Mathematics Specialised

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. Mathematics is also concerned with collecting, analysing, modelling and interpreting data in order to investigate and understand real-world phenomena and solve problems in context. Mathematics provides a framework for thinking and a means of communication that is powerful, logical, concise and precise. It impacts upon the daily life of people everywhere and helps them to understand the world in which they live and work.

Mathematics Specialised is designed for learners with a strong interest in mathematics, including those intending to study mathematics, statistics, all sciences and associated fields, economics or engineering at university. This course provides opportunities, beyond those presented in Mathematics Methods Level 4, to develop rigorous mathematical arguments and proofs, and to use mathematical models more extensively.

Mathematics Specialised aims to develop learners':

• understanding of concepts and techniques drawn from algebraic processes, functions and equation study, complex numbers, matrices, calculus and statistics
• ability to solve applied problems using concepts and techniques drawn from algebraic processes, functions and equation study, complex numbers, matrices, calculus and statistics
• capacity to choose and use technology appropriately
• reasoning in mathematical contexts and interpretation of mathematical information, including ascertaining the reasonableness of solutions to problems
• capacity to communicate in a concise and systematic manner using appropriate mathematical and statistical language.

On successful completion of this course, learners will be able to:

1. be self-directing; be able to plan their study; persevere to complete tasks and meet deadlines
2. demonstrate an understanding of finite and infinite sequences and series
3. demonstrate an understanding of matrices and linear transformations
4. use differential calculus and apply integral calculus to areas and volumes
5. use techniques of integration and solve differential equations
6. demonstrate an understanding of complex numbers
7. choose and use technology appropriately and efficiently.

Programs of study derived from this course need to embrace the range of technological developments that have occurred in relation to mathematics teaching and learning.

Learners should have access to graphics calculators, including algebra-capable (CAS) calculators, and become proficient in their use. Graphics calculators can be used in this course in the development of concepts and as a tool for solving problems as appropriate. Refer to 'What can I take to my exam?' for the current TASC Calculator Policy that applies to Level 3 courses.

The use of computers is recommended as an aid to learners' learning and mathematical development. A range of packages is appropriate and, in particular, spreadsheets should be used.

This course has been assessed as having a complexity level of 4.

In general, courses at this level provide theoretical and practical knowledge and skills for specialised and/or skilled work and/or further learning, requiring:

• broad factual, technical and some theoretical knowledge of a specific area or a broad field of work and learning
• a broad range of cognitive, technical and communication skills to select and apply a range of methods, tools, materials and information to:
  • complete routine and non-routine activities; and
  • provide and transmit solutions to a variety of predictable and sometimes unpredictable problems
• application of knowledge and skills to demonstrate autonomy, judgement and limited responsibility in known or changing contexts and within established parameters.

This Level 4 course has a size value of 15.

This course has an academic mathematical focus.

The course content is presented under topic headings. The topics may be addressed separately but much of the content is inter-related and a more integrated approach is recommended and encouraged. It is also recommended that, where possible, concepts be developed within a context of practical applications. Such an approach provides learners with mathematical experiences that are much richer than a collection of skills. Learners thereby have the opportunity to observe and make connections between related aspects of the course and the real world and to develop further some important abstract ideas.

Through engaging in learning activities derived from this course learners have the opportunity to:

• gain deeper insight into the structure of mathematics
• meet intellectually challenging situations
• develop desirable attitudes towards mathematics.

ROLE OF TECHNOLOGY

It is assumed that learners will be delivered Mathematics Specialised with an extensive range of technological applications and techniques. If appropriately used, these have the potential to enhance the teaching and learning of mathematics. However, learners also need to continue to develop skills that do not depend on technology. The ability to be able to choose when or when not to use some form of technology and to be able to work flexibly with technology are important skills in this subject.

For the content areas of Mathematics Specialised, the proficiency strands – Understanding; Fluency; Problem Solving; and Reasoning – build on learners’ learning in F-10 Australian Curriculum: Mathematics, Mathematics Methods – Foundation Level 2, and Mathematics Methods Level 3. Each of these proficiencies is essential, and all are mutually reinforcing. They are still very much applicable and should be inherent in the study, and applications, of four (4) topics of mathematics:

• Sequences and Series
• Complex Numbers
• Matrices
• Calculus.

