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. Mathematics is also concerned with collecting, analysing, modelling and interpreting data in order to investigate and understand realworld phenomena and solve problems in context. Mathematics provides a framework for thinking and a means of communication that is powerful, logical, concise and precise. It impacts upon the daily life of people everywhere and helps them to understand the world in which they live and work. Mathematics Methods – Foundation Level 3 provides for the study of algebra, functions and their graphs, calculus, probability and statistics. These are necessary prerequisites for the study of Mathematics Methods Level 4 in which the major themes are calculus and statistics. For these reasons this subject provides a foundation for study of Mathematics Methods Level 4 and disciplines in which mathematics has an important role, including engineering, the sciences, commerce, economics, health and social sciences.
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. Mathematics is also concerned with collecting, analysing, modelling and interpreting data in order to investigate and understand realworld phenomena and solve problems in context. Mathematics provides a framework for thinking and a means of communication that is powerful, logical, concise and precise. It impacts upon the daily life of people everywhere and helps them to understand the world in which they live and work.
Mathematics Methods – Foundation Level 3 provides for the study of algebra, functions and their graphs, calculus, probability and statistics. These are necessary prerequisites for the study of Mathematics Methods Level 4 in which the major themes are calculus and statistics. For these reasons this subject provides a foundation for study of Mathematics Methods Level 4 and disciplines in which mathematics has an important role, including engineering, the sciences, commerce, economics, health and social sciences.
Mathematics Methods – Foundation Level 3 aims to develop learners’:
On successful completion of this course, learners will be able to:
Mathematics Methods – Foundation Level 3 is designed for learners whose future pathways may involve the study of further secondary mathematics or a range of disciplines at the tertiary level. It functions as a foundation course for the study of Mathematics Methods Level 4.
It is recommended that learners attempting this course will be concurrently studying Grade 10 Australian Curriculum: Mathematics or have previously achieved at least a ‘B’ grade in that subject.
Programs of study derived from this course need to embrace the range of technological developments that have occurred in relation to mathematics teaching and learning.
Learners must have access to calculator algebraic system (CAS) graphics calculators and become proficient in their use. These calculators can be used in all aspects of this course in the development of concepts and as a tool for solving problems. Refer to 'What can I take to my exam?' for the current TASC Calculator Policy that applies to Level 3 and Level 4 courses.
The use of computer software is also recommended as an aid to students’ learning and mathematical development. A range of packages such as, but not limited to: Wolfram Mathematica, Microsoft Excel, Autograph, Efofex Stat, Graph and Draw are appropriate for this purpose.
This course has a complexity level of 3.
At Level 3, the learner is expected to acquire a combination of theoretical and/or technical and factual knowledge and skills and use judgement when varying procedures to deal with unusual or unexpected aspects that may arise. Some skills in organising self and others are expected. Level 3 is a standard suitable to prepare learners for further study at the tertiary level. VET competencies at this level are often those characteristic of an AQF Certificate III.
This course has a size value of 15.
This course is made up of five (5) areas of study. While each of these is compulsory, the order of delivery is not prescribed:
These areas of mathematics relate to the Assessment Criteria 4 to 8. Assessment Criteria 1 to 3 apply to all five areas of study.
ALGEBRA
Learners will be introduced to relevant conventions relating to algebraic language and notation. They will review then further develop their skills in the algebraic manipulation of simple polynomial functions (linear, quadratic and cubic) in the context of their function study work.
This area of study will include:
Review work:
Quadratic functions:
Powers and Polynomials:
Cubic functions:
Indices:
POLYNOMIAL FUNCTIONS AND GRAPHS
Learners will develop their understanding of the behaviour of a range of functions by sketching and analysing polynomials of degree no higher than three.
This area of study will include:
Linear functions:
Quadratic functions:
Cubic and other polynomial functions:
Language of functions:
EXPONENTIAL, LOGARITHMIC AND CIRCULAR (TRIGONOMETRIC) FUNCTIONS AND GRAPHS
Learners will develop their understanding of the behaviour of exponential, logarithmic and a range of circular (trigonometric) functions.
