Overview
Mathematical Methods Units 1 and 2 cover the areas of study 'Functions, relations and graphs', 'Algebra, number and structure', 'Calculus' and 'Data analysis, probability and statistics'.
Courses based on these units should be implemented so that there is a balanced and progressive development of concepts, knowledge, skills and processes from each of the four areas of study.
Connections between and across the areas of study should be developed consistently throughout both Units 1 and 2, and students should be given the opportunity to apply their learning in practical and theoretical contexts.
In Unit 1 the focus is on the study of algebraic functions, and in Unit 2 the focus is on the study of transcendental functions and the calculus of simple algebraic functions. This is complemented by the study of elementary probability and related statistics in both units.
In undertaking these units, 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, algorithms, 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.
There are a variety of approaches teachers can take to planning and implementing suitable course plans.
The following sample course plan shows one way to sequence content from the areas of study across Units 1 and 2, with indicative time allocations in weeks. It is an example only and teachers may choose to adapt or revise it as appropriate, or to develop their own alternative course plans.
This course plan is based on identifying general and specific features and behaviour of functions graphically, and then developing the corresponding numerical and analytic techniques to locate these features and apply them to model situations and solve related problems.
In Unit 1 the introduction to functions initially involves both graphical and qualitative approaches, leading to a review of techniques, followed by the application of simple familiar functions and transformations of these, and inverse functions, with restricted domains as applicable.
Students then consider an in-depth treatment of polynomial functions of low degree, without calculus, using functionality of technology to identify the approximate location of key features, before being introduced to the concepts of calculus with respect to rates of change. without calculus, in Unit 1, using functionality of technology to identify approximate location of key features.
The work on Probability is covered across the final weeks of Unit 1, and the first couple of weeks of Unit 2, enabling a thorough treatment covering random variables, sample spaces, informal consideration of distributions, experiment and simulation, and the calculation of probabilities of combined events.
Unit 2 of this course plan includes in-depth study of transformations of exponential functions, their properties, behaviour and applications, followed by in-depth study of transformations of the basic circular functions, their properties, behaviour and applications.
Unit 2 concludes with a comprehensive and in-depth treatment of the formal definition of a derivative, numerical approximation, differentiation and anti-differentiations of polynomial functions by rule, and the application to analysis of their graphs, modelling phenomena and solving related problems.
Sample course plan - Unit 1
Functions, relations and graphs and Algebra, number and structure areas of study
Weeks 1 – 3
Functions and function notation, specifying a function, independent and dependent variables, evaluation of
f(x) where
x ∈
R. Domain, including maximal (implied or natural) domain, range and co-domain.
Representation by rule, graph or table, including examples of real-life data represented graphically by a function, such as:
tides,
water storage level,
UV levels or
temperature over a day, exchange rates, tax scales, interest (cash) rate trends, value of stock or other economic data, trend in the number of infections during a pandemic, or data obtained from a scientific experiment.
Qualitative interpretation of data represented graphically by a function:
- domain, corresponding range, co-domain
- key features such as approximate location of axis intercepts, turning points, local and global maxima and minima, points of inflection
- intervals of the domain where the graph is increasing, decreasing or constant, any periodic or asymptotic behaviour, or a sudden change in values
- identifying elements or intervals of the domain when a function takes a particular value, or values within a given subset of its range.
Sample assessment task: Features of graphs of functions
Fitting graphs of functions
f(x) = ax + b, f(x) = ax2 + b, f(x) = ax(x-b), f(x).PNG)
to pairs of points using two simultaneous linear equations in two unknowns to determine coefficients, interpreting the parameters and with respect to these graphs.
Weeks 4–6
Graphing quadratic functions,
f(x) =
ax2 +
bx +
c,
a ≠ 0 domain, range and co-domain, key features (symmetry, intercepts, vertex), interpreting the parameters and with respect to these graphs.
Transforming the graph of
f(x) =
x2 to the graph of
f(x) = a(x+h)2 +
k, interpreting the parameters
a,
b and
c with respect to transformations and key features of the graph.
