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.

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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.

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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.

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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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Glossary

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Achievement standards

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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

Topic 1: Trigonometry

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)

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 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)

Topic 3: Real and complex numbers

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)