This area of study will include:
Exponential functions:
Logarithmic functions:
Circular (Trigonometric) functions:
CALCULUS
Learners will develop an intuitive understanding of instantaneous rates of change through familiar situations, and through a graphical and numerical approach to the measurement of constant, average and instantaneous rates of change. This area of study progresses to the differentiation of polynomials of degree no higher than three. Extension to maximising problems, where the required expression is not given, is not required.
This area of study will include:
PROBABILITY AND STATISTICS
Learners will develop counting techniques required for the calculation of probabilities and gain an understanding of the related concepts of conditional and independent events.
This area of study will include:
Reviewing the fundamentals of probability and the language of events and sets:
Conditional probability and independence:
Combinations:
Criterionbased assessment is a form of outcomes assessment that identifies the extent of learner achievement at an appropriate endpoint of study. Although assessment – as part of the learning program – is continuous, much of it is formative, and is done to help learners identify what they need to do to attain the maximum benefit from their study of the course. Therefore, assessment for summative reporting to TASC will focus on what both teacher and learner understand to reflect endpoint achievement.
The standard of achievement each learner attains on each criterion is recorded as a rating ‘A’, ‘B’, or ‘C’, according to the outcomes specified in the standards section of the course.
A ‘t’ notation must be used where a learner demonstrates any achievement against a criterion less than the standard specified for the ‘C’ rating.
A ‘z’ notation is to be used where a learner provides no evidence of achievement at all.
Providers offering this course must participate in quality assurance processes specified by TASC to ensure provider validity and comparability of standards across all awards. To learn more, see TASC's quality assurance processes and assessment information.
Internal assessment of all criteria will be made by the provider. Providers will report the learner’s rating for each criterion to TASC.
TASC will supervise the external assessment of designated criteria which will be indicated by an asterisk (*). The ratings obtained from the external assessments will be used in addition to internal ratings from the provider to determine the final award.
The following processes will be facilitated by TASC to ensure there is:
Process – TASC gives course providers feedback about any systematic differences in the relationship of their internal and external assessments and, where appropriate, seeks further evidence through audit and requires corrective action in the future.
The external assessment for this course will comprise:
For further information see the current external assessment specifications and guidelines for this course available in the Supporting Documents below.
The assessment for Mathematics Methods  Foundation Level 3 will be based on the degree to which the learner can:
* = denotes criteria that are both internally and externally assessed
The learner:
Rating A  Rating B  Rating C 

presents work that conveys a logical line of reasoning that has been followed between question and answer  presents work that conveys a line of reasoning that has been followed between question and answer  presents work that shows some of the mathematical processes that have been followed between question and answer 
uses mathematical conventions and symbols correctly  uses mathematical conventions and symbols correctly  uses mathematical conventions and symbols. There may be some errors or omissions in doing so. 
presents work with the final answer clearly identified and articulated in terms of the question as required  presents work with the final answer clearly identified  presents work with the final answer apparent 
uses correct units and includes them in an answer for routine and nonroutine problems  uses correct units and includes them in an answer for routine problems  uses correct units and includes them in an answer for routine problems 
presents detailed tables, graphs and diagrams that convey accurate meaning and precise information  presents detailed tables, graphs and diagrams that convey clear meaning  presents tables, graphs and diagrams that include some suitable annotations 
adds a detailed diagram to illustrate and explain a solution  adds a diagram to illustrate a solution  adds a diagram to a solution as directed 
ensures an appropriate degree of accuracy is maintained and communicated throughout a problem.  determines and works to an appropriate degree of accuracy.  works to an appropriate degree of accuracy as directed. 
