The Higher School Certificate in NSW contains a number of mathematics courses catering for a range of abilities. There are four courses offered by NESA for HSC Study:
Mathematics Standard 1 or 2: A basic mathematics course containing precalculus concepts; the course is heavily based on practical mathematics used in everyday life. While the more advanced courses include statistical topics, this is the only course which introduces normal distributions, standard deviations and z-scores. These topics are alluded to in more advanced courses though not formally considered.
Mathematics Advanced: An advanced level calculus-based course with detailed study in probability and statistics, trigonometry, curve sketching, and applications of calculus. It is the highest level non-extension mathematics course. While the calculus is only a single variable in all of year 12 mathematics in NSW, in this course students are able to calculate volumes of revolution for a select class of functions. Computational methods such as the trapezoidal rule and Simpson's rule are encountered for evaluating integrals. The course includes a brief foray into series and sequences, including an application to basic finance through the modelling of compound interest. The nature of lines, circles and parabolas as loci are investigated however these properties are not exploited by the plane geometry coursework. Quadratic equations are studied and students learn techniques to reduce special quintic and exponential equations to quadratics.
Mathematics Extension 1 : A more advanced course building on concepts in calculus, trigonometry, polynomials, basic combinatorics, vectors, and further statistics. Students learn the binomial theorem to extend their knowledge of probability, along with using circle geometry to prove a greater family of statements. The trigonometry component includes double-angle identities and factoring the addition of a sine and cosine function into a single sinusoid. In calculus students are exposed to a greater variety of integration techniques such as substitution. Parametrization of planar curves is introduced, mainly focusing on lines, circles and parabolas. The plotting of cubic equations and solution of specific cases through polynomial long division and the remainder theorem enable a deeper understanding of polynomials.
Mathematics Extension 2 : A highly advanced mathematics course containing an introduction to complex numbers, advanced calculus, motion, and further work with vectors. While NSW Mathematics curricula does not include matrix theory nor group theory, the geometric properties of complex numbers alludes to both of these. The former is hinted at in the multiplicative properties of complex numbers, as students are required to plot the products, sums and quotients of complex numbers on the Argand plane. While group theory is not explicitly mentioned, roots of unity and cyclic groups are extensively studied. With their newfound familiarity with complex numbers, the fundamental theorem of algebra can now be formally stated. Students are now able to exploit this closure to solve even more polynomial equations. Recursive integral sequences, integration by parts and partial fraction decomposition techniques allow the solution to a wider class of problems. Projectile motion is studied in the kinematics module, which surpasses the depth of study found in HSC physics. This course synergizes with HSC Physics, as students are able to apply this knowledge in their Physics exams to arrive at more elegant and efficient solutions. The parametrisation of lines, circles and parabolas in Mathematics Extension 1 is further developed to the entire family of conics, including degenerate cases. Students are exposed to rectangular hyperbolas, however hyperbolic trigonometric functions are not included. Despite this students are expected to adapt to novel material, such as proving properties of the catenary via it's expression in exponential functions.
The defining feature of content progression from Mathematics Advanced through to Extension 2 Mathematics is the level of Mathematical Maturity expected of students. In higher courses students have exposure to a greater breadth and depth of techniques, and are expected to synthesize knowledge from seemingly disparate topics. In Mathematics Advanced exams students may be asked to apply familiar techniques to unfamiliar contexts, such as being given an identity through which they must solve a problem. Further mathematical maturity is vital to success in Extension 2 exams, as assessment focuses on both conceptual understanding and computational abilities. The difficulty in HSC final exam questions generally increases throughout the course of the paper. In one Extension 2 HSC examination, the final question provides students with a series of prompts and smaller questions, which culminate in a proof of the Basel Problem.
The Victorian Certificate of Education mathematics subjects are designed to cater for the varying abilities and aptitudes of Victorian students. There are four courses offered for VCE study:
Foundation Mathematics : Very basic, practical maths skills are taught in this strand, which is only run at a Unit 1 + 2 level.
Further Mathematics : Further Mathematics is considered to be the second-least demanding of the four maths subjects and is the "standard" maths course taken by Victorian students. It deals with topics including basic algebra, matrices, trigonometry, straight line geometry, business-related mathematics and other concepts
Specialist Mathematics : Specialist Mathematics is considered the most advanced high school mathematics subject in Victoria. The subject covers concepts including conic sections, complex numbers, differential equations, kinematics, vector calculus and mechanics. The idea behind Specialist Mathematics is to integrate already learnt concepts of calculus into other fields of mathematics, thus giving Specialist Mathematics a far more practical orientation than standard mathematical subjects.
