Specialist Mathematics (Version 8.4)

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Unit 1
Unit 2
Unit 3
Unit 4

Understand how Specialist Mathematics works

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.

Links to Foundation to Year 10

For all content areas of Specialist Mathematics, the proficiency strands of the F–10 curriculum are still very much applicable and should be inherent in students’ learning of the subject. The strands of Understanding, Fluency, Problem solving and Reasoning are essential and mutually reinforcing.

Representation of General capabilities

The seven general capabilities of Literacy, Numeracy, Information and Communication technology (ICT) capability, 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.

Structure of Specialist Mathematics

Specialist Mathematics is structured over four units. The topics in Unit 1 broaden students’ mathematical experience and provide different scenarios for incorporating mathematical arguments and problem solving.

Glossary

Achievement standards

Unit 3

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. In this unit, three-dimensional vectors are studied and vector equations and vector calculus are introduced, with the latter extending 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.The Cartesian form of complex numbers was introduced in Unit 2, and the study of complex numbers is now 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.

Access to technology to support the computational aspects of these topics is assumed.

Unit 3 Learning Outcomes

By the end of this unit, students will:

understand the concepts and techniques in vectors, complex numbers, functions and graph sketching
apply reasoning skills and solve problems in vectors, complex numbers, functions and graph sketching
communicate their arguments and strategies when solving problems
construct proofs of results
interpret mathematical information and ascertain the reasonableness of their solutions to problems.

Unit 3 Content Descriptions

Topic 1: Complex numbers

Cartesian forms:

review real and imaginary parts $Re\left(z\right)\;$ 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 $\left|z\right|$ 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 $\left|z-3i\right|\leq4$ $\frac\pi4\leq Arg(z)\leq\frac{3\pi}4$, $Re\left(z\right)>Im(z)$ and $\left|z-1\right|=2\vert z-i\vert$ (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. $n^{t h}$ (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)

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 $\left|x\right|$ for the absolute value for the real number $x$ and the graph of $y=\left|x\right|$ (ACMSM098)

examine the relationship between the graph of $y=f(x)$ and the graphs of $y=\frac1{f(x)}$, $y=\vert f\left(x\right)\vert$ and $y=f(\left|x\right|)$ (ACMSM099)

sketch the graphs of simple rational functions where the numerator and denominator are polynomials of low degree. (ACMSM100)

Topic 3: Vectors in three dimensions

The algebra of vectors in three dimensions:

review the concepts of vectors from Unit 1 and extend to three dimensions including introducing the unit vectors i, j and k. (ACMSM101)

prove geometric results in the plane and construct simple proofs in three-dimensions. (ACMSM102)

Vector and Cartesian equations:

introduce Cartesian coordinates for three-dimensional space, including plotting points and the equations of spheres (ACMSM103)

use vector equations of curves in two or three dimensions involving a parameter, and determine a ‘corresponding’ Cartesian equation in the 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, and 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)

determine vector and Cartesian equations of a plane and of regions in a plane. (ACMSM108)

Systems of linear equations:

recognise the general form of a system of linear equations in several variables, and use elementary techniques of elimination to solve a system of linear equations (ACMSM109)

examine the three cases for solutions of systems of equations – a unique solution, no solution, and infinitely many solutions – and the geometric interpretation 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)