The learner:
Rating A  Rating B  Rating C 

selects and applies an appropriate strategy, where several may exist, to solve routine and nonroutine problems  selects and applies an appropriate strategy to solve routine and simple nonroutine problems  identifies an appropriate strategy to solve routine problems 
interprets solutions to routine and nonroutine problems  interprets solutions to routine and simple nonroutine problems  describes solutions to routine problems 
explains the reasonableness of results and solutions to routine problems and nonroutine problems  describes the reasonableness of results and solutions to routine problems  with direction, describes the reasonableness of results and solutions to routine problems 
identifies and describes limitations of simple models  identifies and describes limitations of simple models  identifies limitations of simple models 
uses available technological aids in familiar and unfamiliar contexts  chooses to use available technological aids when appropriate to solve routine problems  uses available technological aids to solve routine problems 
models and solves problems derived from routine and nonroutine scenarios.  models and solves problems derived from routine scenarios. 
The learner:
Rating A  Rating B  Rating C 

uses planning tools and strategies to achieve and manage activities within proposed times  uses planning tools to achieve objectives within proposed times  uses planning tools, with prompting, to achieve objectives within proposed times 
divides a task into appropriate subtasks  divides a task into subtasks  divides a task into subtasks as directed 
selects strategies and formulae to successfully complete routine and nonroutine problems  selects from a range of strategies and formulae to successfully complete routine problems  uses given strategies and formulae to successfully complete routine problems 
plans timelines and monitors and analyses progress towards meeting goals, making adjustments as required  plans timelines and monitors progress towards meeting goals  monitors progress towards meeting goals 
accurately and succinctly addresses all of the required elements of a task  addresses the elements of required tasks  addresses most elements of required tasks 
plans future actions, effectively adjusting goals and plans where necessary.  plans future actions, adjusting goals and plans where necessary.  uses prescribed strategies to adjust goals and plans where necessary. 
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating, the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

uses index laws to simplify expressions in the context of solving problems  uses index laws, including the use of surds (radicals), to simplify expressions with rational indices  uses index laws to simplify expressions with integral indices 
simplifies and manipulates algebraic expressions in the course of solving problems or making calculations  applies routine calculations and manipulations with algebraic fractions to simplify expressions  performs routine calculations and manipulations with algebraic fractions 
selects when to expand and when to leave expressions in factorised form  expands quadratic and cubic expressions from factors  
expands binomial expressions using the binomial theorem  expands binomial expressions using Pascal’s triangle  expands simple binomial expressions using Pascal’s triangle 
factorises algebraic expressions including sums and differences of cubes  factorises algebraic expressions, including methods such as monic and nonmonic quadratics  factorises algebraic expressions involving a combination of methods such as, common factors, difference of squares and factorisation of monic quadratics 
applies knowledge of the discriminant to analyse and interpret problems  uses the discriminant to determine whether or not the real zero of a quadratic is rational or irrational  uses the discriminant to determine whether or not a quadratic expression factorises with real factors 
forms and solves simultaneous equations from applied scenarios in routine and nonroutine problems including simple nonlinear situations  solves simultaneous equations from applied scenarios in routine problems  solves simultaneous equations involving two linear functions 
completes the square in a quadratic expression where the coefficient of the squared term is any integral value  completes the square in a quadratic expression where the coefficient of the squared term is 1  
solves quadratic equations and interprets solutions in the context of the problem posed  solves quadratic equations using the general quadratic formula  solves quadratic equations in factored form using the null factor law 
selects methodologies for factorising cubic polynomials and solving cubic equations in the context of solving problems.  factorises cubic polynomials and solves cubic equations algebraically when one factor is linear, and otherwise using technology.  solves cubic polynomials when presented in factorised form, and otherwise using technology. 
