)
Independent events
Two events are independent if knowing that one occurs tells us nothing about the other. The
concept can be defined formally using probabilities in various ways: events A and B are independent
if P( A ∩ B)=P( A)P (B), if P ( A|B )=P( A) or if P ( B )=P ( B| A ) . For events A and Bwith non-
zero probabilities, any one of these equations implies any other.
Mutually exclusive
Two events are mutually exclusive if there is no outcome in which both events occur.
Point and interval estimates
In statistics estimation is the use of information derived from a sample to produce an estimate of an
unknown probability or population parameter. If the estimate is a single number, this number is
called a point estimate. An interval estimate is an interval derived from the sample that, in some
sense, is likely to contain the parameter.
A simple example of a point estimate of the probability p of an event is the relative frequency f of
the event in a large number of Bernoulli trials. An example of an interval estimate for p is a
confidence interval centred on the relative frequency f .
Relative frequency
If an event E occurs r times when a chance experiment is repeated n times, the relative frequency
of E is r /n .
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
Unit 2
Exponential functions
Index laws
1 −x x y
The index laws are the rules: a x a y =ax+ y, a = xy 0 x x x
x , ( a ) =a , a =1, and ( ab ) =a b , for any real
a
numbers x , y , a and b , with a> 0 and b> 0
Algebraic properties of exponential functions
−x 1
The algebraic properties of exponential functions are the index laws: a x a y =ax+ y, a = ,
ax
y
( a x ) =axy , a 0=1, for any real numbers x , y , and a, with a> 0
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d dy dy
and ( y 1 + y 2 )= 1 + 2
dx dx dx
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A stationary point on the graph y=f (x ) of a differentiable function is a point where f ' ( x )=0.
We say that f (x 0) is a local maximum of the function f (x) if f (x) ≤ f (x 0 ) for all values of x near x 0.
We say that f (x 0) is a global maximum of the function f (x) if f (x) ≤ f (x 0 ) for all values of x in the
domain of f .
We say that f (x 0) is a local minimum of the function f (x) if f (x) ≥ f ( x 0 ) for all values of x near x 0.
We say that f (x 0) is a global minimum of the function f (x) if f (x) ≥ f ( x 0 ) for all values of x in the
domain of f .
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Unit 3
d dv du
and in Leibniz notation: ( uv )=u + v
dx dx dx
Quotient rule
The quotient rule relates the derivative of the quotient of two functions to the functions and their
derivatives
du dv
−u v
and in Leibniz notation: d u dx dx
dx v
=
v 2()
Composition of functions
Chain rule
The chain rule relates the derivative of the composite of two functions to the functions and their
derivatives.
' ' '
If h ( x )=f ∘ g ( x ) then ( f ∘ g ) ( x )=f ( g ( x ) ) g ( x ) ,
dz dz dy
and in Leibniz notation: =
dx dy dx
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Point of inflection
A point P on the graph of y=f (x ) is a point of inflection if the concavity changes at P , i.e. points
near P on one side of P lie above the tangent at P and points near P on the other side of P lie
below the tangent at P
Second derivative test
According to the second derivative test, if f ' ( x )=0 , then f (x) is a local maximum of f if f ' ' ( x ) < 0
and f (x) is a local minimum if f ' ' ( x ) > 0
Integrals
Antidifferentiation
An anti-derivative, primitive or indefinite integral of a function f (x) is a function F (x) whose
derivative is f (x), i.e. F ' (x )=f ( x).
Anti-derivatives are not unique. If F (x) is an anti-derivative of f ( x ) , then so too is the function
F ( x ) +c where c is any number. We write ∫ f ( x ) dx=F ( x ) +c to denote the set of all anti-
derivatives of f ( x ) . The number c is called the constant of integration. For example, since
d 3
( x )=3 x 2 , we can write ∫ 3 x 2 dx=x 3+ c
dx
The linearity property of anti-differentiation
The linearity property of anti-differentiation is summarized by the equations:
b b b
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
The fundamental theorem of calculus relates differentiation and definite integrals. It has two forms:
x b
d
dx (
∫ f ( t ) dt
a
) =f ( x ) and ∫ f ' ( x ) dx=f ( b )−f (a)
a
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
pi=P ( X =x i ) .
∞
If X is continuous, E ( x )=∫ xp ( x ) dx ,where p(x ) is the probability density function of X
−∞
2
If X is discrete, Var ( X )=∑ p i( x i−μ) , where μ= E( X ) is the expected value.
i
∞
2
If X is continuous, Var ( X )= ∫ ( x−μ) p ( x ) dx
−∞
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
Bernoulli trial
A Bernoulli trial is a chance experiment with possible outcomes, typically labeled ‘success’ and
failure’.
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Unit 4
A quantile t α for a continuous random variable X is defined by P ( X >t α )=α , where 0< α <1.
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In the special case where X is a Bernoulli random variable with parameter p , X́ is the sample
proportion ^p , μ= p and σ =√ p (1− p). In this case the Central limit theorem is a statement that as
p^ − p
n → ∞the distribution of approaches the standard normal distribution.
√ p(1− p)/n
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
Margin of error
The margin of error of a confidence interval of the form f −E< p< f + E is E , the half-width of the
confidence interval. It is the maximum difference between f and p if p is actually in the confidence
interval.
Level of confidence
The level of confidence associated with a confidence interval for an unknown population parameter
is the probability that a random confidence interval will contain the parameter.
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ACT BSSS Mathematical Methods 2014-2020 - Board Endorsed November 2014 (Amended October 2015)
College:
Course Title: Mathematical Methods
Classification: T
(integrating Australian Curriculum)
Framework: Mathematics
Condition of 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
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