Senior Secondary Australian Curriculum

• The senior secondary Australian Curriculum was developed by ACARA for English, Mathematics, Science, Humanities, and Social Sciences.
• The curriculum specifies content, achievement standards, knowledge, understanding, and skills for each subject
• Achievement standards describe the quality of learning, depth of understanding, extent of knowledge, and skill sophistication expected.
• The curriculum for each subject is organized into four units, with the last two units being more cognitively challenging.
• Each unit requires approximately 50–60 hours of instruction, including assessment and examinations.
• Units may be studied singly, in pairs (year-long), or as four units over two years.
• State and territory authorities structure and organize senior secondary courses and integrate the Australian Curriculum content and achievement standards.
• They are responsible for the curriculum implementation, assessment, certification, and quality assurance.
• These authorities set assessment and certification specifications for local courses.
• The Australian Curriculum is presented as content and achievement standards for integration into state and territory courses
• It is not intended as a course of study itself.

Senior Secondary Mathematics Subjects

• The Senior Secondary Australian Curriculum: Mathematics consists of four subjects.
• Each subject caters to different learning needs.
• Essential Mathematics focuses on using mathematics to make informed decisions in workplace, personal, further learning, and community settings, preparing students for employment and training.
• General Mathematics uses discrete mathematics to solve problems in financial modeling, network analysis, route and project planning, decision making, and discrete growth and decay.
• It also deals with geometrical problems and statistical investigations.
• Mathematical Methods focuses on calculus and statistical analysis to understand the physical world, model physical processes, and analyze phenomena involving uncertainty and variation.
• Specialist Mathematics develops rigorous mathematical arguments and proofs and extensively uses mathematical models.
• It includes advanced topics in functions, calculus, probability, statistics, vectors, complex numbers, and matrices.
• Specialist Mathematics is designed to be taken in conjunction with Mathematical Methods.

Cross-curriculum priorities

• The mathematics curriculum values the histories, cultures, traditions, and languages of Aboriginal and Torres Strait Islander peoples.
• It integrates their contributions to Australian society and culture and fosters students' understanding and appreciation of their diversity.
• Mathematics provides opportunities to engage with Asian countries and their contributions.
• Analyzing relevant data enhances understanding of Asia's diverse environments and cultures.
• The mathematics subjects promote informed views, discussion, research, and problem-solving. Teachers are encouraged to use contexts connected with sustainability.
• Data analysis allows students to research and discuss sustainability, valuing a wide range of perspectives.

Aims of Specialist Mathematics

• To develop an understanding of concepts and techniques from combinatorics, geometry, trigonometry, complex numbers, vectors, matrices, calculus, and statistics.
• To improve the ability to solve applied problems using concepts and techniques from those mathematical areas.
• To develop the capacity to choose and use technology appropriately.
• To foster reasoning in mathematical and statistical contexts.
• Also, interpretation of information and assessing reasonableness of solutions.
• To communicate in a concise and systematic manner using appropriate mathematical and statistical language.
• To construct proofs.

Links to Foundation to Year 10

• The proficiency strands of the F-10 curriculum - Understanding, Fluency, Problem solving, and Reasoning are applicable and inherent in students' learning the subject.
• Practice enables fluency, allowing students to focus on complex problem solving.
• Formal explanation of reasoning through mathematical proof is important.
• The ability to transfer skills, such as integration, to other fields like biology, kinematics, or statistics, is a vital part of mathematics learning.
• Desirable for students to complete topics from 10A with content descriptions such as ACMG273: Establish the sine, cosine and area rules for any triangle, and solve related problems
• ACMMG274: Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies and ACMNAP266: Investigate the concept of a polynomial, and apply the factor and remainder theorems to solve problems.

General capabilities

• Literacy, Numeracy, ICT, Critical and creative thinking, Personal and social capability, Ethical understanding, and Intercultural understanding are identified where they offer opportunities to add depth and richness to student learning.
• Teachers can incorporate explicit teaching of the capabilities depending on their choice of learning activities.

Literacy in Mathematics

• Literacy skills enable students to express, interpret, and communicate complex mathematical information, ideas, and processes.
• Mathematics provides a specific and rich context for developing the ability to read, write, visualize and talk about complex situations
• Students can apply and further develop their literacy skills and strategies by shifting between verbal, graphic, numerical and symbolic forms.

Numeracy in Mathematics

• Developing students numeracy skills at a more sophisticated level than in Years F to 10.
• The subject will equip students for the ever-increasing demands of the information age.

ICT in Mathematics

• ICT will be used to develop understanding and to apply knowledge to a range of problems
• Software aligned with areas of work and society with which they may be involved such as for statistical analysis, algorithm generation, manipulation, and complex calculation.
• Digital tools will be used to make connections between mathematical theory, practice and application, use data, to address problems, and to operate systems in authentic situations.