* denotes criteria that are both internally and externally assessed
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating,the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

finds the gradient of a straight line function  
determines equations of parallel and perpendicular lines, and chooses to do so in the context of solving a problem  determines equations of parallel lines  given some properties of a line, such as the gradient and point, or two points on the line, determines and interprets its equation and/or sketches its graph 
sketches quadratic functions `y = ax^2 + bx + c` given information about the signs of `a`, `b`, `c` and the discriminant, `b^2  4ac`  determines intercepts, axis of symmetry, turning point, domain and range of a quadratic function  sketches factorised quadratic functions specifying intercepts 
sketches graphs of cubic polynomials requiring factorisation and/or algebraic manipulation  sketches factorised cubic functions which may involve repeated zeros  sketches factorised cubic functions with three distinct linear factors specifying intercepts 
determines and interprets the equation of a quadratic function given in graphical form or via some vital characteristics, in a modelled scenario  interprets the key features of a quadratic function given in graphical form in a modelled scenario  employs a given quadratic function to address questions requiring substitution of values in a modelled scenario 
determines and interprets the equation of a cubic function given in graphical form  interprets the equation of a cubic function given in graphical form  employs a given cubic function to address questions requiring substitution of values 
can identify whether a graph or an equation is a function or a relation  can define the domain and the range of a function and understands the difference between a relation and a function  can interpret given information about the domain and range of a function 
applies knowledge of translations to sketch graphs of polynomials.  applies knowledge of translations to sketch graphs of polynomials involving one translation.  uses key features to distinguish graphs of linear, quadratic and cubic functions. 
* denotes criteria that are both internally and externally assessed
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating, the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

solves more complex equations using index laws  solves routine equations using index laws  solves simple equations using index laws 
solves more complex logarithmic equations and exponential equations using logarithms  solves simple logarithmic equations using log laws  uses the definition of log to change between a log statement and an index statement 
establishes and solves exponential equations modelling real world scenarios  solves routine exponential problems including those requiring some interpretation of real world scenarios  applies knowledge of bases and order conventions to manipulate and solve simple exponential equations 
sketches the graph of an exponential function, including horizontal and/or vertical translations, including asymptote(s) and intercept(s)  sketches the graph of an exponential function, including asymptote and intercepts, and identifies the domain and range  sketches the graph of a simple exponential function using technology 
sketches the graph of a simple logarithmic function, including horizontal and vertical translations  sketches the graph of a simple logarithmic function, including asymptote and intercepts, and identifies the domain and range  sketches the graph of a simple logarithmic function using technology 
determines unknown sides and angles using sine and cosine rules where appropriate  determines side lengths using sine, cosine and tangent ratios in a given right angle triangle and sine and cosine rules in straightforward problems  



applies symmetry properties to general rotations  applies symmetry properties arising from the unit circle for rotations between `0` and `2pi` radians  recalls the unit circle definitions of sine, cosine and tangent 
sketches graphs of these functions, incorporating dilations and reflections and changes in period as applicable, with and without technology.  sketches graphs of these functions with and without technology.  sketches graphs of sine, cosine and tangent functions using technology. 
* denotes criteria that are both internally and externally assessed
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating, the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

elects appropriately to employ the average rate of change in context to solve problems  calculates and interprets the average rate of change from given information using appropriate units  calculates an average rate of change from given information using appropriate units 
applies knowledge of the gradient of a function’s tangent at various points to infer properties of the function  determines and relates the gradient of a function’s tangent to the rate of change of the function  determines the gradient of a function’s tangent with graphical scaffolding 
differentiates polynomials in expanded form with rational powers  differentiates polynomials in expanded or in factored form  differentiates polynomials in expanded form 
sketches a graph of the rate of change of a function whose graph is given  interprets the relationship between two variables by considering rates of change  uses a derivative to determine the instantaneous rate of change at a given value of the independent variable, with suitable units 
uses a derivative to determine the equation of a normal to a curve at a point  uses a derivative to determine the equation of a tangent to a curve at a point  uses a derivative to determine the gradient to a curve at a point 
examines the nature of the turning points of a given function and distinguishes between local and global extreme values, considering behaviours as `x > +oo`  finds, using calculus techniques, and interprets the local maxima and local minima of a given function  finds and interprets the local maxima and local minima of a given function using technology 
constructs and interprets routine and nonroutine displacementtime graphs, with velocity calculations  constructs and interprets routine displacementtime graphs, with velocity calculations  interprets displacementtime graphs 
uses the first principles approach to determine the derivatives of simple quadratic functions.  uses the first principles approach to determine the derivative of `x^n`, for `n = 1, 2, 3` . 