Expanding and factorising quadratic expressions with integer coefficients, including the rational root theorem applied to quadratic functions.
Sample learning activity: Are there rational roots?
Expressing
f(x) = ax2 + bx +
c, a ≠ 0 and
a,
b,
c ∈
Z, in completed square form
f(x) =
a(x+h)2 +
k.
Solving quadratic equations, null factor law, the quadratic formula and discriminant, the algorithm for the
numerical method of
bisection. Graphical interpretation of quadratic equations.
Applications and
modelling with quadratic functions. Solving three simultaneous linear equations in three unknowns to fit the graph of a quadratic function to three points.
Sample assessment task: Investigating different approaches to solving quadratic equations
Weeks 7–9
Graphs of power functions,
f(x)= xn for n ∈ N and
.PNG)
including
transformations of these to the form
f(x) = a(x + b)n + c where a, b, c ∈ R and a ≠ 0.
Sample learning activity: Exploring the graphs of power functions
The use of pattern recognition, transformation of coordinates, or
matrices to describe
transformations and their effects.
Concept of an inverse function, existence,
f -1 notation, graphs of a one-to-one function and its
inverse functions. Relationship of these graphs to the transformation of reflection in the line
y=x.
Use of parameters to represent families of functions, and determination of the rule of a particular function. Applications and modelling with power functions.
Sample assessment task:
Scaling human measures
Weeks 10–13
Polynomial functions
f: R→R, f(x) = an xn + an-1 xn-1 + ⋯ + a2 x2 + a1 x1 + a0 x0, an ≠ 0, graphs of cubic and quartic
polynomial functions, and other polynomial functions of low degree such as
f(x)= x^6-x^4 including approximate location of axis intercepts, stationary points and points of inflection. Constant, linear and quadratic functions as special cases of polynomial functions of degree 0, 1 and 2 respectively.
Sum difference and product of polynomial expressions. Division of a
polynomial expression by a linear factor, and the remainder, factor and rational root theorems. Algebraic manipulation from one form of an expression to an equivalent form.
Sample learning activity:
How many different shapes of graphs are there?
Key features and the effects of transformations on the graphs of cubic and quartic polynomial functions.
Solution of polynomial equations using algebraic, graphical and
numerical methods, including application of the algorithm for
bisection. Verification of solutions over a specified domain.
Sample learning activity:
Bisection for a cubic
Possible assessment: a modelling task finding various polynomial models for data drawn from different contexts, based on solving a set of simultaneous linear equations for a subset of the data.
Sample mathematical investigation: Implementing the bisection algorithm with polynomial functions
Calculus area of study
Weeks 14–15
Average rate of change. Informal treatment of
instantaneous rate of change as the limiting case of average rate of change.
Interpretation of
graphs of empirical data with respect to the rate of change.
Sample learning activity:
Filling vases with water
Use of gradient of the tangent at a point on a graph of a function to describe and measure the instantaneous rate of change of the function. Consideration of positive, negative and zero gradient and relationship of the gradient to the features of the original graph.
Possible assessment: a short graphical modelling task based on a variation of the
Filling vases with water learning activity.
Data analysis, probability and statistics area of study
Week 16
Random experiments (trials), random variables and
simulations, using simple random generators, and pseudo-random generators by technology.
Sample spaces, outcomes, elementary and compound events.
Display and interpretation of results, graphs of proportions in samples.
Sample learning activity:
Experiments, simulation and probability
Sample mathematical investigation:
Exploring long run probabilities
Weeks 17–18
Addition and multiplication principles for counting.
Combinations including
computation of
ⁿCr and the application of counting techniques to computing probabilities.
Sample course plan - Unit 2
Data analysis, probability and statistics area of study
Weeks 1–2
Use of lists, grids, tables, tree diagrams and Venn diagrams to represent probabilities for elementary and compound events.
Sample learning activity:
Data representations and conditional probability
The addition rule and mutually exclusive events.
Conditional probability, reduced sample space, relations, the law of total probability for two events, representation of conditional probability using
Karnaugh maps and tree diagrams.
Pairwise independent events, simulation of events involving selection with and without replacement.
