# Specialist Mathematics (Version 8.4)

### 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.

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## Unit 1

### Unit 1 Description

Unit 1 of Specialist Mathematics contains three topics – ‘Combinatorics’, ‘Vectors in the plane’ and ‘Geometry’ – that complement the content of Mathematical Methods. The proficiency strand, Reasoning, of the F–10 curriculum is continued explicitly in ‘Geometry’ through a discussion of developing mathematical arguments. While these ideas are illustrated through deductive Euclidean geometry in this topic, they recur throughout all of the topics in Specialist Mathematics. ‘Geometry’ also provides the opportunity to summarise and extend students’ studies in Euclidean Geometry. An understanding of this topic is of great benefit in the study of later topics in the course, including vectors and complex numbers.

‘Vectors in the plane’ provides new perspectives for working with two-dimensional space, and serves as an introduction to techniques that will be extended to three-dimensional space in Unit 3.

‘Combinatorics’ provides techniques that are useful in many areas of mathematics including probability and algebra. All these topics develop students’ ability to construct mathematical arguments.

These three topics considerably broaden students’ mathematical experience and therefore begin an awakening to the breadth and utility of the subject. They also enable students to increase their mathematical flexibility and versatility.

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

### Unit 1 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques in combinatorics, geometry and vectors
• apply reasoning skills and solve problems in combinatorics, geometry and vectors
• communicate their arguments and strategies when solving problems
• construct proofs in a variety of contexts including algebraic and geometric
• interpret mathematical information and ascertain the reasonableness of their solutions to problems.

### Unit 1 Content Descriptions

#### Permutations (ordered arrangements):

solve problems involving permutations  (ACMSM001)

use the multiplication principle (ACMSM002)

use factorial notation (ACMSM003)

solve problems involving permutations and restrictions with or without repeated objects (ACMSM004)

#### The inclusion-exclusion principle for the union of two sets and three sets:

determine and use the formulas for finding the number of elements in the union of two and the union of three sets.  (ACMSM005)

#### The pigeon-hole principle:

solve problems and prove results using the pigeon-hole principle. (ACMSM006)

#### Combinations (unordered selections):

solve problems involving combinations (ACMSM007)

use the notation $$\begin{pmatrix}n\\r\end{pmatrix}$$ or $${}^nC_r$$ (ACMSM008)

derive and use simple identities associated with Pascal’s triangle. (ACMSM009)

#### Representing vectors in the plane by directed line segments:

examine examples of vectors including displacement and velocity (ACMSM010)

define and use the magnitude and direction of a vector (ACMSM011)

represent a scalar multiple of a vector  (ACMSM012)

use the triangle rule to find the sum and difference of two vectors. (ACMSM013)

#### Algebra of vectors in the plane:

use ordered pair notation and column vector notation to represent a vector (ACMSM014)

define and use unit vectors and the perpendicular unit vectors i and j (ACMSM015)

express a vector in component form using the unit vectors i and j (ACMSM016)

examine and use addition and subtraction of vectors in component form (ACMSM017)

define and use multiplication by a scalar of a vector in component form (ACMSM018)

define and use scalar (dot) product  (ACMSM019)

apply the scalar product to vectors expressed in component form (ACMSM020)

examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular (ACMSM021)

define and use projections of vectors  (ACMSM022)

solve problems involving displacement, force and velocity involving the above concepts.  (ACMSM023)

#### The nature of proof:

use implication, converse, equivalence, negation, contrapositive (ACMSM024)

use the symbols for implication ($$\Rightarrow$$), equivalence ($$\Longleftrightarrow$$), and equality ($$=$$) (ACMSM026)

use the quantifiers ‘for all’ and ‘there exists’  (ACMSM027)

use examples and counter-examples.  (ACMSM028)

#### Circle properties and their proofs including the following theorems:

An angle in a semicircle is a right angle (ACMSM029)

The angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc (ACMSM030)

Angles at the circumference of a circle subtended by the same arc are equal (ACMSM031)

The opposite angles of a cyclic quadrilateral are supplementary  (ACMSM032)

Chords of equal length subtend equal angles at the centre and conversely chords subtending equal angles at the centre of a circle have the same length (ACMSM033)

The alternate segment theorem (ACMSM034)

When two chords of a circle intersect, the product of the lengths of the intervals on one chord equals the product of the lengths of the intervals on the other chord (ACMSM035)