* denotes criteria that are both internally and externally assessed
This criterion is both internally and externally assessed.
Rating 'A'
In addition to the standards for a 'C' and a 'B' rating, the learner:
Rating 'B'
In addition to the standards for a 'C' rating, the learner:
Rating 'C'
The learner:
Rating A  Rating B  Rating C 

distinguishes between random selection and random events  
designs and carries out simple probability experiments and simulations, recognising that randomness confers longterm order and predictability  designs and carries out simple probability experiments and simulations, identifying cases where inherent bias disrupts randomness  draws conclusions from experiments and simulations where relative frequencies provide point estimates of probabilities 
calculates probabilities by constructing and using Venn diagrams and trees diagrams, where appropriate, in routine and nonroutine problems  calculates probabilities by constructing Venn diagrams and tree diagrams in routine problems  calculates probabilities from data presented in lists, tables, Venn diagrams and tree diagrams 
identifies situations where complementarity and mutually exclusivity apply  employs complementarity to simplify calculations, and mutually exclusivity to reduce the addition rule  distinguishes complementary and mutually exclusive events, and employs the addition rule for probability 
identifies when conditional probability applies, including in more complex situations  applies the conditional probability formula `P(AB) = (n(A nn B))/(n(B))` in straightforward situations  interprets conditional probabilities in straightforward situations 
constructs diagrams that illustrate independent threestage events and nonindependent twostage events to calculate probabilities  constructs diagrams that illustrate independent twostage events to calculate probabilities  uses given diagrams that illustrate two stage events to calculate probabilities 
determines the relevance of and calculates combinations in more complex situations.  uses the combination formula, ^{n}C_{r} or `((n),(r))`, to determine probability in a straightforward problems.  uses the combination formula, ^{n}C_{r} or `((n),(r))`, to determine the number of possibilities in straightforward combination problems. 
* denotes criteria that are both internally and externally assessed
Mathematics Methods – Foundation Level 3 (with the award of):
EXCEPTIONAL ACHIEVEMENT
HIGH ACHIEVEMENT
COMMENDABLE ACHIEVEMENT
SATISFACTORY ACHIEVEMENT
PRELIMINARY ACHIEVEMENT
The final award will be determined by the Office of Tasmanian Assessment, Standards and Certification from 13 ratings (8 from the internal assessment, 5 from the external assessment).
The minimum requirements for an award in Mathematics Methods – Foundation Level 3 are as follows:
EXCEPTIONAL ACHIEVEMENT (EA)
11 ‘A’ ratings, 2 ‘B’ ratings (4 ‘A’ ratings and 1 ‘B’ rating from external assessment)
HIGH ACHIEVEMENT (HA)
5 ‘A’ ratings, 5 ‘B’ ratings, 3 ‘C’ ratings (2 ‘A’ ratings, 2 ‘B’ ratings and 1 ‘C’ rating from external assessment)
COMMENDABLE ACHIEVEMENT (CA)
7 ‘B’ ratings, 5 ‘C’ ratings (2 ‘B’ ratings and 2 ‘C’ ratings from external assessment)
SATISFACTORY ACHIEVEMENT (SA)
11 ‘C’ ratings (3 ‘C’ ratings from external assessment)
PRELIMINARY ACHIEVEMENT (PA)
6 ‘C’ ratings
A learner who otherwise achieves the ratings for a CA (Commendable Achievement) or SA (Satisfactory Achievement) award but who fails to show any evidence of achievement in one or more criteria (‘z’ notation) will be issued with a PA (Preliminary Achievement) award.
The Department of Education’s Curriculum Services will develop and regularly revise the curriculum. This evaluation will be informed by the experience of the course’s implementation, delivery and assessment.
In addition, stakeholders may request Curriculum Services to review a particular aspect of an accredited course.
Requests for amendments to an accredited course will be forwarded by Curriculum Services to the Office of TASC for formal consideration.