Each topic is compulsory and their content relates to Criteria 4 – 8. Criteria 1 – 3 apply to all four topics of mathematics.

SEQUENCES AND SERIES

The intention of this section of the course is that learners will gain experience with a range of sequences and series by learning about their properties, meet some of the important uses of them, begin to understand the ideas of convergence and divergence and develop some methods of proof.

This area of study will include:

• arithmetic and geometric sequences and series, including the development of formulae for the `n^"th"` term and the sum to `n` terms
• the “sum of infinity” of geometric series, and the conditions under which it exists
• the definition of a sequence as a function defined on the natural numbers
• the formal definition of a convergent sequence, and the broad definition of a divergent sequence as one which does not converge
• the formal definition of a sequence which diverges to either plus infinity or minus infinity, and an informal consideration of sequences which oscillate finitely or infinitely
• simple applications of the formal definitions to establish the convergence or divergence of given sequences
• consideration of the special sequences `{x^n}` and `{(1 + 1/n)^n}`
• recursive definitions and sigma notation
• mathematical induction applied to series
• the results `sum_(r=1)^nr=(n(n + 1))/2`, `sum_(r=1)^nr^2 = (n(n + 1)(2n + 1))/6` and `sum_(r=1)^nr^3 = (n^2(n + 1)^2)/4` established by a “method of differences” or by mathematical induction, and the sums of series utilising these results
• “methods of differences” applied to series such as `sum_(r=1)^nr(r + 1)(r + 2)`, `sum_(r=1)^n1/((2r - 1)(2r + 1)(2r + 3))` and `sum_(r=1)^oo1/((2r - 1)(2r + 1)(2r + 3))` 
• MacLaurin series for simple functions such as `(1 + x)^n`, `sin ax`, `cos ax`, `e^(ax)` and `ln(x + 1)`, with an informal consideration of interval of convergence.

COMPLEX NUMBERS

The intention of this section of the course is to introduce learners to a different class of numbers, to appreciate that such numbers can be represented in several ways and to understand that their use allows factorisation to be carried out more fully than was previously possible.

This area of study will include:

• the Cartesian form of a complex number `a + ib`, its real and imaginary parts, and fundamental operations involving complex numbers in this form
• representation of a complex number on the Argand plane as a point or vector
• the polar form of a complex number `r(cos theta + i sin theta)` or `r cis theta`
• Euler’s formula `e^(i theta) = cos theta + i sin theta`, justified using MacLaurin series or differential equations
• the modulus `|z|`, argument `arg(z)` and principal argument `Arg(z)` of a complex number
• multiplication and division of complex numbers in polar form, including the use of De Moivre’s theorem
• the results `cis theta + cis (-theta) = 2 cos theta`, `cis theta - cis (-theta) = 2i sin theta` and `(z - cis theta)(z - cis (-theta)) = z^2 - 2 cos theta.z + 1`
• the conjugate `bar z` of a complex number, and the result that `z.bar z = |z|^2`
• De Moivre’s theorem and its proof for rational exponents
• application of De Moivre’s theorem to solving equations of the form `z^n = p` where `n` is a positive integer, including sketching the solutions set on the Argand plane
• application of De Moivre’s theorem to factorising a polynomial into linear and real (quadratic) factors, and to simplifying expressions such as `(1 + i)^5(sqrt3 - i)^4`
• regions of the Argand plane satisfying simple functions – straight lines, polynomials, hyperbola, truncus and square roots as well as circles and ellipses and combinations of these, e.g. `|z| >= 4`, `pi/6 < Arg(z) <= (2pi)/3[Im(z - 1)]^2 + [Re(z + 1)]^2 = 1`, not including locus problems
• informal treatment of the Fundamental Theorem of Algebra and the conjugate root theorem.

MATRICES

The intention of this section of the course is to introduce learners to new mathematical structures and to help them appreciate some of the ways in which these structures can be put to use. Trigonometric identities should be used to develop ideas in matrices and linear transformations, but will not be externally assessed in isolation.