Critical and creative thinking

• Students compare predictions with observations when evaluating a theory.
• Check the extent to which theory-based predictions match observations, assess whether observations and predictions don't match.
• Revise, or reapply their theory more skillfully.

Personal and social capability in Mathematics

• Students develop competence in Mathematics through setting and monitoring goals.
• Taking initiative, building adaptability, communication, teamwork, and decision-making.
• The personal and social competence relevant includes the application of mathematical skills for their decision-making, life-long learning, citizenship and self-management.

Ethical understanding in Mathematics

• Students develop ethical understanding through decision-making connected with dilemmas that arise when engaged in mathematical calculation and the dissemination of results, they also learn ethics through teamwork and attribution of input.

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.
• They 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.

Structure of Specialist Mathematics

• Specialist Mathematics is structured over four units, it presents different scenarios for incorporating mathematical arguments and problem solving.
• It provides a blending of algebraic and geometric thinking.
• There is a progression of content, applications, level of sophistication and abstraction, for example, vectors in the plane are introduced in Unit 1 and then in Unit 3 they are studied for three-dimensional space.
• The 'Vectors in three dimensions' topic leads to the establishment of the equations of lines and planes, and prepares students for solving simultaneous equations in three variables.

Unit Descriptions

• Unit 1: Combinatorics, Vectors in the plane, Geometry.
• Unit 2: Trigonometry, Matrices, Real and complex numbers.
• Unit 3: Complex numbers, Functions and sketching graphs, Vectors in three dimensions.
• Unit 4: Integration and applications of integration, Rates of change and differential equations, Statistical inference.

Unit 1

• Contains three topics that complement the content of Mathematical Methods.
• The proficiency strand, 'Reasoning', of the F-10 curriculum continued in the topic 'Geometry'.
• This topic also provides the opportunity to summarise and extend students' studies in Euclidean Geometry, knowledge which is of great benefit in the later study of topics such as vectors and complex numbers.
• The topic ‘Combinatorics' provides techniques that are useful in many areas of mathematics, including probability and algebra.
• The topic ‘Vectors in the plane' provides new perspectives on working with two-dimensional space and serves as an introduction to techniques which can be extended to three-dimensional space in Unit 3.

Unit 2

• Contains three topics, 'Trigonometry', 'Matrices' and 'Real and complex numbers'.
• 'Matrices' provides new perspectives for working with two-dimensional space.
• 'Real and complex numbers' provides a continuation of the study of numbers.
• All of these topics develop students' ability to construct mathematical arguments.
• The technique of proof by the principle of mathematical induction is introduced

Unit 3

• Contains three topics, 'Complex numbers', 'Vectors in three dimensions', and 'Functions and sketching graphs'.
• The Cartesian form of complex numbers was introduced in Unit 2, and in Unit 3 the study of complex numbers is extended to the polar form.
• The study of functions and techniques of calculus begun in Mathematical Methods is extended and utilised in the sketching of graphs and the solution of problems involving integration.
• The study of vectors begun in Unit 1 focused on vectors in one- and two-dimensional space and it extended in Unit 3 to three-dimensional vectors, vector equations and vector calculus, building on students' knowledge of calculus from Mathematical Methods.
• Cartesian and vector equations, together with equations of planes, enables students to solve geometric problems and to solve problems involving motion in three-dimensional space.

Unit 4

• Contains three topics: 'Integration and applications of integration', 'Rates of change and differential equations' and 'Statistical inference'.
• In Unit 4, the study of differentiation and integration of functions is continued, and the techniques developed are applied to the area of simple differential equations, in particular in biology and kinematics.
• Topics demonstrate the applicability of the mathematics learnt throughout this course.
• The students' previous experience working with probability and statistics is drawn together in the study of the distribution of sample means and confidence intervals for sample means demonstrates the utility and power of statistics.

Achievement standards

• The two dimensions in achievement standards in Mathematics are, 'Concepts and Techniques' and 'Reasoning and Communication'.
• Senior secondary achievement standards provide an indication of typical performance at five different levels (corresponding to grades A to E) following the completion of study of senior secondary Australian Curriculum content for a pair of units.
• They are structured to reflect key dimensions of the content of the relevant learning area and be eventually accompanied by illustrative and annotated samples of student work/ performance/ responses.
• The achievement standards will be refined empirically through an analysis of samples of student work and responses to assessment tasks.

Role of technology

• Students will be taught subjects with an extensive range of technological applications and techniques
• These have the potential to enhance the teaching and learning of mathematics.
• Students also need to continue to develop skills that do not depend on technology.