Such requests for amendment will be considered in terms of the likely improvements to the outcomes for learners, possible consequences for delivery and assessment of the course, and alignment with Australian Curriculum materials.
A course is formally analysed prior to the expiry of its accreditation as part of the process to develop specifications to guide the development of any replacement course.
The statements in this section, taken from documents endorsed by Education Ministers as the agreed and common base for course development, are to be used to define expectations for the meaning (nature, scope and level of demand) of relevant aspects of the sections in this document setting out course requirements, learning outcomes, the course content and standards in the assessment.
For the content areas of Mathematics Methods – Foundation, the proficiency strands – Understanding; Fluency; Problem Solving; and Reasoning – build on learners’ learning in F10 Australian Curriculum: Mathematics. Each of these proficiencies is essential, and all are mutually reinforcing. They are still very much applicable and should be inherent in the five areas of study.
MATHEMATICAL METHODS
Unit 1 – Topic 1: Functions and Graphs
Lines and linear relationships:
Review of quadratic relationships:
Inverse proportion:
Powers and polynomials:
Graphs of relations:
Functions:
Unit 1 – Topic 2: Trigonometric Functions
Cosine and sine rules:
Circular measure and radian measure:
Trigonometric functions:
Unit 1 – Topic 3: Counting and Probability
Combinations:
Language of events and sets:
Review of the fundamentals of probability:
Conditional probability and independence:
Unit 2 – Topic 1: Exponential Functions
Indices and the index laws:
Exponential functions:
Unit 2 – Topic 3: Introduction to Differential Calculus
Rates of change:
The concept of the derivative:
Computation of derivatives:
Properties of derivatives:
Applications of derivatives:
Unit 4 – Topic 1: The Logarithmic Function
Logarithmic functions:
The accreditation period for this course has been renewed from 1 January 2019 until 31 December 2021.
During the accreditation period required amendments can be considered via established processes.
Should outcomes of the Years 912 Review process find this course unsuitable for inclusion in the Tasmanian senior secondary curriculum, its accreditation may be cancelled. Any such cancellation would not occur during an academic year.
Version 1 – Accredited on 17 August 2016 for use from 1 January 2017. This course replaces Mathematics Methods – Foundation (MTM215116) that expired on 31 December 2016.
Version 1.1 – Renewal of accreditation on 13 August 2017 for use in 2018.
Version 1.1.a – Change of subheading 'Exponential and Logarithmic Functions' to 'Indices' and move of three content points from that place to 'Logarithmic functions'. 22 December 2017.
Accreditation renewed on 22 November 2018 for the period 1 January 2019 until 31 December 2021.
Accreditation renewed on 14 July 2021 for the period 1 January 2022 until 31 December 2023.
GLOSSARY
Algebraic properties of exponential functions
The algebraic properties of exponential functions are the index laws: `a^x a^y = a^(x + y)`, `a^(x) = 1/(a^x)`, `(a^x)^y = a^(xy)`, `a^0 = 1`, where `x`, `y` and `a` are real.
Algebraic properties of logarithms
The algebraic properties of logarithms are the rules: `log_a (xy) = log_a x + log_a y`, `log_a (1/x) = log_a x`, and `log_a 1 = 0`, for any positive real numbers `x`, `y` and `a`.
Asymptote
A straight line is an asymptote of the function `y = f(x)` if graph of `y = f(x)` gets arbitrarily close to the straight line. An asymptote can be horizontal, vertical or oblique. For example, the line with equation `x = pi/2` is a vertical asymptote to the graph of `y = tan x`, and the line with equation `y = 0` is a horizontal asymptote to the graph of `y = 1/x`.
Binomial distribution
The expansion `(x + y)^n = x^n + ((n),(1))x^(n  1)y + ... + ((n),(r))x^(n  r)y^r + ... + y^n` is known as the binomial theorem. The numbers `((n),(r)) = (n!)/(r!(nr)!) = (n xx (n  1) xx ... xx (n  r + 1))/(r xx (r  1) xx ... xx 2 xx 1)` are called binomial coefficients.