This are of study will include:

• addition and multiplication of matrices, including the concepts of identities, inverses, associativity and commutativity
• determinant of a 2 x 2 matrix, and the idea of singularity or non-singularity
• solutions of two equations in two unknowns or three equations in three unknowns, including the process of Gauss-Jordan reduction, and the use of technology to solve larger systems (applications should be included)
• the definition of a linear transformation (with an emphasis on non-singular transformations), and its representation by a matrix
• the image of a point, the “unit square”, a straight line, a circle or a curve under a non-singular linear transformation
• a study of dilation, shear, rotation (about the origin) and reflection (in the line `y = tan (alpha).x`) transformations and composites of these
• the relationship between the determinant of a matrix and the area of an image
• composition of transformations used to develop the addition theorems for `cos(A +- B)`, `sin(A +- B)` and `tan(A +- B)`
• the “double angle” formulae for `cos 2A`, `sin 2A` and `tan 2A`
• “sums to products” formulae
• “products to sums” formulae.

CALCULUS

The intention of this section of the course is to extend learners’ existing knowledge and understanding of a very important branch of mathematics by developing a greater capacity for integrating functions and by introducing learners to simple differential equations and their uses.

This section of the course develops and extends the ideas introduced in the Mathematics Methods course.

This area of study will include:

• implicit differentiation and its use in finding tangents and normals to curves
• a review of rules for differentiating functions as described in Mathematics Methods, derivatives of inverse trigonometric functions, `a^x` and `log_a x`, and compositions of these
• application of first and second derivatives to curve sketching, including stationary points and points of inflection
• review of the Fundamental Theorem of Calculus
• properties of definite integrals:

• applications of definite integrals to finding areas under or between curves and to finding volumes of solids of revolution about either the x-axis or the y-axis
• the trapezoidal rule and its use to approximate the area under a curve of a non-integrable function
• techniques of integration, using trigonometric identities (“double angle” formulae and "products to sums"), partial fractions, change of variable and integration by parts
• first order linear differential equations of the type:

• applications of differential equations, but not including input/output problems.

APPLICATIONS

Learners will be given opportunities to analyse situations that reinforce skills and concepts studied in this course. Extended problems, investigations and applications of technology are strongly recommended. Examples include the following:

• graphs of functions in polar form
• graphs of functions in parametric form
• coding with matrices
• matrices in the real world, such as Markov chains or Leslie Matrices
• translations
• length of a curve
• mathematical induction involving inequalities or recursively defined sequences and series
• monotonicity and boundedness of sequences
• logistic equations
• Mandelbrot set
• simple harmonic motion
• Correolis component of acceleration (as affects our weather)
• Spring–Mass systems
• balancing of a 6-cylinder petrol engine
• centripetal acceleration (derivative of equation).

Criterion-based assessment is a form of outcomes assessment that identifies the extent of learner achievement at an appropriate end-point of study. Although assessment – as part of the learning program - is continuous, much of it is formative, and is done to help learners identify what they need to do to attain the maximum benefit from their study of the course. Therefore, assessment for summative reporting to TASC will focus on what both teacher and learner understand to reflect end-point achievement.

The standard of achievement each learner attains on each criterion is recorded as a rating ‘A’, ‘B’, or ‘C’, according to the outcomes specified in the standards section of the course.

A ‘t’ notation must be used where a learner demonstrates any achievement against a criterion less than the standard specified for the ‘C’ rating.

A ‘z’ notation is to be used where a learner provides no evidence of achievement at all.

Providers offering this course must participate in quality assurance processes specified by TASC to ensure provider validity and comparability of standards across all awards. For further information, see TASC's quality assurance and assessment processes.

Internal assessment of all criteria will be made by the provider. Providers will report the learner’s rating for each criterion to TASC.

TASC will supervise the external assessment of designated criteria which will be indicated by an asterisk (*). The ratings obtained from the external assessments will be used in addition to internal ratings from the provider to determine the final award.

The following processes will be facilitated by TASC to ensure there is:

• a match between the standards of achievement specified in the course and the skills and knowledge demonstrated by learners
• community confidence in the integrity and meaning of the qualification.

Process – TASC gives course providers feedback about any systematic differences in the relationship of their internal and external assessments and, where appropriate, seeks further evidence through audit and requires corrective action in the future.