Unit 3 Description

• Unit 3 of Specialist Mathematics contains three topics vectors in three dimensions, complex numbers and functions and sketching graphs.
• The study of vectors was introduced in Unit 1 with a focus on vectors in two-dimensional space
• Three-dimensional vectors are studied Students' knowledge of calculus from Mathematical Methods is extended by introducing vector calculus.
• Cartesian and vector equations, together with equations of planes, enable geometric and motion problems to be solved in three-dimensional space.
• The form of complex numbers was introduced in Unit 1, and study of complex numbers is extended to the polar form.
• The study of functions and techniques of graph sketching, begun in Mathematical Methods, is extended and applied in sketching graphs and solving problems involving integration.

Unit 3 Learning Outcomes

• Students will understand the concepts and techniques in vectors, complex numbers, functions and graph sketching, they will apply reasoning skills in these areas.
• Communicate arguments and strategies when solving problems.
• Construct proofs of results interpreting mathematical information and ascertain solutions.

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) and 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,|z−3i|≤4, π4<Arg(z)≤3π,4,Re(z)>Im(z) and |z−1|=2|z−i| (ACMSM086)π4<Arg(z)≤3π,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 nth 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) with solving simple polynomial equations (ACMSM091).

Topic 2: Functions and sketching graphs

• Functions: determine when the composition of two functions is defined (ACMSM092)
• Find the composition of two functions (ACMSM093)
• Determine if a function is one-to-one (ACMSM094)
• Consider inverses of one-to-one function (ACMSM095)
• Examine the reflection property of the graph of a function and the graph of its inverse (ACMSM096).

Sketching graphs

• Use and apply the notation |𝑥||x| for the absolute value for the real number 𝑥x and the graph of 𝑦=|𝑥|y=|x| (ACMSM098)
• Examine the relationship between the graph of 𝑦=𝑓(𝑥)y=f(x) and the graphs of 𝑦=1𝑓(𝑥)y=1f(x), 𝑦=|𝑓(𝑥)|y=|f(x)| and 𝑦=𝑓(|𝑥|)y=f(|x|) (ACMSM099)

Topic 3: Vectors in three dimensions

• Review the concepts of vectors from Unit 1 and extend to three dimensions including introducing the unit vectors 𝑖,𝑗and 𝑘.(ACMSM101)
• Prove geometric results in the plane and construct simple proofs in three-dimensions (ACMSM102)

Vector and Cartesian equations

• Three dimensional space includes plotting points and the equations of spheres (ACMSM103)
• Use vector equations of curves in two or three dimensions involving a parameter Cartesian equation corresponding in two dimensional case (ACMSM104)
• 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 (ACMSM105)
• Examine the position of two particles each described as a vector function of time Determine if their paths cross or if the particles meet (ACMSM106)
• Use the cross product to determine a vector normal to a given plane (ACMSM107)
• Determining vector and Cartesian equations of a plane and of regions in a plane (ACMSM108)

Systems of linear equations

• The form of a system of linear equations in several variables and solve a system of linear equations with elimination (ACMSM109)
• Examine unique solution, no solution, and infinitely many solutions and the geometric of a solution of a system of equations with three variables(ACMSM110)

Vector calculus

• Consider position of vectors as a function of time (ACMSM111)
• Derive the Cartesian equation of a path given as a vector equation in two dimensions including ellipses and hyperbolas (ACMSM112)
• Differentiate and integrate a vector function with respect to time (ACMSM113)
• Determine equations of motion of a particle travelling in a straight line with both constant and variable acceleration (ACMSM114)
• Apply vector calculus to motion in a plane including projectile and circular motion (ACMSM115)

Unit 4 Description

Unit 4 contains three main topics: Integration and applications of integration, rates of change and different equations, and statistical inference. Topics demonstrate the real world applications of the mathematical leaned

Unit 4 Learning Outcomes

• Students will understand the concepts and techniques in applications of calculus and statistical reasoning and solve problems.
• Communicate arguments and strategies when problems solving, constructing proofs and interpret what it means when the solutions

Topic 1: Integration and applications of integration

Integration Techniques

• Using trigonometric identities
• Sin^2(x) = 1/2(1 - cos(2x)), cos^2(x) = 1/2(1 + cos(2x)) use u = g(x) to integrate from f(g(x)) g’ (x)
• The formula integral(1/x) dx = ln |x| + c for all
• Find the derivatives of the inverse trigonometric functions as well as the arcsine and tangent.

Applications of integral calculus

• Calculating the areas between curves and the curves of the function
• Determining volumes of revolving acids

Topic 2: Rates of change and differential equations

• Use implicit differentiation to determine when the gradient of curves equations are provided
• Examining the rates of change as an instance of the formula

Topic 3: Statistical inference

• Examining the x in random sample
• Simulating repeated random from a variety of distributions for each number.
• Sample sizes can give the properties of the distribution of x across samples of the size including its mean.