Possible assessment: a test comprising a collection of short-answer and extended response questions.
Functions, relations and graphs and Algebra, number and structure areas of study
Weeks 3–5
Exponential functions
f:R→R,f(x)=Aakx+c ,a∈R+,A≠0,A,k,c∈R, their
graphs and key features, shape, axis intercepts and asymptotes.
Transformation from the graph of
f(x)= ax to the graph of
f(x)=Aakx+c, and interpretation of the parameters
A, k and
c with respect to transformations.
Applications and
modelling, such as
radioactive decay,
cooling,
consumption, compound interest and depreciation, and spread of disease with exponential functions. Interpretation of initial value, rate of growth or decay, doubling time, half-life and long run value in these contexts and their relationship to the parameters
A, k and
c.
Exponent laws and
logarithm laws,
y = loga (x) as the inverse function of
y = ax,a > 1 and their graphs, the relationships
aloga(x) = x and
loga (ax) = x, and the solution of exponential equations, including finding doubling time or half-life.
Possible assessment: a modelling task based on using exponential functions to model contexts such as a cooling liquid, population growth, compound interest and depreciation, residential density from city centre or the initial spread of a disease, and the solution of related equations.
Weeks 6–8
The unit circle at the origin
x2 + y2 = 1,
radians and arc length.
Sine (sin), cosine (cos) and tangent (tan) as functions of a real variable, and the relationships sin(x)
≈ x for
x small, sin2(x)
+ cos2(x) = 1, and tan(x) =
.PNG)
.
Symmetry properties, complementary relationships and periodicity properties of
sine, cosine and tangent functions.
Exact and approximate values of sine, cosine and tangent for multiples of .PNG)
and multiples of .PNG)
Sample learning activity:
Estimating radians, sine and cosine on a unit circle
Solving simple equations involving circular functions, including the use of inverse functions (sin-1, cos-1, tan-1) and transformations to solve equations of the form
y = Af(nx) + c and
f is sine, cosine or tangent, using exact or approximate values over a given interval.
Weeks 9–10
Circular functions of the form
y = Af(nx) + c, a, n ≠ 0, a, n, c ∈ R and their
graphs, where
f is the sine, cosine or tangent function.
Interpretation of transformation from the graph of
y = f(x) to the graph
y = Af(nx) + c with respect to the parameters
a, n and
c
Applications and modelling with circular functions. Interpretation of period, amplitude and mean value in these contexts and their relationships to the parameters a, n and c.
Possible assessment: an investigation of the use of transformed circular functions and simple sums of such functions to model periodic data such as tides, temperatures, or pollution levels, including the solution of related equations and inequalities.
Sample mathematical investigation:
Impact of rising sea levels on coastal towns
Calculus area of study
Week 11
The derivative as the gradient of the graph of a function at a point and its representation by the gradient function, using the limit definition.
Notations for the derivative or gradient function:
Left secant .PNG)
, right secant .PNG)
and central difference .PNG)
.
approximations for small values of
h, central difference as the average of left secant and right secant approximations.
Sample learning activity:
Numerical approximation of derivatives
Weeks 12–13
Informal concepts of
limit, continuity and differentiability.
Establishing from definition that the derivative of
xn, n ∈ N is
nxn-1 and extension to the differentiation of polynomial functions.
Differentiation of
polynomial functions by rule, and application to the analysis of graphs of polynomial functions, stationary values of functions including local maxima and minima, the sign of the gradient at or near a point, and
points of inflection
Weeks 14–16
Applications of differentiation,
instantaneous rates of change,
straight line or
rectilinear motion.
Sample learning activity:
Simulating rectilinear motion
Sample mathematical investigation:
Simulating motion along a straight line
Solving
maximum and minimum problems, including consideration of
endpoint values.
Sample assessment task:
Perimeter and area of a rectangle
Application of the
Newton's method algorithms to find a
numerical approximation to the root of a cubic polynomial function.
Sample learning activity: Linear equations for cubic roots
Weeks 17–18
Anti-differentiation as the inverse process to differentiation, and identification of families of polynomial functions with the same gradient function.