The assessment for Mathematics Specialised Level 4 will be based on the degree to which the learner can:

• 1. communicate mathematical ideas and information
• 2. analysis: demonstrate mathematical reasoning, analysis and strategy in problem solving situations
• 3. plan, organise and complete mathematical tasks
• 4. demonstrate an understanding of finite and infinite sequences and series*
• 5. demonstrate an understanding of matrices and linear transformations*
• 6. use differential calculus and apply integral calculus to areas and volumes*
• 7. use techniques of integration and solve differential equations*
• 8. demonstrate an understanding of complex numbers*

* = denotes criteria that are both internally and externally assessed

Mathematics Specialised Level 4 (with the award of):

EXCEPTIONAL ACHIEVEMENT

HIGH ACHIEVEMENT

COMMENDABLE ACHIEVEMENT

SATISFACTORY ACHIEVEMENT

PRELIMINARY ACHIEVEMENT

The final award will be determined by the Office of Tasmanian Assessment, Standards and Certification from 13 ratings (8 from the internal assessment, 5 from the external assessment).

The minimum requirements for an award in Mathematics Specialised Level 4 are as follows:

EXCEPTIONAL ACHIEVEMENT (EA)
11 ‘A’ ratings, 2 ‘B’ ratings (4 ‘A’ ratings and 1 ‘B’ rating in the external assessment)

HIGH ACHIEVEMENT (HA)
5 ‘A’ ratings, 5 ‘B’ ratings, 3 ‘C’ ratings (2 ‘A’ ratings, 2 ‘B’ ratings and 1 ‘C’ rating in the external assessment)

COMMENDABLE ACHIEVEMENT (CA)
7 ‘B’ ratings, 5 ‘C’ ratings (2 ‘B’ ratings and 2 ‘C’ ratings in the external assessment)

SATISFACTORY ACHIEVEMENT (SA)
11 ‘C’ ratings (3 ‘C’ ratings in the external assessment)

PRELIMINARY ACHIEVEMENT (PA)
6 ‘C’ ratings

A learner who otherwise achieves the ratings for a CA (Commendable Achievement) or SA (Satisfactory Achievement) award but who fails to show any evidence of achievement in one or more criteria (‘z’ notation) will be issued with a PA (Preliminary Achievement) award.

The Department of Education’s Curriculum Services will develop and regularly revise the curriculum. This evaluation will be informed by the experience of the course’s implementation, delivery and assessment.

In addition, stakeholders may request Curriculum Services to review a particular aspect of an accredited course.

Requests for amendments to an accredited course will be forwarded by Curriculum Services to the Office of TASC for formal consideration.

Such requests for amendment will be considered in terms of the likely improvements to the outcomes for learners, possible consequences for delivery and assessment of the course, and alignment with Australian Curriculum materials.

A course is formally analysed prior to the expiry of its accreditation as part of the process to develop specifications to guide the development of any replacement course.

The statements in this section, taken from documents endorsed by Education Ministers as the agreed and common base for course development, are to be used to define expectations for the meaning (nature, scope and level of demand) of relevant aspects of the sections in this document setting out course requirements, learning outcomes, the course content and standards in the assessment.

Unit 2 – Topic 2: Matrices

Matrix arithmetic:

• understand the matrix definition and notation (ACMSM051)
• define and use addition and subtraction of matrices, scalar multiplication, matrix multiplication, multiplicative identity and inverse (ACMSM052)
• 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 (ACMSM053)

Transformations in the plane:

• translations and their representation as column vectors (ACMSM054)
• define and use basic linear transformations: dilations of the form `(x,y) -> (lambda_1 x, lambda_2 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 (ACMSM055)
• apply these transformations to points in the plane and geometric objects (ACMSM056)
• define and use composition of linear transformations and the corresponding matrix products (ACMSM057)
• define and use inverses of linear transformations and the relationship with the matrix inverse (ACMSM058)
• examine the relationship between the determinant and the effect of a linear transformation on area (ACMSM059)
• establish geometric results by matrix multiplications; for example, show that the combined effect of two reflections in lines through the origin is a rotation (ACMSM060)

Unit 2 – Topic 3: Real and Complex Numbers

An introduction to proof by mathematical induction:

• understand the nature of inductive proof including the ‘initial statement’ and inductive step (ACMSM064)
• prove results for sums, such as `1 + 4 + 9 + ... + n^2 = (n(n + 1)(2n + 1))/6` for any positive integer `n` (ACMSM065)

