Mathematical Methods Units 1 & 2 provide an introductory study of simple elementary functions of a single real variable, algebra, calculus, probability and statistics and their applications in a variety of practical and theoretical contexts. They are designed as preparation for Mathematical Methods Units 3 & 4 and contain assumed knowledge and skills for these units.

The focus of Unit 1 is the study of simple algebraic functions, and the areas of study are ‘Functions and graphs’, ‘Algebra’, ‘Calculus’ and ‘Probability and statistics’. At the end of Unit 1, students are expected to have covered the content outlined in each area of study, with the exception of ‘Algebra’ which extends across Units 1 & 2. This content should be presented so that there is a balanced and progressive development of skills and knowledge from each of the four areas of study with connections between and across the areas of study being developed consistently throughout both Units 1 & 2.

In undertaking this unit, students are expected to be able to apply techniques, routines and processes involving rational and real arithmetic, sets, lists and tables, diagrams and geometric constructions, algebraic manipulation, equations, graphs and differentiation with and without the use of technology. They should have facility with relevant mental and by-hand approaches to estimation and computation. The use of numerical, graphical, geometric, symbolic and statistical functionality of technology for teaching and learning mathematics, for working mathematically, and in related assessment, is to be incorporated throughout the unit as applicable.

OUTCOMES

For this unit the student is required to demonstrate achievement of three outcomes. As a set these outcomes encompass all of the areas of study for the unit.

Outcome 1

On completion of this unit the student should be able to define and explain key concepts as specified in the content from the areas of study, and apply a range of related mathematical routines and procedures. To achieve this outcome the student will draw on knowledge and skills outlined in all the areas of study.

Key skills

determine by hand the length of a line segment and the coordinates of its midpoint, the equation of a straight line given two points or one point and gradient, and the gradient and equation of lines parallel and perpendicular to a given line through some other point
specify the rule, domain (including maximal, natural or implied domain), co-domain, and range of a relation and identify whether or not a relation is a function
substitute integer, simple rational and irrational numbers in exact form into expressions, including rules of functions and relations, and evaluate these by hand
re-arrange and solve simple algebraic equations and inequalities by hand
expand and factorise linear and simple quadratic expressions with integer coefficients by hand
express ax2 + bx + c in completed square form where a, b, c ∈ Z and a ≠ 0, by hand
express a cubic polynomial p (x), with integer coefficients
use algebraic, graphical and numerical approaches, including the factor theorem and the bisection method, to determine and verify solutions to equations over a specified interval
apply distributive and index (exponent) laws to manipulate and simplify expressions involving polynomial and power function, by hand in simple cases
set up and solve systems of simultaneous linear equations involving up to four unknowns, including by hand for a system of two equations in two unknowns
sketch by hand graphs of linear, quadratic and cubic polynomial functions, and quartic polynomial functions in factored form (approximate location of stationary points only for cubic and quartic functions), including cases where an x-axis intercept is a touchpoint or a stationary point of inflection
draw graphs of polynomial functions of low degree, simple power functions and simple relations that are not functions
describe the effect of transformations on the graphs of relations and functions and apply matrix transformations, by hand in simple cases
use graphical, numerical and algebraic approaches to find an approximate value or the exact value
(as appropriate) for the gradient of a secant or tangent to a curve at a given point
set up probability simulations, and describe the notion of randomness, variability and its relation to events
calculate probabilities for compound events using rules and tree diagrams, by hand in simple cases
solve probability problems involving Karnaugh maps and tree diagrams, by hand in simple cases.

Outcome 2

On completion of this unit the student should be able to apply mathematical processes in non-routine contexts, including situations requiring problem-solving, modelling or investigative techniques or approaches, and analyse and discuss these applications of mathematics. To achieve this outcome the student will draw on knowledge and skills outlined in one or more areas of study.

Key skills

specify the relevance of key mathematical content from one or more areas of study to the investigation of various questions in a given context
develop mathematical formulations of specific and general cases used to derive results for analysis within a given context for investigation
use a variety of techniques to verify results
make inferences from analysis and use these to draw valid conclusions related to a given context for investigation
communicate conclusions using both mathematical expression and everyday language, in particular, the interpretation of mathematics with respect to the context for investigation.

Outcome 3

On completion of this unit the student should be able to select and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches. To achieve this outcome the student will draw on knowledge and skills outlined in all the areas of study.

Key skills

distinguish between exact and approximate presentations of mathematical results produced by technology, and interpret these results to a specified degree of accuracy
use technology to carry out numerical, graphical and symbolic computation as applicable
produce results using a technology which identify examples or counter-examples for propositions
produce tables of values, families of graphs and collections of other results using technology, which support general analysis in problem-solving, investigative and modelling contexts
use appropriate domain and range specifications to illustrate key features of graphs of functions and relations
identify the relation between numerical, graphical and symbolic forms of information about functions and equations and the corresponding features of those functions and equations
specify the similarities and differences between formal mathematical expressions and their representation by technology, in particular, equivalent forms of symbolic expressions
select an appropriate functionality of technology in a variety of mathematical contexts, and provide a rationale for these selections
apply suitable constraints and conditions, as applicable, to carry out required computations
relate the results from a particular technology application to the nature of a particular mathematical task (investigative, problem solving or modelling) and verify these results
specify the process used to develop a solution to a problem using technology, and communicate the key stages of mathematical reasoning (formulation, solution, interpretation) used in this process.