SACE Stage 1:
Mathematics

Functions, Graphs and Polynomials in SACE Stage 1 Mathematics develops the formal function theory and algebraic reasoning skills that underpin the entire Mathematics subject and provide the foundation for Stage 2 Mathematical Methods. Topics include the formal definition of a function, function notation, domain and range, the distinction between a function and a general relation, composite functions, inverse functions and their graphical relationships, and the study of key function families including linear, quadratic, cubic, and higher-degree polynomial functions — developing students’ ability to sketch, transform, and analyse these graphically and algebraically. The Polynomials component covers polynomial arithmetic including addition, subtraction, and multiplication, the division of polynomials, the remainder theorem and its applications, the factor theorem and its use in fully factorising polynomials, and roots and zeros of polynomial functions. Students are expected to work fluently between graphical and algebraic representations throughout this topic, building the function fluency that is directly assessed in school-based assessment tasks and carried forward into Stage 2.

Matrices, Real and Complex Numbers in SACE Stage 1 Mathematics introduces two algebraic structures that are central to the Stage 1 Mathematics subject and that both extend significantly in Stage 2 Specialist Mathematics. The Matrices component covers matrix notation, matrix addition, subtraction, scalar multiplication, and matrix multiplication including the conditions for conformability, the identity matrix, the determinant and inverse of a 2×2 matrix, and the use of matrix equations to solve systems of simultaneous linear equations in two variables. The Real and Complex Numbers component develops the real number system and its properties, introduces complex numbers in Cartesian form as an extension of the real numbers, covers arithmetic of complex numbers including addition, subtraction, multiplication, and division in Cartesian form, the modulus and argument of a complex number, the conjugate and its algebraic properties, and the geometric representation of complex numbers on the Argand diagram. Together these topics provide the algebraic and structural foundation for the more advanced work in matrices, complex numbers, and proof at Stage 2 Specialist Mathematics level.

Vectors in the Plane and Further Trigonometry in SACE Stage 1 Mathematics develops two foundational areas that are distinctive to the Mathematics subject at Stage 1 and that are directly continued in Stage 2 Specialist Mathematics. The Vectors in the Plane component introduces vector notation in component form using i and j, vector addition, subtraction, and scalar multiplication geometrically and algebraically, the magnitude and direction of a vector, unit vectors, the dot product of two vectors and its geometric interpretation including the angle between vectors and the condition for perpendicularity, and applications of vectors to geometric proof and to describing motion and position in the plane. The Further Trigonometry component extends standard trigonometry to include the compound angle identities for sin(A ± B), cos(A ± B), and tan(A ± B), the double angle formulae, using these identities to prove trigonometric results and solve more complex trigonometric equations over specified domains, and an introduction to the inverse circular functions sin⁻¹x, cos⁻¹x, and tan⁻¹x including their domains and ranges. Together these topics provide the geometric and trigonometric foundation required for Stage 2 Specialist Mathematics.

Geometry, Sequences and Series in SACE Stage 1 Mathematics develops geometric reasoning and discrete mathematics skills that are applied and extended across Stage 2. The Geometry component covers circle properties including the angle in a semicircle, the angle at the centre and circumference theorem, cyclic quadrilateral properties, tangent-chord angles, and the application of these properties to geometric proof, as well as similarity and congruence in triangles and their application to solving problems. The Sequences and Series component introduces arithmetic sequences and series, the general term formula tⁿ = a + (n − 1)d, the sum formula Sⁿ = n/2(2a + (n−1)d), geometric sequences and series, the general term formula tⁿ = arⁿ⁻¹, the sum formula Sⁿ = a(rⁿ − 1)/(r − 1), the limiting sum of a geometric series for |r| < 1, and applications of arithmetic and geometric sequences to practical modelling situations including simple interest, compound interest, population growth, and depreciation. This topic directly supports the Stage 2 content in both Mathematical Methods and Specialist Mathematics.

Trigonometry, Counting and Statistics in SACE Stage 1 Mathematics covers three distinct areas that each provide essential preparation for Stage 2 Mathematics subjects. The Trigonometry component develops a formal understanding of trigonometric functions including radian measure, the unit circle and its use in defining sine, cosine, and tangent for all angles, the graphs of y = sin x, y = cos x, and y = tan x and their key features including amplitude, period, and phase, transformations of trigonometric graphs, and the Pythagorean identity sin²θ + cos²θ = 1 and related identities. The Counting component introduces the fundamental counting principle, permutations and combinations using ⁿPᵣ and ⁿCᵣ, and the binomial theorem including Pascal’s triangle and the general term of a binomial expansion. The Statistics component develops data literacy through classifying data, constructing and interpreting appropriate displays, calculating and interpreting measures of centre and spread, working with correlation and lines of best fit for bivariate data, and applying basic probability including the addition rule and conditional probability. Together these topics support the full breadth of Stage 2 Mathematics, Mathematical Methods, and Specialist Mathematics content.

Differential Calculus, Growth and Decay in SACE Stage 1 Mathematics introduces the foundational concepts of differential calculus and their application to modelling, and is the most directly connected topic to Stage 2 Mathematical Methods and Specialist Mathematics. The Differential Calculus component covers the concept of a limit and the derivative as the gradient of the tangent to a curve, differentiation from first principles for polynomial functions, differentiating polynomial functions using the power rule, differentiating sums and differences, and the chain rule, product rule, and quotient rule applied to polynomial and power function compositions, finding equations of tangents and normals to curves, identifying and classifying stationary points including local maxima, local minima, and stationary points of inflection, and applying differentiation to optimisation and rates of change problems in practical contexts. The Growth and Decay component covers exponential functions and the natural base e, logarithmic functions and the natural logarithm, the laws of logarithms, solving exponential and logarithmic equations, modelling exponential growth and decay, and differentiating exponential and logarithmic functions as a bridge to the full Stage 2 calculus content. Differential Calculus, Growth and Decay is the most extensively assessed topic in school-based assessment tasks and provides the direct gateway to Stage 2 Mathematical Methods.