When a secant (meeting the circle at $$A$$ and $$B$$) and a tangent (meeting the circle at $$T$$) are drawn to a circle from an external point $$M$$, the square of the length of the tangent equals the product of the lengths to the circle on the secant. $$(AM\times BM=TM^2)$$.  (ACMSM036)

Suitable converses of some of the above results  (ACMSM037)

Solve problems finding unknown angles and lengths and prove further results using the results listed above. (ACMSM038)

#### Geometric proofs using vectors in the plane including:

The diagonals of a parallelogram meet at right angles if and only if it is a rhombus (ACMSM039)

Midpoints of the sides of a quadrilateral join to form a parallelogram (ACMSM040)

The sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides.  (ACMSM041)

## Unit 2

### Unit 2 Description

Unit 2 of Specialist Mathematics contains three topics – ‘Trigonometry’, ‘Real and complex numbers’ and ‘Matrices’…

‘Trigonometry’ contains techniques that are used in other topics in both this unit and Unit 3. ‘Real and complex numbers’ provides a continuation of students’ study of numbers, and the study of complex numbers is continued in Unit 3. This topic also contains a section on proof by mathematical induction. The study of matrices is undertaken, including applications to linear transformations of the plane.

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

### Unit 2 Learning Outcomes

By the end of this unit, students:

• understand the concepts and techniques in trigonometry, real and complex numbers, and matrices
• apply reasoning skills and solve problems in trigonometry, real and complex numbers, and matrices
• 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 2 Content Descriptions

#### The basic trigonometric functions:

find all solutions of $$\mathrm f\left(\mathrm a\left(\mathrm x-\mathrm b\right)\right)=\mathrm c$$ where $$f$$ is one of $$\sin$$, $$\cos$$ or $$\tan$$ (ACMSM042)

graph functions with rules of the form $$\mathrm y=\mathrm f(\mathrm a\left(\mathrm x-\mathrm b\right))$$ where $$f$$ is one of $$\sin$$, $$\cos$$ or $$\tan$$ (ACMSM043)

#### Compound angles:

prove and apply the angle sum, difference and double angle identities. (ACMSM044)

#### The reciprocal trigonometric functions, secant, cosecant and cotangent:

define the reciprocal trigonometric functions, sketch their graphs, and graph simple transformations of them. (ACMSM045)

#### Trigonometric identities:

prove and apply the Pythagorean identities  (ACMSM046)

prove and apply the identities for products of sines and cosines expressed as sums and differences (ACMSM047)

convert sums $$\mathrm a\cos\mathrm x+\mathrm b\;\sin\mathrm x$$ to $$\mathrm R\;\cos{(\mathrm x\pm\mathrm\alpha)}$$ or $$\mathrm R\sin{(\mathrm x\pm\mathrm\alpha)}$$ and apply these to sketch graphs, solve equations of the form $$\mathrm a\cos\mathrm x+\mathrm b\sin\mathrm x=\mathrm c$$ and solve problems (ACMSM048)

prove and apply other trigonometric identities such as $$\cos3\mathrm x=4\;\mathrm c\mathrm o\mathrm s^{3\;}\mathrm x-3\cos\mathrm x$$ (ACMSM049)

#### Applications of trigonometric functions to model periodic phenomena:

model periodic motion using sine and cosine functions and understand the relevance of the period and amplitude of these functions in the model. (ACMSM050)

#### 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 2x2 matrices and solve matrix equations of the form AX=B , where A is a 2x2 matrix and 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 $$(\mathrm x,\mathrm y)\longrightarrow({\mathrm\lambda}_1\mathrm x,{\mathrm\lambda}_2\mathrm y)$$ , rotations about the origin and reflection in a line which passes through the origin, and the representations of these transformations by 2x2 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)

#### Proofs involving numbers:

prove simple results involving numbers.  (ACMSM061)

#### Rational and irrational numbers:

express rational numbers as terminating or eventually recurring decimals and vice versa (ACMSM062)

prove irrationality by contradiction for numbers such as $$\sqrt[{}]2$$ and $$\log_25$$ (ACMSM063)

#### 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 $$1+4+9\dots+n^2=\frac{n(n+1)(2n+1)}6$$ for any positive integer n (ACMSM065)

prove divisibility results, such as $$3^{2n+4}-2^{2n}$$  is divisible by 5 for any positive integer n.

(ACMSM066)

#### 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)

examine addition of complex numbers as vector addition in the complex plane (ACMSM072)

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

### 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

#### 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. (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)

#### 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)

#### 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)