In Queensland, the senior curriculum is divided into three parts: Maths A, Maths B, and Maths C.
Mathematics A
Maths A covers more practical topics than Maths B and C, but it is still OP eligible. There are considerably fewer algebraic concepts in this subject, and it is suitable for students who either struggled with mathematics in Year 10, or who do not require a knowledge of abstract mathematics in the future. Maths A is designed to help students to develop an appreciation of the value of Mathematics to humanity. Students learn how mathematical concepts may be applied to a variety of life situations including business and recreational activities. The skills encountered are relevant to a vast array of careers. Assessments in the subject include both formative and summative written tests, assignments and practical work. It is assessed in the categories: Knowledge & Procedures ; Modelling & Problem Solving ; Communication & Justification. Although Maths A is not a pre-requisite subject, but it is sufficient for entrance to many tertiary courses. The course is divided into four semesters. The skills learned in each semester are as follows: Semester 1 :
Data Analysis
Managing Money
Applied Geometry
Linking 2 and 3 Dimensions
Semester 2 :
Land Measurement
Applied Geometry
Statistics
Managing Money
Semester 3 :
Managing Money
Land Measurement
Data Analysis
Operations Research
Semester 4 :
Statistics
Land Measurement
Navigation
and an elective topic on Data
Mathematics B
Maths B is considerably more theoretical than Maths A, requiring advanced algebra skills to successfully complete. It is a common prerequisite for science and engineering courses at Queensland Universities. Maths B can be studied at the same time with either Maths A or Maths C, but not both. Maths B gives students an understanding of the methods and principles of mathematics and the ability to apply them in everyday situations and in purely mathematical contexts; the capacity to model actual situations and deduce properties from the model; an interest and ability in framing and testing mathematical hypotheses; the ability to express and communicate any results obtained; some knowledge of the history of mathematics; encouragement to think independently and creatively. Assessments are similar as those of Maths A, which includes both formative and summative written tests, assignments and post-assignment tests. It is also assessed in the three categories Knowledge & Procedures ; Modelling & Problem Solving ; Communication & Justification. Maths B is a pre-requisite for any tertiary course which deals with or uses math and/or science. According to the Queensland Studies Authority, in 2010, 93% of students who studied Maths B were OP eligible. The course is divided into four semesters. The skills learned each semester are as follows: Semester 1 :
Functions
Periodic Functions
Applied Statistics
Applied Statistics 2
Semester 2 :
Exploring Data / Statistics
Indices and Logarithms/ Exponential Functions
Limits and Differential Calculus 1
Semester 3 :
Exponential and Log Functions
Optimization Using Derivatives
Integration
Integral Calculus
Semester 4 :
Applied Statistical Analysis
Integration
Differential Calculus 2
Optimisation
Mathematics C
Maths C extends the topics taught in Maths B, and covers additional pure-maths topics. Although not necessarily more difficult, it must be studied in conjunction with Maths B. Maths C gives the students an understanding of the methods and principles of mathematics and the ability to apply them in everyday situations and in purely mathematical contexts; the capacity to model actual situations and deduce properties from the model; an interest and ability in framing and testing mathematical hypotheses; the ability to express and communicate any results obtained; some knowledge of the history of mathematics; encouragement to think independently and creatively. Assessments are in the same as the other two courses, formative and summative written tests, assignments and practical work. The student is assessed in the areas of Knowledge & Procedures ; Modelling & Problem Solving ; Communication & Justification. Maths C can be a pre-requisite to tertiary courses with a heavy maths/science basis. Some skills learned in Maths C would be found in business and economics degrees. The course is divided into four semesters. The areas learned are in the following: Semester 1 :
New WACE mathematics courses were introduced for Year 11 students in 2015 to replace previous mathematics courses and being the Western Australian course in line with the Australian Curriculum. The new WACE mathematics courses consist of four units. Each unit is studied over one semester. Therefore, Unit 1 & 2 is studied in Year 11, and Unit 3 & 4 is studied in Year 12. The new WACE mathematics courses are:
Mathematics Preliminary General
Mathematics Foundation General
Mathematics Essential General
Mathematics Applications ATAR
Mathematics Methods ATAR
Mathematics Specialist ATAR
ATAR mathematics courses are for university-bound students, whereas general courses are for non-ATAR students. Syllabus information is available from the School Curriculum and Standards Authority website.
South Australia
In South Australia the mathematics courses are split into six levels:
Numeracy for Work and Community Life
Essential Mathematics/Mathematical Pathways
General Mathematics/Mathematical Applications
Mathematical Methods/Mathematical Studies
Specialist Mathematics — more advanced topics that complement and are taken concurrently with Mathematical Studies