Cartesian plane
The Cartesian plane is a plane consisting of a set of two lines intersecting each other at right angles. The horizontal line is the `x`axis and the vertical one is the `y`axis, and the point of their intersection is called the origin with the coordinates `(0, 0)`.
Circular measure
A rotation, typically measured in radians or degrees.
Completing the square
The quadratic expression `ax^2 + bx + c` can be rewritten as `a(x + b/(2a))^2 + (c  (b^2)/(4a))`. Rewriting it in this way is called completing the square.
Conditional probability
The probability of an event `A` occurring when it is known that some event `B` has already occurred, is given by `P(AB) = (P(A nn B))/(P(B))`.
Discriminant
The discriminant (`Delta`) of the quadratic expression `ax^2 + bx + c` is the quantity `b^2  4ac`.
Function
A function `f` is a rule such that for each `x`value there is only one corresponding `y`value. This means that if `(a, b)` and `(a, c)` are ordered pairs, then `b = c`.
Gradient (Slope)
The gradient of the straight line passing through points `(x_1, y_1)` and `(x_2, y_2)` is the ratio `(y_2  y_1)/(x_2  x_1)`. Slope is a synonym for gradient.
Graph of a function
The graph of a function `f` is the set of all points `(x, y)` in the Cartesian plane where `x` is in the domain of `f` and `y = f(x)`.
Independent events
Two events are independent if knowing that one occurs tells us nothing about the other. The concept can be defined formally using probabilities in various ways: events `A` and `B` are independent if `P(A nn B) = P(A)P(B)`, if `P(AB) = P(A)` or if `P(B) = P(BA)`. For events `A` and `B` with nonzero probabilities, any one of these equations implies any other.
Index laws
The index laws are the rules: `a^xa^y = a^(x + y)`, `a^(x) = 1/(a^x)`, `(a^x)^y = a^(xy)`, `a^0 = 1`, and `(ab)^x = a^xb^x`, where `a`, `b`, `x` and `y` are real numbers.
Length of an arc
The length of an arc of a circle is given by `l = r theta`, where `l` is the arc length, `r` is the radius and `theta` is the angle subtended at the centre, measured in radians. This is simply a rearrangement of the formula defining the radian measure of an angle.
Linearity property of the derivative
The linearity property of the derivative is summarized by the equations:
`d/dx(ky) = kdy/dx` for any constant `k`, and `d/dx(y_1 + y_2) = (dy_1)/dx + (dy_2)/dx`.
Local and global maximum and minimum
A stationary point on the graph `y = f(x)` of a differentiable function is a point where `f'(x) = 0`.
We say that `f(x_0)` is a local maximum of the function `f(x)` if `f(x) <= f(x_0)` for all values of `x` near `x_0`.
We say that `f(x_0)` is a global maximum of the function `f(x)` if `f(x) <= f(x_0)` for all values of `x` in the domain of `f`.
We say that `f(x_0)` is a local minimum of the function `f(x)` if `f(x) >= f(x_0)` for all values of `x` near `x_0`.
We say that `f(x_0)` is a global minimum of the function `f(x)` if `f(x) >= f(x_0)` for all values of `x` in the domain of `f`.
Mutually exclusive
Two events are mutually exclusive if there is no outcome in which both events occur.
Nonroutine problems
Problems solved using procedures not regularly encountered in learning activities.
Pascal’s triangle
Pascal’s triangle is a triangular arrangement of binomial coefficients. The `n^"th"` row consists of the binomial coefficients `((n),(r))`, for `0 <= r <= n`; each interior entry is the sum of the two entries above it, and the sum of the entries in the `n^"th"` row is `2^n`.
Period of a function
The period of a function `f(x)` is the smallest positive number `p` with the property that `f(x + p) = f(x)` for all `x`. The functions `sin x` and `cos x` both have period `2pi`, and `tan x` has period `pi`.
Point of inflection
A point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.