Complex numbers:

• define the imaginary number `i` as a root of the equation `x^2 = -1` (ACMSM067)
• use complex numbers in the form `a + bi` where `a` and `b` are the real and imaginary parts (ACMSM068)
• determine and use complex conjugates (ACMSM069)
• perform complex-number arithmetic: addition, subtraction, multiplication and division (ACMSM070)

The complex plane:

• consider complex numbers as points in a plane with real and imaginary parts as Cartesian coordinates (ACMSM071)
• understand and use location of complex conjugates in the complex plane (ACMSM073)

Roots of equations:

• use the general solution of real quadratic equations (ACMSM074)
• determine complex conjugate solutions of real quadratic equations (ACMSM075)
• determine linear factors of real quadratic polynomials (ACMSM076)

Unit 3 – Topic 1: Complex Numbers

Cartesian forms:

• review real and imaginary parts `Re(z)` and `Im(z)` of a complex number `z` (ACMSM077)
• review Cartesian form (ACMSM078)
• review complex arithmetic using Cartesian forms (ACMSM079).

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 (ACMSM080)
• convert between Cartesian and polar form (ACMSM081)
• define and use multiplication, division, and powers of complex numbers in polar form and the geometric interpretation of these (ACMSM082)
• prove and use De Moivre’s theorem for integral powers (ACMSM083).

The complex plane (the Argand plane):

• examine and use addition of complex numbers as vector addition in the complex plane (ACMSM084)
• examine and use multiplication as a linear transformation in the complex plane (ACMSM085)
• identify subsets of the complex plane determined by relations such as `|z - 3i| <= 4`, `pi/4 <= Arg(z) <= (3pi)/4`, `Re(z) > Im(z)` and `|z - 1| = 2|z - i|` (ACMSM086).

Roots of complex numbers:

• determine and examine the `n^"th"` roots of unity and their location on the unit circle (ACMSM087)
• determine and examine the `n^"th"` roots of complex numbers and their location in the complex plane (ACMSM088).

Factorisation of polynomials:

• prove and apply the factor theorem and the remainder theorem for polynomials (ACMSM089)
• consider conjugate roots for polynomials with real coefficients (ACMSM090)
• solve simple polynomial equations (ACMSM091).

Unit 4 – Topic 1: Integration and Applications of Integration

Integration techniques:

• integrate using the trigonometric identities `sin^2 x = 1/2 (1 - cos 2x)`, `cos^2 x = 1/2 (1 - cos2x)` and `1 + tan^2 x = sec^2 x`  (ACMSM116)
• use substitution `u = g(x)` to integrate expressions of the form `f(g(x)) g'(x)` (ACMSM117)
• establish and use the formula `int 1/x dx = ln|x| + c`, for `x!= 0` ACMSM118)
• find and use the inverse trigonometric functions: arcsine, arccosine and arctangent (ACMSM119)
• find and use the derivative of the inverse trigonometric functions: arcsine, arccosine and arctangent (ACMSM120)
• integrate expressions of the form `(+- 1)/sqrt(alpha^2 - x^2)` and `alpha/(alpha^2 + x^2)`  (ACMSM121)
• use partial fractions where necessary for integration in simple cases (ACMSM122)
• integrate by parts (ACMSM123).

Applications of integral calculus:

• calculate areas between curves determined by functions (ACMSM124)
• determine volumes of solids of revolution about either axis (ACMSM125)
• use numerical integration using technology (ACMSM126).

Unit 4 – Topic 2: Rates of Change and Differential Equations

• use implicit differentiation to determine the gradient of curves whose equations are given in implicit form (ACMSM128)
• solve simple first-order differential equations of the form `dy/dx = f(x)`, differential equations of the form `dy/dx = g(y)` and, in general, differential equations of the form `dy/dx = f(x) g(y)` using separation of variables (ACMSM130).

Version 1 – Accredited on 9 September 2013 for use in 2014 to 2017. This course replaces Mathematics Specialised (MTS315109) that expired on 31 December 2013.

Version 2 – As per TASC decision on 3 December 2014, this course has been assigned TASC Level 4 (not 3) in complexity. The level and complexity section of this course document has been updated and the code changed to MTS415109.