Notations for the anti-derivative function:
F(x), ∫ f(x)dx.
Use of a boundary condition (which may be an initial condition) to determine a specific anti-derivative function of a given polynomial function.
Applications of anti-differentiation of polynomial functions, including solving simple problems involving straight line motion.
Possible assessment: a test comprising a collection of multiple-choice, short-answer and extended response questions including applications to distance travelled.
Overview
Mathematical Methods Units 3 and 4 cover the areas of study ‘Functions, relations and graphs’, ‘Algebra, number and structure’, ‘Calculus’ and ‘Data analysis, probability and statistics’, and extend the introductory study of simple elementary functions of a single real variable from Units 1 and 2 to include combinations of these functions and related graphs, algebra and calculus.
Courses based on these units should be implemented so that there is a balanced and progressive development of concepts, knowledge, skills and processes from each of the four areas of study.
Connections between and across the areas of study should be developed consistently throughout both Units 3 and 4, and students should be given the opportunity to apply their learning in practical and theoretical contexts.
In Unit 3 the focus is on the study of transformed and combined functions, related graphs and algebra, and differentiation of these functions. In Unit 4 the focus is on anti-differentiation and integration, and the study of random variables, probability distributions and their application to simple statistical inference with proportions.
In undertaking these units, students learn to apply techniques, routines and processes involving rational and real arithmetic, sets, lists and tables, diagrams and geometric constructions, algorithms, algebraic manipulation, equations, graphs and differentiation, anti-differentiation, and integration, with and without the use of technology. They should develop their 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 units as applicable.
There are a variety of approaches teachers can use to plan, develop and implement suitable course plans.
The following sample course plan shows one way to sequence content from the areas of study across Units 3 and 4, with indicative time allocations in weeks. It is an example only and teachers may choose to adapt or revise it as appropriate, or to develop their own alternative course plans. This course plan takes place over 27 weeks across Terms 1–3; however, the timing of topics and assessments may vary from year to year.
Within this course plan, the treatment of elementary functions of a single real variable and their transformations is extended to combinations of these functions and related graphs, algebra, calculus, problems and applications. It also covers some distributions of discrete and continuous random variables, sample proportions, problems and applications.
The application task would typically be conducted from the end of Term 1 through to early or mid-Term 2, depending on when the content required for the task is covered in class and the length of Term 1 in a given year.
The two modelling or problem-solving tasks can be conducted in any order during Unit 4, depending on whether distributions of continuous random variables are included in the task that addresses the Data analysis, probability and statistics area of study, and when related content on anti-differentiation and integration is covered.
Sample course plan
Polynomials and power functions including differential calculus – 5 weeks
2 weeks
Review of
polynomial functions and their
graphs, solving equations (graphically, numerically and algebraically),
derivatives of polynomial functions, stationary points and points of inflection, intervals over which a polynomial function is strictly increasing and decreasing. Simple piecewise defined (hybrid) functions based on polynomial functions, informal consideration of continuity and differentiability.
Sample learning activity:
Piecing things together with polynomials
Composition of functions and application of the chain rule for differentiation to polynomial functions with rules of the form
f(x)= (ax+b)n and other simple cases of
(p(x))n where
p(x) is polynomial.
3 weeks
Review of simple power functions, generalisation to rational powers, and transformations of power functions. Consideration of domain and rang, and related inverse functions and solving equations.
Differentiation and anti-differentiation of power functions, including those with rules of the form 
Sums, differences, composites and products of polynomial and power functions and their graphs, solving equations, differentiation of these functions, related modelling and
maximum/minimum problems and other applications.
Exponential and logarithmic functions including differential calculus – 3 weeks
1 week
Review of
exponential and logarithmic functions,
growth and decay and transformations of these functions and their graphs, the
inverse relation and equation solving. The exponential function
f:R → R, f (x) = ex and its inverse function
f -1(x): R+ → R, f -1(x)= loge(x) and the relation
ax = (eloge(a))x. The relation
.PNG)
and differentiation of
ekx, the relation
.PNG)
and differentiation of
loge(kx).