Quadratic formula
If `ax^2 + bx + c = 0` with `a != 0`, then `x = (b + sqrt(b^2  4ac))/(2a)`. This formula for the roots is called the quadratic formula.
Radian measure
The radian measure `theta` of an angle in a sector of a circle is defined by `theta = l/r`, where `r` is the radius and `l` is the arc length. Thus an angle whose degree measure is `180^"o"` has radian measure of `pi`.
Random variable
A random variable is a numerical quantity, the value of which depends on the outcome of a chance experiment. For example, the proportion of heads observed in 100 tosses of a coin.
A discrete random variable is one which can only take a countable number of value, usually whole numbers.
A continuous random variable is one whose set of possible values are all of the real numbers in some interval.
Relative frequency
If an event `E` occurs `r` times in `n` trials of a chance experiment, the relative frequency of `E` is `r/n`.
Routine problems
Problems solved using procedures regularly encountered in learning activities.
Secant
A secant of the graph of a function is a straight line passing through two points on the graph. The line segment between the two points is called a chord.
Sine and cosine functions
In the unit circle definition of `"cosine"` and `"sine"`, `cos theta` and `sin theta` are the `x` and `y`coordinates of the point on the unit circle corresponding to the angle `theta` measured as a rotation from the ray `OX`. If `theta` is measured in the counterclockwise direction, then it is said to be positive; otherwise it is said to be negative.
Sine rule and cosine rule
The lengths of the sides of a triangle are related to the sines of its angles by the equations
`a/(sinA) = b/(sinB) = c/(sinC)`
This is known as the sine rule.
The lengths of the sides of a triangle are related to the cosine of one of its angles by the equation
`c^2 = a^2 + b^2  2ab cos C`
This is known as the cosine rule.
Tangent line
The tangent line (or simply the tangent) to a curve at a given point `P` can be described intuitively as the straight line that has the same gradient at a curve where they meet. In this sense it is the best straightline approximation to the curve at the point `P`.
Vertical line test
A relation between two real variables `x` and `y` is a function, and `y = f(x)` for some function `f`, if and only if each vertical line, i.e. each line parallel to the `y`axis, intersects the graph of the relation in at most one point. This test to determine whether a relation is, in fact, a function is known as the vertical line test.
LINE OF SIGHT: Mathematics Method – Foundation Level 3
Learning Outcomes  Separating out content from skills in 4 of LOs  Criteria and Elements  Content  
explain key concepts and techniques used in solving problems  algebra  key concepts problem solving interpret and evaluate select and use appropriate technology/tools 
C4 E110 C4 E110 C4 E2, 3, 6, 10 C2 E1, 5 
Algebra 
solve problems using algebra, functions, graphs, calculus, probability and statistics 
polynomial functions and graphs 
key concepts problem solving interpret and evaluate select and use appropriate technology/tools 
C5 E18 C5 E18 C5 E58 C2 E1, 5 
Polynomial functions and graphs 
interpret and evaluate mathematical information and ascertain the reasonableness of solutions to problems 
exponential, logarithmic and circular functions 
key concepts problem solving interpret and evaluate select and use appropriate technology/tools 
C6 E19 C6 E19 C6 E3, 7, 8, 9 C2 E1, 5 C6 9 
Exponential, logarithmic and circular functions 
choose when or when not to use technology when solving problems 
calculus 
key concepts problem solving interpret and evaluate select and use appropriate technology/tools 
C7 E18 C7 E18 C7 E6, 7 C2 E1, 5 
Calculus 
probability and statistics 
key concepts problem solving interpret and evaluate select and use appropriate technology/tools 
C8 E17 C8 E17 C8 E6, 7 C2 E1, 5 C7 E7 
Probability and Statistics 

communicate their arguments and strategies when solving problems 
C1 E17 
All content areas 

apply reasoning skills in the context of algebra, functions, graphs, calculus, probability and statistics 
C2 E16 
All content areas 

organise and undertake activities including practical tasks 
C3 E16 
All content areas 