1 week
Sums, differences, composites and products of exponential, logarithmic, polynomial and power functions,
rates of change and
differentiation, involving these functions, and their graphs, including the case of
in preparation for the normal distribution
Sample learning activity:
product functions
1 week
Consideration of these combinations of functions in related modelling and
maximum/minimum problems and other applications.
Circular functions including differential calculus – 3 weeks
2 weeks
Review of
circular functions sin, cos and tan, transformations of these functions and their graphs,
inverse functions on restricted domains and equation solving,
differentiation involving sine and cosine functions and combinations with other functions.
1 week
Consideration of circular functions and
combinations of functions in related modelling, maximum / minimum problems such as
Varying rates or
Viewing angle optimisation, and other applications.
Application task – 2 weeks
This could be held earlier or later depending on when the concepts, skills and processes relevant to the task are covered.
Sample application task –
Bezier curves
Sample application task –
Investigating some polynomial functions
Sample application task –
Splining a pathway
Sample application task –
Product functions and pendulum clocks
Sample application task –
Graphs of products of polynomial functions
Sample application task –
Sample application task – drug concentrations
The
VCE Mathematical Methods study webpage includes a series of videos on how to develop an application task and related assessment processes, advice on performance criteria, and other support materials for school-assessed coursework.
Calculus applications – 3 weeks
1 week
Composition of functions involving all of the functions covered and graphs of these functions. Special cases of
f(ax+b) and transformation
f(x) → Af(ax+b) + B. Differentiation of composite functions
f ∘ g (chain rule).
Differentiation of sum,
f + g, difference
f - g, product
f g, and quotient,
f / g of two functions. (Note: while the
technique of differentiation of quotients, and its application to relevant functions and expressions such as tan(
x) and
.PNG)
is part of VCE Mathematical Methods, the general study of quotient functions is part of VCE Specialist Mathematics.)
2 weeks
Consideration of these combinations of functions and their derivatives in related graphing, modelling, maximum/minimum problems and other
applications.
Probability and statistics (1) – 5 weeks
2 weeks
Random variables, Bernoulli trials and
discrete probability distributions, including the
binomial distribution,
graphs of binomial distributions, mean, variance and standard deviation, problems and applications.
1 week
The function
.PNG)
the standard
normal distribution, its
probability density function, transformation between standard normal and other normal distribution by.PNG)
and
X = σZ + μ, the cumulative density function, intervals and
areas, problems and applications.
2 weeks
Random sampling, statistical inference, the use of binomial and normal distributions, and
sample proportions.
Modelling or problem solving (1) – 1 week
This task addresses the Data analysis, probability and statistics area of study.
sample modelling or problem-solving task –
Close to normal
sample modelling or problem-solving task –
Proportions of popularity
The
VCE Mathematical Methods study webpage includes a series of videos on how to develop a modelling or problem-solving task and related assessment processes, advice on performance criteria, and other support materials for school-assessed coursework.
Anti-differentiation and integral calculus – 3 weeks
Anti-differentiation, the fundamental theorem of calculus,
integration, including numerical integration and the trapezium method approximation, problems and applications. Differentiation of expressions such as
xloge(x) to find an antiderivative for
loge(x).
Sample learning activity:
Anti-derivatives from derivatives
Sample learning activity:
Exploring approximation of a definite integral
Probability and statistics (2) – 1 week
Other
continuous probability distributions including those with piecewise defined probability density functions, integration and intervals, mean, variance and standard deviation, problems and applications.
Modelling or problem solving (2) – 1 week
Differentiation or anti-differentiation and integration context.
sample modelling or problem-solving task –
Traversing terrains
sample modelling or problem-solving task –
Wall and window
The
VCE Mathematical Methods study webpage includes a series of videos on how to develop a modelling or problem-solving task and related assessment processes, advice on performance criteria, and other support materials for school-assessed coursework.
Consolidation, review, practice and examination preparation
Schools and teachers will have their own programs to support students in consolidation, review and practice work as part of their examination preparation.
Past VCE Mathematics examinations and reports are available on the
VCAA website